AoPS.preload_topics = {};AoPS.preload_topics[3]={"3535283":{"num_posts":15,"posts_data":[{"post_id":34373030,"topic_id":3535283,"poster_id":1118831,"post_rendered":"Ill be posting questions once in a while. Here's the first question:<br>\n<br>\nWhat fraction of numbers from <img src=\"\/\/latex.artofproblemsolving.com\/d\/c\/e\/dce34f4dfb2406144304ad0d6106c5382ddd1446.png\" class=\"latex\" alt=\"$1$\" style=\"vertical-align: -1px\" width=\"11\" height=\"14\" > to <img src=\"\/\/latex.artofproblemsolving.com\/6\/e\/e\/6ee927e1332358c96c62c277441c907c4f51057f.png\" class=\"latex\" alt=\"$1000$\" style=\"vertical-align: -1px\" width=\"38\" height=\"14\" > have the digit <img src=\"\/\/latex.artofproblemsolving.com\/e\/0\/a\/e0a0db32027a732ac57d37ef2ae9bb150f65b108.png\" class=\"latex\" alt=\"$7$\" style=\"vertical-align: -1px\" width=\"11\" height=\"14\" > and are divisible by <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/7\/c\/d\/7cde695f2e4542fd01f860a89189f47a27143b66.png\" class=\"latex\" alt=\"$3$\" style=\"vertical-align: -1px\" width=\"11\" height=\"14\" >?<\/span>","post_canonical":"Ill be posting questions once in a while. Here's the first question:\n\nWhat fraction of numbers from $1$ to $1000$ have the digit $7$ and are divisible by $3$?","username":"Soupboy0","reported":false,"is_thanked":false,"is_nothanked":false,"attachment":false,"thanks_received":0,"nothanks_received":0,"thankers":null,"deleted":false,"post_number":1,"post_time":1742946025,"num_edits":0,"post_format":"bbcode","last_edit_time":0,"last_editor_username":"","last_edit_reason":"","admin":false,"avatar":"\/\/avatar.artofproblemsolving.com\/avatar_1118831.png?t=1740690504","num_posts":215,"editable":false,"deletable":true,"show_from_start":true,"show_from_end":false}],"topic_id":3535283,"category_id":3,"category_name":"Middle School Math","category_main_color":"#f90","category_secondary_color":"#fff5d4","topic_title":"real math problems","topic_type":"forum","is_public":true,"roles":{"38516":"mod","53544":"mod","242520":"mod"},"first_post_id":34373030,"first_poster_id":1118831,"first_poster_name":"Soupboy0","first_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_1118831.png?t=1740690504","first_post_time":1742946025,"last_post_id":34399182,"last_poster_id":1118831,"last_poster_name":"Soupboy0","last_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_1118831.png?t=1740690504","last_post_time":1743203980,"last_update_time":1743203980,"comment_count":15,"num_deleted":1,"num_reports":0,"num_views":661,"category_num_users":16,"category_num_topics":51240,"category_num_posts":713278,"poll_id":0,"source":"","tags":[{"tag_id":2699651,"tag_text":"is 3 divisible by 7","is_visible":true},{"tag_id":2699652,"tag_text":"is 7 divisible by 3","is_visible":true}],"can_have_source":false,"locked":false,"forum_locked":0,"announce_type":"none","announce_through":"","announce_factor":0,"preview":"Ill be posting questions once in a while. Here's the first question:<br>\n<br>\nWhat fraction of numbers from <img src=\"\/\/latex.artofproblemsolving.com\/d\/c\/e\/dce34f4dfb2406144304ad0d6106c5382ddd1446.png\" class=\"latex\" alt=\"$1$\" style=\"vertical-align: -1px\" width=\"11\" height=\"14\" > to <img src=\"\/\/latex.artofproblemsolving.com\/6\/e\/e\/6ee927e1332358c96c62c277441c907c4f51057f.png\" class=\"latex\" alt=\"$1000$\" style=\"vertical-align: -1px\" width=\"38\" height=\"14\" > have the digit <img src=\"\/\/latex.artofproblemsolving.com\/e\/0\/a\/e0a0db32027a732ac57d37ef2ae9bb150f65b108.png\" class=\"latex\" alt=\"$7$\" style=\"vertical-align: -1px\" width=\"11\" height=\"14\" > and are divisible by <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/7\/c\/d\/7cde695f2e4542fd01f860a89189f47a27143b66.png\" class=\"latex\" alt=\"$3$\" style=\"vertical-align: -1px\" width=\"11\" height=\"14\" >?<\/span>","target_url":"","target_url_application":null,"target_text":"","state":"none","cat_can_target":0,"has_thanks":true,"has_nothanks":false,"is_bookmarked":false,"in_feed":false,"is_watched":false},"3537433":{"num_posts":2,"posts_data":[{"post_id":34399070,"topic_id":3537433,"poster_id":1155512,"post_rendered":"I'm in fifth grade and got an 18 on AMC 8, and 72 on the 10B. What should my score trend look like over the next 4-5 years to maybe make MOP. Just so that I have a pathway to compare it to.","post_canonical":"I'm in fifth grade and got an 18 on AMC 8, and 72 on the 10B. What should my score trend look like over the next 4-5 years to maybe make MOP. Just so that I have a pathway to compare it to.","username":"RabtejKalra","reported":false,"is_thanked":false,"is_nothanked":false,"attachment":false,"thanks_received":0,"nothanks_received":0,"thankers":null,"deleted":false,"post_number":1,"post_time":1743203102,"num_edits":0,"post_format":"bbcode","last_edit_time":0,"last_editor_username":"","last_edit_reason":"","admin":false,"avatar":"\/\/avatar.artofproblemsolving.com\/avatar_1155512.png","num_posts":54,"editable":false,"deletable":false,"show_from_start":true,"show_from_end":false}],"topic_id":3537433,"category_id":3,"category_name":"Middle School Math","category_main_color":"#f90","category_secondary_color":"#fff5d4","topic_title":"MOP Qual","topic_type":"forum","is_public":true,"roles":{"38516":"mod","53544":"mod","242520":"mod"},"first_post_id":34399070,"first_poster_id":1155512,"first_poster_name":"RabtejKalra","first_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_1155512.png","first_post_time":1743203102,"last_post_id":34399112,"last_poster_id":1232590,"last_poster_name":"hashbrown2009","last_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_1232590.png?t=1741788770","last_post_time":1743203483,"last_update_time":1743203483,"comment_count":2,"num_deleted":0,"num_reports":0,"num_views":26,"category_num_users":16,"category_num_topics":51240,"category_num_posts":713278,"poll_id":0,"source":"","tags":[{"tag_id":55,"tag_text":"AMC 8","is_visible":true}],"can_have_source":false,"locked":false,"forum_locked":0,"announce_type":"none","announce_through":"","announce_factor":0,"preview":"I'm in fifth grade and got an 18 on AMC 8, and 72 on the 10B. What should my score trend look like over the next 4-5 years to maybe make MOP. Just so that I have a pathway to compare it to.","target_url":"","target_url_application":null,"target_text":"","state":"none","cat_can_target":0,"has_thanks":true,"has_nothanks":false,"is_bookmarked":false,"in_feed":false,"is_watched":false},"3536869":{"num_posts":10,"posts_data":[{"post_id":34393436,"topic_id":3536869,"poster_id":1064974,"post_rendered":"Hi, does anyone happen to have MathCounts National round test problems from 2018-2024? I found 2000-2017 online but can't find anything past that year.","post_canonical":"Hi, does anyone happen to have MathCounts National round test problems from 2018-2024? I found 2000-2017 online but can't find anything past that year.","username":"UberPiggy","reported":false,"is_thanked":false,"is_nothanked":false,"attachment":false,"thanks_received":0,"nothanks_received":0,"thankers":null,"deleted":false,"post_number":1,"post_time":1743139872,"num_edits":0,"post_format":"bbcode","last_edit_time":0,"last_editor_username":"","last_edit_reason":"","admin":false,"avatar":"\/\/avatar.artofproblemsolving.com\/avatar_1064974.png","num_posts":22,"editable":false,"deletable":false,"show_from_start":true,"show_from_end":false}],"topic_id":3536869,"category_id":3,"category_name":"Middle School Math","category_main_color":"#f90","category_secondary_color":"#fff5d4","topic_title":"MathCounts National Sets","topic_type":"forum","is_public":true,"roles":{"38516":"mod","53544":"mod","242520":"mod"},"first_post_id":34393436,"first_poster_id":1064974,"first_poster_name":"UberPiggy","first_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_1064974.png","first_post_time":1743139872,"last_post_id":34399074,"last_poster_id":1232590,"last_poster_name":"hashbrown2009","last_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_1232590.png?t=1741788770","last_post_time":1743203145,"last_update_time":1743203145,"comment_count":10,"num_deleted":0,"num_reports":0,"num_views":257,"category_num_users":16,"category_num_topics":51240,"category_num_posts":713278,"poll_id":0,"source":"","tags":[{"tag_id":1,"tag_text":"MATHCOUNTS","is_visible":true}],"can_have_source":false,"locked":false,"forum_locked":0,"announce_type":"none","announce_through":"","announce_factor":0,"preview":"Hi, does anyone happen to have MathCounts National round test problems from 2018-2024? I found 2000-2017 online but can't find anything past that year.","target_url":"","target_url_application":null,"target_text":"","state":"none","cat_can_target":0,"has_thanks":true,"has_nothanks":false,"is_bookmarked":false,"in_feed":false,"is_watched":false},"3533298":{"num_posts":12,"posts_data":[{"post_id":34348636,"topic_id":3533298,"poster_id":1159780,"post_rendered":"Iowa, 38 on chapter, first in written and countdown and I\u2019m just another person asking for nats chances.","post_canonical":"Iowa, 38 on chapter, first in written and countdown and I\u2019m just another person asking for nats chances.","username":"iwillregretthisnamelater","reported":false,"is_thanked":false,"is_nothanked":false,"attachment":false,"thanks_received":0,"nothanks_received":0,"thankers":null,"deleted":false,"post_number":1,"post_time":1742695115,"num_edits":0,"post_format":"bbcode","last_edit_time":0,"last_editor_username":"","last_edit_reason":"","admin":false,"avatar":"\/\/avatar.artofproblemsolving.com\/avatar_1159780.png","num_posts":13,"editable":false,"deletable":false,"show_from_start":true,"show_from_end":false}],"topic_id":3533298,"category_id":3,"category_name":"Middle School Math","category_main_color":"#f90","category_secondary_color":"#fff5d4","topic_title":"Chances at nats? Mathcounts","topic_type":"forum","is_public":true,"roles":{"38516":"mod","53544":"mod","242520":"mod"},"first_post_id":34348636,"first_poster_id":1159780,"first_poster_name":"iwillregretthisnamelater","first_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_1159780.png","first_post_time":1742695115,"last_post_id":34399071,"last_poster_id":1159780,"last_poster_name":"iwillregretthisnamelater","last_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_1159780.png","last_post_time":1743203110,"last_update_time":1743203110,"comment_count":12,"num_deleted":0,"num_reports":0,"num_views":450,"category_num_users":16,"category_num_topics":51240,"category_num_posts":713278,"poll_id":0,"source":"","tags":[{"tag_id":1,"tag_text":"MATHCOUNTS","is_visible":true}],"can_have_source":false,"locked":false,"forum_locked":0,"announce_type":"none","announce_through":"","announce_factor":0,"preview":"Iowa, 38 on chapter, first in written and countdown and I\u2019m just another person asking for nats chances.","target_url":"","target_url_application":null,"target_text":"","state":"none","cat_can_target":0,"has_thanks":true,"has_nothanks":false,"is_bookmarked":false,"in_feed":false,"is_watched":false}};AoPS.preload_topics[4]={"3536748":{"num_posts":13,"posts_data":[{"post_id":34391994,"topic_id":3536748,"poster_id":1190801,"post_rendered":"If I have a solved Rubik's cube, and I make a finite sequence of (legal) moves repeatedly, prove that I will eventually resolve the puzzle.<br>\n<br>\n(this wording is kinda goofy but i hope its sorta intuitive)","post_canonical":"If I have a solved Rubik's cube, and I make a finite sequence of (legal) moves repeatedly, prove that I will eventually resolve the puzzle.\n\n(this wording is kinda goofy but i hope its sorta intuitive)","username":"ilikejam","reported":false,"is_thanked":false,"is_nothanked":false,"attachment":false,"thanks_received":0,"nothanks_received":0,"thankers":null,"deleted":false,"post_number":1,"post_time":1743123486,"num_edits":0,"post_format":"bbcode","last_edit_time":0,"last_editor_username":"","last_edit_reason":"","admin":false,"avatar":"\/\/avatar.artofproblemsolving.com\/avatar_1190801.png","num_posts":25,"editable":false,"deletable":false,"show_from_start":true,"show_from_end":false}],"topic_id":3536748,"category_id":4,"category_name":"High School Math","category_main_color":"#e75400","category_secondary_color":"#ffe7cc","topic_title":"Rubik's cube problem","topic_type":"forum","is_public":true,"roles":{"38516":"mod","242520":"mod","53544":"mod"},"first_post_id":34391994,"first_poster_id":1190801,"first_poster_name":"ilikejam","first_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_1190801.png","first_post_time":1743123486,"last_post_id":34399104,"last_poster_id":407095,"last_poster_name":"kred9","last_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_407095.png?t=1607403485","last_post_time":1743203418,"last_update_time":1743203418,"comment_count":13,"num_deleted":0,"num_reports":0,"num_views":212,"category_num_users":8,"category_num_topics":120201,"category_num_posts":636112,"poll_id":0,"source":"","tags":[{"tag_id":48,"tag_text":"geometry","is_visible":true},{"tag_id":146,"tag_text":"3D geometry","is_visible":true},{"tag_id":240,"tag_text":"graph theory","is_visible":true}],"can_have_source":false,"locked":false,"forum_locked":0,"announce_type":"none","announce_through":"","announce_factor":0,"preview":"If I have a solved Rubik's cube, and I make a finite sequence of (legal) moves repeatedly, prove that I will eventually resolve the puzzle.<br>\n<br>\n(this wording is kinda goofy but i hope its sorta intuitive)","target_url":"","target_url_application":null,"target_text":"","state":"none","cat_can_target":0,"has_thanks":true,"has_nothanks":false,"is_bookmarked":false,"in_feed":false,"is_watched":false},"3537130":{"num_posts":3,"posts_data":[{"post_id":34396598,"topic_id":3537130,"poster_id":402075,"post_rendered":"source: some old mathdash monday, mine<br>\nLet <img src=\"\/\/latex.artofproblemsolving.com\/4\/3\/a\/43af0b10f7710738c2511d2395fbd714d617f48b.png\" class=\"latex\" alt=\"$f(n)=\\frac{n^3}{3}+\\frac{n^2}{2}+\\frac{n}{6}.$\" style=\"vertical-align: -13px\" width=\"171\" height=\"41\" > Compute <img src=\"\/\/latex.artofproblemsolving.com\/8\/7\/1\/87150724c5c83a6c5b21c75ab608758217dda9a9.png\" class=\"latex\" alt=\"$\\sum_{i=0}^{2023}(f(2024)-f(i))\\pmod{1000}.$\" style=\"vertical-align: -23px\" width=\"262\" height=\"55\" >","post_canonical":"source: some old mathdash monday, mine\nLet $f(n)=\\frac{n^3}{3}+\\frac{n^2}{2}+\\frac{n}{6}.$ Compute $\\sum_{i=0}^{2023}(f(2024)-f(i))\\pmod{1000}.$","username":"fruitmonster97","reported":false,"is_thanked":false,"is_nothanked":false,"attachment":false,"thanks_received":0,"nothanks_received":0,"thankers":null,"deleted":false,"post_number":1,"post_time":1743186079,"num_edits":0,"post_format":"bbcode","last_edit_time":0,"last_editor_username":"","last_edit_reason":"","admin":false,"avatar":"\/\/avatar.artofproblemsolving.com\/avatar_402075.jpg?t=1631296469","num_posts":2415,"editable":false,"deletable":false,"show_from_start":true,"show_from_end":false}],"topic_id":3537130,"category_id":4,"category_name":"High School Math","category_main_color":"#e75400","category_secondary_color":"#ffe7cc","topic_title":"dumb and bad","topic_type":"forum","is_public":true,"roles":{"38516":"mod","242520":"mod","53544":"mod"},"first_post_id":34396598,"first_poster_id":402075,"first_poster_name":"fruitmonster97","first_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_402075.jpg?t=1631296469","first_post_time":1743186079,"last_post_id":34398968,"last_poster_id":859268,"last_poster_name":"joeym2011","last_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_859268.jpg?t=1717795995","last_post_time":1743202183,"last_update_time":1743202183,"comment_count":3,"num_deleted":0,"num_reports":0,"num_views":81,"category_num_users":8,"category_num_topics":120201,"category_num_posts":636112,"poll_id":0,"source":"","tags":[{"tag_id":31381,"tag_text":"Summation","is_visible":true},{"tag_id":298,"tag_text":"function","is_visible":true}],"can_have_source":false,"locked":false,"forum_locked":0,"announce_type":"none","announce_through":"","announce_factor":0,"preview":"source: some old mathdash monday, mine<br>\nLet <img src=\"\/\/latex.artofproblemsolving.com\/4\/3\/a\/43af0b10f7710738c2511d2395fbd714d617f48b.png\" class=\"latex\" alt=\"$f(n)=\\frac{n^3}{3}+\\frac{n^2}{2}+\\frac{n}{6}.$\" style=\"vertical-align: -13px\" width=\"171\" height=\"41\" > Compute <img src=\"\/\/latex.artofproblemsolving.com\/8\/7\/1\/87150724c5c83a6c5b21c75ab608758217dda9a9.png\" class=\"latex\" alt=\"$\\sum_{i=0}^{2023}(f(2024)-f(i))\\pmod{1000}.$\" style=\"vertical-align: -23px\" width=\"262\" height=\"55\" >","target_url":"","target_url_application":null,"target_text":"","state":"none","cat_can_target":0,"has_thanks":true,"has_nothanks":false,"is_bookmarked":false,"in_feed":false,"is_watched":false},"3535124":{"num_posts":3,"posts_data":[{"post_id":34371055,"topic_id":3535124,"poster_id":779544,"post_rendered":"Three mutually tangent non-degenerate spheres rest on a plane. Let their centers be <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/4\/1\/8\/418d43d60810a2d318ec566fe2fdedaf702c56fa.png\" class=\"latex\" alt=\"$C_1, C_2$\" style=\"vertical-align: -3px\" width=\"47\" height=\"16\" >,<\/span> and <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/8\/5\/1\/85160260d550bf33be28c0a722237c3bc60e8c02.png\" class=\"latex\" alt=\"$C_3$\" style=\"vertical-align: -2px\" width=\"19\" height=\"15\" >.<\/span> The spheres with centers <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/4\/1\/8\/418d43d60810a2d318ec566fe2fdedaf702c56fa.png\" class=\"latex\" alt=\"$C_1, C_2$\" style=\"vertical-align: -3px\" width=\"47\" height=\"16\" >,<\/span> and <img src=\"\/\/latex.artofproblemsolving.com\/8\/5\/1\/85160260d550bf33be28c0a722237c3bc60e8c02.png\" class=\"latex\" alt=\"$C_3$\" style=\"vertical-align: -2px\" width=\"19\" height=\"15\" > touch the plane at <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/d\/9\/d\/d9d5126d7d1f6b2eb9460290573a1e28afdc2454.png\" class=\"latex\" alt=\"$P_1, P_2$\" style=\"vertical-align: -3px\" width=\"44\" height=\"16\" >,<\/span> and <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/9\/0\/9\/9090f2e6bd5da465fea10a8c755d64e9185f61f1.png\" class=\"latex\" alt=\"$P_3$\" style=\"vertical-align: -2px\" width=\"17\" height=\"15\" >,<\/span> respectively. Prove that <img src=\"\/\/latex.artofproblemsolving.com\/d\/7\/5\/d751b767e7cebde1922aaeaaae8e047ebe0568fe.png\" class=\"latexcenter\" alt=\"$$\\frac{(P_1P_2)(P_2P_3)(P_1P_3)}{(C_1P_1)(C_2P_2)(C_3P_3)}=8$$\" width=\"196\" height=\"43\" >and <img src=\"\/\/latex.artofproblemsolving.com\/3\/e\/0\/3e09d32e08c196ef1911c82751df4cfb132a6080.png\" class=\"latexcenter\" alt=\"$$(C_1C_2)^2+(C_1C_3)^2+(C_2C_3)2+\\frac{1}{(C_1P_1)^2}+\\frac{1}{(C_2P_2)^2}+\\frac{1}{(C_3P_3)^2} \\ge 6.$$\" width=\"531\" height=\"41\" ><br>\nSource: Own","post_canonical":"Three mutually tangent non-degenerate spheres rest on a plane. Let their centers be $C_1, C_2$, and $C_3$. The spheres with centers $C_1, C_2$, and $C_3$ touch the plane at $P_1, P_2$, and $P_3$, respectively. Prove that $$\\frac{(P_1P_2)(P_2P_3)(P_1P_3)}{(C_1P_1)(C_2P_2)(C_3P_3)}=8$$ and $$(C_1C_2)^2+(C_1C_3)^2+(C_2C_3)2+\\frac{1}{(C_1P_1)^2}+\\frac{1}{(C_2P_2)^2}+\\frac{1}{(C_3P_3)^2} \\ge 6.$$\n\nSource: Own","username":"ReticulatedPython","reported":false,"is_thanked":false,"is_nothanked":false,"attachment":false,"thanks_received":0,"nothanks_received":0,"thankers":null,"deleted":false,"post_number":1,"post_time":1742933528,"num_edits":7,"post_format":"bbcode","last_edit_time":1743091365,"last_editor_username":"ReticulatedPython","last_edit_reason":"","admin":false,"avatar":"\/\/avatar.artofproblemsolving.com\/avatar_779544.jpeg?t=1712157413","num_posts":549,"editable":false,"deletable":false,"show_from_start":true,"show_from_end":false}],"topic_id":3535124,"category_id":4,"category_name":"High School Math","category_main_color":"#e75400","category_secondary_color":"#ffe7cc","topic_title":"3D Geometry Problem","topic_type":"forum","is_public":true,"roles":{"38516":"mod","242520":"mod","53544":"mod"},"first_post_id":34371055,"first_poster_id":779544,"first_poster_name":"ReticulatedPython","first_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_779544.jpeg?t=1712157413","first_post_time":1742933528,"last_post_id":34398913,"last_poster_id":859268,"last_poster_name":"joeym2011","last_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_859268.jpg?t=1717795995","last_post_time":1743201831,"last_update_time":1743201831,"comment_count":3,"num_deleted":1,"num_reports":0,"num_views":156,"category_num_users":8,"category_num_topics":120201,"category_num_posts":636112,"poll_id":0,"source":"","tags":[{"tag_id":48,"tag_text":"geometry","is_visible":true},{"tag_id":146,"tag_text":"3D geometry","is_visible":true},{"tag_id":149,"tag_text":"sphere","is_visible":true}],"can_have_source":false,"locked":false,"forum_locked":0,"announce_type":"none","announce_through":"","announce_factor":0,"preview":"Three mutually tangent non-degenerate spheres rest on a plane. Let their centers be <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/4\/1\/8\/418d43d60810a2d318ec566fe2fdedaf702c56fa.png\" class=\"latex\" alt=\"$C_1, C_2$\" style=\"vertical-align: -3px\" width=\"47\" height=\"16\" >,<\/span> and <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/8\/5\/1\/85160260d550bf33be28c0a722237c3bc60e8c02.png\" class=\"latex\" alt=\"$C_3$\" style=\"vertical-align: -2px\" width=\"19\" height=\"15\" >.<\/span> The spheres with centers <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/4\/1\/8\/418d43d60810a2d318ec566fe2fdedaf702c56fa.png\" class=\"latex\" alt=\"$C_1, C_2$\" style=\"vertical-align: -3px\" width=\"47\" height=\"16\" >,<\/span> and <img src=\"\/\/latex.artofproblemsolving.com\/8\/5\/1\/85160260d550bf33be28c0a722237c3bc60e8c02.png\" class=\"latex\" alt=\"$C_3$\" style=\"vertical-align: -2px\" width=\"19\" height=\"15\" > touch the plane at <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/d\/9\/d\/d9d5126d7d1f6b2eb9460290573a1e28afdc2454.png\" class=\"latex\" alt=\"$P_1, P_2$\" style=\"vertical-align: -3px\" width=\"44\" height=\"16\" >,<\/span> and <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/9\/0\/9\/9090f2e6bd5da465fea10a8c755d64e9185f61f1.png\" class=\"latex\" alt=\"$P_3$\" style=\"vertical-align: -2px\" width=\"17\" height=\"15\" >,<\/span> respectively. Prove that <img src=\"\/\/latex.artofproblemsolving.com\/d\/7\/5\/d751b767e7cebde1922aaeaaae8e047ebe0568fe.png\" class=\"latexcenter\" alt=\"$$\\frac{(P_1P_2)(P_2P_3)(P_1P_3)}{(C_1P_1)(C_2P_2)(C_3P_3)}=8$$\" width=\"196\" height=\"43\" >and <img src=\"\/\/latex.artofproblemsolving.com\/3\/e\/0\/3e09d32e08c196ef1911c82751df4cfb132a6080.png\" class=\"latexcenter\" alt=\"$$(C_1C_2)^2+(C_1C_3)^2+(C_2C_3)2+\\frac{1}{(C_1P_1)^2}+\\frac{1}{(C_2P_2)^2}+\\frac{1}{(C_3P_3)^2} \\ge 6.$$\" width=\"531\" height=\"41\" ><br>\nSource: Own","target_url":"","target_url_application":null,"target_text":"","state":"none","cat_can_target":0,"has_thanks":true,"has_nothanks":false,"is_bookmarked":false,"in_feed":false,"is_watched":false},"3534955":{"num_posts":2,"posts_data":[{"post_id":34368819,"topic_id":3534955,"poster_id":1212513,"post_rendered":"A set <img src=\"\/\/latex.artofproblemsolving.com\/8\/1\/c\/81c236418d5594564911b2be04456679e4d5bb18.png\" class=\"latex\" alt=\"\\( A \\subset \\mathbb{R} \\)\" style=\"vertical-align: -2px\" width=\"53\" height=\"15\" > is called a <img src=\"\/\/latex.artofproblemsolving.com\/d\/6\/8\/d689e7f2cc1ed6ed498aea80872d7d63bccbe897.png\" class=\"latex\" alt=\"$\\textit{nice}$\" width=\"29\" height=\"11\" > if it satisfies the following conditions:<br>\n<img src=\"\/\/latex.artofproblemsolving.com\/f\/5\/6\/f5655b34d3ed271b063522a7cbca0ff3865961e0.png\" class=\"latex\" alt=\"$i)$\" style=\"vertical-align: -4px\" width=\"12\" height=\"18\" > <img src=\"\/\/latex.artofproblemsolving.com\/3\/0\/8\/3084f7330c45acbc2642845f420d9618cf2faaef.png\" class=\"latex\" alt=\"\\( A \\)\" style=\"vertical-align: -1px\" width=\"15\" height=\"14\" > contains at least two elements.<br>\n<img src=\"\/\/latex.artofproblemsolving.com\/e\/3\/0\/e306d3df061bd255de2d15429decdf8192753b7b.png\" class=\"latex\" alt=\"$ii)$\" style=\"vertical-align: -4px\" width=\"18\" height=\"18\" > For all <img src=\"\/\/latex.artofproblemsolving.com\/1\/d\/1\/1d1d2eedd3890536dae47e99e2dc6f86a9c9f715.png\" class=\"latex\" alt=\"\\( x, y \\in A \\)\" style=\"vertical-align: -4px\" width=\"66\" height=\"18\" > with <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/b\/0\/5\/b057529d44058cf8c43560d5a6920a64e64f898e.png\" class=\"latex\" alt=\"\\( x \\neq y \\)\" style=\"vertical-align: -4px\" width=\"46\" height=\"18\" >,<\/span> we have <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/d\/6\/f\/d6f6f38896a8f3b2c731ce1f9ae7b51e9f247e7e.png\" class=\"latex\" alt=\"\\( xy(x+y) \\neq 0 \\)\" style=\"vertical-align: -5px\" width=\"111\" height=\"20\" >,<\/span> and among the two numbers <img src=\"\/\/latex.artofproblemsolving.com\/1\/8\/4\/184499e539c31fc66c8f38e733def74669087352.png\" class=\"latex\" alt=\"\\( x+y \\)\" style=\"vertical-align: -4px\" width=\"44\" height=\"16\" > and <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/6\/b\/b\/6bbdee607da2e2578ff4dba7994e50bd0a622e99.png\" class=\"latex\" alt=\"\\( xy \\)\" style=\"vertical-align: -4px\" width=\"22\" height=\"13\" >,<\/span> exactly one is rational.<br>\n<img src=\"\/\/latex.artofproblemsolving.com\/5\/2\/8\/528d903396faf28af16a6a4340b29a3584acce19.png\" class=\"latex\" alt=\"$iii)$\" style=\"vertical-align: -4px\" width=\"25\" height=\"18\" > For all <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/8\/4\/5\/845977bf6745fb9948a07bcdc187382f42f5dd41.png\" class=\"latex\" alt=\"\\( x \\in A \\)\" style=\"vertical-align: -2px\" width=\"48\" height=\"15\" >,<\/span> <img src=\"\/\/latex.artofproblemsolving.com\/8\/d\/8\/8d85f295db843ebbd3a453d72696a84253cb9e8e.png\" class=\"latex\" alt=\"\\( x^2 \\)\" style=\"vertical-align: -1px\" width=\"20\" height=\"16\" > is irrational.<br>\nWhat is the maximum number of elements that <img src=\"\/\/latex.artofproblemsolving.com\/3\/0\/8\/3084f7330c45acbc2642845f420d9618cf2faaef.png\" class=\"latex\" alt=\"\\( A \\)\" style=\"vertical-align: -1px\" width=\"15\" height=\"14\" > can have?","post_canonical":"A set \\( A \\subset \\mathbb{R} \\) is called a $\\textit{nice}$ if it satisfies the following conditions:\n$i)$ \\( A \\) contains at least two elements.\n $ii)$ For all \\( x, y \\in A \\) with \\( x \\neq y \\), we have \\( xy(x+y) \\neq 0 \\), and among the two numbers \\( x+y \\) and \\( xy \\), exactly one is rational.\n $iii)$ For all \\( x \\in A \\), \\( x^2 \\) is irrational.\nWhat is the maximum number of elements that \\( A \\) can have?\n \n\n","username":"toanrathay","reported":false,"is_thanked":false,"is_nothanked":false,"attachment":false,"thanks_received":0,"nothanks_received":0,"thankers":null,"deleted":false,"post_number":1,"post_time":1742917001,"num_edits":0,"post_format":"bbcode","last_edit_time":0,"last_editor_username":"","last_edit_reason":"","admin":false,"avatar":"\/\/avatar.artofproblemsolving.com\/avatar_1212513.png","num_posts":25,"editable":false,"deletable":false,"show_from_start":true,"show_from_end":false}],"topic_id":3534955,"category_id":4,"category_name":"High School Math","category_main_color":"#e75400","category_secondary_color":"#ffe7cc","topic_title":"Problem of set","topic_type":"forum","is_public":true,"roles":{"38516":"mod","242520":"mod","53544":"mod"},"first_post_id":34368819,"first_poster_id":1212513,"first_poster_name":"toanrathay","first_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_1212513.png","first_post_time":1742917001,"last_post_id":34398386,"last_poster_id":209000,"last_poster_name":"alexheinis","last_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_209000.png","last_post_time":1743197321,"last_update_time":1743197321,"comment_count":2,"num_deleted":0,"num_reports":0,"num_views":103,"category_num_users":8,"category_num_topics":120201,"category_num_posts":636112,"poll_id":0,"source":"","tags":[{"tag_id":177,"tag_text":"number theory","is_visible":true},{"tag_id":309,"tag_text":"combinatorics","is_visible":true}],"can_have_source":false,"locked":false,"forum_locked":0,"announce_type":"none","announce_through":"","announce_factor":0,"preview":"A set <img src=\"\/\/latex.artofproblemsolving.com\/8\/1\/c\/81c236418d5594564911b2be04456679e4d5bb18.png\" class=\"latex\" alt=\"\\( A \\subset \\mathbb{R} \\)\" style=\"vertical-align: -2px\" width=\"53\" height=\"15\" > is called a <img src=\"\/\/latex.artofproblemsolving.com\/d\/6\/8\/d689e7f2cc1ed6ed498aea80872d7d63bccbe897.png\" class=\"latex\" alt=\"$\\textit{nice}$\" width=\"29\" height=\"11\" > if it satisfies the following conditions:<br>\n<img src=\"\/\/latex.artofproblemsolving.com\/f\/5\/6\/f5655b34d3ed271b063522a7cbca0ff3865961e0.png\" class=\"latex\" alt=\"$i)$\" style=\"vertical-align: -4px\" width=\"12\" height=\"18\" > <img src=\"\/\/latex.artofproblemsolving.com\/3\/0\/8\/3084f7330c45acbc2642845f420d9618cf2faaef.png\" class=\"latex\" alt=\"\\( A \\)\" style=\"vertical-align: -1px\" width=\"15\" height=\"14\" > contains at least two elements.<br>\n<img src=\"\/\/latex.artofproblemsolving.com\/e\/3\/0\/e306d3df061bd255de2d15429decdf8192753b7b.png\" class=\"latex\" alt=\"$ii)$\" style=\"vertical-align: -4px\" width=\"18\" height=\"18\" > For all <img src=\"\/\/latex.artofproblemsolving.com\/1\/d\/1\/1d1d2eedd3890536dae47e99e2dc6f86a9c9f715.png\" class=\"latex\" alt=\"\\( x, y \\in A \\)\" style=\"vertical-align: -4px\" width=\"66\" height=\"18\" > with <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/b\/0\/5\/b057529d44058cf8c43560d5a6920a64e64f898e.png\" class=\"latex\" alt=\"\\( x \\neq y \\)\" style=\"vertical-align: -4px\" width=\"46\" height=\"18\" >,<\/span> we have <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/d\/6\/f\/d6f6f38896a8f3b2c731ce1f9ae7b51e9f247e7e.png\" class=\"latex\" alt=\"\\( xy(x+y) \\neq 0 \\)\" style=\"vertical-align: -5px\" width=\"111\" height=\"20\" >,<\/span> and among the two numbers <img src=\"\/\/latex.artofproblemsolving.com\/1\/8\/4\/184499e539c31fc66c8f38e733def74669087352.png\" class=\"latex\" alt=\"\\( x+y \\)\" style=\"vertical-align: -4px\" width=\"44\" height=\"16\" > and <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/6\/b\/b\/6bbdee607da2e2578ff4dba7994e50bd0a622e99.png\" class=\"latex\" alt=\"\\( xy \\)\" style=\"vertical-align: -4px\" width=\"22\" height=\"13\" >,<\/span> exactly one is rational.<br>\n<img src=\"\/\/latex.artofproblemsolving.com\/5\/2\/8\/528d903396faf28af16a6a4340b29a3584acce19.png\" class=\"latex\" alt=\"$iii)$\" style=\"vertical-align: -4px\" width=\"25\" height=\"18\" > For all <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/8\/4\/5\/845977bf6745fb9948a07bcdc187382f42f5dd41.png\" class=\"latex\" alt=\"\\( x \\in A \\)\" style=\"vertical-align: -2px\" width=\"48\" height=\"15\" >,<\/span> <img src=\"\/\/latex.artofproblemsolving.com\/8\/d\/8\/8d85f295db843ebbd3a453d72696a84253cb9e8e.png\" class=\"latex\" alt=\"\\( x^2 \\)\" style=\"vertical-align: -1px\" width=\"20\" height=\"16\" > is irrational.<br>\nWhat is the maximum number of elements that <img src=\"\/\/latex.artofproblemsolving.com\/3\/0\/8\/3084f7330c45acbc2642845f420d9618cf2faaef.png\" class=\"latex\" alt=\"\\( A \\)\" style=\"vertical-align: -1px\" width=\"15\" height=\"14\" > can have?<br>\n<br>\n<br>\n","target_url":"","target_url_application":null,"target_text":"","state":"none","cat_can_target":0,"has_thanks":true,"has_nothanks":false,"is_bookmarked":false,"in_feed":false,"is_watched":false}};AoPS.preload_topics[5]={"3536664":{"num_posts":52,"posts_data":[{"post_id":34391094,"topic_id":3536664,"poster_id":964376,"post_rendered":"\n<div class=\"bbcode_center\" style=\"text-align:center\">\n<br>\nMe (PatTheKing806) and EaZ_Shadow have created a <b>AMC 10\/AIME Study Forum!<\/b> Hopefully, this forum wont die quickly. To signup, do \/join or \\join.<br>\n<br>\n<a href=\"https:\/\/artofproblemsolving.com\/community\/c4273272_amc_10aime_study_forum\" class=\"bbcode_url\" target=\"_blank\">Click here to join! (or do some pushups) <img src=\"\/assets\/images\/smilies\/tongue.gif\" alt=\":P\" title=\":P\" class=\"bbcode_smiley\" \/><\/a><br>\n<br>\nPeople should join this forum if they are wanting to do well on the AMC 10 next year, trying get into AIME, or <b>loves<\/b> math!\n<\/div>\n","post_canonical":"[center]\n\nMe (PatTheKing806) and EaZ_Shadow have created a [b]AMC 10\/AIME Study Forum![\/b] Hopefully, this forum wont die quickly. To signup, do \/join or \\join.\n\n[url=https:\/\/artofproblemsolving.com\/community\/c4273272_amc_10aime_study_forum]Click here to join! (or do some pushups) :P[\/url]\n\nPeople should join this forum if they are wanting to do well on the AMC 10 next year, trying get into AIME, or [b]loves[\/b] math!","username":"PatTheKing806","reported":false,"is_thanked":false,"is_nothanked":false,"attachment":false,"thanks_received":0,"nothanks_received":0,"thankers":null,"deleted":false,"post_number":1,"post_time":1743118468,"num_edits":2,"post_format":"bbcode","last_edit_time":1743118528,"last_editor_username":"PatTheKing806","last_edit_reason":"","admin":false,"avatar":"\/\/avatar.artofproblemsolving.com\/avatar_964376.png?t=1728169699","num_posts":1005,"editable":false,"deletable":false,"show_from_start":true,"show_from_end":false}],"topic_id":3536664,"category_id":5,"category_name":"Contests & Programs","category_main_color":"#008fd5","category_secondary_color":"#d9effd","topic_title":"AMC 10\/AIME Study Forum","topic_type":"forum","is_public":true,"roles":{"53544":"mod","93494":"mod","86424":"mod","1662":"mod","38516":"mod","242520":"mod","159507":"mod","565384":"mod"},"first_post_id":34391094,"first_poster_id":964376,"first_poster_name":"PatTheKing806","first_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_964376.png?t=1728169699","first_post_time":1743118468,"last_post_id":34398940,"last_poster_id":1155512,"last_poster_name":"RabtejKalra","last_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_1155512.png","last_post_time":1743201986,"last_update_time":1743201986,"comment_count":52,"num_deleted":4,"num_reports":0,"num_views":859,"category_num_users":6,"category_num_topics":34573,"category_num_posts":570355,"poll_id":0,"source":"","tags":[{"tag_id":30366,"tag_text":"Forum","is_visible":true}],"can_have_source":true,"locked":false,"forum_locked":0,"announce_type":"none","announce_through":"","announce_factor":0,"preview":"[center]<br>\n<br>\nMe (PatTheKing806) and EaZ_Shadow have created a <b>AMC 10\/AIME Study Forum!<\/b> Hopefully, this forum wont die quickly. To signup, do \/join or \\join.<br>\n<br>\n<a href=\"https:\/\/artofproblemsolving.com\/community\/c4273272_amc_10aime_study_forum\" class=\"bbcode_url\">Click here to join! (or do some pushups) :P<\/a><br>\n<br>\nPeople should join this forum if they are wanting to do well on the AMC 10 next year, trying get into AIME, or <b>loves<\/b> math!","target_url":"","target_url_application":null,"target_text":"","state":"none","cat_can_target":0,"has_thanks":true,"has_nothanks":false,"is_bookmarked":false,"in_feed":false,"is_watched":false},"347279":{"num_posts":81,"posts_data":[{"post_id":1860753,"topic_id":347279,"poster_id":61204,"post_rendered":"Let <img src=\"\/\/latex.artofproblemsolving.com\/e\/2\/a\/e2a559986ed5a0ffc5654bd367c29dfc92913c36.png\" class=\"latex\" alt=\"$ABC$\" width=\"42\" height=\"13\" > be a triangle with <span style=\"white-space:nowrap;\"><img src=\"\/\/latex.artofproblemsolving.com\/e\/a\/4\/ea494684dc60997279257d5a54e877a1fb28ca60.png\" class=\"latex\" alt=\"$\\angle A = 90^{\\circ}$\" width=\"74\" height=\"13\" >.<\/span> Points <img src=\"\/\/latex.artofproblemsolving.com\/9\/f\/f\/9ffb448918db29f2a72f8f87f421b3b3cad18f95.png\" class=\"latex\" alt=\"$D$\" width=\"15\" height=\"12\" > and <img src=\"\/\/latex.artofproblemsolving.com\/f\/a\/2\/fa2fa899f0afb05d6837885523503a2d4df434f9.png\" class=\"latex\" alt=\"$E$\" width=\"14\" height=\"12\" > lie on sides <img src=\"\/\/latex.artofproblemsolving.com\/a\/1\/7\/a179ff2638e4799cadd820db205c2beff6299ce9.png\" class=\"latex\" alt=\"$AC$\" width=\"27\" height=\"13\" > and <span style=\"white-space:nowrap;\"><img src=\"\/\/latex.artofproblemsolving.com\/5\/7\/e\/57ee5125358c0606c9b588580ddfa66f83e607b7.png\" class=\"latex\" alt=\"$AB$\" width=\"27\" height=\"13\" >,<\/span> respectively, such that <img src=\"\/\/latex.artofproblemsolving.com\/8\/a\/b\/8abe9cd198b78eb776b95d375c73a0553fa224e9.png\" class=\"latex\" alt=\"$\\angle ABD = \\angle DBC$\" width=\"138\" height=\"13\" > and <span style=\"white-space:nowrap;\"><img src=\"\/\/latex.artofproblemsolving.com\/9\/2\/7\/92795aed81f4bb5f45bd951633bfe47843d98778.png\" class=\"latex\" alt=\"$\\angle ACE = \\angle ECB$\" width=\"135\" height=\"13\" >.<\/span> Segments <img src=\"\/\/latex.artofproblemsolving.com\/8\/9\/f\/89fff82bb65d0215e49c8c91cb7c553da52205e2.png\" class=\"latex\" alt=\"$BD$\" width=\"29\" height=\"12\" > and <img src=\"\/\/latex.artofproblemsolving.com\/1\/3\/c\/13c970cbf5e2e7029089517d5e8efbb8cade6d8a.png\" class=\"latex\" alt=\"$CE$\" width=\"28\" height=\"12\" > meet at <span style=\"white-space:nowrap;\"><img src=\"\/\/latex.artofproblemsolving.com\/0\/2\/7\/027f4a11d6090f9eac0ce2488df6384dad1263ea.png\" class=\"latex\" alt=\"$I$\" width=\"9\" height=\"12\" >.<\/span> Determine whether or not it is possible for segments <span style=\"white-space:nowrap;\"><img src=\"\/\/latex.artofproblemsolving.com\/5\/7\/e\/57ee5125358c0606c9b588580ddfa66f83e607b7.png\" class=\"latex\" alt=\"$AB$\" width=\"27\" height=\"13\" >,<\/span> <span style=\"white-space:nowrap;\"><img src=\"\/\/latex.artofproblemsolving.com\/a\/1\/7\/a179ff2638e4799cadd820db205c2beff6299ce9.png\" class=\"latex\" alt=\"$AC$\" width=\"27\" height=\"13\" >,<\/span> <span style=\"white-space:nowrap;\"><img src=\"\/\/latex.artofproblemsolving.com\/3\/5\/b\/35bb3f3bcadf5dc5c045b80fc24d014c0b969cd4.png\" class=\"latex\" alt=\"$BI$\" width=\"24\" height=\"12\" >,<\/span> <span style=\"white-space:nowrap;\"><img src=\"\/\/latex.artofproblemsolving.com\/f\/0\/b\/f0bfbf26e6ab98a461a5cb0964c6778a20122f07.png\" class=\"latex\" alt=\"$ID$\" width=\"24\" height=\"12\" >,<\/span> <span style=\"white-space:nowrap;\"><img src=\"\/\/latex.artofproblemsolving.com\/a\/8\/d\/a8d37413bae1973358eebff164711f292d16be56.png\" class=\"latex\" alt=\"$CI$\" width=\"24\" height=\"12\" >,<\/span> <img src=\"\/\/latex.artofproblemsolving.com\/5\/1\/f\/51f69010cc07adcd7f79449395facf8bfc06b6e5.png\" class=\"latex\" alt=\"$IE$\" width=\"23\" height=\"12\" > to all have integer lengths.","post_canonical":"Let $ABC$ be a triangle with $\\angle A = 90^{\\circ}$. Points $D$ and $E$ lie on sides $AC$ and $AB$, respectively, such that $\\angle ABD = \\angle DBC$ and $\\angle ACE = \\angle ECB$. Segments $BD$ and $CE$ meet at $I$. Determine whether or not it is possible for segments $AB$, $AC$, $BI$, $ID$, $CI$, $IE$ to all have integer lengths.","username":"BarbieRocks","reported":false,"is_thanked":false,"is_nothanked":false,"attachment":false,"thanks_received":7,"nothanks_received":0,"thankers":"Davi-8191, OlympusHero, samrocksnature, HWenslawski, megarnie, Adventure10, Mango247","deleted":false,"post_number":1,"post_time":1272546747,"num_edits":0,"post_format":"bbcode","last_edit_time":0,"last_editor_username":"","last_edit_reason":"","admin":false,"avatar":"\/\/avatar.artofproblemsolving.com\/avatar_61204.png","num_posts":1102,"editable":false,"deletable":false,"show_from_start":true,"show_from_end":false}],"topic_id":347279,"category_id":5,"category_name":"Contests & Programs","category_main_color":"#008fd5","category_secondary_color":"#d9effd","topic_title":"A lot of integer lengths: JMO #6 or USAMO Problem 4","topic_type":"forum","is_public":true,"roles":{"53544":"mod","93494":"mod","86424":"mod","1662":"mod","38516":"mod","242520":"mod","159507":"mod","565384":"mod"},"first_post_id":1860753,"first_poster_id":61204,"first_poster_name":"BarbieRocks","first_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_61204.png","first_post_time":1272546747,"last_post_id":34398788,"last_poster_id":871434,"last_poster_name":"Maximilian113","last_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_871434.png","last_post_time":1743200729,"last_update_time":1743200729,"comment_count":81,"num_deleted":6,"num_reports":0,"num_views":12540,"category_num_users":6,"category_num_topics":34573,"category_num_posts":570355,"poll_id":0,"source":"","tags":[{"tag_id":3,"tag_text":"AMC","is_visible":true},{"tag_id":174,"tag_text":"USA(J)MO","is_visible":true},{"tag_id":175,"tag_text":"USAMO","is_visible":true},{"tag_id":194,"tag_text":"trigonometry","is_visible":true},{"tag_id":48,"tag_text":"geometry","is_visible":true},{"tag_id":120,"tag_text":"incenter","is_visible":true},{"tag_id":121,"tag_text":"inradius","is_visible":true}],"can_have_source":true,"locked":false,"forum_locked":0,"announce_type":"none","announce_through":"","announce_factor":0,"preview":"Let <img src=\"\/\/latex.artofproblemsolving.com\/e\/2\/a\/e2a559986ed5a0ffc5654bd367c29dfc92913c36.png\" class=\"latex\" alt=\"$ABC$\" width=\"42\" height=\"13\" > be a triangle with <span style=\"white-space:nowrap;\"><img src=\"\/\/latex.artofproblemsolving.com\/e\/a\/4\/ea494684dc60997279257d5a54e877a1fb28ca60.png\" class=\"latex\" alt=\"$\\angle A = 90^{\\circ}$\" width=\"74\" height=\"13\" >.<\/span> Points <img src=\"\/\/latex.artofproblemsolving.com\/9\/f\/f\/9ffb448918db29f2a72f8f87f421b3b3cad18f95.png\" class=\"latex\" alt=\"$D$\" width=\"15\" height=\"12\" > and <img src=\"\/\/latex.artofproblemsolving.com\/f\/a\/2\/fa2fa899f0afb05d6837885523503a2d4df434f9.png\" class=\"latex\" alt=\"$E$\" width=\"14\" height=\"12\" > lie on sides <img src=\"\/\/latex.artofproblemsolving.com\/a\/1\/7\/a179ff2638e4799cadd820db205c2beff6299ce9.png\" class=\"latex\" alt=\"$AC$\" width=\"27\" height=\"13\" > and <span style=\"white-space:nowrap;\"><img src=\"\/\/latex.artofproblemsolving.com\/5\/7\/e\/57ee5125358c0606c9b588580ddfa66f83e607b7.png\" class=\"latex\" alt=\"$AB$\" width=\"27\" height=\"13\" >,<\/span> respectively, such that <img src=\"\/\/latex.artofproblemsolving.com\/8\/a\/b\/8abe9cd198b78eb776b95d375c73a0553fa224e9.png\" class=\"latex\" alt=\"$\\angle ABD = \\angle DBC$\" width=\"138\" height=\"13\" > and <span style=\"white-space:nowrap;\"><img src=\"\/\/latex.artofproblemsolving.com\/9\/2\/7\/92795aed81f4bb5f45bd951633bfe47843d98778.png\" class=\"latex\" alt=\"$\\angle ACE = \\angle ECB$\" width=\"135\" height=\"13\" >.<\/span> Segments <img src=\"\/\/latex.artofproblemsolving.com\/8\/9\/f\/89fff82bb65d0215e49c8c91cb7c553da52205e2.png\" class=\"latex\" alt=\"$BD$\" width=\"29\" height=\"12\" > and <img src=\"\/\/latex.artofproblemsolving.com\/1\/3\/c\/13c970cbf5e2e7029089517d5e8efbb8cade6d8a.png\" class=\"latex\" alt=\"$CE$\" width=\"28\" height=\"12\" > meet at <span style=\"white-space:nowrap;\"><img src=\"\/\/latex.artofproblemsolving.com\/0\/2\/7\/027f4a11d6090f9eac0ce2488df6384dad1263ea.png\" class=\"latex\" alt=\"$I$\" width=\"9\" height=\"12\" >.<\/span> Determine whether or not it is possible for segments <span style=\"white-space:nowrap;\"><img src=\"\/\/latex.artofproblemsolving.com\/5\/7\/e\/57ee5125358c0606c9b588580ddfa66f83e607b7.png\" class=\"latex\" alt=\"$AB$\" width=\"27\" height=\"13\" >,<\/span> <span style=\"white-space:nowrap;\"><img src=\"\/\/latex.artofproblemsolving.com\/a\/1\/7\/a179ff2638e4799cadd820db205c2beff6299ce9.png\" class=\"latex\" alt=\"$AC$\" width=\"27\" height=\"13\" >,<\/span> <span style=\"white-space:nowrap;\"><img src=\"\/\/latex.artofproblemsolving.com\/3\/5\/b\/35bb3f3bcadf5dc5c045b80fc24d014c0b969cd4.png\" class=\"latex\" alt=\"$BI$\" width=\"24\" height=\"12\" >,<\/span> <span style=\"white-space:nowrap;\"><img src=\"\/\/latex.artofproblemsolving.com\/f\/0\/b\/f0bfbf26e6ab98a461a5cb0964c6778a20122f07.png\" class=\"latex\" alt=\"$ID$\" width=\"24\" height=\"12\" >,<\/span> <span style=\"white-space:nowrap;\"><img src=\"\/\/latex.artofproblemsolving.com\/a\/8\/d\/a8d37413bae1973358eebff164711f292d16be56.png\" class=\"latex\" alt=\"$CI$\" width=\"24\" height=\"12\" >,<\/span> <img src=\"\/\/latex.artofproblemsolving.com\/5\/1\/f\/51f69010cc07adcd7f79449395facf8bfc06b6e5.png\" class=\"latex\" alt=\"$IE$\" width=\"23\" height=\"12\" > to all have integer lengths.","target_url":"","target_url_application":null,"target_text":"","state":"none","cat_can_target":0,"has_thanks":true,"has_nothanks":false,"is_bookmarked":false,"in_feed":false,"is_watched":false},"3487694":{"num_posts":14,"posts_data":[{"post_id":33835452,"topic_id":3487694,"poster_id":550672,"post_rendered":"<b><span style=\"color:#674EA7\"><span class=\"bbfont-double\">Join the 2025 Math and AI 4 Girls Competition for a chance to win up to $1,000!<\/span><\/span><\/b><br>\n<br>\nHey Everyone, I'm pleased to announce the dates for the 2025 MA4G Competition are set!<br>\nApplications will open on<b> March 22nd, 2025<\/b>, and they will close on <b>April 26th, 2025<\/b> (@ 11:59pm PST).<br>\n<br>\nApplicants will have one month to fill out an application with <u>prizes for the top 50 contestants & <b>cash prizes for the top 20 contestants (including $1,000 for the winner!)<\/b><\/u>. More details below!<br>\n<br>\n<b>Eligibility:<\/b><br>\nThe competition is <b>free to enter<\/b>, and open to middle school female students living in the US (5th-8th grade).<br>\nAward recipients are selected based on their aptitude, activities and aspirations in STEM.<br>\n<br>\n<b>Event dates:<\/b><br>\nApplications will open on<b> March 22nd, 2025<\/b>, and they will close on <b>April 26th, 2025<\/b> (by 11:59pm PST)<br>\nWinners will be announced on <b>June 28, 2025<\/b> during an online award ceremony.<br>\n<br>\n<b>Application requirements:<\/b><br>\nComplete a 12 question problem set on math and computer science\/AI related topics<br>\nWrite 2 short essays<br>\n<br>\n<b><span class=\"bbfont-one-five\"><span style=\"color:#674EA7\">Prizes:<br>\n1st place: $1,000 Cash prize<br>\n2nd place: $500 Cash prize<br>\n3rd place: $300 Cash prize<br>\n4th-10th: $100 Cash prize each<br>\n11th-20th: $50 Cash prize each<br>\nTop 50 contestants: Over $50 worth of gadgets and stationary<\/span><\/span><\/b><br>\n<br>\n<i>Many thanks to our current and past sponsors and partners: Hudson River Trading, MATHCOUNTS, Hewlett Packard Enterprise, Automation Anywhere, JP Morgan Chase, D.E. Shaw, and AI4ALL.<\/i><br>\n<br>\n<i><b><span class=\"bbfont-regular\"><span style=\"color:#8E7CC3\">Math and AI 4 Girls is a nonprofit organization aiming to encourage young girls to develop an interest in math and AI by taking part in STEM competitions and activities at an early age. The organization will be hosting an inaugural Math and AI 4 Girls competition to identify talent and encourage long-term planning of academic and career goals in STEM.<\/span><\/span><\/b><\/i><br>\n<br>\n<b>Contact:<\/b><br>\n<a href=\"https:\/\/mathandAI4girls@yahoo.com\" class=\"bbcode_url\" target=\"_blank\" rel=\"nofollow\">mathandAI4girls@yahoo.com<\/a><br>\n<br>\n<b>For more information on the competition:<\/b><br>\n<a href=\"https:\/\/www.mathandai4girls.org\/math-and-ai-4-girls-competition\" class=\"bbcode_url\" target=\"_blank\" rel=\"nofollow\">https:\/\/www.mathandai4girls.org\/math-and-ai-4-girls-competition<\/a><br>\n<br>\n<b>More information on how to register will be posted on the website. If you have any questions, please ask here!<\/b>","post_canonical":"[b][color=#674EA7][size=250]Join the 2025 Math and AI 4 Girls Competition for a chance to win up to \\$1,000![\/size][\/color][\/b]\n\nHey Everyone, I'm pleased to announce the dates for the 2025 MA4G Competition are set! \nApplications will open on[b] March 22nd, 2025[\/b], and they will close on [b]April 26th, 2025[\/b] (@ 11:59pm PST).\n\nApplicants will have one month to fill out an application with [u]prizes for the top 50 contestants & [b]cash prizes for the top 20 contestants (including \\$1,000 for the winner!)[\/b][\/u]. More details below!\n\n[b]Eligibility:[\/b]\nThe competition is [b]free to enter[\/b], and open to middle school female students living in the US (5th-8th grade).\nAward recipients are selected based on their aptitude, activities and aspirations in STEM.\n\n[b]Event dates:[\/b]\nApplications will open on[b] March 22nd, 2025[\/b], and they will close on [b]April 26th, 2025[\/b] (by 11:59pm PST)\nWinners will be announced on [b]June 28, 2025[\/b] during an online award ceremony.\n\n[b]Application requirements:[\/b]\nComplete a 12 question problem set on math and computer science\/AI related topics\nWrite 2 short essays\n\n[b][size=125][color=#674EA7]Prizes:\n1st place: \\$1,000 Cash prize\n2nd place: \\$500 Cash prize\n3rd place: \\$300 Cash prize\n4th-10th: \\$100 Cash prize each\n11th-20th: \\$50 Cash prize each\nTop 50 contestants: Over \\$50 worth of gadgets and stationary[\/color][\/size][\/b]\n\n[i]Many thanks to our current and past sponsors and partners: Hudson River Trading, MATHCOUNTS, Hewlett Packard Enterprise, Automation Anywhere, JP Morgan Chase, D.E. Shaw, and AI4ALL.[\/i]\n\n[i][b][size=115][color=#8E7CC3]Math and AI 4 Girls is a nonprofit organization aiming to encourage young girls to develop an interest in math and AI by taking part in STEM competitions and activities at an early age. The organization will be hosting an inaugural Math and AI 4 Girls competition to identify talent and encourage long-term planning of academic and career goals in STEM.[\/color][\/size][\/i][\/b]\n\n[b]Contact:[\/b]\n[url]mathandAI4girls@yahoo.com[\/url]\n\n[b]For more information on the competition:[\/b]\n[url]https:\/\/www.mathandai4girls.org\/math-and-ai-4-girls-competition[\/url]\n\n[b]More information on how to register will be posted on the website. If you have any questions, please ask here![\/b]\n\n\n","username":"audio-on","reported":false,"is_thanked":false,"is_nothanked":false,"attachment":false,"thanks_received":9,"nothanks_received":0,"thankers":"stuffedmath, christinaa.luu, manaleec, KevinYang2.71, EveningDawn, KnowingAnt, crazyeyemoody907, elasticwealth, clarkculus","deleted":false,"post_number":1,"post_time":1737855273,"num_edits":0,"post_format":"bbcode","last_edit_time":0,"last_editor_username":"","last_edit_reason":"","admin":false,"avatar":"\/\/avatar.artofproblemsolving.com\/avatar_550672.png?t=1710035923","num_posts":507,"editable":false,"deletable":false,"show_from_start":true,"show_from_end":false}],"topic_id":3487694,"category_id":5,"category_name":"Contests & Programs","category_main_color":"#008fd5","category_secondary_color":"#d9effd","topic_title":"2025 Math and AI 4 Girls Competition: Win Up To $1,000!!!","topic_type":"forum","is_public":true,"roles":{"53544":"mod","93494":"mod","86424":"mod","1662":"mod","38516":"mod","242520":"mod","159507":"mod","565384":"mod"},"first_post_id":33835452,"first_poster_id":550672,"first_poster_name":"audio-on","first_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_550672.png?t=1710035923","first_post_time":1737855273,"last_post_id":34398495,"last_poster_id":548685,"last_poster_name":"mkwhe","last_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_548685.png?t=1690238974","last_post_time":1743198226,"last_update_time":1743198227,"comment_count":14,"num_deleted":5,"num_reports":0,"num_views":2267,"category_num_users":6,"category_num_topics":34573,"category_num_posts":570355,"poll_id":0,"source":"","tags":[{"tag_id":31581,"tag_text":"Math Competitions","is_visible":true},{"tag_id":29385,"tag_text":"math contests","is_visible":true},{"tag_id":182825,"tag_text":"middle school math contests","is_visible":true},{"tag_id":757199,"tag_text":"Middle School contests","is_visible":true},{"tag_id":30457,"tag_text":"AI","is_visible":true},{"tag_id":2638691,"tag_text":"artificial intel","is_visible":true},{"tag_id":29357,"tag_text":"computer science","is_visible":true}],"can_have_source":true,"locked":false,"forum_locked":0,"announce_type":"none","announce_through":"","announce_factor":0,"preview":"<b>Join the 2025 Math and AI 4 Girls Competition for a chance to win up to $1,000!<\/b><br>\n<br>\nHey Everyone, I'm pleased to announce the dates for the 2025 MA4G Competition are set!<br>\nApplications will open on<b> March 22nd, 2025<\/b>, and they will close on <b>April 26th, 2025<\/b> (@ 11:59pm PST).<br>\n<br>\nApplicants will have one month to fill out an application with <u>prizes for the top 50 contestants & <b>cash prizes for the top 20 contestants (including $1,000 for the winner!)<\/b><\/u>. More details below!<br>\n<br>\n<b>Eligibility:<\/b><br>\nThe competition is <b>free to enter<\/b>, and open to middle school female students living in the US (5th-8th grade).<br>\nAward recipients are selected based on their aptitude, activities and aspirations in STEM.<br>\n<br>\n<b>Event dates:<\/b><br>\nApplications will open on<b> March 22nd, 2025<\/b>, and they will close on <b>April 26th, 2025<\/b> (by 11:59pm PST)<br>\nWinners will be announced on <b>June 28, 2025<\/b> during an online award ceremony.<br>\n<br>\n<b>Application requirements:<\/b><br>\nComplete a 12 question problem set on math and computer science\/AI related topics<br>\nWrite 2 short essays<br>\n<br>\n<b>Prizes:<br>\n1st place: $1,000 Cash prize<br>\n2nd place: $500 Cash prize<br>\n3rd place: $300 Cash prize<br>\n4th-10th: $100 Cash prize each<br>\n11th-20th: $50 Cash prize each<br>\nTop 50 contestants: Over $50 worth of gadgets and stationary<\/b><br>\n<br>\n<i>Many thanks to our current and past sponsors and partners: Hudson River Trading, MATHCOUNTS, Hewlett Packard Enterprise, Automation Anywhere, JP Morgan Chase, D.E. Shaw, and AI4ALL.<\/i><br>\n<br>\n<i><b>Math and AI 4 Girls is a nonprofit organization aiming to encourage young girls to develop an interest in math and AI by taking part in STEM competitions and activities at an early age. The organization will be hosting an inaugural Math and AI 4 Girls competition to identify talent and encourage long-term planning of academic and career goals in STEM.<\/b><\/i><br>\n<br>\n<b>Contact:<\/b><br>\n<a href=\"https:\/\/mathandAI4girls@yahoo.com\" class=\"bbcode_url\" rel=\"nofollow\">mathandAI4girls@yahoo.com<\/a><br>\n<br>\n<b>For more information on the competition:<\/b><br>\n<a href=\"https:\/\/www.mathandai4girls.org\/math-and-ai-4-girls-competition\" class=\"bbcode_url\" rel=\"nofollow\">https:\/\/www.mathandai4girls.org\/math-and-ai-4-girls-competition<\/a><br>\n<br>\n<b>More information on how to register will be posted on the website. If you have any questions, please ask here!<\/b><br>\n<br>\n<br>\n","target_url":"","target_url_application":null,"target_text":"","state":"none","cat_can_target":0,"has_thanks":true,"has_nothanks":false,"is_bookmarked":false,"in_feed":false,"is_watched":false},"3536878":{"num_posts":17,"posts_data":[{"post_id":34393546,"topic_id":3536878,"poster_id":1235771,"post_rendered":"Practice AMC 12A<br>\n<br>\n1. Find the sum of the infinite geometric series 1\/2 + 7\/36 + 49\/648 + \u2026<br>\nA - 18\/11, B - 9\/22, C - 9\/11, D - 18\/7, E - 9\/14<br>\n<br>\n2. What is the first digit after the decimal point in the square root of 420?<br>\nA - 1, B - 2, C - 3, D - 4, E - 5<br>\n<br>\n3. Two circles with radiuses 47 and 96 intersect at two points A and B. Let P be the point 82% of the way from A to B. A line is drawn through P that intersects both circles twice. Let the four intersection points, from left to right be W, X, Y, and Z. Find (PW\/PX)*(PY\/PZ).<br>\nA - 50\/5863, B - 47\/96, C - 1, D - 96\/47, E - 5863\/50<br>\n<br>\n4. What is the largest positive integer that cannot be expressed in the form 6a + 9b + 4c + 20d, where a, b, c, and d are positive integers?<br>\nA - 29, B - 38, C - 43, D - 76, E - 82<br>\n<br>\n5. What is the absolute difference of the probabilities of getting at least 6\/10 on a 10-question true or false test and at least 3\/5 on a 5-question true or false test?<br>\nA - 0, B - 1\/504, C - 1\/252, D - 1\/126, E - 1\/63<br>\n<br>\n6. How many arrangements of the letters in the word \u201cginger\u201d are there such that the two vowels have an even number of letters (remember 0 is even) between them (including the original \u201cginger\u201d)?<br>\nA - 72, B - 108, C - 144, D - 216, E - 432<br>\n<br>\n7. After opening his final exam, Jason does not know how to solve a single question. So he decides to pull out his phone and search up the answers. Doing this, Jason has a success rate of anywhere from 94-100% for any given question he uses his phone on. However, if the teacher sees his phone at any point during the test, then Jason gets a 0.5 multiplier on his final test score, as well as he must finish the rest of the test questions without his phone. (Assume Jason uses his phone on every question he does until he finishes the test or gets caught.) Every question is a 5-choice multiple choice question. Jason has a 90% chance of not being caught with his phone. What is the expected value of Jason\u2019s test score, rounded to the nearest tenth of a percent?<br>\nA - 89.9%, B - 90.0%, C - 90.1%, D - 90.2%, E - 90.3%<br>\n<br>\n8. A criminal is caught by a police officer. Due to a lack of cooperation, the officer calls in a second officer so they can start the arrest smoothly. Officer 1 takes 26:18 to arrest a criminal, and officer 2 takes 13:09 to arrest a criminal. With these two police officers working together, how long should the arrest take?<br>\nA - 4:23, B - 5:26, C - 8:46, D - 17:32, E - 19:44<br>\n<br>\n9. Statistics show that people in Memphis who eat at KFC n days a week have a (1\/10)(n+2) chance of liking kool-aid, and the number of people who eat at KFC n days a week is directly proportional to 8 - n (Note that n can only be an integer from 0 to 7, inclusive). A random person in Memphis is selected. Find the probability that they like kool-aid.<br>\nA - 13\/30, B - 17\/30, C - 19\/30, D - 23\/30, E - 29\/30<br>\n<br>\n10 (Main). PM me for problem (I copied over this problem from the 10A but just found out a \u201csheriff\u201d removed it for some reason so I don\u2019t want to take any risks)<br>\nA - 51, B - 52, C - 53, D - 54, E - 55<br>\n<br>\n10 (Alternate). Suppose that on the coordinate grid, the x-axis represents economic freedom, and the y-axis represents social freedom, where -1 <= x, y <= 1 and a higher number for either coordinate represents more freedom along that particular axis. Accordingly, the points (0, 0), (1, 1), (-1, 1), (-1, -1), and (1, -1) represent democracy, anarchy, socialism, communism, and fascism, respectively. A country is classified as whichever point it is closest to. Suppose a theoretical new country is selected by picking a random point within the square bounded by anarchy, socialism, communism, and fascism as its vertices. What is the probability that it is fascist?<br>\nA - 1 - (1\/4)pi, B - 1\/5, C - (1\/16)pi, D - 1\/4, E - 1\/8<br>\n<br>\n11. Two congruent towers stand near each other. Both take the shape of a right rectangular prism. A plane that cuts both towers into two pieces passes through the vertical axes of symmetry of both towers and does not cross the floor or roof of either tower. Let the point that the plane crosses the axis of symmetry of the first tower be A, and the point that the plane crosses the axis of symmetry of the second tower be B. A is 81% of the way from the floor to the roof of the first tower, and B is 69% of the way from the floor to the roof of the second tower. What percent of the total mass of both towers combined is above the plane?<br>\nA - 19%, B - 25%, C - 50%, D - 75%, E - 81%<br>\n<br>\n12. On an analog clock, the minute hand makes one full revolution every hour, and the hour hand makes one full revolution every 12 hours. Both hands move at a constant rate. During which of the following time periods does the minute hand pass the hour hand?<br>\nA - 7:35 - 7:36, B - 7:36 - 7:37, C - 7:37 - 7:38, D - 7:38 - 7:39, E - 7:39 - 7:40<br>\n<br>\n13. How many axes of symmetry does the graph of (x^2)(y^2) = 69 have?<br>\nA - 2, B - 3, C - 4, D - 5, E - 6<br>\n<br>\n14. Let f(n) be the sum of the positive integer divisors of n. Find the sum of the digits of the smallest odd positive integer n such that f(n) is greater than 2n.<br>\nA - 15, B - 18, C - 21, D - 24, E - 27<br>\n<br>\n15. A basketball has a diameter of 9 inches, and the hoop has a diameter of 18 inches. Peter decides to pick up the basketball and make a throw. Given that Peter has a 1\/4 chance of accidentally hitting the backboard and missing the shot, but if he doesn\u2019t, he is guaranteed that the frontmost point of the basketball will be within 18 inches of the center of the hoop at the moment when a great circle of the basketball crosses the plane containing the rim. No part of the ball will extend behind the backboard at any point during the throw, and the rim is attached directly to the backboard. What is the probability that Peter makes a green FN?<br>\nA - 3\/128, B - 3\/64, C - 3\/32, D - 3\/16, E - 3\/8<br>\n<br>\n16. Martin decides to rob 6 packages of Kool-Aid from a store. At the store, they have 5 packages each of 5 different flavors of Kool-Aid. How many different combinations of Kool-Aid could Martin rob?<br>\nA - 180, B - 185, C - 195, D - 205, E - 210<br>\n<br>\n17. Find the area of a cyclic quadrilateral with side lengths 6, 9, 4, and 2, rounded to the nearest integer.<br>\nA - 16, B - 19, C - 22, D - 25, E - 28<br>\n<br>\n18. Find the slope of the line tangent to the graph of y = x^2 + x + 1 at the point (2, 7).<br>\nA - 2, B - 3, C - 4, D - 5, E - 6<br>\n<br>\n19. Suppose that the strength of a protest is measured in \u201ceffectiveness points\u201d. Malcolm gathers 2048 people for a protest. During the first hour of the protest, all 2048 people protest with an effectiveness of 1 point per person. At the start of each hour of the protest after the first, half of the protestors will leave, but the ones remaining will gain one effectiveness point per person. For example, that means that during the second hour, there will be 1024 people protesting at 2 effectiveness points each, during the third hour, there will be 512 people protesting at 3 effectiveness points each, and so on. The protest will conclude at the end of the twelfth hour. After the protest is over, how many effectiveness points did it earn in total?<br>\nA - 8142, B - 8155, C - 8162, D - 8169, E - 8178<br>\n<br>\n20. Find the sum of all positive integers n greater than 1 and less than 16 such that (n-1)! + 1 is divisible by n.<br>\nA - 41, B - 44, C - 47, D - 50, E - 53<br>\n<br>\n21. Scientific research suggests that Stokely Carmichael had an IQ of 30. Given that IQ ranges from 1 to 200, inclusive, goes in integer increments, and the chance of having an IQ of n is proportional to n if n <= 100 and to 201 - n if n >= 101, what is the sum of the numerator and denominator of the probability that a random person is smarter than Stokely Carmichael, when expressed as a common fraction in lowest terms?<br>\nA - 1927, B - 2020, C - 2025, D - 3947, E - 3952<br>\n<br>\n22. In Alabama, Jim Crow laws apply to anyone who has any positive amount of Jim Crow ancestry, no matter how small the fraction, as long as it is greater than zero. In a small town in Alabama, there were initially 9 Non-Jim Crows and 3 Jim Crows. Denote this group to be the first generation. Then those 12 people would randomly get into 6 pairs and reproduce, making the second generation, consisting of 6 people. Then the process repeats for the second generation, where they get into 3 pairs. Of the 3 people in the third generation, what is the probability that exactly one of them is Non-Jim Crow?<br>\nA - 8\/27, B - 1\/3, C - 52\/135, D - 11\/27, E - 58\/135<br>\n<br>\n23. Goodman, Chaney, and Schwerner each start at the point (0, 0). Assume the coordinate axes are in miles. At t = 0, Goodman starts walking along the x-axis in the positive x direction at 0.6 miles per hour, Chaney starts walking along the y-axis in the positive y direction at 0.8 miles per hour, and Schwerner starts walking along the x-axis in the negative x direction at 0.4 miles per hour. However, a clan that does not like them patrols the circumference of the circle x^2 + y^2 = 1. Three knights of the clan, equally spaced apart on the circumference of the circle, walk counterclockwise along its circumference and make one revolution every hour. At t = 0, one of the knights of the clan is at (1, 0). Any of Goodman, Chaney, and Schwerner will be caught by the clan if they walk within 50 meters of one of their 3 knights. How many of the three will be caught by the clan?<br>\nA - 0, B - 1, C - 2, D - 3, E - Not enough info to determine<br>\n<br>\n24.<br>\nA list of 9 positive integers consists of 100, 112, 122, 142, 152, and 160, as well as a, b, and c, with a <= b <= c. The range of the list is 70, both the mean and median are multiples of 10, and the list has a unique mode. How many ordered triples (a, b, c) are possible?<br>\nA - 1, B - 2, C - 3, D - 4, E - 5<br>\n<br>\n25. What is the integer closest to the value of tan(83)? (The 83 is in degrees)<br>\nA - 2, B - 3, C - 4, D - 6, E - 8","post_canonical":"Practice AMC 12A\n\n1. Find the sum of the infinite geometric series 1\/2 + 7\/36 + 49\/648 + \u2026\nA - 18\/11, B - 9\/22, C - 9\/11, D - 18\/7, E - 9\/14\n\n2. What is the first digit after the decimal point in the square root of 420?\nA - 1, B - 2, C - 3, D - 4, E - 5\n\n3. Two circles with radiuses 47 and 96 intersect at two points A and B. Let P be the point 82% of the way from A to B. A line is drawn through P that intersects both circles twice. Let the four intersection points, from left to right be W, X, Y, and Z. Find (PW\/PX)*(PY\/PZ).\nA - 50\/5863, B - 47\/96, C - 1, D - 96\/47, E - 5863\/50\n\n4. What is the largest positive integer that cannot be expressed in the form 6a + 9b + 4c + 20d, where a, b, c, and d are positive integers?\nA - 29, B - 38, C - 43, D - 76, E - 82\n\n5. What is the absolute difference of the probabilities of getting at least 6\/10 on a 10-question true or false test and at least 3\/5 on a 5-question true or false test?\nA - 0, B - 1\/504, C - 1\/252, D - 1\/126, E - 1\/63\n\n6. How many arrangements of the letters in the word \u201cginger\u201d are there such that the two vowels have an even number of letters (remember 0 is even) between them (including the original \u201cginger\u201d)?\nA - 72, B - 108, C - 144, D - 216, E - 432\n\n7. After opening his final exam, Jason does not know how to solve a single question. So he decides to pull out his phone and search up the answers. Doing this, Jason has a success rate of anywhere from 94-100% for any given question he uses his phone on. However, if the teacher sees his phone at any point during the test, then Jason gets a 0.5 multiplier on his final test score, as well as he must finish the rest of the test questions without his phone. (Assume Jason uses his phone on every question he does until he finishes the test or gets caught.) Every question is a 5-choice multiple choice question. Jason has a 90% chance of not being caught with his phone. What is the expected value of Jason\u2019s test score, rounded to the nearest tenth of a percent?\nA - 89.9%, B - 90.0%, C - 90.1%, D - 90.2%, E - 90.3%\n\n8. A criminal is caught by a police officer. Due to a lack of cooperation, the officer calls in a second officer so they can start the arrest smoothly. Officer 1 takes 26:18 to arrest a criminal, and officer 2 takes 13:09 to arrest a criminal. With these two police officers working together, how long should the arrest take?\nA - 4:23, B - 5:26, C - 8:46, D - 17:32, E - 19:44\n\n9. Statistics show that people in Memphis who eat at KFC n days a week have a (1\/10)(n+2) chance of liking kool-aid, and the number of people who eat at KFC n days a week is directly proportional to 8 - n (Note that n can only be an integer from 0 to 7, inclusive). A random person in Memphis is selected. Find the probability that they like kool-aid.\nA - 13\/30, B - 17\/30, C - 19\/30, D - 23\/30, E - 29\/30\n\n10 (Main). PM me for problem (I copied over this problem from the 10A but just found out a \u201csheriff\u201d removed it for some reason so I don\u2019t want to take any risks)\nA - 51, B - 52, C - 53, D - 54, E - 55\n\n10 (Alternate). Suppose that on the coordinate grid, the x-axis represents economic freedom, and the y-axis represents social freedom, where -1 <= x, y <= 1 and a higher number for either coordinate represents more freedom along that particular axis. Accordingly, the points (0, 0), (1, 1), (-1, 1), (-1, -1), and (1, -1) represent democracy, anarchy, socialism, communism, and fascism, respectively. A country is classified as whichever point it is closest to. Suppose a theoretical new country is selected by picking a random point within the square bounded by anarchy, socialism, communism, and fascism as its vertices. What is the probability that it is fascist?\nA - 1 - (1\/4)pi, B - 1\/5, C - (1\/16)pi, D - 1\/4, E - 1\/8\n\n11. Two congruent towers stand near each other. Both take the shape of a right rectangular prism. A plane that cuts both towers into two pieces passes through the vertical axes of symmetry of both towers and does not cross the floor or roof of either tower. Let the point that the plane crosses the axis of symmetry of the first tower be A, and the point that the plane crosses the axis of symmetry of the second tower be B. A is 81% of the way from the floor to the roof of the first tower, and B is 69% of the way from the floor to the roof of the second tower. What percent of the total mass of both towers combined is above the plane?\nA - 19%, B - 25%, C - 50%, D - 75%, E - 81%\n\n12. On an analog clock, the minute hand makes one full revolution every hour, and the hour hand makes one full revolution every 12 hours. Both hands move at a constant rate. During which of the following time periods does the minute hand pass the hour hand?\nA - 7:35 - 7:36, B - 7:36 - 7:37, C - 7:37 - 7:38, D - 7:38 - 7:39, E - 7:39 - 7:40\n\n13. How many axes of symmetry does the graph of (x^2)(y^2) = 69 have?\nA - 2, B - 3, C - 4, D - 5, E - 6\n\n14. Let f(n) be the sum of the positive integer divisors of n. Find the sum of the digits of the smallest odd positive integer n such that f(n) is greater than 2n.\nA - 15, B - 18, C - 21, D - 24, E - 27\n\n15. A basketball has a diameter of 9 inches, and the hoop has a diameter of 18 inches. Peter decides to pick up the basketball and make a throw. Given that Peter has a 1\/4 chance of accidentally hitting the backboard and missing the shot, but if he doesn\u2019t, he is guaranteed that the frontmost point of the basketball will be within 18 inches of the center of the hoop at the moment when a great circle of the basketball crosses the plane containing the rim. No part of the ball will extend behind the backboard at any point during the throw, and the rim is attached directly to the backboard. What is the probability that Peter makes a green FN?\nA - 3\/128, B - 3\/64, C - 3\/32, D - 3\/16, E - 3\/8\n\n16. Martin decides to rob 6 packages of Kool-Aid from a store. At the store, they have 5 packages each of 5 different flavors of Kool-Aid. How many different combinations of Kool-Aid could Martin rob?\nA - 180, B - 185, C - 195, D - 205, E - 210\n\n17. Find the area of a cyclic quadrilateral with side lengths 6, 9, 4, and 2, rounded to the nearest integer.\nA - 16, B - 19, C - 22, D - 25, E - 28\n\n18. Find the slope of the line tangent to the graph of y = x^2 + x + 1 at the point (2, 7).\nA - 2, B - 3, C - 4, D - 5, E - 6\n\n19. Suppose that the strength of a protest is measured in \u201ceffectiveness points\u201d. Malcolm gathers 2048 people for a protest. During the first hour of the protest, all 2048 people protest with an effectiveness of 1 point per person. At the start of each hour of the protest after the first, half of the protestors will leave, but the ones remaining will gain one effectiveness point per person. For example, that means that during the second hour, there will be 1024 people protesting at 2 effectiveness points each, during the third hour, there will be 512 people protesting at 3 effectiveness points each, and so on. The protest will conclude at the end of the twelfth hour. After the protest is over, how many effectiveness points did it earn in total?\nA - 8142, B - 8155, C - 8162, D - 8169, E - 8178\n\n20. Find the sum of all positive integers n greater than 1 and less than 16 such that (n-1)! + 1 is divisible by n.\nA - 41, B - 44, C - 47, D - 50, E - 53\n\n21. Scientific research suggests that Stokely Carmichael had an IQ of 30. Given that IQ ranges from 1 to 200, inclusive, goes in integer increments, and the chance of having an IQ of n is proportional to n if n <= 100 and to 201 - n if n >= 101, what is the sum of the numerator and denominator of the probability that a random person is smarter than Stokely Carmichael, when expressed as a common fraction in lowest terms?\nA - 1927, B - 2020, C - 2025, D - 3947, E - 3952\n\n22. In Alabama, Jim Crow laws apply to anyone who has any positive amount of Jim Crow ancestry, no matter how small the fraction, as long as it is greater than zero. In a small town in Alabama, there were initially 9 Non-Jim Crows and 3 Jim Crows. Denote this group to be the first generation. Then those 12 people would randomly get into 6 pairs and reproduce, making the second generation, consisting of 6 people. Then the process repeats for the second generation, where they get into 3 pairs. Of the 3 people in the third generation, what is the probability that exactly one of them is Non-Jim Crow?\nA - 8\/27, B - 1\/3, C - 52\/135, D - 11\/27, E - 58\/135\n\n23. Goodman, Chaney, and Schwerner each start at the point (0, 0). Assume the coordinate axes are in miles. At t = 0, Goodman starts walking along the x-axis in the positive x direction at 0.6 miles per hour, Chaney starts walking along the y-axis in the positive y direction at 0.8 miles per hour, and Schwerner starts walking along the x-axis in the negative x direction at 0.4 miles per hour. However, a clan that does not like them patrols the circumference of the circle x^2 + y^2 = 1. Three knights of the clan, equally spaced apart on the circumference of the circle, walk counterclockwise along its circumference and make one revolution every hour. At t = 0, one of the knights of the clan is at (1, 0). Any of Goodman, Chaney, and Schwerner will be caught by the clan if they walk within 50 meters of one of their 3 knights. How many of the three will be caught by the clan?\nA - 0, B - 1, C - 2, D - 3, E - Not enough info to determine\n\n24. \nA list of 9 positive integers consists of 100, 112, 122, 142, 152, and 160, as well as a, b, and c, with a <= b <= c. The range of the list is 70, both the mean and median are multiples of 10, and the list has a unique mode. How many ordered triples (a, b, c) are possible?\nA - 1, B - 2, C - 3, D - 4, E - 5\n\n25. What is the integer closest to the value of tan(83)? (The 83 is in degrees)\nA - 2, B - 3, C - 4, D - 6, E - 8","username":"freddyfazbear","reported":true,"is_thanked":false,"is_nothanked":false,"attachment":false,"thanks_received":3,"nothanks_received":0,"thankers":"fake123, Alex-131, sixoneeight","deleted":false,"post_number":1,"post_time":1743143704,"num_edits":2,"post_format":"bbcode","last_edit_time":1743194926,"last_editor_username":"freddyfazbear","last_edit_reason":"Fixed a typo in problem 20","admin":false,"avatar":"\/\/avatar.artofproblemsolving.com\/avatar_1235771.jpeg?t=1743181660","num_posts":27,"editable":false,"deletable":false,"show_from_start":true,"show_from_end":false}],"topic_id":3536878,"category_id":5,"category_name":"Contests & Programs","category_main_color":"#008fd5","category_secondary_color":"#d9effd","topic_title":"Practice AMC 12A","topic_type":"forum","is_public":true,"roles":{"53544":"mod","93494":"mod","86424":"mod","1662":"mod","38516":"mod","242520":"mod","159507":"mod","565384":"mod"},"first_post_id":34393546,"first_poster_id":1235771,"first_poster_name":"freddyfazbear","first_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_1235771.jpeg?t=174318166","first_post_time":1743143704,"last_post_id":34398479,"last_poster_id":1096417,"last_poster_name":"blueprimes","last_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_1096417.png","last_post_time":1743198092,"last_update_time":1743198092,"comment_count":17,"num_deleted":1,"num_reports":1,"num_views":479,"category_num_users":6,"category_num_topics":34573,"category_num_posts":570355,"poll_id":0,"source":"","tags":[{"tag_id":3,"tag_text":"AMC","is_visible":true},{"tag_id":37,"tag_text":"AMC 12","is_visible":true},{"tag_id":185,"tag_text":"AMC 12 A","is_visible":true}],"can_have_source":true,"locked":false,"forum_locked":0,"announce_type":"none","announce_through":"","announce_factor":0,"preview":"Practice AMC 12A<br>\n<br>\n1. Find the sum of the infinite geometric series 1\/2 + 7\/36 + 49\/648 + \u2026<br>\nA - 18\/11, B - 9\/22, C - 9\/11, D - 18\/7, E - 9\/14<br>\n<br>\n2. What is the first digit after the decimal point in the square root of 420?<br>\nA - 1, B - 2, C - 3, D - 4, E - 5<br>\n<br>\n3. Two circles with radiuses 47 and 96 intersect at two points A and B. Let P be the point 82% of the way from A to B. A line is drawn through P that intersects both circles twice. Let the four intersection points, from left to right be W, X, Y, and Z. Find (PW\/PX)*(PY\/PZ).<br>\nA - 50\/5863, B - 47\/96, C - 1, D - 96\/47, E - 5863\/50<br>\n<br>\n4. What is the largest positive integer that cannot be expressed in the form 6a + 9b + 4c + 20d, where a, b, c, and d are positive integers?<br>\nA - 29, B - 38, C - 43, D - 76, E - 82<br>\n<br>\n5. What is the absolute difference of the probabilities of getting at least 6\/10 on a 10-question true or false test and at least 3\/5 on a 5-question true or false test?<br>\nA - 0, B - 1\/504, C - 1\/252, D - 1\/126, E - 1\/63<br>\n<br>\n6. How many arrangements of the letters in the word \u201cginger\u201d are there such that the two vowels have an even number of letters (remember 0 is even) between them (including the original \u201cginger\u201d)?<br>\nA - 72, B - 108, C - 144, D - 216, E - 432<br>\n<br>\n7. After opening his final exam, Jason does not know how to solve a single question. So he decides to pull out his phone and search up the answers. Doing this, Jason has a success rate of anywhere from 94-100% for any given question he uses his phone on. However, if the teacher sees his phone at any point during the test, then Jason gets a 0.5 multiplier on his final test score, as well as he must finish the rest of the test questions without his phone. (Assume Jason uses his phone on every question he does until he finishes the test or gets caught.) Every question is a 5-choice multiple choice question. Jason has a 90% chance of not being caught with his phone. What is the expected value of Jason\u2019s test score, rounded to the nearest tenth of a percent?<br>\nA - 89.9%, B - 90.0%, C - 90.1%, D - 90.2%, E - 90.3%<br>\n<br>\n8. A criminal is caught by a police officer. Due to a lack of cooperation, the officer calls in a second officer so they can start the arrest smoothly. Officer 1 takes 26:18 to arrest a criminal, and officer 2 takes 13:09 to arrest a criminal. With these two police officers working together, how long should the arrest take?<br>\nA - 4:23, B - 5:26, C - 8:46, D - 17:32, E - 19:44<br>\n<br>\n9. Statistics show that people in Memphis who eat at KFC n days a week have a (1\/10)(n+2) chance of liking kool-aid, and the number of people who eat at KFC n days a week is directly proportional to 8 - n (Note that n can only be an integer from 0 to 7, inclusive). A random person in Memphis is selected. Find the probability that they like kool-aid.<br>\nA - 13\/30, B - 17\/30, C - 19\/30, D - 23\/30, E - 29\/30<br>\n<br>\n10 (Main). PM me for problem (I copied over this problem from the 10A but just found out a \u201csheriff\u201d removed it for some reason so I don\u2019t want to take any risks)<br>\nA - 51, B - 52, C - 53, D - 54, E - 55<br>\n<br>\n10 (Alternate). Suppose that on the coordinate grid, the x-axis represents economic freedom, and the y-axis represents social freedom, where -1 <= x, y <= 1 and a higher number for either coordinate represents more freedom along that particular axis. Accordingly, the points (0, 0), (1, 1), (-1, 1), (-1, -1), and (1, -1) represent democracy, anarchy, socialism, communism, and fascism, respectively. A country is classified as whichever point it is closest to. Suppose a theoretical new country is selected by picking a random point within the square bounded by anarchy, socialism, communism, and fascism as its vertices. What is the probability that it is fascist?<br>\nA - 1 - (1\/4)pi, B - 1\/5, C - (1\/16)pi, D - 1\/4, E - 1\/8<br>\n<br>\n11. Two congruent towers stand near each other. Both take the shape of a right rectangular prism. A plane that cuts both towers into two pieces passes through the vertical axes of symmetry of both towers and does not cross the floor or roof of either tower. Let the point that the plane crosses the axis of symmetry of the first tower be A, and the point that the plane crosses the axis of symmetry of the second tower be B. A is 81% of the way from the floor to the roof of the first tower, and B is 69% of the way from the floor to the roof of the second tower. What percent of the total mass of both towers combined is above the plane?<br>\nA - 19%, B - 25%, C - 50%, D - 75%, E - 81%<br>\n<br>\n12. On an analog clock, the minute hand makes one full revolution every hour, and the hour hand makes one full revolution every 12 hours. Both hands move at a constant rate. During which of the following time periods does the minute hand pass the hour hand?<br>\nA - 7:35 - 7:36, B - 7:36 - 7:37, C - 7:37 - 7:38, D - 7:38 - 7:39, E - 7:39 - 7:40<br>\n<br>\n13. How many axes of symmetry does the graph of (x^2)(y^2) = 69 have?<br>\nA - 2, B - 3, C - 4, D - 5, E - 6<br>\n<br>\n14. Let f(n) be the sum of the positive integer divisors of n. Find the sum of the digits of the smallest odd positive integer n such that f(n) is greater than 2n.<br>\nA - 15, B - 18, C - 21, D - 24, E - 27<br>\n<br>\n15. A basketball has a diameter of 9 inches, and the hoop has a diameter of 18 inches. Peter decides to pick up the basketball and make a throw. Given that Peter has a 1\/4 chance of accidentally hitting the backboard and missing the shot, but if he doesn\u2019t, he is guaranteed that the frontmost point of the basketball will be within 18 inches of the center of the hoop at the moment when a great circle of the basketball crosses the plane containing the rim. No part of the ball will extend behind the backboard at any point during the throw, and the rim is attached directly to the backboard. What is the probability that Peter makes a green FN?<br>\nA - 3\/128, B - 3\/64, C - 3\/32, D - 3\/16, E - 3\/8<br>\n<br>\n16. Martin decides to rob 6 packages of Kool-Aid from a store. At the store, they have 5 packages each of 5 different flavors of Kool-Aid. How many different combinations of Kool-Aid could Martin rob?<br>\nA - 180, B - 185, C - 195, D - 205, E - 210<br>\n<br>\n17. Find the area of a cyclic quadrilateral with side lengths 6, 9, 4, and 2, rounded to the nearest integer.<br>\nA - 16, B - 19, C - 22, D - 25, E - 28<br>\n<br>\n18. Find the slope of the line tangent to the graph of y = x^2 + x + 1 at the point (2, 7).<br>\nA - 2, B - 3, C - 4, D - 5, E - 6<br>\n<br>\n19. Suppose that the strength of a protest is measured in \u201ceffectiveness points\u201d. Malcolm gathers 2048 people for a protest. During the first hour of the protest, all 2048 people protest with an effectiveness of 1 point per person. At the start of each hour of the protest after the first, half of the protestors will leave, but the ones remaining will gain one effectiveness point per person. For example, that means that during the second hour, there will be 1024 people protesting at 2 effectiveness points each, during the third hour, there will be 512 people protesting at 3 effectiveness points each, and so on. The protest will conclude at the end of the twelfth hour. After the protest is over, how many effectiveness points did it earn in total?<br>\nA - 8142, B - 8155, C - 8162, D - 8169, E - 8178<br>\n<br>\n20. Find the sum of all positive integers n greater than 1 and less than 16 such that (n-1)! + 1 is divisible by n.<br>\nA - 41, B - 44, C - 47, D - 50, E - 53<br>\n<br>\n21. Scientific research suggests that Stokely Carmichael had an IQ of 30. Given that IQ ranges from 1 to 200, inclusive, goes in integer increments, and the chance of having an IQ of n is proportional to n if n <= 100 and to 201 - n if n >= 101, what is the sum of the numerator and denominator of the probability that a random person is smarter than Stokely Carmichael, when expressed as a common fraction in lowest terms?<br>\nA - 1927, B - 2020, C - 2025, D - 3947, E - 3952<br>\n<br>\n22. In Alabama, Jim Crow laws apply to anyone who has any positive amount of Jim Crow ancestry, no matter how small the fraction, as long as it is greater than zero. In a small town in Alabama, there were initially 9 Non-Jim Crows and 3 Jim Crows. Denote this group to be the first generation. Then those 12 people would randomly get into 6 pairs and reproduce, making the second generation, consisting of 6 people. Then the process repeats for the second generation, where they get into 3 pairs. Of the 3 people in the third generation, what is the probability that exactly one of them is Non-Jim Crow?<br>\nA - 8\/27, B - 1\/3, C - 52\/135, D - 11\/27, E - 58\/135<br>\n<br>\n23. Goodman, Chaney, and Schwerner each start at the point (0, 0). Assume the coordinate axes are in miles. At t = 0, Goodman starts walking along the x-axis in the positive x direction at 0.6 miles per hour, Chaney starts walking along the y-axis in the positive y direction at 0.8 miles per hour, and Schwerner starts walking along the x-axis in the negative x direction at 0.4 miles per hour. However, a clan that does not like them patrols the circumference of the circle x^2 + y^2 = 1. Three knights of the clan, equally spaced apart on the circumference of the circle, walk counterclockwise along its circumference and make one revolution every hour. At t = 0, one of the knights of the clan is at (1, 0). Any of Goodman, Chaney, and Schwerner will be caught by the clan if they walk within 50 meters of one of their 3 knights. How many of the three will be caught by the clan?<br>\nA - 0, B - 1, C - 2, D - 3, E - Not enough info to determine<br>\n<br>\n24.<br>\nA list of 9 positive integers consists of 100, 112, 122, 142, 152, and 160, as well as a, b, and c, with a <= b <= c. The range of the list is 70, both the mean and median are multiples of 10, and the list has a unique mode. How many ordered triples (a, b, c) are possible?<br>\nA - 1, B - 2, C - 3, D - 4, E - 5<br>\n<br>\n25. What is the integer closest to the value of tan(83)? (The 83 is in degrees)<br>\nA - 2, B - 3, C - 4, D - 6, E - 8","target_url":"","target_url_application":null,"target_text":"","state":"none","cat_can_target":0,"has_thanks":true,"has_nothanks":false,"is_bookmarked":false,"in_feed":false,"is_watched":false}};AoPS.preload_topics[6]={"3537411":{"num_posts":5,"posts_data":[{"post_id":34398885,"topic_id":3537411,"poster_id":814169,"post_rendered":"W\/o the diagram, is the answer the first option?","post_canonical":"W\/o the diagram, is the answer the first option?","username":"RenheMiResembleRice","reported":false,"is_thanked":false,"is_nothanked":false,"attachment":true,"thanks_received":0,"nothanks_received":0,"thankers":null,"deleted":false,"post_number":1,"post_time":1743201621,"num_edits":1,"post_format":"bbcode","last_edit_time":1743201680,"last_editor_username":"RenheMiResembleRice","last_edit_reason":"","admin":false,"avatar":"\/\/avatar.artofproblemsolving.com\/avatar_814169.png","num_posts":252,"editable":false,"deletable":false,"attachments":[{"name":"Screenshot 2025-03-29 at 9.39.20 am.png","href":"https:\/\/cdn.artofproblemsolving.com\/attachments\/2\/a\/fb2d93361e955a8d1ee645b7de71b664e70e70.png","url":"2\/a\/fb2d93361e955a8d1ee645b7de71b664e70e70.png","size":208,"mimetype":"image\/png"}],"show_from_start":true,"show_from_end":false}],"topic_id":3537411,"category_id":6,"category_name":"High School Olympiads","category_main_color":"#029386","category_secondary_color":"#e1fff2","topic_title":"Complex roots intuitive problem","topic_type":"forum","is_public":true,"roles":{"2":"mod","432":"mod","38516":"mod","53544":"mod","285":"mod","93494":"mod","67223":"mod","56597":"mod","242520":"mod","159507":"mod"},"first_post_id":34398885,"first_poster_id":814169,"first_poster_name":"RenheMiResembleRice","first_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_814169.png","first_post_time":1743201621,"last_post_id":34399133,"last_poster_id":814169,"last_poster_name":"RenheMiResembleRice","last_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_814169.png","last_post_time":1743203630,"last_update_time":1743203630,"comment_count":5,"num_deleted":0,"num_reports":0,"num_views":45,"category_num_users":13,"category_num_topics":311321,"category_num_posts":1360718,"poll_id":0,"source":"","tags":[{"tag_id":169,"tag_text":"algebra","is_visible":true}],"can_have_source":true,"locked":false,"forum_locked":0,"announce_type":"none","announce_through":"","announce_factor":0,"preview":"W\/o the diagram, is the answer the first option?","target_url":"","target_url_application":null,"target_text":"","state":"none","cat_can_target":0,"has_thanks":true,"has_nothanks":false,"is_bookmarked":false,"in_feed":false,"is_watched":false},"3537078":{"num_posts":3,"posts_data":[{"post_id":34395985,"topic_id":3537078,"poster_id":1133540,"post_rendered":"Determine with proof all functions <img src=\"\/\/latex.artofproblemsolving.com\/6\/5\/e\/65edf4e07e54fbb2a62d0429efd621867fee7266.png\" class=\"latex\" alt=\"$f: \\mathbb{R} \\to \\mathbb{R}$\" style=\"vertical-align: -3px\" width=\"79\" height=\"16\" > such that for all real <img src=\"\/\/latex.artofproblemsolving.com\/0\/a\/4\/0a4cf8a66a511c0a090860461307ebbc045b24d9.png\" class=\"latex\" alt=\"$x,y:$\" style=\"vertical-align: -3px\" width=\"36\" height=\"11\" ><br>\n<img src=\"\/\/latex.artofproblemsolving.com\/1\/e\/2\/1e2fef35ee53cd3224527f672b9edbb7a5293fad.png\" class=\"latexcenter\" alt=\"\\[f(x+f(y))=f(x+y)+y\\]\" width=\"214\" height=\"18\" >","post_canonical":"Determine with proof all functions $f: \\mathbb{R} \\to \\mathbb{R}$ such that for all real $x,y:$\n\\[f(x+f(y))=f(x+y)+y\\]","username":"John_Mgr","reported":false,"is_thanked":false,"is_nothanked":false,"attachment":false,"thanks_received":0,"nothanks_received":0,"thankers":null,"deleted":false,"post_number":1,"post_time":1743182066,"num_edits":0,"post_format":"bbcode","last_edit_time":0,"last_editor_username":"","last_edit_reason":"","admin":false,"avatar":"\/\/avatar.artofproblemsolving.com\/avatar_1133540.png","num_posts":54,"editable":false,"deletable":false,"show_from_start":true,"show_from_end":false}],"topic_id":3537078,"category_id":6,"category_name":"High School Olympiads","category_main_color":"#029386","category_secondary_color":"#e1fff2","topic_title":"f(x+f(y))=f(x+y)+y","topic_type":"forum","is_public":true,"roles":{"2":"mod","432":"mod","38516":"mod","53544":"mod","285":"mod","93494":"mod","67223":"mod","56597":"mod","242520":"mod","159507":"mod"},"first_post_id":34395985,"first_poster_id":1133540,"first_poster_name":"John_Mgr","first_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_1133540.png","first_post_time":1743182066,"last_post_id":34399108,"last_poster_id":1133540,"last_poster_name":"John_Mgr","last_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_1133540.png","last_post_time":1743203449,"last_update_time":1743203449,"comment_count":3,"num_deleted":1,"num_reports":0,"num_views":103,"category_num_users":13,"category_num_topics":311321,"category_num_posts":1360718,"poll_id":0,"source":"","tags":[{"tag_id":45468,"tag_text":"Function equations","is_visible":true},{"tag_id":169,"tag_text":"algebra","is_visible":true}],"can_have_source":true,"locked":false,"forum_locked":0,"announce_type":"none","announce_through":"","announce_factor":0,"preview":"Determine with proof all functions <img src=\"\/\/latex.artofproblemsolving.com\/6\/5\/e\/65edf4e07e54fbb2a62d0429efd621867fee7266.png\" class=\"latex\" alt=\"$f: \\mathbb{R} \\to \\mathbb{R}$\" style=\"vertical-align: -3px\" width=\"79\" height=\"16\" > such that for all real <img src=\"\/\/latex.artofproblemsolving.com\/0\/a\/4\/0a4cf8a66a511c0a090860461307ebbc045b24d9.png\" class=\"latex\" alt=\"$x,y:$\" style=\"vertical-align: -3px\" width=\"36\" height=\"11\" ><br>\n<img src=\"\/\/latex.artofproblemsolving.com\/1\/e\/2\/1e2fef35ee53cd3224527f672b9edbb7a5293fad.png\" class=\"latexcenter\" alt=\"\\[f(x+f(y))=f(x+y)+y\\]\" width=\"214\" height=\"18\" >","target_url":"","target_url_application":null,"target_text":"","state":"none","cat_can_target":0,"has_thanks":true,"has_nothanks":false,"is_bookmarked":false,"in_feed":false,"is_watched":false},"3537435":{"num_posts":1,"posts_data":[{"post_id":34399101,"topic_id":3537435,"poster_id":1230209,"post_rendered":"find the minimum value of positive integer which when multiplied by 2024, its digits are only 1 or 0.","post_canonical":"find the minimum value of positive integer which when multiplied by 2024, its digits are only 1 or 0.","username":"Hip1zzzil","reported":false,"is_thanked":false,"is_nothanked":false,"attachment":false,"thanks_received":0,"nothanks_received":0,"thankers":null,"deleted":false,"post_number":1,"post_time":1743203401,"num_edits":0,"post_format":"bbcode","last_edit_time":0,"last_editor_username":"","last_edit_reason":"","admin":false,"avatar":"\/\/avatar.artofproblemsolving.com\/avatar_1230209.png","num_posts":4,"editable":false,"deletable":false,"show_from_start":true,"show_from_end":true}],"topic_id":3537435,"category_id":6,"category_name":"High School Olympiads","category_main_color":"#029386","category_secondary_color":"#e1fff2","topic_title":"Tricky NT","topic_type":"forum","is_public":true,"roles":{"2":"mod","432":"mod","38516":"mod","53544":"mod","285":"mod","93494":"mod","67223":"mod","56597":"mod","242520":"mod","159507":"mod"},"first_post_id":34399101,"first_poster_id":1230209,"first_poster_name":"Hip1zzzil","first_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_1230209.png","first_post_time":1743203401,"last_post_id":34399101,"last_poster_id":1230209,"last_poster_name":"Hip1zzzil","last_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_1230209.png","last_post_time":1743203401,"last_update_time":1743203401,"comment_count":1,"num_deleted":0,"num_reports":0,"num_views":5,"category_num_users":13,"category_num_topics":311321,"category_num_posts":1360718,"poll_id":0,"source":"2024 Korea","tags":[{"tag_id":177,"tag_text":"number theory","is_visible":true}],"can_have_source":true,"locked":false,"forum_locked":0,"announce_type":"none","announce_through":"","announce_factor":0,"preview":"find the minimum value of positive integer which when multiplied by 2024, its digits are only 1 or 0.","target_url":"","target_url_application":null,"target_text":"","state":"none","cat_can_target":0,"has_thanks":true,"has_nothanks":false,"is_bookmarked":false,"in_feed":false,"is_watched":false},"3537274":{"num_posts":2,"posts_data":[{"post_id":34397880,"topic_id":3537274,"poster_id":1133804,"post_rendered":"Find the biggest positive integer <img src=\"\/\/latex.artofproblemsolving.com\/1\/7\/4\/174fadd07fd54c9afe288e96558c92e0c1da733a.png\" class=\"latex\" alt=\"$n$\" style=\"vertical-align: -1px\" width=\"13\" height=\"10\" > such that both <img src=\"\/\/latex.artofproblemsolving.com\/7\/e\/a\/7ea580d3cd2e8e5bb0838524808795e1016edfc8.png\" class=\"latex\" alt=\"$n^3-n$\" width=\"51\" height=\"15\" > and <img src=\"\/\/latex.artofproblemsolving.com\/7\/0\/a\/70a393ae27282270e83325392b512653044d779e.png\" class=\"latex\" alt=\"$(n+1)^3-(n+1)$\" style=\"vertical-align: -4px\" width=\"140\" height=\"19\" > are divisible by <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/7\/8\/d\/78db23e86da8fe319c63bdb5eaf6d3fd8fefa17b.png\" class=\"latex\" alt=\"$2025$\" width=\"35\" height=\"12\" >.<\/span>","post_canonical":"Find the biggest positive integer $n$ such that both $n^3-n$ and $(n+1)^3-(n+1)$ are divisible by $2025$.","username":"nAalniaOMliO","reported":false,"is_thanked":false,"is_nothanked":false,"attachment":false,"thanks_received":0,"nothanks_received":0,"thankers":null,"deleted":false,"post_number":1,"post_time":1743194149,"num_edits":0,"post_format":"bbcode","last_edit_time":0,"last_editor_username":"","last_edit_reason":"","admin":false,"avatar":"\/\/avatar.artofproblemsolving.com\/avatar_1133804.png","num_posts":280,"editable":false,"deletable":false,"show_from_start":true,"show_from_end":false}],"topic_id":3537274,"category_id":6,"category_name":"High School Olympiads","category_main_color":"#029386","category_secondary_color":"#e1fff2","topic_title":"Two numbers divisible by 2025","topic_type":"forum","is_public":true,"roles":{"2":"mod","432":"mod","38516":"mod","53544":"mod","285":"mod","93494":"mod","67223":"mod","56597":"mod","242520":"mod","159507":"mod"},"first_post_id":34397880,"first_poster_id":1133804,"first_poster_name":"nAalniaOMliO","first_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_1133804.png","first_post_time":1743194149,"last_post_id":34399013,"last_poster_id":279587,"last_poster_name":"RagvaloD","last_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_279587.png","last_post_time":1743202574,"last_update_time":1743202574,"comment_count":2,"num_deleted":0,"num_reports":0,"num_views":40,"category_num_users":13,"category_num_topics":311321,"category_num_posts":1360718,"poll_id":0,"source":"Belarusian National Olympiad 2025","tags":[{"tag_id":177,"tag_text":"number theory","is_visible":true}],"can_have_source":true,"locked":false,"forum_locked":0,"announce_type":"none","announce_through":"","announce_factor":0,"preview":"Find the biggest positive integer <img src=\"\/\/latex.artofproblemsolving.com\/1\/7\/4\/174fadd07fd54c9afe288e96558c92e0c1da733a.png\" class=\"latex\" alt=\"$n$\" style=\"vertical-align: -1px\" width=\"13\" height=\"10\" > such that both <img src=\"\/\/latex.artofproblemsolving.com\/7\/e\/a\/7ea580d3cd2e8e5bb0838524808795e1016edfc8.png\" class=\"latex\" alt=\"$n^3-n$\" width=\"51\" height=\"15\" > and <img src=\"\/\/latex.artofproblemsolving.com\/7\/0\/a\/70a393ae27282270e83325392b512653044d779e.png\" class=\"latex\" alt=\"$(n+1)^3-(n+1)$\" style=\"vertical-align: -4px\" width=\"140\" height=\"19\" > are divisible by <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/7\/8\/d\/78db23e86da8fe319c63bdb5eaf6d3fd8fefa17b.png\" class=\"latex\" alt=\"$2025$\" width=\"35\" height=\"12\" >.<\/span>","target_url":"","target_url_application":null,"target_text":"","state":"none","cat_can_target":0,"has_thanks":true,"has_nothanks":false,"is_bookmarked":false,"in_feed":false,"is_watched":false}};AoPS.preload_topics[7]={"3537379":{"num_posts":1,"posts_data":[{"post_id":34398646,"topic_id":3537379,"poster_id":548549,"post_rendered":"Let <img src=\"\/\/latex.artofproblemsolving.com\/e\/f\/f\/eff43e84f8a3bcf7b6965f0a3248bc4d3a9d0cd4.png\" class=\"latex\" alt=\"$R$\" style=\"vertical-align: -1px\" width=\"16\" height=\"14\" > be a ring. Let <img src=\"\/\/latex.artofproblemsolving.com\/7\/c\/a\/7ca53e78b06848e441caa5bb6fa4ab1531559e61.png\" class=\"latex\" alt=\"$x,y\\in R$\" style=\"vertical-align: -3px\" width=\"64\" height=\"16\" > such that <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/5\/0\/2\/5026dae4f97e7580363285d44343506f0d29dc04.png\" class=\"latex\" alt=\"$x^2=y^2=0$\" style=\"vertical-align: -4px\" width=\"94\" height=\"20\" >.<\/span> Prove that if <img src=\"\/\/latex.artofproblemsolving.com\/4\/a\/0\/4a0a9c4135f5cbc72b0d643ac08499027fc7835b.png\" class=\"latex\" alt=\"$x+y-xy$\" style=\"vertical-align: -4px\" width=\"86\" height=\"16\" > is nilpotent, so is <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/3\/f\/7\/3f79c3676b62fb47243e0d357c87115b6f3395e4.png\" class=\"latex\" alt=\"$xy$\" style=\"vertical-align: -3px\" width=\"19\" height=\"11\" >.<\/span><br>\n<br>\n<i>Janez \u0160ter<\/i>","post_canonical":"Let $R$ be a ring. Let $x,y\\in R$ such that $x^2=y^2=0$. Prove that if $x+y-xy$ is nilpotent, so is $xy$.\n\n[i]Janez \u0160ter[\/i]","username":"AndreiVila","reported":false,"is_thanked":false,"is_nothanked":false,"attachment":false,"thanks_received":0,"nothanks_received":0,"thankers":null,"deleted":false,"post_number":1,"post_time":1743199559,"num_edits":0,"post_format":"bbcode","last_edit_time":0,"last_editor_username":"","last_edit_reason":"","admin":false,"avatar":"\/\/avatar.artofproblemsolving.com\/avatar_548549.png","num_posts":207,"editable":false,"deletable":false,"show_from_start":true,"show_from_end":true}],"topic_id":3537379,"category_id":7,"category_name":"College Math","category_main_color":"#511e8f","category_secondary_color":"#f2e6fe","topic_title":"Gheorghe \u021ai\u021beica 2025 Grade 12 P4","topic_type":"forum","is_public":true,"roles":{"432":"mod","93494":"mod","67223":"mod","9049":"mod","242520":"mod"},"first_post_id":34398646,"first_poster_id":548549,"first_poster_name":"AndreiVila","first_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_548549.png","first_post_time":1743199559,"last_post_id":34398646,"last_poster_id":548549,"last_poster_name":"AndreiVila","last_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_548549.png","last_post_time":1743199559,"last_update_time":1743199559,"comment_count":1,"num_deleted":0,"num_reports":0,"num_views":22,"category_num_users":1,"category_num_topics":114003,"category_num_posts":443403,"poll_id":0,"source":"Gheorghe \u021ai\u021beica 2025","tags":[{"tag_id":29450,"tag_text":"Ring Theory","is_visible":true},{"tag_id":29490,"tag_text":"abstract algebra","is_visible":true}],"can_have_source":true,"locked":false,"forum_locked":0,"announce_type":"none","announce_through":"","announce_factor":0,"preview":"Let <img src=\"\/\/latex.artofproblemsolving.com\/e\/f\/f\/eff43e84f8a3bcf7b6965f0a3248bc4d3a9d0cd4.png\" class=\"latex\" alt=\"$R$\" style=\"vertical-align: -1px\" width=\"16\" height=\"14\" > be a ring. Let <img src=\"\/\/latex.artofproblemsolving.com\/7\/c\/a\/7ca53e78b06848e441caa5bb6fa4ab1531559e61.png\" class=\"latex\" alt=\"$x,y\\in R$\" style=\"vertical-align: -3px\" width=\"64\" height=\"16\" > such that <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/5\/0\/2\/5026dae4f97e7580363285d44343506f0d29dc04.png\" class=\"latex\" alt=\"$x^2=y^2=0$\" style=\"vertical-align: -4px\" width=\"94\" height=\"20\" >.<\/span> Prove that if <img src=\"\/\/latex.artofproblemsolving.com\/4\/a\/0\/4a0a9c4135f5cbc72b0d643ac08499027fc7835b.png\" class=\"latex\" alt=\"$x+y-xy$\" style=\"vertical-align: -4px\" width=\"86\" height=\"16\" > is nilpotent, so is <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/3\/f\/7\/3f79c3676b62fb47243e0d357c87115b6f3395e4.png\" class=\"latex\" alt=\"$xy$\" style=\"vertical-align: -3px\" width=\"19\" height=\"11\" >.<\/span><br>\n<br>\n<i>Janez \u0160ter<\/i>","target_url":"","target_url_application":null,"target_text":"","state":"none","cat_can_target":0,"has_thanks":true,"has_nothanks":false,"is_bookmarked":false,"in_feed":false,"is_watched":false},"3537377":{"num_posts":1,"posts_data":[{"post_id":34398626,"topic_id":3537377,"poster_id":548549,"post_rendered":"Let <img src=\"\/\/latex.artofproblemsolving.com\/6\/5\/3\/653f07268b36bc014788d5ac245519cad5ed5bc0.png\" class=\"latex\" alt=\"$\\mathcal{P}_n$\" style=\"vertical-align: -2px\" width=\"20\" height=\"15\" > be the set of all real monic polynomial functions of degree <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/1\/7\/4\/174fadd07fd54c9afe288e96558c92e0c1da733a.png\" class=\"latex\" alt=\"$n$\" style=\"vertical-align: -1px\" width=\"13\" height=\"10\" >.<\/span> Prove that for any <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/7\/1\/0\/710e3e4f18d7c0270ab719a00adc61fa12363b1e.png\" class=\"latex\" alt=\"$a<b$\" style=\"vertical-align: 0px\" width=\"42\" height=\"13\" >,<\/span> <img src=\"\/\/latex.artofproblemsolving.com\/c\/c\/1\/cc10a49d7bba92b0918f759ce5aeec1d52088a51.png\" class=\"latexcenter\" alt=\"$$\\inf_{P\\in\\mathcal{P}_n}\\int_a^b |P(x)|\\, dx >0.$$\" width=\"175\" height=\"44\" ><br>\n<i>Cristi S\u0103vescu<\/i>","post_canonical":"Let $\\mathcal{P}_n$ be the set of all real monic polynomial functions of degree $n$. Prove that for any $a<b$, $$\\inf_{P\\in\\mathcal{P}_n}\\int_a^b |P(x)|\\, dx >0.$$\n\n[i]Cristi S\u0103vescu[\/i]","username":"AndreiVila","reported":false,"is_thanked":false,"is_nothanked":false,"attachment":false,"thanks_received":0,"nothanks_received":0,"thankers":null,"deleted":false,"post_number":1,"post_time":1743199453,"num_edits":0,"post_format":"bbcode","last_edit_time":0,"last_editor_username":"","last_edit_reason":"","admin":false,"avatar":"\/\/avatar.artofproblemsolving.com\/avatar_548549.png","num_posts":207,"editable":false,"deletable":false,"show_from_start":true,"show_from_end":true}],"topic_id":3537377,"category_id":7,"category_name":"College Math","category_main_color":"#511e8f","category_secondary_color":"#f2e6fe","topic_title":"Gheorghe \u021ai\u021beica 2025 Grade 12 P3","topic_type":"forum","is_public":true,"roles":{"432":"mod","93494":"mod","67223":"mod","9049":"mod","242520":"mod"},"first_post_id":34398626,"first_poster_id":548549,"first_poster_name":"AndreiVila","first_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_548549.png","first_post_time":1743199453,"last_post_id":34398626,"last_poster_id":548549,"last_poster_name":"AndreiVila","last_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_548549.png","last_post_time":1743199453,"last_update_time":1743199453,"comment_count":1,"num_deleted":0,"num_reports":0,"num_views":22,"category_num_users":1,"category_num_topics":114003,"category_num_posts":443403,"poll_id":0,"source":"Gheorghe \u021ai\u021beica 2025","tags":[{"tag_id":170,"tag_text":"polynomial","is_visible":true},{"tag_id":298,"tag_text":"function","is_visible":true},{"tag_id":427,"tag_text":"real analysis","is_visible":true}],"can_have_source":true,"locked":false,"forum_locked":0,"announce_type":"none","announce_through":"","announce_factor":0,"preview":"Let <img src=\"\/\/latex.artofproblemsolving.com\/6\/5\/3\/653f07268b36bc014788d5ac245519cad5ed5bc0.png\" class=\"latex\" alt=\"$\\mathcal{P}_n$\" style=\"vertical-align: -2px\" width=\"20\" height=\"15\" > be the set of all real monic polynomial functions of degree <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/1\/7\/4\/174fadd07fd54c9afe288e96558c92e0c1da733a.png\" class=\"latex\" alt=\"$n$\" style=\"vertical-align: -1px\" width=\"13\" height=\"10\" >.<\/span> Prove that for any <span style=\"white-space:pre;\"><img src=\"\/\/latex.artofproblemsolving.com\/7\/1\/0\/710e3e4f18d7c0270ab719a00adc61fa12363b1e.png\" class=\"latex\" alt=\"$a<b$\" style=\"vertical-align: 0px\" width=\"42\" height=\"13\" >,<\/span> <img src=\"\/\/latex.artofproblemsolving.com\/c\/c\/1\/cc10a49d7bba92b0918f759ce5aeec1d52088a51.png\" class=\"latexcenter\" alt=\"$$\\inf_{P\\in\\mathcal{P}_n}\\int_a^b |P(x)|\\, dx >0.$$\" width=\"175\" height=\"44\" ><br>\n<i>Cristi S\u0103vescu<\/i>","target_url":"","target_url_application":null,"target_text":"","state":"none","cat_can_target":0,"has_thanks":true,"has_nothanks":false,"is_bookmarked":false,"in_feed":false,"is_watched":false},"3537375":{"num_posts":1,"posts_data":[{"post_id":34398605,"topic_id":3537375,"poster_id":548549,"post_rendered":"Let <img src=\"\/\/latex.artofproblemsolving.com\/9\/d\/0\/9d00bb0f11e69022dded4a8baba1b496f66d272e.png\" class=\"latex\" alt=\"$f:[0,1]\\rightarrow\\mathbb{R}$\" style=\"vertical-align: -5px\" width=\"103\" height=\"18\" > be a continuous function. Prove that <img src=\"\/\/latex.artofproblemsolving.com\/5\/c\/2\/5c20914674d4386a0e2cb2eef69b115e3b02fa24.png\" class=\"latexcenter\" alt=\"$$\\int_0^{\\pi\/2}f(\\sin(2x))\\sin x\\, dx = \\int_0^{\\pi\/2} f(\\cos^2 x)\\cos x\\, dx.$$\" width=\"390\" height=\"45\" >","post_canonical":"Let $f:[0,1]\\rightarrow\\mathbb{R}$ be a continuous function. Prove that $$\\int_0^{\\pi\/2}f(\\sin(2x))\\sin x\\, dx = \\int_0^{\\pi\/2} f(\\cos^2 x)\\cos x\\, dx.$$","username":"AndreiVila","reported":false,"is_thanked":false,"is_nothanked":false,"attachment":false,"thanks_received":0,"nothanks_received":0,"thankers":null,"deleted":false,"post_number":1,"post_time":1743199286,"num_edits":0,"post_format":"bbcode","last_edit_time":0,"last_editor_username":"","last_edit_reason":"","admin":false,"avatar":"\/\/avatar.artofproblemsolving.com\/avatar_548549.png","num_posts":207,"editable":false,"deletable":false,"show_from_start":true,"show_from_end":true}],"topic_id":3537375,"category_id":7,"category_name":"College Math","category_main_color":"#511e8f","category_secondary_color":"#f2e6fe","topic_title":"Gheorghe \u021ai\u021beica 2025 Grade 12 P2","topic_type":"forum","is_public":true,"roles":{"432":"mod","93494":"mod","67223":"mod","9049":"mod","242520":"mod"},"first_post_id":34398605,"first_poster_id":548549,"first_poster_name":"AndreiVila","first_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_548549.png","first_post_time":1743199286,"last_post_id":34398605,"last_poster_id":548549,"last_poster_name":"AndreiVila","last_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_548549.png","last_post_time":1743199286,"last_update_time":1743199286,"comment_count":1,"num_deleted":0,"num_reports":0,"num_views":26,"category_num_users":1,"category_num_topics":114003,"category_num_posts":443403,"poll_id":0,"source":"Gheorghe \u021ai\u021beica 2025","tags":[{"tag_id":298,"tag_text":"function","is_visible":true},{"tag_id":427,"tag_text":"real analysis","is_visible":true},{"tag_id":117,"tag_text":"integration","is_visible":true}],"can_have_source":true,"locked":false,"forum_locked":0,"announce_type":"none","announce_through":"","announce_factor":0,"preview":"Let <img src=\"\/\/latex.artofproblemsolving.com\/9\/d\/0\/9d00bb0f11e69022dded4a8baba1b496f66d272e.png\" class=\"latex\" alt=\"$f:[0,1]\\rightarrow\\mathbb{R}$\" style=\"vertical-align: -5px\" width=\"103\" height=\"18\" > be a continuous function. Prove that <img src=\"\/\/latex.artofproblemsolving.com\/5\/c\/2\/5c20914674d4386a0e2cb2eef69b115e3b02fa24.png\" class=\"latexcenter\" alt=\"$$\\int_0^{\\pi\/2}f(\\sin(2x))\\sin x\\, dx = \\int_0^{\\pi\/2} f(\\cos^2 x)\\cos x\\, dx.$$\" width=\"390\" height=\"45\" >","target_url":"","target_url_application":null,"target_text":"","state":"none","cat_can_target":0,"has_thanks":true,"has_nothanks":false,"is_bookmarked":false,"in_feed":false,"is_watched":false},"3537060":{"num_posts":2,"posts_data":[{"post_id":34395803,"topic_id":3537060,"poster_id":1127103,"post_rendered":"I have an equation involving a function f(x) and its inverse f\u207b\u00b9(x):<br>\n<br>\nf(x) * (a + bx) + f\u207b\u00b9(x) * (a - bx) = 2<br>\n<br>\nwhere a and b are constants. I have tried different approaches to find f(x) but haven't reached a solution. Are there any known techniques or transformations that could help solve this equation? Any insights would be appreciated.","post_canonical":"I have an equation involving a function f(x) and its inverse f\u207b\u00b9(x):\n\nf(x) * (a + bx) + f\u207b\u00b9(x) * (a - bx) = 2\n\nwhere a and b are constants. I have tried different approaches to find f(x) but haven't reached a solution. Are there any known techniques or transformations that could help solve this equation? Any insights would be appreciated.\n\n","username":"Takitaki","reported":false,"is_thanked":false,"is_nothanked":false,"attachment":false,"thanks_received":0,"nothanks_received":0,"thankers":null,"deleted":false,"post_number":1,"post_time":1743180845,"num_edits":0,"post_format":"bbcode","last_edit_time":0,"last_editor_username":"","last_edit_reason":"","admin":false,"avatar":"\/\/avatar.artofproblemsolving.com\/avatar_1127103.png","num_posts":7,"editable":false,"deletable":false,"show_from_start":true,"show_from_end":false}],"topic_id":3537060,"category_id":7,"category_name":"College Math","category_main_color":"#511e8f","category_secondary_color":"#f2e6fe","topic_title":"Equation with f(x) and f\u207b¹(x) – need help finding f(x)","topic_type":"forum","is_public":true,"roles":{"432":"mod","93494":"mod","67223":"mod","9049":"mod","242520":"mod"},"first_post_id":34395803,"first_poster_id":1127103,"first_poster_name":"Takitaki","first_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_1127103.png","first_post_time":1743180845,"last_post_id":34398600,"last_poster_id":1244096,"last_poster_name":"paxtonw","last_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_1244096.png","last_post_time":1743199235,"last_update_time":1743199235,"comment_count":2,"num_deleted":0,"num_reports":0,"num_views":62,"category_num_users":1,"category_num_topics":114003,"category_num_posts":443403,"poll_id":0,"source":"","tags":[{"tag_id":300,"tag_text":"functional equation","is_visible":true}],"can_have_source":true,"locked":false,"forum_locked":0,"announce_type":"none","announce_through":"","announce_factor":0,"preview":"I have an equation involving a function f(x) and its inverse f\u207b\u00b9(x):<br>\n<br>\nf(x) * (a + bx) + f\u207b\u00b9(x) * (a - bx) = 2<br>\n<br>\nwhere a and b are constants. I have tried different approaches to find f(x) but haven't reached a solution. Are there any known techniques or transformations that could help solve this equation? Any insights would be appreciated.<br>\n<br>\n","target_url":"","target_url_application":null,"target_text":"","state":"none","cat_can_target":0,"has_thanks":true,"has_nothanks":false,"is_bookmarked":false,"in_feed":false,"is_watched":false}};AoPS.preload_topics[10]={"3525880":{"num_posts":22,"posts_data":[{"post_id":34255473,"topic_id":3525880,"poster_id":895173,"post_rendered":"Summary of the problem: Pressing go down to last post button always creates a gray box overlapping last post<br>\nPage URL: any forum<br>\nSteps to reproduce:<br>\n1. Go to any topic in a forum<br>\n2. The gray box at the bottom overlaps part of the first post<br>\nExpected behavior: Should not show a gray box<br>\nFrequency: 100% of the time<br>\nOperating system(s): Linux HP EliteBook 835 G8 Notebook PC<br>\nBrowser(s), including version: Chrome 133.0.6943.142 (Official Build) (64-bit) (cohort: Stable)<br>\nAdditional information: It works on any other device, on my iPhone XR, a MacOS, and my iPad. Took the screenshot a month ago. The gray box still appears","post_canonical":"Summary of the problem: Pressing go down to last post button always creates a gray box overlapping last post\nPage URL: any forum\nSteps to reproduce:\n1. Go to any topic in a forum\n2. The gray box at the bottom overlaps part of the first post\nExpected behavior: Should not show a gray box\nFrequency: 100% of the time\nOperating system(s): Linux HP EliteBook 835 G8 Notebook PC\nBrowser(s), including version: Chrome 133.0.6943.142 (Official Build) (64-bit) (cohort: Stable)\nAdditional information: It works on any other device, on my iPhone XR, a MacOS, and my iPad. Took the screenshot a month ago. The gray box still appears","username":"Craftybutterfly","reported":false,"is_thanked":false,"is_nothanked":false,"attachment":true,"thanks_received":2,"nothanks_received":0,"thankers":"Exponent11, PikaPika999","deleted":false,"post_number":1,"post_time":1741756292,"num_edits":2,"post_format":"bbcode","last_edit_time":1741756366,"last_editor_username":"Craftybutterfly","last_edit_reason":":3","admin":false,"avatar":"\/\/avatar.artofproblemsolving.com\/avatar_895173.jpeg?t=1743019491","num_posts":259,"editable":false,"deletable":false,"attachments":[{"name":"Screenshot 2025-02-18 101303.png","href":"https:\/\/cdn.artofproblemsolving.com\/attachments\/0\/e\/537a5820938169614328ed48e4be832406f523.png","url":"0\/e\/537a5820938169614328ed48e4be832406f523.png","size":313,"mimetype":"image\/png"}],"show_from_start":true,"show_from_end":false}],"topic_id":3525880,"category_id":10,"category_name":"Site Support","category_main_color":"#a90008","category_secondary_color":"#ffe4e1","topic_title":"Pressing &#039;go down button&#039; always creates a gray box on the last post","topic_type":"forum","is_public":true,"roles":{"470423":"mod","481507":"mod","473145":"mod"},"first_post_id":34255473,"first_poster_id":895173,"first_poster_name":"Craftybutterfly","first_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_895173.jpeg?t=1743019491","first_post_time":1741756292,"last_post_id":34399072,"last_poster_id":944981,"last_poster_name":"jlacosta","last_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_944981.jpeg?t=1675270607","last_post_time":1743203116,"last_update_time":1743203117,"comment_count":22,"num_deleted":1,"num_reports":0,"num_views":887,"category_num_users":2,"category_num_topics":18564,"category_num_posts":155777,"poll_id":0,"source":"","tags":[{"tag_id":30593,"tag_text":"posts","is_visible":true},{"tag_id":30722,"tag_text":"Forums","is_visible":true},{"tag_id":2700772,"tag_text":"donotputmoretags","is_visible":true},{"tag_id":5,"tag_text":"\/closed","is_visible":false}],"can_have_source":false,"locked":false,"forum_locked":0,"announce_type":"none","announce_through":"","announce_factor":0,"preview":"Summary of the problem: Pressing go down to last post button always creates a gray box overlapping last post<br>\nPage URL: any forum<br>\nSteps to reproduce:<br>\n1. Go to any topic in a forum<br>\n2. The gray box at the bottom overlaps part of the first post<br>\nExpected behavior: Should not show a gray box<br>\nFrequency: 100% of the time<br>\nOperating system(s): Linux HP EliteBook 835 G8 Notebook PC<br>\nBrowser(s), including version: Chrome 133.0.6943.142 (Official Build) (64-bit) (cohort: Stable)<br>\nAdditional information: It works on any other device, on my iPhone XR, a MacOS, and my iPad. Took the screenshot a month ago. The gray box still appears","target_url":"","target_url_application":null,"target_text":"","state":"closed","cat_can_target":0,"has_thanks":true,"has_nothanks":false,"is_bookmarked":false,"in_feed":false,"is_watched":false},"3537333":{"num_posts":8,"posts_data":[{"post_id":34398319,"topic_id":3537333,"poster_id":977200,"post_rendered":"They added something new on community!!!!","post_canonical":"They added something new on community!!!!","username":"jb2015007","reported":false,"is_thanked":false,"is_nothanked":false,"attachment":false,"thanks_received":0,"nothanks_received":0,"thankers":"","deleted":false,"post_number":1,"post_time":1743196982,"num_edits":0,"post_format":"bbcode","last_edit_time":0,"last_editor_username":"","last_edit_reason":"","admin":false,"avatar":"\/\/avatar.artofproblemsolving.com\/avatar_977200.png","num_posts":1730,"editable":false,"deletable":false,"show_from_start":true,"show_from_end":false}],"topic_id":3537333,"category_id":10,"category_name":"Site Support","category_main_color":"#a90008","category_secondary_color":"#ffe4e1","topic_title":"They added something new on community!!!!","topic_type":"forum","is_public":true,"roles":{"470423":"mod","481507":"mod","473145":"mod"},"first_post_id":34398319,"first_poster_id":977200,"first_poster_name":"jb2015007","first_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_977200.png","first_post_time":1743196982,"last_post_id":34398537,"last_poster_id":536807,"last_poster_name":"eg4334","last_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_536807.jpeg?t=1677450752","last_post_time":1743198587,"last_update_time":1743202064,"comment_count":8,"num_deleted":1,"num_reports":0,"num_views":87,"category_num_users":2,"category_num_topics":18564,"category_num_posts":155777,"poll_id":0,"source":"","tags":[{"tag_id":5,"tag_text":"\/closed","is_visible":false}],"can_have_source":false,"locked":true,"forum_locked":0,"announce_type":"none","announce_through":"","announce_factor":0,"preview":"They added something new on community!!!!","target_url":"","target_url_application":null,"target_text":"","state":"closed","cat_can_target":0,"has_thanks":true,"has_nothanks":false,"is_bookmarked":false,"in_feed":false,"is_watched":false},"3426367":{"num_posts":12,"posts_data":[{"post_id":33002059,"topic_id":3426367,"poster_id":883067,"post_rendered":"Summary of the problem: When using <span style=\"font-family:monospace\">exit()<\/span> (or <span style=\"font-family:monospace\">quit()<\/span>) in the Python windows on AoPS the "Description" and "To fix" options show up as "undefined"<br>\n<span class=\"cmty-hide-heading faux-link\" onclick=\"AoPS.Community.Utils.clickHide($(this));\" href=\"#\">sample program<\/span><div class=\"cmty-hide-content\" style=\"display:none\"><div id=\"pywindow0b865955c6e04aabf82931f0c33ecb9d\" class=\"pywindow\" style=\"z-index:0\"><div style=\"display: block\"><div class=\"pywindow-temp-preview-code pywindow-one-digit-lines\" onclick=\"pythonTool.addCodeSourceAndClick(event);\"><pre class=\"python\" style=\"font-family:monospace;\"><ol><li style=\"font-weight: normal; vertical-align:top;\"><div style=\"font: normal normal 1em\/1.2em monospace; margin:0; padding:0; background:none; vertical-align:top;\">exit<span style=\"color: black;\">(<\/span><span style=\"color: #483d8b;\">"Goodbye world"<\/span><span style=\"color: black;\">)<\/span><\/div><\/li><\/ol><\/pre><\/div><textarea rows=\"1\" id=\"pywindow0b865955c6e04aabf82931f0c33ecb9d_code\" style=\"display:none;\" class=\"active-code\" prefixcode=\"undefined\">exit("Goodbye world")<\/textarea><\/div><div><button type=\"button\" class=\"btn btn-run\" onclick=\"pythonTool.addCodeSourceAndRun('pywindow0b865955c6e04aabf82931f0c33ecb9d', false);\">Run<\/button><button type=\"button\" class=\"btn btn-reset\" onclick=\"pythonTool.addCodeSource('pywindow0b865955c6e04aabf82931f0c33ecb9d', false);\">Reset<\/button><button type=\"button\" class=\"btn btn-link btn-pop\" onclick=\"pythonTool.addCodeSourceAndPop('pywindow0b865955c6e04aabf82931f0c33ecb9d', false);\">Pop Out <span class=\"aops-font\">\/<\/span><\/button><div style=\"clear:both\"><\/div><\/div><div id=\"pywindow0b865955c6e04aabf82931f0c33ecb9d_error\"><\/div><div style=\"text-align: center\"><canvas id=\"pywindow0b865955c6e04aabf82931f0c33ecb9d_canvas\" class=\"ac-canvas\" height=\"400px\" width=\"400px\" style=\"border-style: solid; display: none; text-align: center\"><\/canvas><\/div><pre id=\"pywindow0b865955c6e04aabf82931f0c33ecb9d_pre\" class=\"active-out active-out-hidden\"><\/pre><div id=\"pywindow0b865955c6e04aabf82931f0c33ecb9d_files\" class=\"ac-files ac-files-hidden\"><\/div><\/div><\/div><br>\nPage URL: N\/A<br>\nSteps to reproduce:<br>\n1. Use the AoPS Python module to execute the <span style=\"font-family:monospace\">exit()<\/span> or <span style=\"font-family:monospace\">quit()<\/span> functions in a program. (try running the sample program)<br>\nExpected behavior: The "Description" and "To fix" sections give a description of SystemExit<br>\nFrequency: Always<br>\nOperating system(s): Windows 11 Home<br>\nBrowser(s), including version: Microsoft Edge 130.0.2849.46<br>\nAdditional information: N\/A","post_canonical":"Summary of the problem: When using [font=monospace]exit()[\/font] (or [font=monospace]quit()[\/font]) in the Python windows on AoPS the \"Description\" and \"To fix\" options show up as \"undefined\"\n[hide=sample program][pywindow]exit(\"Goodbye world\")[\/pywindow][\/hide]\nPage URL: N\/A\nSteps to reproduce:\n1. Use the AoPS Python module to execute the [font=monospace]exit()[\/font] or [font=monospace]quit()[\/font] functions in a program. (try running the sample program)\nExpected behavior: The \"Description\" and \"To fix\" sections give a description of SystemExit\nFrequency: Always\nOperating system(s): Windows 11 Home\nBrowser(s), including version: Microsoft Edge 130.0.2849.46\nAdditional information: N\/A","username":"SoaringHigh","reported":false,"is_thanked":false,"is_nothanked":false,"attachment":true,"thanks_received":0,"nothanks_received":0,"thankers":null,"deleted":false,"post_number":1,"post_time":1729566605,"num_edits":2,"post_format":"bbcode","last_edit_time":1729606276,"last_editor_username":"SoaringHigh","last_edit_reason":"","admin":false,"avatar":"\/\/avatar.artofproblemsolving.com\/avatar_883067.jpg?t=1732247041","num_posts":282,"editable":false,"deletable":false,"attachments":[{"name":"Screenshot 2024-10-21 201115.png","href":"https:\/\/cdn.artofproblemsolving.com\/attachments\/e\/3\/081a2231791412ae5034cab1b6e73cabbbad2a.png","url":"e\/3\/081a2231791412ae5034cab1b6e73cabbbad2a.png","size":55,"mimetype":"image\/png"}],"show_from_start":true,"show_from_end":false}],"topic_id":3426367,"category_id":10,"category_name":"Site Support","category_main_color":"#a90008","category_secondary_color":"#ffe4e1","topic_title":"Python exit() module decriptions appear as "undefined"","topic_type":"forum","is_public":true,"roles":{"470423":"mod","481507":"mod","473145":"mod"},"first_post_id":33002059,"first_poster_id":883067,"first_poster_name":"SoaringHigh","first_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_883067.jpg?t=1732247041","first_post_time":1729566605,"last_post_id":34397995,"last_poster_id":785184,"last_poster_name":"bpan2021","last_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_785184.png?t=1733178742","last_post_time":1743194735,"last_update_time":1743202076,"comment_count":12,"num_deleted":0,"num_reports":0,"num_views":505,"category_num_users":2,"category_num_topics":18564,"category_num_posts":155777,"poll_id":0,"source":"","tags":[{"tag_id":298,"tag_text":"function","is_visible":true},{"tag_id":29406,"tag_text":"Python","is_visible":true},{"tag_id":5,"tag_text":"\/closed","is_visible":false}],"can_have_source":false,"locked":false,"forum_locked":0,"announce_type":"none","announce_through":"","announce_factor":0,"preview":"Summary of the problem: When using exit() (or quit()) in the Python windows on AoPS the "Description" and "To fix" options show up as "undefined"<br>\n<span class=\"cmty-hide-heading faux-link\" onclick=\"AoPS.Community.Utils.clickHide($(this));\" href=\"#\">sample program<\/span><div class=\"cmty-hide-content\" style=\"display:none\"><pre>exit("Goodbye world")<\/pre><\/div><br>\nPage URL: N\/A<br>\nSteps to reproduce:<br>\n1. Use the AoPS Python module to execute the exit() or quit() functions in a program. (try running the sample program)<br>\nExpected behavior: The "Description" and "To fix" sections give a description of SystemExit<br>\nFrequency: Always<br>\nOperating system(s): Windows 11 Home<br>\nBrowser(s), including version: Microsoft Edge 130.0.2849.46<br>\nAdditional information: N\/A","target_url":"","target_url_application":null,"target_text":"","state":"closed","cat_can_target":0,"has_thanks":true,"has_nothanks":false,"is_bookmarked":false,"in_feed":false,"is_watched":false},"3535483":{"num_posts":15,"posts_data":[{"post_id":34375474,"topic_id":3535483,"poster_id":895173,"post_rendered":"This is very wrong","post_canonical":"This is very wrong","username":"Craftybutterfly","reported":false,"is_thanked":false,"is_nothanked":false,"attachment":true,"thanks_received":0,"nothanks_received":0,"thankers":null,"deleted":false,"post_number":1,"post_time":1742963561,"num_edits":0,"post_format":"bbcode","last_edit_time":0,"last_editor_username":"","last_edit_reason":"","admin":false,"avatar":"\/\/avatar.artofproblemsolving.com\/avatar_895173.jpeg?t=1743019491","num_posts":259,"editable":false,"deletable":false,"attachments":[{"name":"Screenshot 2025-03-25 213142.png","href":"https:\/\/cdn.artofproblemsolving.com\/attachments\/e\/0\/c319addb2d78c9d214d1292b405ea5e5c49d36.png","url":"e\/0\/c319addb2d78c9d214d1292b405ea5e5c49d36.png","size":48,"mimetype":"image\/png"}],"show_from_start":true,"show_from_end":false}],"topic_id":3535483,"category_id":10,"category_name":"Site Support","category_main_color":"#a90008","category_secondary_color":"#ffe4e1","topic_title":"How is this possible","topic_type":"forum","is_public":true,"roles":{"470423":"mod","481507":"mod","473145":"mod"},"first_post_id":34375474,"first_poster_id":895173,"first_poster_name":"Craftybutterfly","first_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_895173.jpeg?t=1743019491","first_post_time":1742963561,"last_post_id":34397212,"last_poster_id":895173,"last_poster_name":"Craftybutterfly","last_poster_avatar":"\/\/avatar.artofproblemsolving.com\/avatar_895173.jpeg?t=1743019491","last_post_time":1743190346,"last_update_time":1743202080,"comment_count":15,"num_deleted":0,"num_reports":0,"num_views":431,"category_num_users":2,"category_num_topics":18564,"category_num_posts":155777,"poll_id":0,"source":"","tags":[{"tag_id":30722,"tag_text":"Forums","is_visible":true},{"tag_id":282620,"tag_text":"I don t know what is going on","is_visible":true},{"tag_id":5,"tag_text":"\/closed","is_visible":false}],"can_have_source":false,"locked":false,"forum_locked":0,"announce_type":"none","announce_through":"","announce_factor":0,"preview":"This is very wrong","target_url":"","target_url_application":null,"target_text":"","state":"closed","cat_can_target":0,"has_thanks":true,"has_nothanks":false,"is_bookmarked":false,"in_feed":false,"is_watched":false}};