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Difference between revisions of "2023 AIME II Problems"
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{̶̨͇̜̩̦̞͎̀͋̀͒̚͝ͅ{̵̜͈͕̻̻͉͚̙͐̇̍̆A̸̻͓̓̍̾̑̋͝Ḭ̵͍͓͈̮͎̠̑̀͝M̴̨̦̮͚͕͎͈̃̄̓͠Ė̷̺̯̯͍̃̈́̂͠ ̵͚̓̿̈P̷̙̰̣̼̳͕̓̋̍r̵̗̫͙͈̹̥͈̣̋̈͆͌̀͌̇ǫ̸̹͈͉̣̠̽̽̂̅̍̒b̸̧̜͖̃̄̽͗͐͋̑̚l̵̜̹͇̘̀̊̒̌̈́̒͝e̵̞͆̓͊̃m̸̬̮̾s̷̨̘̬̘̍̀͐̓̒͒̈͝|̴̜̈́̐͗̂͝y̸̡̞̞̥̞͗̓͐̓̅̿͆ȇ̶̛̫̙͙̆̈́ă̴̖͇̞̳̙̿̒̓̅r̴̛̛̹͓͚͗͊̉͐̄͘=̴̨̙̻̳̘̗͉̒͜2̵̨̡̡͍̖͇͎̪͑̓́̔͂̒͝͠0̸̧͌̿̎͌͑͌̚2̷̦̟̝̫̐̑͗3̷̛̛͍͚̤͕͇̜̂̂͑̏͝|̵̡̘͓̳́͋͊̀͒̂͝n̸͖͔͛̊́͑=̴̦̠̳̠̼̈́͂̅̿ͅỈ̷̘̙̙I̴̡͉͇̞̗̟͈͒̎́͘͠}̸͈͖̝̦̟͘͠}̶̖̊́̀͗̕
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{{AIME Problems|year=2023|n=II}}
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̵̤̺̺̹̟̗̝̉͐͛̃̉̚͝=̶̹̭̈=̴̧̺̥͂̏̕͜Ṕ̷̨̭̦̙̊͋͊̅͝͝ř̷̡̪̉̐̋͝ò̴͙͔̬͇̭͜b̴̡̢̯͖͍͚̝͓͗̂̿̓l̶͓͑e̷̡̼̐͝͝m̶̞͙͖̥͍͓͊̉͆̀́ ̴͙̟͒͐͌̊͂͜ͅ1̵̧̢̦̹̹̼̳̏͂̏͜=̷̠͍̜͍̭̓̀̆̃̃́͘͜͠≠̛̞͚͈̑̄̋͗
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==Problem 1==
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̵̢͋̈̊̂
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̵̢̗̞̗̠̙̂T̴̩̜́̾͜h̷͍͍̘̔̆͗ę̶̗̹̻͎̥̖̇͒͝ͅ ̸͈̻̬̼͙͗̌̐͂̀̋́͝n̸̪͗ǘ̵̘̆̃̿̀̍m̷̯͚̺̪͖̰̃͝ͅb̶̗̣̰̯̥̰̱̀͒́͝e̴̤̐ŗ̴̛̖̇̚͝s̶͇̗͖͕͋̀̊̈́͆͝ ̸̦̘̭̹͉͈̜̓o̸̧̧̭̱̦͖͖͆̈́͒͛f̴̡͙̝̼̈́̈́̉̓͂̓̒͜ͅ ̵̡͙̩̜̜̣͈͗́͛̑a̷͓̠̣̙͂̄̽̌̓̌̈͌͜ṕ̸͚͎͍͌͒̄͜͝ͅp̶̛͓̜͕͐̐̐ļ̴̰̲̳̥̙̙͌͛̉ḙ̸̡̏̈́̍̊͠s̴̛̘̜̼̲̼͔͚̮͒͐̋̑̇̇ ̶̝͔͐̌̇͘̕g̵̜̻̱͙̲̿̎͆r̵̲̩̗̪̥̰̪̚͝ǒ̷̧͍̖͕͇̎̌̀̃̔͜w̶̧̱̮̫̮̦̒į̴͍̻̼̩̪̰̊̄͊̀n̵͙̖̼̜̖͓͊̈́̃̈̉͒̑͝g̵͇̙̼͕̞̒̓̈́̕ ̶̺̞̯̬̐ō̷͍̆͑̈́̏͆̃n̷̛̦̥̳͒̔͆̒̍ ̵̨̛̦͙͍̆̅̓̕͜͝͝e̷̛̺͋͂̑̌́̈́ä̴̫́ċ̴̜̥̠ḧ̵̛͓̠̭͉̝́̊̒̐̕ ̶̡̜̭̫͈̮̘̓̄͊̉̓̓͜o̴̤͕̺̐̀͌f̵̞͈̤̀̓ ̴̡̛̬̪̗̩̍̓̈́́ş̴̱̮͖̹̳̪̩͒̄͘i̵̘̱̱̩̺̖̅x̷̮͍̫̒̈̽͝ ̷͎̟͇̗̲̥̙̑̅̓̑̕a̵͓͉̓p̶̧͙̻̓̀̐̀́͂p̴̤̅̆̔̽̕̚͘͝l̸̼̣̻̳̭̹͈̤͂̈̂̂̾̆̀e̴̡̲̫̮͚͗̕ͅ ̸̡̱͙̱̐̍̄͐̂̕t̶͔͍̾̿́̓̃͘r̵̢̡̠̱̒̿̅̊̃e̶̼̫͙͚̪͓̤̾͒̐́͛̄̑̚ͅe̷̖̠͐͝ś̵̡̤̭̈́̀̔̓͆̂ ̴̮͉͍̫͒͐̌f̴̱̫̗̭͕̽ͅȍ̷̘̹̳̯̞͝r̵̢̙̰͖̦͐́͘m̴̦̩͂̿ ̷̢͍͓͖͙̖̩̏̃͊͗̓̊ǡ̴̡̢̯͇̖̰̕ͅṇ̴̡̭͚̗̤̲͊̽͒̊͒͂̍ ̵̢̨̜̣̪͍̣̉͐́̾͂̿̾̕a̶̹͖͍̒r̶̡̘̘̻̿̓̅́̍͑͂ī̸̛͍̼̤̈́͋̿̃t̶̩̤͔͋ḥ̵̨̢̅̊̓͋͠͠ḿ̵̗̙͖̈́͛͊͂̍͗ȩ̷̖̮̞̼̒̌̑͑̏͜͝t̵̠̀ị̴̑͌̉͘͜c̸̗̙̭͉͇͑̐ ̶̧̝̟͓͇͌͛͑̀ś̷̻̤̮̹̂͌͐͋̃͗̃e̷̼̬̽̀̀̽̊͆͜q̶̘͊̿̀̄̔̊̋̚ů̶̢̹̹̱͚͎̏̾̊̎̇͌̂e̶͖͓̩̜̍͗̈́̚̚ͅn̸̨͈̦̽́͐̈́͘͝c̸̨͚̗̫͙̘͐̎̂̍̿ͅe̵̙̯̫͉̔͋̀̀̆̋ ̷̡̮̹̱͚̪̭̳͗̃̍̀̈̔w̶̠͙͗̊͐̎̈́̔ĥ̷̜͓̤̱̜́ḛ̸̹̲̬̗̝̎̽͠ͅr̷̪̠͋̌ȩ̴̍ ̷̤̘̘̥̜̠̚t̸̙̟̬͕̦̞̔̎̓͋̈́̾̚h̴͕͓̞͖̞͚̠̱̋̐̒e̵̤͈̜͎̜̲͒ ̸̥̈͘g̴̱̙̭̺̜͔̈̂͊͗̊̑͘r̵̪̪̩͖̤̥̋͊ȩ̷̭̤̗̙̝̤̈́ã̵̰͎͈̯̥̃́̀̓͗̂t̶͇̻̖͈̳͐̑̈̏̋̀͂͐ê̴̮̬͓͚̥͛̓͗͒̏s̷̟̀t̴̨̳̯̣̼͓̺̕ ̷͚̖͈̬̪̈́̅͊̃̒̔̂͘n̶̺̱͈͖̺̓͋̽̈́̈͝u̸̠͗m̶̨̨̝͈̞̜̠͕͂̄̆͘b̴̠̖͎̲͎͛̀̿͘ë̴̜̦̭̽̉̉̾r̵̛̝̬̠̪̬̓̋̓̎̆̾͋ ̸̡̼̱͔͛̂͑͊͂͌̓o̷̮͓̗͎̞̭̿̆̍̈̆̚̕͜f̷̞͖͎̙̼̆̆͑͗̾͝ͅ ̷̧̧̬̠̹͕͎̑͐͘a̷̛̗͕̦̪͙̰̎̇͂͑̄̋͘͜p̷̩̫̩̊͑̿̓͂p̸̠̃l̷̗̹͈͔̲̍̀̚ͅe̴̩̤̞̤̭̔̾̒s̷̛̝̳͍̺̗̘̀͝ ̵͔̟̫̖̹̆͂g̸̱͎̽͛͗̐̃̃r̷͉̊̌̏͌̉͝͠o̶̧͈̹̝̖̥͈͈̅̿̋̋̇͐͠͠w̶̛̹̹̲̩̭̠͊͐̓̽͑̓i̴̢̛͍̙͚͛͛͗̎͂͋̚n̸͎͚̬̈́̒̇̅̿͐̔ǧ̸̬̅̕ ̷̘̯͎͇̜̈̎͐͑ͅo̷̧̡͚̗͓̺͍͋̓̓ň̴̜͒̆͗ ̵̡̯̣͍̭̱̈́̓̎́͑̄̚a̸̹͎̾͂̈́̀ṅ̷̗͕̲͈̺͉̝̿̄̀y̵̧̳͉̦͛́͊̒ ̷̦̦̟̊̌̃ȍ̸͖͕̲͕̝̮̤f̷̳̗̙͇̓̏ ̸̡͉̟̘͇̤̲̍͌́͆̎̍t̶͔͚͔̮̹̀̅̈́̿͐͜ẖ̵͍̗̰̈̈́́̉̅̔͘͜͝ͅe̸̢̲̘͂ ̸̡͍̤̺͚̲͑͑s̶̡̟̼͖̖͚͉̭͌̑̀͒ḭ̷̜̬͚͕̱̥̈́͊́̋́̕͝x̶̛̱͑̆͆̌̎̅̚ ̸̞͉̼̬̣͚̞̘̾̑̇͋̚t̸̨̫̥͆̚ṟ̴͊̄̏̊̈ẻ̷̜̞̲̠͇̮͑͛̀̋̔͋͂ę̷̘̘͕̗̈̀͊̚͜͝s̴͍̖͙̹͈̣̝̑ ̵̭̼͋̓̈́͝ǐ̸̝̈́̎͊̓̈́́̚s̶̪̘̮͈̙͓͕̲̑̊̍ ̶̛̱̖̭̠̹̙̰͋̈̑̄ͅd̷̺̯̫͕͈̯͌̍̇͜ö̶̱́͛͌̾͜ͅụ̵̹̻̫̥͊͆b̴̖̞̱̠͈̗̲͕͛̓̑̋͠l̵̠̼̘̯̭̊̍̈́͛́ͅĕ̶̯͊͜ ̸̱̘̝͛̆̔͛̊̕t̶̥̲̯͋̅͊̓͜h̴͔͕̝̞͖̞͉̾͛͜e̸̡͍͙͚̹͔̖̍͂̋ ̶̣͎̙͋͌̋͐̉l̷̨̪̞̮̰͍̜̆͌̀ė̸̥̼̦͇̝̓̊̂̀̂̄͘͜a̵̬̮̅̇̆̐̏͝s̶̢͚̖͔͙̀̂̿ͅt̸̙͌͂͗ ̶̧̬̲͙̼͌́͌͛̏̀́̆ͅn̶̢͍̠͇̺̽ù̶͚͍̼̘̈̅́̒̕̚͘m̵̛͓̀̎͋̆̂͐̕b̷̫͒ę̵̥̲͈̱̩̌͒͐̃ͅr̶̟̝̮̪̺̈́̌̂́̕ ̸̹̍͑͜͝ô̵̘͐f̸̛̘̟̟͇̺͎̪̌̾̒͘ ̴̪̦̇̅͛̏̈́̔̒͜͝ä̸̛̦̖͕̣͑̃́̆͐̕ͅp̸̘̪̗̋̎̄͋p̴̨̛̟͇͖̲̳̹̟̉͂͐̕l̶̡̼͒̂̏̑͊̏e̵͇͔̦͕̠͆̀̒̎͐ͅͅs̵̻̫͖̤͇͚͖͊̎̈́͘ ̴͔͕̯̞̦͆̾̒g̴̫̙͚̫̏̀̉̿͊̈́r̶̢̖̫̮̈̾͐͝͝͝o̶̧͉̩̦͋͊̆̂͂͗̃̀w̵͓͗̔̿̚į̵̛̥̣̳͉̬͈͌́̿̀̄̿͜n̵̫̬̱͍̮̪͗̾g̶̪̜̳̬͔̫̞͌̔̎͊͛͠ ̸̧͇͉̺̎̆̒̂͂͂́͘ǫ̸̣̞͖͓̼̘̉͋͑̍̇̕n̶̨̥͙̦͕̜͗́͗̐͌̋̎̔ ̵̗̱̭͔͆̋̏͒̆̿͑͘ä̷̡͉͎̼̣̺̲n̷̨͎͒̿̍̌ỳ̶̛͎͕͐̋̈͂͠͠ ̵̛̣͐̂̈͆͘͘ǫ̵̦̋̿̓̾̆f̷̥̩͎̣̜̯̦͆ ̴̧͙̦̲̦̞͛̊͘̕t̷̛̻̐̍̈́h̷̤͕̥̤̗̺̟̲̋̃͌̿ȩ̵̖̹͉̖̺͊̓̀̊͘ ̷̰̘͛͊̔̈͋̑̚s̵̢̨̢̱̱̙̥͋̒͒͠ì̶̠̲x̸̢͉̹̳̙̣̩͐́̒̏̊̿̚ ̴̦̭̋̈̀̏̈́ţ̷͙̗̦̩̖̞̟̀̂̑͋̉̕ŕ̶̘͔͖̪̮̊e̶̝̖̘̗̊̐͘ͅe̵͇̰̝͙̻͑̒̈̅s̵̛̫̍.̷̢̙̹͛̈́̍́̓̌͑ ̶̗͔̳̝̰͓̒͋T̸͖̦̣͎͒ḧ̷̛͙̹̼̱͍̼̭̏̓ë̴̛̙̪̙̦̳͉͆̆̀ ̸̛̜͙͍͖̼̟̱t̴̨̰͕͈͎̎̋̽̒̅̚ó̴̧̝̘̮̹͑̿͊̌̇ͅt̴͉̞̟̣̲͆à̸̗͖̪̺̋̃̃̋ļ̸͔͙̖͗̓͗̊̕͘͝ͅ ̸͇̜̗̬̑̈́̊͜n̸̝͇̳͎̰̍͊͛̆̈̐͜u̴̫͓͙͈̝̻̙͒͋͂͜m̶͔͕̹͎̄̔b̶̖͉̅̀̊ͅe̵̜͆̉͛ŗ̵̛̞̇́̚ ̴̨̞̦̳͙͓̹̙̀̐́̀̾̕͝ỡ̸͍̟͈̺̦̫̳̾f̸͕̰͕̻͚̩̬͕̊̾́͑̊͠ ̵̥͕̼͐̇a̸̡͇̫̜̺̔̒̊́̆ṕ̸̭͙͙͍̰͜p̸̡̪̗̰̠̳͐͋̋̉͌̿̐l̶͈͙͎͓̞̞͓̓͆̆͜e̴̱͕̙̟̱̳͐͗̇̌̚͠s̵̫͊ ̴̨̜̀̍̀̂͛͂̀͗g̸̜̯̑͘r̸̝̱͓͎̯̺̈͊o̸̰̜͆̽̀̇̎̚͝w̵͖͛͋̎̓ỉ̶̼̗͉̫͙̗̯͇̐n̶͇͇͍̦͆̍́͛̽̒͘͝ͅg̵̟̳̓̈́͆͘ ̵͔̰̫̞̥͆o̸̞̍̔̓͗̇͝n̸̨̡͚̤͓͘ ̷̮̯͎͎͎̺̅͛͑̈́̿͘a̴̱̞̺̯͋̽̀͒͝l̴̟͗l̵͙̳̥̦͒́̎́̽͂̽̅ ̴̡̹̏͠s̶̞̘̙͖̖͎͚͓͂̄̇̏͒̃ĩ̶̱̮̠̬̪̲̿̅̆͘x̶̡͕̝̑̾͝ ̸̜̩̉̽̏ͅt̸̨͓̪̿̀̚͜r̵̮̦̭̟͓̺̐͐̌̂͆e̶̢̡͔̯̗͎͙̻̐́̉͝ȩ̶̢̝̤̯̫̯͒͐̏́́͒̓̀s̸̡͍̯̤̋̆̾̅͋̍͜ͅͅ ̴̡̺̣̗͔̠̗̬͂i̷̧̫͓͆̓̋͝s̴̫̲̠̖͍̦͌̀̊͜ ̵̧̛̲̖̱̓<math>̵̙͕̼̖̱̰͎̘̑͋̕9̴̲̫̞̖̞̪̔́̏̿͝9̶̼̠̣̩͗0̶̡̞̯̮̗̱̇͊͐̆́̐̔.̵̡̟̥̘̔̒̎</math>̶͕̮̲͚͓̺̆̎͝ ̸̦͔͊̅̌̕F̴̡͍̦̜̭̘̄͛̆̌ī̶͔̯͕̺̟͉͆̄ǹ̵̨̥͉͖͔̥͖͜͝d̸̩͈̯̆̈͊͐̈́̈́̚ ̵̟̹̥̐̑̓͠ͅt̶̥̘̜͈͆ḩ̸̧̙̭͉̠̆̂ë̴̬̬̦̞̹͛̈̄̑̕ ̵̼̖̣̠̭́͜ͅg̶̣͑r̵̭͕̯̜̔͗͐͊e̵̢̺̦̣̦͋̓́̀̿̀̕͝a̸̳̤͓͔͌́̈t̸̥̹͙͍͑̈̓͌ͅe̵̺̟̠͍͊͊͛̓̆͐s̷͓̥̹͕̲̉̔̈͂ṱ̴̨͛́͌̑̀͝ ̴̲̞͒̚͠n̸̡̺̙͇͙͖̮͙͐̓́̒̂̓͒͂ủ̴̢̻͖̇͑͐̒̓̆͘m̴̛̬̱͓͉̈́͝b̵̙͔̤̠̮͚̑ͅe̷̛͓̭͐̒̀͘r̸̺͒̆̾͛̀̀͝ ̵̰̮̼͙̦͕̭͚̉ơ̷͎̂̅̋̓̕f̷̨͕̪̙͔̃͛̈́͜͝ ̴̳̗͔̝̬̿͒͂̐́̓a̷̛̞̖̯̦̗͙̺͐͑̓̀͆͝p̴̳̭̻͚̮͚̍̓̕p̷͙͓̫̀ļ̴͓̻̼̼̗̻͙̆̅̏̓͗̿͝e̷̛̫̓̔̋̄̕s̷̨̲̞̟̳̬̳͂͒̎͒̾ ̶͔̀́̋͒̚g̸̡̞̖̩͉̳̐̊͆̔̽͂r̶̨͚̞͈̝̝͕̼̉̍̈́̾̄̚͝ǫ̷͎͖̞̰͗͜w̷̟͍̻͍̺̄̄͗̈́͝i̴̹̎͛̅͐̔̋n̶͈̻̗̐͒̀̏͐g̸̡̧͉̼͚̞̞̜͐̂̍͗͊̔̋͝ ̵̮̋̔o̴̡̡̻̻͔͓̟͊̃͆͌ṇ̵̡̨͇͓̠͆̄ ̵̦͉̩͚̙̥̏͆͊̚a̸̺̩̱̜͑̅͒̑̍͝͝ņ̵̦̜̫̜̊̓͆̀̃͜ỹ̶̭͖̞͓͔̍͛͂͜ ̷̢͖̻̾ͅo̷̧͔̥̹͐̅f̷̢̟̱͖̱̑̎̿́͐̉͂͝ ̶͓̆t̵̨̙͖̄̍̅͒͐́͝͝ẖ̴̖͓̦̖̔̚e̷͙̱̗͍̝͒̀̇̀͂̀̊̉ ̴̧̫͍̻̮̃̉̑͆̈́͘͘͠s̷̞̩̏̾̿͠i̶̭͙͔̙̳̫̥͆̆̇̏x̶͓͈̫̌͘ ̸͔͙͎̩͘t̷͕̟̫̎̆̅ṙ̵̫̬̍͠ȇ̸̳͔̦͇̅͐̈́̈́̄̚ę̵̲͗̆̎͒̈̂s̷̰̭̤̭͋̊̂͌̽.̸͓̩̣̥̖̪̔̃
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The numbers of apples growing on each of six apple trees form an arithmetic sequence where the greatest number of apples growing on any of the six trees is double the least number of apples growing on any of the six trees. The total number of apples growing on all six trees is <math>990.</math> Find the greatest number of apples growing on any of the six trees.
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̷͔͕̼̼̜̖̎́͒̈́͂̚
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̷̛͎̤̫͍̱̓̒͋͗̓̉̿͜[̷͈̲̤̪̉͝[̷̡̈́̅̚2̴̦̻̩̤̔͋͒̌̈́0̶̻͈̙͙͛͛2̷̡̲͍̣̺̻͕̂̒̒̍3̵͈̮̰̟͇̫͖̻̈̒͛̉ ̵̧̘̗͎̠̹͚͊͝͠A̵̧̯̪̟͔̙͐̂̉̈́̇̕ͅḬ̸̻̼̭̈́̀̌͛̏̚͝M̵̝̦̬̙̅͗̉̐͆̀̕Ė̷̢͚̅ ̸̙̫̤̗̮̖̂̓Ị̷̡̟̙̮̽Į̷̙́͑̅ ̵̣̍̒̓͌̽̃Ṕ̸̢̯̖̽̐͂͂͋͝r̷̼̦̤͎̝̈́͛̑̒̌͠͠o̶̙̮͆́̿̍̒̚b̸̛̼̜̔̐̆ļ̷̲̻̗̓̉͛͌̈́́̂ͅĕ̷͈̦̟͓̠̋̍̓́͛͂͝m̴̰̯̗̙̎͒ͅs̵͚̳̯̉͐͂̾͗̈́̈̈́/̵̱̺̐P̵̬̫̄̀͆͑̐͐r̵̤͕̞̗̱̥̱̓̄̓̇̄o̶͚̪͑́̿͝͝͝͝b̵̞̉l̵̨͓͎̠̳̊̄̿̈́̉́͌ę̷̥̩̏͌ṁ̸̡͉͍̊̆ ̷̣̪̊1̷̹̱̟̩͇̇́̒̽̐̆̕|̶̢̛̬͍̟̗͖̓̏̓̏̈́̍S̴̜̀̋̆͝ô̸̘̙̝̻̭̠̈̈́l̸̜̹̮̣̜̬̏̋̂̓͋̅͝͝ù̴̙̥̮̩͓̈́͒̃̈́͌͘ṯ̷̺̳̊̐͊͊̑͘͘i̸͈͚̗̩̋̍̔̂̓ở̷͇̤͓̈́̒̾͜͝n̵̠͖̙̾̈́̆̄͐͘]̶̨͈̳̙͔̐]̸̘̲̫͑̀͒̅̀͋̾̾
+
[[2023 AIME II Problems/Problem 1|Solution]]
−
̷̙̯̩̹̙͔̍͐͆̈̔͋͆̆
+
 
−
̵̡̨̱̘͙̜͍̇͘͜=̷̢͍͇̪̿̔̄̔̄͐̔̓ͅ=̵̨̟͔̦̯̀ͅP̵̳͉̤̩̫̯̑͘ͅŗ̷̂͒o̷̱̺̕b̴̨̡̛̛̙̱̺̾̏͆͒͛̚l̶̨͕͖̳͙̹̜̈́̓̌͘̕e̸̛̛̬͖̬̟̜̿͑̌̂ͅm̸̢̠̀̈͋ ̸̼̮̈̎̎̋͂̀̚2̷̢̜̾̅͘=̶͕͎͈̘̟̳̰͊̈́=̵͙͔̺̲͛̀̿̈́̕͠
+
==Problem 2==
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̸̨͙̬̪̘͉͇̪̾̎͗
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̶̦̞̔R̸̢͎̙̯͇̒̌̾̇ȅ̶̼̒̇̋̆͠c̶̣͕̋͝a̷̦̼̮͌l̷̡̛͉͋͌̀̽͝l̶͇͕̻͇̳͑͊͘ͅͅ ̴̛̠̯̬̊̎̌͠ͅt̷̪̤̾̆̔͛͋ḩ̵̹̫͓͚̖̯̃̀a̷̤̽̐ẗ̶̢̟̬̞̣́͛ ̸̟͓͈̞͉̍̾́̎̒̌̊a̶̢̡̖̱̣̦̤̻͆͑̋ ̴̠̯̈́̎p̵̧̥̫̺̦̦͒ȁ̴̢̹̱̯̰̖ͅl̵̢̳͓̽̈́́̇͝i̵̠͍̞͖̊̑͛͐̌̚n̵̫̹̖̹̬̯͛̿̃̍̚͝d̵̛̮͍̤̬̿͊͋̂̐̊r̸͕̈́͐̿͑o̵͔̥̘͚̓͂̈́̍͘͜͠͠m̵̡̬͉̹̎͜ẻ̶̢̢͎͍͓͈̥̣̄́ ̶͔̹̤̪̯͓̾̑͝͠i̵̱͙̳̅͑͒̔͛̿s̵̡̨̬̟̳̝̾̃͌͝ ̸̨͍͙̙͍̇̓̍̓̈ą̶̲̘͈̖͈̒̉̈́̈̎͠ ̸͎̒̚n̶͍̼͙̟̼͖̪̆̂͛̑̎͝ů̶͉̱̰̬̦̺͔̆͆̒͗͝m̷̨̧̥̫̮̭̏̒̂͛̽́͠ḃ̵̙̹̘͕̞́̑͝ȩ̸͂̾͛̈́̍̀̈́r̶̠̤̬̦̾͗́̅̈͒ ̸̢̜̝͇̹̗̪̦̀t̴̹̥̆͊͝h̵̦͊͒̅ą̷̛͙̝̣̒̍̐̈͐͗͌t̶̡̮̟̖͗̊͌̃̑͊ ̸͉̔̐̆̊̈́r̵͎͖̟̬̟͌͋̍͘e̷̡̲̼̊̈́̍̅̈̋̐͝ā̸̢̦̗͚̖̱̿̚ͅḋ̶͎̞̰̪͇̮̓̅̑͋̈́̎͝s̸̫̺͕̘̉́̎͌̈́ ̵̘̯͉͙͎̥͎̓͌̓͂̍͑͋͝t̸̡̰̤̹͎̆͊͐͋̈́͊́͝h̶͕̱̬͊̎̃͌́̆͛͠é̶͔͔̭̦͛ ̴̼̠̾̈̐͑͜͠ş̶̻̩̥̳̋̈͊͋͗̉͠ạ̷͖̭̈́̈́̈́m̶͔͇̻̋͐̌͑e̴̦̱̪͈̣̿̐̆̃̈́̎͠͝ͅ ̴̖̹̱̮̮̩̳̍̆͐͊̆̈́͗̚ͅf̷̲͓̯̙̮́̈́̒̍̓̂̿ő̸̩̪r̶̟̎͗̇̃̽̋̈́ẁ̵̫̐̇͋ạ̷̡͚̪̓́r̵͈̗̞̀̚d̶̖̦̔̑̈́̊͘ ̶̨̥̳̻̤͊̍ä̸̻̜̫́̌͗̃͆̈́̕͘n̴̥̻̠̈́͛̽͑d̶̢̧̺̰̞̭͌̏̀͝͝ ̷̨̛̙̦͙͐b̴͕͚̘͖̐̔̄̑a̸̞̹̬̜̍̊͒͒̎c̴̮̪̪̥̣̜͒̋̐͗̽̾̎͗ͅk̶̢̡̦̳̟͓̻͈̎́͌̉͗̑̚w̸̭͓͑̈́͒̒̕͠͝͠a̸̢̦̽̿̇͋̎̍̚r̶̯̭̩̜̓̋̿̏̿͌̈́͠ͅd̸͈͖̾̍̽.̴̨̢͚̘̙̲̈́̂̀͒ ̸̖̮̪͚͎̏̅̇̈͆F̷̡͔͔͙̪̈́i̶̧̺̭̅̽̏͊́̂͝ͅn̵̡̦̆̀̄̈́͠d̴̡̰̰̙̖̒̽͗̃̾͆͜ ̴̝͚̺̙̎̀̂́͒͜t̴̖́͒̂̈́h̷̡̻̘̞̦̖̓͗e̴͔̟̗̽́͒ ̶̡̍g̴̨̮̦̱̥̹͕̤͛̈̌̂̔̋͝r̵̛̥̗̼͈̪̤̹͖̂͑̉̈́͗́e̵̥͈̾̓a̵̘͍͆̈́͗͛̏̀ͅẗ̵̡̪̅̾ḛ̶̺̯͖̟̳̦̌̃̕ś̵̩̻̜̣̦͉̈́̿̿̅̈́̿͘t̴̢̘̳̩͎͍̃̎ ̵̺̽̔̐i̷̱̱̥̣̫̱̽̄ņ̴̹̪͚̻̹̙̲̇̿̐̃̚̚t̸͚̟̻̼̦̫̺̒̈́̈́̎̐́ę̴̟͘͝g̶̯̳̹̹̅̃e̴̬͉̘̱͓̔̀̄̆̑̿̆͑r̶̟̰̞̽̃͊͗̉͆̂ ̵̢̻̼̀l̷̜̮͎̽̊̓̓̾͝e̷͍̜̝̤͍̝͊͌̑̐͂̚s̷̢̯͍̗̤͙̩̄ͅs̵̨͔̜̺͇̞͑͆̀̕͘͘͠ͅ ̶̧̨̘̖͍͙̳̂̍͋̽̿̀̇͛t̵̲͚͉̣̉̌̒̈́͆̉̕͠h̷̛̼͊͒́͒̀͋ā̶̬̪̥͚͚͕̗̜͒̌̌̽̒͝n̶͕̿̐̿̀͛́̋͠ ̸̨̖͔͎̬̬̯͉̄̄͒̄<math>̴̲̂͌̽̔̽͝1̴̫͐̈́̇͒̒͆0̶͔̀͆̅̈0̶̢̱͕͈͂͝0̵̧͖͗̄̓̈́͆</math>̵̱̹̥̦̄̂̿̚ ̶̼͎̂̈́̈́̀̆͐̇͘t̸̰͕͋́̓̑̉͋̂̌ḫ̵͋̓̃̈́̕͝ạ̷̖̟̹̟̭̜͒̿́̌̈́͂̇̎t̵̗͎̭̜̙̙̦̔͛͌̍̑͋̀ ̴̧̠̻̘̤͕̜̄͝i̷̢̠͇̗̻͙̬͖͋̈̒̎͛̊̋s̵̙͐ ̵͇̬͚̠̲͇̝̜̏̽̏͝ả̸̦ ̶̨̜̖͖̆̓p̴̢͚̜̘̭̄̈́͂̓͋͌̓́a̴̞̯̲̜͉͎̟̞͑̅͆̄͗ļ̸͎̦̳͖̋ḯ̶̳̆̋̔̿͛͝n̸͓͗̾̆̽̓̌̈́̃ͅd̷̞̼̲̩͐̈́̆̈́̄̉̚r̶̲͉̟̘̂̀ǫ̴̧̢̨̛͇͖͎̟́̽̔̍͝͠͝m̶̧̰͚̦̿̆́̎͘͘͝è̸̢̫͖͉̤̬̹̔͑͗̀̚ͅ ̸̨̛͙͙̟͔̥̇̌͂̆̊͠b̵̨̡̲̳̱̟̰͂͗́̾̑͠ŏ̶̞͙̺̜͖̗̭̇t̸̞͎͙͆̇ḣ̵̟͇̥͎̒̍̇̊̅́ ̵̧̜̠͚͔̈́̿̽̃w̵̱̄͑̐̀͑͆̂͌ḧ̵̤́̋̌͋̀́̚ẹ̷̠͎̞͐̅̈̄̃̏ņ̷̫̫̜̯̯̍̔̒͆̏ ̷̧͔̺͎̇̌͜w̴̖̺̾̔͂̊̾̽͝r̸̼̩̆̾͑̚̚ȋ̸͍̥t̴̥̰̯̰̗̤̉͊͜t̶̻̠̻͕̯͍̅̿̂͑̊̕͝͝ḙ̶͇̠̖̫͋̔͌͜ǹ̷̦̭͉͉̳̼̊͂̊͒̇ ̴̘́̒͋͠͝i̷̘̰̭̯͂͗̉n̵̜̳͎̔̈́̏̀̇̕̚͝ ̵͙͈͓̲̉͝b̸̠̫̋a̵̗͂̅͆s̴̡̫̀͌͌ͅé̷̫̗̭ ̷̘͓͂͐̐͆́̕ṫ̴̩̜͔̜̤͇̈e̸̢̡̘͔̓͌̎̍̂͑n̸̠̣̖̙̹͇̺̬̎̿̈́̄̓ ̸̯͈̹͕͙̓a̴̭͖̖̱̲̪̍̈́̚ͅn̵̨͎̘̼͕̪̹̒͗̅̌̎͠d̷̬̘̻̘͈̮̾͛̊̽͛͑͝͝ ̸̳͉͙́̄͑w̵̝̺̜̙̃̓̒ḩ̸̬̫̲̜̺̱̠̋̍̅͒̌̈́͑̐e̷̟̰̥̞̫̥͝͝ͅņ̵͖̞̳͙͓͍̣̆̄ ̷̢̞̮̝̀̂́̋̀̽̏ẅ̸̹̜͓̺̥̜́͋̾̑r̵̰̭̖̘̃̇͊͐̃ǐ̶̭͚̪̯̳̼̣̬̄͋̂̔̌͒͝t̸̖̱̰̮̞̓t̷̼̮͑͆̎̇͒̇͝e̶̪͒̃̄n̵̤̍ ̶͙̔͂̎͂͋͠ĭ̷͍̩̥̤͖͖̊̊̽͘͝͝͝n̸̰̈ ̵̢̣̫̣́̂̈́̍̊̄̂̍ͅb̴̟̯͖̖͈̝̈́͒̈̆ą̸̖̠̜̖͍̹͛̄ͅŝ̶͓̥͇̮̖͙̃̚͝e̴̮̝̲͖͌̚͘̕ͅ ̵̙̭̙͔̀̓̽ë̵̯̼̞̮́͛͛͗i̵͕̪̹̪̪͌͌̕͝g̷͓̎͝h̴͈̰͇͉̥͛̅̽t̸̢̪͔͔͙̩̽͝͝,̸̘̜̺̠̰̞̅̆̿͒̄̉ ̵̢̬̠͙̩͇́͝s̸͙̜̻̘͕̬͍̤̔̏͒̃͊̈͝͝ụ̸̝̱͔̬̤͇̅̀͗͗̈́č̸̮̠̖̼̈́͒̋̀͒́͠h̸̘̩̃̈ ̸̨̨̫͎͔̱̅̐̚̕͜ȁ̷͉͎̥̖͔̲͓̝̉̃̈́s̴̟̺̯̬̈́̇͋͜ ̵̼̼̹̜̜̜͕̂̈̊<math>̷̣͈̻̘͑̄̊͝2̵̢̳̳̪̲̺͙̇̎̌́͊́͝ͅ9̴̙̘̞̳̬̃̍̿̒̈͘2̴͕̬͓͙̙̪̄̿̊͛̋͒͠͝ ̷͇̳͉̲̔͌̌=̶̜̘̤̯̯͊͆̿ ̸̥̞̳̱͎͔̙̉̍̿͐͐͝4̷͕̩͎̯̗̔͗̃̏̽4̷̧͚͔̎̏͑͜4̵̢̞͔̯̈́̀͜_̴̣̩̠̤̖̓̊̀͜{̸̡̗̥̻̂͛̓̅̎̎̐\̷͚̫̙̣͊̿̅̂̉̀͜͝t̸̟̗̱̭̬̲͎͛͌e̴͕̬̅̈́̋͛́̈x̵̧̖̞͎͍̂ͅt̵͚̯̦̠̉͐{̸̧̙̻͈̟̞̽̓̑8̵̖̹͇̄͊̇͆}̶̝̘̩͔̲̺͆}̶̮̥̎̊̄͝.̵̢̧̧̹̙͕̦̜̾̉͆̓̅͗͌̒</math>̴̛̼͕͗͒̏̉̀͛
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Recall that a palindrome is a number that reads the same forward and backward. Find the greatest integer less than <math>1000</math> that is a palindrome both when written in base ten and when written in base eight, such as <math>292 = 444_{\text{eight}}.</math>
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̸͔͇̟̓̎
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̴̻̲̬̹͍̓͜ͅ[̴̭͈͍̝͊̅͒̇[̴̬̻̺̣̞͚͓͖̽̕͝͝2̸̛̦̗̻̬̬͐0̸̨̡̛̛̛͖̺̱̂́̐͛͜͝2̶̹͎͕̘̝͍̤̗̔̐̽͊͊̈́̓3̷͍̬̠̫̘̅͐͗͋͝ ̴̥̤̗̐͒̂̆͗̈͝Ḁ̸̺͓̼̬̉̇̓ͅI̵̳͔̼̙͗̚M̷̹͈̭̼̉̽̈̋͠Ē̵̞̪͚̉̄͂̀́̄͝ ̷̞̪̬̼͉̜̲͕̒̀̽̏̅̔͆̉Ì̶̮̜̻̥̰̝̲̘̉̋͋̉̀̚͠Í̴̩͕̻͚̗́͆̀͘ ̵̠̫̔̑͛̾̈̃̿̋P̷̧̣̜͈͓̲͌̐̆̾̒͊r̵̞͎̲̥̻̯̹̳̔̓̄ō̷͍͈̜̪͚̟͇̅b̴͖̭̱̝̑̚ͅl̷͇̟͈̮̝͆̎͊̑͜e̷͔͙͗̂m̷̡͇̰̱̭͈̝̪̋̍̿͌s̸̭̬̤̰̩̿͛͒̑́̕͜ͅ/̴͎̩̘̍͋̈́̿P̴̛͓͉͊͒͒͜ȓ̶̜̜̑̍̌̚̕͝ó̴̮͕̣̜̖̭͆b̶̬̪̟̰̺̓́͌ĺ̶̯̀̿͛̕͠͠ě̶̢̨̻̝̜̮̕͠m̴̡̛͈̫͖̄͊̈͒ ̷̧͓͂́̐̑́2̸̨̧̛̯͓̝̰̋̋̓̊͌|̴̬̣̾͒̄̎Ṣ̸̛͎͓̬̰̏̈́̔̀͘o̵̢̯̟̦͗́͛̓͊̆͒̇l̸̢̡̬͈̱͍̯̝̇ụ̵̉̆̏͗́ṭ̸̯̦̰̺͔̂̓̉͜͜ȉ̶̢͉̻͙̲̲̱ǫ̶̰͚̀̀̊̈́̈́͜ͅņ̸̧̭͙̳̍͆̑͂̄]̶̨̡̡̼̪̉̆͛͝]̶̱́
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[[2023 AIME II Problems/Problem 2|Solution]]
−
̶̢̳̿̈́
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̷̢̬̮̱̿≠̡̗̰̦̦̺̗͂̌̀̊̈́̉̓=̷̨̧̺͙̉̀͐̐͠͠P̷̹̘̂̎̒̌̎͘͠r̴͕̬͐͋̈́͐̏̑͝o̸̧̡͙̝̦͠b̵̮̣̻̜̺̫̈́l̷͎̘̻̖̯͛̇̐̊͛͆̈́ę̷̧̘͇͍͔̻̃̉m̷̢̛̺͓̘̹̆̀ ̶͕̏̏̊̎͊̿͂3̷̖̂͊̃́̕͘̚͝=̵̖͓̮̭̟͇͗̕=̶̱̣͐̒̈́̚
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==Problem 3==
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̴̧͕̀̀̉̽͛̾
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̴͍̈́̓̀̓̈́́̄͝L̸͙͋̊̄͒̑̋̀ę̷̺̟̞̞̰̮̓̎̀ṯ̶͈̬̻̿̅̎͆ ̷͈̤̮́͆͜<math≯̻͖͉̤͙͈̅̍\̴̖̒͆̂̾͝t̶̳̲̥̣̞̩͉̄̋͑̐̇ͅr̷͎̦̟̩̖͑̏̚̕͜͝ḭ̷̢̘̭̺̬́à̶̡̨̛̠̞͌̒ň̴̦͓̩̱̿͛̋̏̈͜͠ͅĝ̶̙̞͚̤͖͌̄̓͝ḷ̸͎̺̺͖̝̽͂̈́̽̂͒͜ê̵̢̧̩̝͕̘̻̓̌͂̆͛̍̚ ̸̗͎̦̥̈̅̾̈́̉̆ͅĄ̸̲̙̯̗͖͔̋̒̋͛B̷̭̞̐͊͐̍͐̈͝C̵̰̭̬͈̒̈́̄̋́̚͠</math>̴̨͆̈́̋̍̕͜ ̷̧̠̩̆̌̇͑̀́̎͠b̸̨̧̼̞̣͛͋e̷̻̦̱̿̔͑̅͋̀͝ ̷͇̥̜̣̙̦͖̹̑͂̓̑̀̾ä̴̢̛̞̩̖͙̥͗̀̆̓̄ͅǹ̴̛̝̱͙̪̋̑ ̴̭͔̻̓͛ī̵̬̲̘̣̆̐̍͌̎̕͝s̶̲̬̙̼̆̍̅̆̋̉̀͝o̵̥͈̺͂̚s̸̡̧̨̠͓̱̀͊̔̄c̵̰͎͍̦͕͔̥̈́̄̔͋e̸̢̧̠̣̩̗͗̋͌̀̎̾͜͜͝l̶̛̠͐̒̿̔̆̃͝ḙ̶̌̕̚s̶̨̻̠̟̒̀̂͗͝ ̶̘̣̭̘͆͋̚͝͠t̴̡̛͔͔̗̻̃̔̀͐͛̀̃r̶͙͉̻̣̈́i̷̢̟͖̦͔̬͕̊̈́͛͂͊ḁ̶̡̌́̈͋̕n̶̦̹̋̏ģ̵̤̙̯̞̙̠̅͋̓͒̑́̕l̵͚̣̤͕̎̑̒̆͝͝͠e̴̡̠̭̖̥͊͂̎̑̒͋̆͜͜ ̵̯̫̥̬͉͋͐͌̈͠͠w̸̟͓̄̐i̵̠̥̞͉͗t̴͓̔̅̾h̶͍͓̥͈̦̮̬̾ ̸͈̫̞̳͈͙͝<math≯͔̭̳̻̈́̊̒͆̏̇̎̒\̸̯̹̦͎̫͓͍̰̈́̇̓͑͆̐̏̒á̵͖̬͕͉̥̱̎̂n̸̻̉̓͒g̴̢̧̬̝̯̗͎̋̂́̋͜ļ̵̭̹̦͈̠̫͂̐̈́͒͝ę̸̧̘̙͎͗̓͐͘̕͝ ̸̣͇̿̍̿̀̌̐̈́̕A̵̳̱̮͔̤͛ ̶͙̰̪̫͇̪̭̦̌͛̕=̷̢̀̐͂͛͊ ̷̛̪̺͚̯̟́̔̾9̴̢̦̱̼̟̯̿͜0̵̘̹̫̐̒̆̈́̂͋̈́^̵͈̂\̸̨̧͎̲̱̳͓̔̾̐͑ċ̵͎̦͊̃̑͝i̴̢̯̙͓͎͎̞͗͂̆ͅr̴̥͚̓͒͊͌͊̋̈͜͝c̶̱̑̓̆̾͐̿̇̚.̸̛̫̺͖̭̰̞͉̣̂͐͒̅͘͝</math>̵̒̾̅̄̅̈́̂͠ͅ ̵̛̤͚͊̔̓̈́̉̆̔Ť̸̼̲̪̦̃́͗̾h̵̢̨̗̙̯̫͗̔̐̄͆̌͠e̸̻̅͋̈̑͊̓r̴̪̲̺̮̈́̎̍̈́̒ȩ̶̞͍̾́͛͂͜͝ ̸̧̬̒̾̅e̶̜̲̬̎̅͘x̸̛̳̫̀͂̉͐į̶̧̩̓͊̽s̸̺̜͐͜t̵̡͈̩̬͇̥͉̠͂̄̎͆s̵̗̘̞̩̙̐̑ ̵̨̧̣̹̣̆̈́̍̉̀̕͜͝a̴̛͇͉̞̘̤̹̙̝̾̅ ̶̻̅̏̉̃̉̆͆̒p̴̨̻͚̙͈̳̻̓͜o̵̰̭͖̟̍́̋͂̉̊̕͠i̸̘͇̥̥̯̙̿̇́͝ṅ̴̘̭͍̦͑̿̅͝͝t̷̹͖͓̕ͅ ̸̢̛̫͔̙͐̉̂̒̓͜<math>̵̛͚̰͕̥͂̽͘̕P̵̡̦̣̠̈́̆̏̂̈͋̚</math≯̨̞̠̜̰͚͌̿̚ ̶̘̼͔̳̎̍i̶̧̬̖̮̥̠͆̀̓̊̀ň̴̨͓͖͍̼̽̾̀͜ͅş̸̣͕̹̿̈́͑̈́͘͠ͅi̸̱̲̼̅͑̀̀͛̅̍ḏ̵̥̯̠̹̔͛̿́̾͑͋e̶͉͕̝͉͊͗̃̕͠͠͝ ̷̛̫̖̺͎͓̱͌̊̀̀̕͘͜<math>̶̛̼͉̏̃̓̀̀͝\̵̪͆͂̈́̎̈́͜͝t̶̺̱̝̔͛́̒͋́̒̆r̵̘̟̺͚͔͆̋̐̊͊̈͠ì̶̜̖̙͙̮͇̖͐̃͒̾͂̌ą̸͆̿͒̆̐̀̕n̵̗̮̺̟̯̼̆͑g̵̢͚̦̲͖̩̅́l̶̡̆e̶̟̮̙͙͊́̀́̎̕͝ ̵̪̰͕̤̳̪̒́̕͜A̷̛͂͜ͅB̴̼̓̎C̵̭̦̞̞͖̹̼̊̏̓̓̈́̈́̀͘ͅ</math>̷̝̓͘ ̴̧̜͕͎̯͕͓̒͋̃̓͊ͅs̸̫̎̂̋̃ù̷͎͕̳̗͜c̷̺̗̙̠̒̎̅̈̾͜h̷̢̦͚̏̄̏̐͗͠͝ ̵̢̮̼͔̊̈́͆͊̓̌ṯ̸̮̖͍̩̹̹͖̒̾͊̾̑͠h̴͕̦̙͖̳̦̱̬̍̇͛ą̸̬͚̩͍̠̅͝t̷͉̟̫͐͗͒̐̇̃ ̶̩̓͊͑̔̌̓<math≯̧̻̩̠̲́̈͆̈͜͜\̶̟̱͕̘̪̦̥̜͗̂̽̃̐͐a̸̛͍͚̱̲̥̣͎͂̽͒n̶̲̠̤̦͋̿̍g̷͔͉̣̰̰̑l̴̢͈͖̩͔͈̖̘͊̇̀͆̕ê̷͚͗̽̐̒̈́̓͋ ̵̡̻̭̫̦͐͂̌̃P̶͍̞̼̓̍̎À̸̡̨̢̮̠̦͓̖̓̈́͝B̸̛͙̯̬̣͑̚ ̴͉͖̞̯̰̟͑͂̿̑=̵̘̝̼̘̙̥̅͝ ̴̞͑̉̈͗̇̾͠\̶̛̛̲͖͓̘͉͔͋̆͌̆͂͠a̵̧̫̟̳̎̿̓̋̊͗͠͝n̶̲̐̄̂͊̀̉͒̅g̶͔̣̩̥͌̈̆l̶̥̇̈̿̆̈́͗e̸͉̱̱̻̊͂̅̑̇́̚͜ ̷͕̦̬̭̞͕̀́̓̿͘͝P̷̨̗͕̬̠̘̻̓́̉̀B̷̗͙̆͛Ċ̴͕̟̯̏ ̴̜̭̹̀̿̈́͊͐=̵̟̬̽ ̶̠͚̹̀͐͛̋͝͠\̵̢̻͙͚̍̃̆̂̂à̶̡̮̹͕̝ͅn̵̙̗̭̥̱̤͇̿̆̊͊g̵̡̡͉͚̪͉̈́͌ͅl̶̮̮̲̈́͂̈́́̃͌̿e̴̟͖̻͇͋̏͝ ̴͔̔̂͘͝P̴̙̳͍͕̈́̽̑̈́̎̔̂͘Ç̶̲̹͕̾Ā̴̧̦̗͈̮̱̯͔</math>̵̖̋͂̂̐ ̸̡̯̮̝̟͍͖̮́̈́̽ä̸̞̪̫̣̳́͒͑̓̉̕n̷̺̗̺̩̟̍̑̅̒̌d̸̳͈̩̼̖̗̆͝ ̵̨̜̗̞̍̀̄̄͠͝<math>̵͉̒̅̉͝A̶̫̺̰̯̙̙̫̱̔P̴̗͎͎͙̟̓͊ ̴̼͊̑̑́̚͝͝=̴̛̻̟͚͌͆͊̆͘͝ ̶̧̺͕̺͉̘̈͂̀̃1̸̧̮͍̠̘̺͉͓̄̏̆0̶͉̦̠̖̄̓͂̓̈́̀̓̚ͅ.̶̡͉͔͑</math>̵̨̠͙̹̜͓̖̹͊͛͐̔͒ ̴̗̽̒F̴̧͈̭́̀̍i̵̬̭̞̩͚̩̭͓͆̓͝n̵̪͖̬͓̠͌͛̇̓͐̎́͝ḑ̸̗̤̜̽͌ ̷̛͈̫͎̼̮͋̈͂̽́͠t̷̼̦̲̬͚̘̞̟̽ḧ̸̢̦̭̖̜̣́͆̈́̓̕̚e̸̱͉̻̭͑̈̓̾͠ ̴̨̬͕̫͚̉͑̎̓̚͜ǎ̸̺̟̺̟̬̹͛̓̄̋̋͜r̷͚͂̏̅͛̒̿͘ͅḛ̶̛̞͆͆̊̚ą̸̬͔͋ ̷̜̺̃̌̅̍͑ó̶̧̲̬͎̗͋̓͘͜ͅf̸̖̻̹́̄͘ ̴̜͖͗̔̏̿̄͊̐<math>̶̛̠̓̿̆̔̚̚͝\̵͈̘̞͉͕̤̽̈t̸̛͔̔̏̔̎̋̕ṟ̷̝̬̓͗͆̏í̸̢̺̼͈̙̜̒͒̊̉à̸̧̙́̀̈̀̑͠͠n̵̳̜̱̉̐g̶̡̧̭̳̖̓̽͛͗́l̸̩͔̔͜͜é̵͈̓̽̔͌̑ ̸̩͈̱̈́̊̈́̄̄̉̌͝A̷̩̣͆̾̿̀̚B̷̰̠̓̽͛C̶̢̺̖̜̻̺͉̏̌̒̐͂̈́̓͜͝.̸̢̝̬̖̑̋̋̀̓̈́̆</math>̴͚͙͔͖̅
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Let <math>\triangle ABC</math> be an isosceles triangle with <math>\angle A = 90^\circ.</math> There exists a point <math>P</math> inside <math>\triangle ABC</math> such that <math>\angle PAB = \angle PBC = \angle PCA</math> and <math>AP = 10.</math> Find the area of <math>\triangle ABC.</math>
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̷̡̗̣͕͉̣̭̥̈́̎͌
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̵̛̛͕͈̺̜́͋̄̽͐ͅ[̶̡̻͉̿̈̐͒[̴̡͑́̋̀̿̒̍̾2̴̧̞̇͐0̷̟̹̀̎̀͋͑2̶̫̙͈̬͚̰͂̓3̸̺͔̈́̓̌͜ ̷̠̙̚Á̷̢̨̢̮͍̻̟͝I̴̞̘͛͂̌̅̆͘M̷̺͌̈́̒͗̎̉E̵̡̧͍̗̤̬̪͗̀̇ ̸̻͕̥̪̬͉̖̱̓̃I̷̢̜̠̤͎͕͚̎͋I̷̤͌̃͠ ̸͚̥̣͍̙͔̙͚́P̷̡̀̆̔̒̈́ṙ̸̟̣̣̍o̸̡̮̔̚b̴̡̢̻̜͓̼̝͐̏͗̅͗͂̓͜l̶̞̱̳͕̗̤͍̮̇̆́͆̅ȩ̴̛͍̳͙̎ṁ̷̱̂͗̃̋͝s̸̤͔̼̘̽͐͊͛/̵̠͖͊̕P̵̻̜͍̠͂̈́̑́͒́͑̍r̵̮̫̬̺͍̔̌̄͑͝o̸̮̹̽͆̌̉̋͒̐͘b̷̻̙̦͓̱͎͇͖̾̎̇̐́̈́͘͝ļ̸̨̬̈͑͒̈́̔̕͠͝ͅͅe̸̽̽̍̌̃͒̀̌ͅm̶̨̘̣͘ ̴̻͚̰͈͖̓̊̀̐3̸̯̲͊̇̅̆̑|̵̛̪̅͋S̸̠̬̋̂̓̕͜o̷̼̠̦̜̍̓̊̓̀͊͑l̴͈̖̰̜̪̖̦̉͌̈͘͠ȗ̶̦ţ̷̤̙͈̥̝̿͑͜ͅi̶̙͙͚͂͒͝o̶̧̡͕̞̜͎̠̱̐̿̾͑͠ņ̶̩̱͎̩̪̣̒̒́ͅ]̸̛̞̤͉͔̟̦̭̈́͛͋͌̈́̚͜]̷̦̏̽͋
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[[2023 AIME II Problems/Problem 3|Solution]]
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̵̯̪̖̩͚͙̓̓
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̵̧̗͖͇̗̳̱͗̿̾̽̂͊̀=̶̹̣͋͂͋̋͛̄͆=̷̡͍̝͚̬̼̒̄͂̒͒̌̓̆P̶̮̥̐̄͛̇͜r̷̢̢͉̺̬̍̉͂̇̀̇̓̈͜o̴̢̱̩̖͋͂̃̈̀b̸̭̮͋͛͒̂̀̿͐̚͜l̶̢̼̝̱̺̯͇̓̽̀̒͐̈́̏͒ḝ̶̣̳͕͕̹͐͋m̴̯̀̒̈́͊͋ ̵̢̣͙̤̙̪̖̬̑͛͑4̶͎͋=̵̭̥̾͆̌̊͝=̵̖͍̹̟͑̆́͛̕͘
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==Problem 4==
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̸̛̞̰̎͆̀̌̍̿̏
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̴͙̄̈́̇̀̉̏͝L̶̮̹̖̑̈́̈̽͛̽ȩ̶̢̡̻͈͗̈́̚ẗ̴̖͈̹̰͑͜ ̶̢̗̞̻̎͊͒͑̅̇͘͝<math>̴̙̞͕̟̬̣̙̒̎͆̾̄͘ͅx̶̢̺̗̘̪͙̍̉̃̂͂̿,̶̨͍̮͑̌́͑̿y̶̗̽,̴̛̟̪̟͓̦</math>̷̞̏̌́͑̈́̋͗ ̶͈͎̟̝̂͆̒̀͆ǎ̷̢͔̹͇͇͌͋͜͝ͅn̵̗̣͔͋ḋ̸̡͖̦̱̭̠͂̆͌͌̅͜͝ ̸͕̯̪̻̹̦́͐͌͗͛͘͘<math≯̡͇̘̯̟́̐͑͝z̶̗͚̫̠̠̈̔͜͠͝ͅ</math>̶͎͑͛͂̈́̋̽́͋ ̷͈̞̟͖̌̂̀̉b̷̨͕̄́̒̒e̸̟͚̦̞͒̽͆̒̚ ̷̧͓͈̈́̓̌̏͒r̷̯̾̅͘e̷͓͖̮̝̺͑̿͒͝ą̶̹̣̱͖̠͊̅̃̄͑̂̂͘l̴̪̿̋̾̆̈͐̈́̕ ̴̗̏͐̂̾͆̌̚n̵̥͎̯͎̥̥͍̓́͊̃̌͋u̸͖͓̞̎̽͑̔̀͋m̷͍̖͚͈̤̩̤̋̈̑̇̓̓̊b̸͚̩͙̈́͋͋͗̐͋̓͠e̵͍̯͓̥̫̮̙̾r̷̻̾͆̓̀̑͝s̷̠͔͈̱͑̓̏̒ ̵̛͕͌̉̅s̶̨̗̪̠̥̏̈̏́͘͜a̸̢̩͚̘̋̀̊̋͝͝t̸͖̥̗̩̿i̴̳͕͓͓̯̻͙͊̉ͅs̶̫͈̱͆͐͐̏̒͜͠f̸̟̄̈͋̄́̚ͅy̷̢̤̲̯̒̔̽ͅḯ̶͕̖̬͖͙͊̾͜͠n̷̠͌̈́̇̏̎͑g̸̡̰͓̖̰͕̮̎̒̄̒ ̶͖͉͚̩͕͒̌ẗ̵̛͓̩̩̰́́̒͒͘̚ȟ̵͖̹̱̒́̉̈̕e̶̢̘͖̪̽̋̿̈͗̇̔̚ ̶̩͔̹̊͊̕s̷̙̯̝̽͗̅͝͝ͅẏ̷̫̲̗̄͐s̶̢̟̤̙̱̊̽̇t̶̨̞͖̜͕̀͌̋͝͠e̶̼̘͊͌̚m̷̙̭̞͈̟̣̥̈́̃ ̷͓̍̎̏̈̇͜͝͠o̶̗̜̠͉̗̫̭̽̓͆ͅf̴̨̳̜̘͔̱̾ ̸̡̩̯͇̫̃̀̊é̸͖̔͜q̸̯̯̹̫͚̬̣͖̈́̄͂̓ự̸͖̟͈̫͍̀̅̃̚̚a̷̧͖͚͋̿͗̽̒̋̽ţ̷̹͓̘̻͓̩͝ỉ̸̢̖͙̠̒͑͒͠͠ỏ̵̞͖̫͕̂̑͂n̷̖̜͈͚͈̟͊̊́̈́͝s̶̛͈̐̎̂̿͑͗͝ͅ
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Let <math>x,y,</math> and <math>z</math> be real numbers satisfying the system of equations
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̴̢̮̦̠̬͉̍̊́ͅ<math>̴̧̧̥͉̗͖̮̜͑̈</math>̶̢͉͓͔̹̜̎\̴̙̣̘̮͚̫̞̑b̸̛̪̬̘̫̻̩͘e̵̡̦̼̣͉̲̤͋̒̄͂͊̔͜͝g̸̞̙̟̠̈͝i̵̛̩̫n̶͚̰̓͊͌̎̊̕͝{̵̖͕͉̯̑́̓̐̉̈͠ä̵̡̧̧̩͕͖̯́̃̑̔ͅĺ̶͈̫̼͕͎̋̏̆ͅĩ̸͚͎̖͙̝̃̔͂̓ĝ̴͙̗̫̝̫́̎̍͐̒́̚ñ̷̢̤͓̲̹̂*̶̛̲̘̗̌̑̕}̷͓̮̗̠̖̩̙͕̈́̆̄͂̅̈͋͠
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<cmath>\begin{align*}
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̸̯͆̏̈͆͂x̷͓̯͕̩͖͕̦͈̂̾͊͌͝y̸͍̎͘ ̴̢͉̼͕̘͍̟͊̀̽͒̐̏̉̊+̶̝͈͙͖̥͕̙̈́̋͊̿̿̅̀͘ ̴̖̆̂̋̍̀̚̚4̷̛͍̱̩̣̪̯̄̽̈́͊͂͌͝z̷͙͉͔̦̫̑̔̀͑͐̇͘͜͜ ̴͕̻̩̇͂̅̑̑̂͊̒&̷̡̢̖̖͉̬̪̠̀͊̎̄̒͊̈́≠̜̬͐̒͗̕ ̶̧̗̰̬͔̀̍́͂̌́̕6̸̜̠̫͓̍́̏͋͜͝͠0̷̢͉̺̭͆̆̈́͊̾͂̚͝ ̸̡̥̫͓̺̟͕͝\̶̩̅͂\̸̹̩̲́̄͗̈́͊͘͜͝
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xy + 4z &= 60 \\
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̸͖̭͍̮̳̜̌̋̓̿ͅỳ̸̞͒͛̑̊̌̆̐z̶̧̨͖̥̬̭̬͐ͅ ̶̠̥͖̋̊̍͆͘̚͜+̷̛̣́ ̷̱̜̼̌̀̽̈́͝͝4̶̤͊͒̈͋̏̀̍͠x̷̞̒͋̃ ̷̨̫̳̠̭̥̰̜̀̈́̒̎͘&̶̜̤̈́̈́̅̾̀̈̏̑=̵̣͈̓̈̿̑́̽̓͜ ̵̨̘̰̱̦͔̜̟́̃̎͋̎͂͘6̷̢͎̒͋̈͑̓́̈͘0̷̲͙̭̳̘̭̳͓̀̂̀̏̀̾̕ ̷̧̡̳̳̱͍͐̔͑͑͘\̵̨̜̙̄͌͜\̴̺̞̓
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yz + 4x &= 60 \\
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̶̥̎̐̊̂̓̽ͅz̶͓͓̐̕x̵̨̤͍̅̆̀͝ ̸̡̰͔͕͍̜̤̑͐+̷͓̭͌ ̴̭̲̠̰̬̻̜̲̀̆̈̾4̵͔̻̌͒̿̇̉͝y̴͉̞͔̹̯̪̟͊̅̊͛̂͒̕͝ͅ ̸̛̝̞͙͚͇̬̆͌̂͆̂̀͘ͅ&̶̘̗̪̩͗̆͘=̵̨̛̗̱͇̠̠̭̋̽̃̀̏̓ ̶̛̝͋̈́̇6̶̩̩̼͆̕0̸̞͊.̴̥̻͕͒̌̋̕̕͜
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zx + 4y &= 60.
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̴̢̨̫͉̹̠̩́̔̾͗͜\̴̼̥̈́̇̕e̵̢̢̧̅̄̈̽̌̎͜n̶̢̛͖̪͖̳͕̽̓̇́̑̚d̵͖̗̯̋̍͂̄̚{̵̩͉̣̹̓a̵̬͝ļ̷̦͚̱̞͍̹̓͋̒̀̀̔͜ị̵͕̫̪̉͊̎̈́̈̅g̸̢̙͙̀n̸̯͔͈̱̖͒́͊̽*̴̗͇̹̠̲̥͓̎́}̷̡̛̤̺̟̗́̃͐́͜͝<math≯͔̑͆͒̈́̍̕͝</math>̵̻̠̮̪̼̮̈́̒̇̐
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\end{align*}</cmath>
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̸̨̛̻͇̥͚̖̘̫͂́͂̂̀̈́͘Ḽ̴̬͋́̔̑ͅe̸͉̲͒͌̈́͗̋t̸͔̗͖̀͛̽͋ͅ ̷̨̛̇̈́̒<math≯͍̥̯̪̼̚͜͝ͅS̶̗̔̽́́̉͒</math>̵̡̢͖̰̭͉̤̒̑͌͠ ̵̨̩̆̈̆̃͗̈́̆͝b̶̻͚̀͐̎̋̓̓͘e̷̬̺̲̭͇͗̇̀̑̈́͘͜ͅ ̶̡̨̧̛̬̰̼̞̾́̍ẗ̸̺̭́͝h̵̨̨̰̲͋̽̓̍͑̂͠e̸͉͍͛̾ ̷̢̨̭͎̘̤͇͛̀̐̔͝s̶̡̱̮̲̭̺̏̀ẹ̴̠̲͎͓͔̓̆͒̈t̵̻̘̬͆̐͊̓͘ ̶̛̖̞̈̀̍o̶̧̢̡̤͔̦̿͜ͅf̴̡̺̜͆͐ ̵̧̅͐̐́̈́̔̚͝p̸̳̭̦̥̾̊̃̐ō̵̫͈̪̮̝̮͎̻̈ș̵̹͈̣͚̼̩͔͑́͌̈́͂s̷̛̠̖̦͎̱̜̳̃̒̒̓͗͑ì̶̡͕̥̱̙͕̘͒͊̕b̶͚̗̞̳͙̬̣͔͊̏̉̆̄̓̕̚l̶̡̰͔̗̼̜̫̑̽̽͜͠ẽ̷̛͍̥͖͉̭̭̋̔̈́͝͠ ̵̟͕̯̤̦́̿̏̀͒͝v̶̡̩̜͈͊͐̉̄̈̈́̓͘å̵̼̲̮͕̖̗̗̰̆̓̓l̷̯͙̺̖͕̜͑͋͐͂̿̉͆̕u̸̘͚̲͖͊̾̄̓ȩ̶̛̥̥̦͚̈́͒͂̌̔s̸͖͛́ ̸̯͓̯̭̪̎͊o̶̧̹̜̟̥͎̿̀̅̉̓͠f̷̡̟̖͇̞̙̿͛̔̄́̕ͅ ̸̧̥͚̰̥̻̩̒͂̕͝<math>̷̲̱̉̓̎͛̈́͘͝͝ẍ̵̣̓̅͑ͅ.̷̡̣͎̲̝̺̝̒̏̈̊̋͜͠</math≯̛̻͖̉̈́̈̋͝͝ ̸̧̟̻̹͙͂̀ͅF̴̪̻̫͇́ͅį̶̭͍̉͋͐͑͠n̸̪̙̪͉̎͛͗̐̕d̵̖͇̞̖̣͊̓̓͊̅̓͑͜ ̴̺̞͚̩̗̒̔́͐̈́̚͠t̷̻̤̠͖̣̼̽̈̒̈ͅḣ̴̼̯̙̟͙̒́ȩ̸̠̜̀͒̔̂̀͋̎͋ ̴̻͌̏̈́̂̕͝ș̵̢̢͈̤̅͛̋̓͊̕u̵̠̱̥͓̒̀̆́̔͠m̵̢̱̭̠͐̓͌̂͆ ̴̨̮͔̟͋õ̸̡̡̠̲̜̱̓̏̐͐f̴̟͂ ̸͕͑̅ţ̶̠̪̣̗̼̯̽̂̃̑̇̒h̶͕̋̓̿̿̊̇̚̕ė̴͖̝̙̩͈̦̘͑͜ ̵͌̈͆ͅs̵̳̫̭͕̜̊̾q̷̛͓́̽̓̅̚͠u̸̡͙͇̼͉͋̄ạ̸̧̢͋͜ř̶̜̲̻̫͗̇͋͗̓̊e̵̙̙̮͓̣̓̓̉̿̾̀͜s̴̖͚͖̰͆͋̉̌͑̉̀͝ ̸̐̏̇͆͋̑̊̾ͅǫ̶̯̪͕̫̩̽͑͑́̃͆͜f̴̗͍̥̙͚͙͔̄͛̚ ̴̨̡̧̛̬̙͔̲̪t̵̡̲̠̬͆̌̓ḥ̷̟̹͕̖͔͉̀̃͐é̷̻̮̪̭̤̈́͂̒̇̐̒̚ ̸̨̢̞̩̏̈́̀̚é̶̺̰̬̖̳̬̤̿̈́̀ͅl̷͉̳̙͕̈́̚͝ͅe̸̙̱̻̣̫͖͌͒͐͑̊̋̎̀͜m̶͎̗̝̘̒͛̔̇̕ͅe̸͖͖̾̋͗ņ̸̹̳̥̣̜̼̫̂͂t̶̗͈͗́͐̈́̉́̕͠ṣ̸̨̗̟́́̿̈͛̿ͅ ̸̼̤͚̖͎̥͕͚̓͒̌̉͝ǒ̵̱͂f̴͇͍̜̝̝̜̽̀̅͜ ̸̰̟͙͕̻͈͂͗̊̿͗̕͝<math>̵̍͌͋̓ͅS̸͉̅.̶͓̳̻̦̀̚͠</math≯̾͒̆ͅ
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Let <math>S</math> be the set of possible values of <math>x.</math> Find the sum of the squares of the elements of <math>S.</math>
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̷̣̰̮̞͍̓̂͆
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̴̭̦̠͋̓[̸̥̩̚[̴̡̰̖̖͗̈̿̾̑2̵̖̲̻̎̈́̓̋̎͂͐0̸͈͔̪͓͊̐̆̇͠2̵͎̏̒̄̃̀̕͝͝3̴̧̡̠͖͍̬͎̳͑̅͑̍͝ ̵̺̭̼̃̂̃̓͂̈́͝A̸̫͙̱̻̰̅̾̽̈́ͅI̴̢̧͉̣͍̯͓̠̽̈́̓̒M̷̨̛̝E̷̤̯͚͎̺̽͘ ̷̧̟̤͉̓̂Ì̴̜̲͇͉͉̲̦̇͗̒̏̚I̸̧̳̬̜͓͖̊͐̎̒͑͋̎̍ͅ ̸͔̬̓̄̀͊̍̃̒̕P̴̟̩̻̹͍̌̕͠r̷̤̓̅̈́̏̚ȏ̶̱͕̪̦̜̬̂̈́̐̅̀̓͌b̸̗̠̰̄̏l̸̲̀͑̀͛͘͝͠ë̶̳̩̭́m̵͕̾̇̈̿̕ś̸̬̺͍̻̦̥̹̝̆̈́̍/̸͈͉̦̲̜̦̈́̈͆́͌͘P̷̡̮͎̦̻̀̚r̵̜̦̩̻̰͓͚̯̽́͒̉o̴̡͍̲̫̳̤͐̈́̄͌͛͂͝b̵̨̡͈͚͓͒̽͂̅̑́̔̃ͅl̶̨̬͇͚͊͑̽̍͝ḛ̷̡͎͍̞͙̯̀́́̈m̴̱̄ ̷̳̳͕̣̗̖̾͂̐̒̈́̔̀4̷̨̭̫̾̏̇͝|̶̗̮͛̽͝S̸̺̭̯͔̙͑́́͂͗ͅö̷̡̡̪͉́̿ļ̷̛͖̮͈͐͐͑͠ừ̵̡̤̙̼̮̎́̓͆̅͗t̸͚͓̆̈i̸̢̛͂͒͒͐̿̕͝o̷̡̱̽̈́̽̓n̴͙̤̙̩̆̅̇̑̾]̶̢̺͎͖̖̯̗̈́͛]̴͙͚̮͐͐
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[[2023 AIME II Problems/Problem 4|Solution]]
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̸̣͕͉͙̈́͆̑̓̆̉͆͊
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̶̪̄̀͛̈́́̕=̷̮̮̫͙͐̂́̂͂̓=̵̺͇̤͖̲̤̫̑͗̈́͑̇̾̏P̴̤̼̯̭͎̗͍̖̑̎r̸̡̢̦͕̯͚̮̿̏̋̆̾̅͠ó̸̪̀͌̈͆͑̊͑b̷͔͓͑̎̍̂̀͂͊͜͠l̵̻͖̪̈́̂͆e̵̻̱̜̠͚̱͑͂͌͘͝m̴̧̬̼̠͖̜̜̠̊̽̆́ ̵̡̞̖̗̙̝͆̌ͅͅ5̷̢̢̘͉̣̙̟̰̐̂̀͐́=̴̥͚͙̯̦͓̬̆͠=̴̜̭̠̮̞͎͍̪̒̑͠
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==Problem 5==
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̶̡̊͐̕
+
 
−
̸̜͙̳͕͗̽̃͐Ḻ̴͔̖̳͔͍̉̃e̴̛͎͍͘͠ț̶̥̹̔ ̴̛͙̳͉̺̮͚͙̟̀̆̇́̂<math>̴̨̠̙̥̖̿S̵̨̡̟͇̠̥͉͉̚</math>̶͓͎̀̃̀̒͝ ̷̨̬̮̱̣͎̲̎̀̀̔̌̀̓b̴͓̲͎͇͕͐̐̿̿͑͊̐e̵̤̠͕͍̦͍̎͌̂̏͑̌͝ ̵̨̬̘̬̲̿̋t̸͍͖͈͙̞̗͙͑͒̊̌ḩ̶͒̑̿̅̿͋̕ě̴̢̠̺̹ ̵͕̪̤͈̘͊͗̒͐͠͝s̸̯͈̮̯͝e̴̘͔͔̰̤͍̤͌͑̽͘ț̸̰̩̠͙̫͕̑͐̈́͑͜ ̵̥̣͍͖̝̻͚̇̋́̆ǭ̸̬̘̍͌̓f̴̨̢̨̖̞́͊͌̀͌͗̕͜ ̸̛͎̙͛̈͑͛a̸̯̒l̵͖̩͕̰̱̙̘͉͋͗l̸̛̗͚̳̅̂̋͂̔̅ ̷͙̱͈͕̺͈̲͚̓̽̒̏́͝p̶̙̹͔͚̄̔͆ö̷̯͚́̾̓͋͠s̵̛̭͎͈͎͉͌̅̇̕͜͝i̶̗͖̪͊̀́͊͝ͅț̴̮͚̏͊́̃͆͊i̵̮̪̱̘̥̠̮̼͒v̴͚̗͚̞̤͇̽́̎̃͒͊͐ͅͅȇ̷̹͎̝͇̅͐́̾̊ͅ ̵̢̡̲͜͝r̶͈̞͝a̵̱͚͖̅̀͛́t̵̙̼̯̫͒̍̋͝i̷͔͓̮̲̩̰̩̔̋̓̏͆ŏ̵̺̥̣̣͎̱̔ǹ̷̠͗̀̑̾͝͠ā̴̜̱͓͖͚̥͎̊̈́l̴̡͚̦̳̬̯͈̔́ ̵̨͍̹̟̖͛̕n̴̛͚̫̮̠̣̖̪͛͆̂̂̊̚ͅu̸̻̓ͅm̸̨̡͎͇̓̐̅ͅb̵͖͎̻̤̉̕e̶̡̼̩̅̓́͘̕ř̶͓̫̭̜̽͂̈̚̚͘s̵̪̑̇̈ ̴̧̤͕̟̯͔̟̀̍ͅ<math>̴̧̛̳̤̗͉̤͉͉̿̌r̶̜͖̬̺͗͂̀̽̈</math>̵̙̻̘̖̭̺̭͘ ̷̢̦̱̗̲͕̑͋̅͋͝͝s̶̭͝ṵ̵̡̖̀̿c̸̠͑̎͂̽̚h̷̢̲̪̻͎̙͈̿̈́ ̴̢̒̋̍́̈́͛̚͝ţ̸̯̪͔͇͂̆̑̿̈́ḧ̴̡̖̱̲̥̯͒̾̾ͅa̴̜̲̎͆̈̎t̸̜̝̦͍̍̃͊̉͌̊ ̷̰̃̀͂̌͜w̵̛̠̗̩̱̹̗̬̍̀̓́̽̐̿͜h̴̗͛̈͆̽ë̵̡͔͇̿̈́̈́n̶͙̫̳̪̦̲̉͂̀̈́͐̍̚͘ ̷̭̽̀͒͑t̴̅ͅh̷̛̰̮̓̉͊̓̕ȩ̶̠̫̟̑̐̓͗̄̕ ̴̨̃͂́̿͗̒͘͠t̶̰̞̻͚̟̗͎̓͑̀́̿̔͝͝ẉ̸̨̨̗̰̬͒̉̓̏̚͜ͅŏ̸̧͌̿̆͌̃̽ ̵̢̙̈́̊̒ņ̶̮̘͍̠͒ű̵̧͓͔̙͖͔̌͑m̵̧͍͍̱̻͖̺̆̃́͒̚͝b̷͕̟̝̩̍ë̷̤̮̬̩̔͝r̶̭͚̳͊͊̀̂̀̿͝ṣ̶̛̛̎͂̐̄̊̚ ̷̛̜͖̦̞̪͎̫̈́̈́̈́͝<math>̶̛̳̟͖̺͍̦͕̜̇̐̾̊͆̍̋ṛ̵̥̭͇̪̖͙̭͒͛̃͛</math>̷̥̖̳̓̍͐͋̌̄̾̓ ̷̨̛͚̦͇̜̈́͑̔͆̽̐̕͜ả̶̲̽̋̎̀̓̿̚ṉ̷͙̙͍̮̻̪́͂̽͝ͅd̸̡̓́̚ ̵̧͖̦͉̙̽͌<math>̵̢͍͗͑̊͌̄͋͛͘5̷̟̻̘̼͖͑̈́̋͌͆5̵̫̋́̀̅̚r̵̙͗̽͝</math>̷̦̞̣͚̖̤̻̔͑̇̑̊ͅ ̶̭͎̜͈̝̄ǎ̸̧̧͎̏̆́͝r̴̻̦͓̓̊̊̍͆̌͘è̴̳͙̟͔͜ ̶̰̾́́͘ẅ̴̲̣̟͔̦̼̈́͆̚͠͝͠r̵̨̧͉̖̭̤̺͂͋̅̍i̷̺͎̩̪͍͆̑̈́̆̎͒̀̕t̶̼̩̘̣͇̝͉̆ț̸͖͌̏͒e̴̛͙͈̪̱̓͋͐̎̍ǹ̴̢̜̺͖͉̊͆͊͐ ̸̟̖̭̖͓̍͛̚̕͝ą̸͖̹̠̩͚̉̂̓̈͒̿̚͠s̸̡̜̰͚̀͑ ̴̻̝̪̀ḟ̴̨̛͕͈̳̩̮̺͒͊̉́͂̚r̸̡̠̽̏̋̐̓a̷̧̨̪̭̣̪͒̐̾͗̕c̸̨̻̗͉̍̀̂͐t̸̢͖̙͔̺̲̥̐̈́̏͜͝i̷͈̘͑ơ̷̢̔͑̿̑͐̔͘n̸̨̢͇̰̤̣͕͗̈́̾̉̄̅̏s̵̨͖͍̩̃ ̷̭̱͙͕̜͖̼̄̐͂̌̂̍̽͗͜ḯ̸̧̡̮͕͋̋̀͝n̶͇͈̒̾ ̷͉̰̬̲͇̱̼̄̇͌̎̇l̴̨̳̞͙̘͓̘̤̔͆͑̈̈ơ̶̟̮̙͠w̶̧̖̪̺͋͛͗͗̆͜ͅė̵̲͓͇̲͔̈́͝ș̸̛̙͒̎̄͗́̍t̵̨̗̜̳̝̺̣̟͆͆̉͂̐͛͆ ̸̼͕̹͋͛̆̉t̴̻̹̤̼̀̌͝e̶̟̪̞͇͉̐̊̀̅̄̒r̶̢̨͓̟̟̍̈́̈́͊͛̈m̴̦̈́̈́̓͐ṡ̶̯̖͇̣͕͇̉̓͑͑͘͝,̷̢̨͍̹̦̝̀̊ ̵̳̰͚͛͌̒̉̑́̽͘t̴̪̲̪̆͆̄̆h̴̞̅̑̿̕e̸͉͊̌͋̏̔̚͜͝ ̴̢̰̞̻͇̲̀͝s̴̘͇̪̓̄̅́̒͝ȗ̶̫̯͎̣m̴̬̜̲͉̆͑̽̄̾̑͘͜͜͝ ̵̜͙̦̥̠̠͔̄̈́͗̓͘o̶̡̰̣͖̖̜̟̗̊͑͋f̶̧̛͚̯͔̠͉̬͆̽͑̂͂̎͠ ̷͚̠̹̤͊͑̒̃t̷̡̘̓̚h̷̛͎̭̐̃̎͛͜ë̸͓͈͈́̏͒̈́̏̈͘͝ ̸̢̠͔͉̤̘̳̫̾̇͘n̶̮̬̈̿̑̓̉̉u̶̧͍̝̟͉͐̏͒̔̊͝m̴̠̼͍̰̈ė̴̜̻̞͔͙͙r̸̨̯̪̟̎̔̂̃̋a̵̻̍̽͛ť̷̢͚̓́ő̸̪͔̗̭̯̤͉̩́̂̈́͂̕r̶͍̬̼͓̳̹͑̓ ̴̛̛̩̮̘̤͚̍̅̈̚͝a̶̛͇̒̏͋̈̔ņ̴͙̰͉͖͉̎̅͂̊ḑ̷̧̙̯͎͍̞͆̆̓͒̕̚ͅ ̸̞̻̙͈̞͆͗͘d̷̨̝̘̩̥̺̠̥̀e̶̢̝͚̩̞̳̣̖̽̑͋̿͗̓͠n̶͙͆ő̴̘͇̑̿́̾͠m̷͚̗̭̯̊i̶͕͍̙̅̾n̷̡̧̠̜͙̱͋̚ͅa̷̬͔̼̞̣͎͙͂́̅̋ţ̶̼̫̗̫̪̅̔̏o̸̰͒̇̎́̈́̋̂̄r̷̭̗͎̰̓͛̒̅͆ ̴̢̰̪̥͙̑̑͗͛͌̏ȏ̷̬̘͈̞͖̹̝͒̐͜f̸̧͈̗̭̖̫̭̣̈́̊̅̑̕͘ ̶̙̥͉̥̗̐͛̾̎̔͒o̷̺̪͇͉̘̲͝͠n̴͍̊͌̈̈̉ĕ̵̱̝̟̎̊̈́̽̌͘ ̸̨̠͇͔̠̰͓̿̿̅͋̇̓̚f̷̡̢͍͇̞̼͝ŗ̸̥́̾͐̀́̚͝a̴͇͗̓͊̋̋̔̃̓c̷͈͔̳̈́̒̽̐͘ṫ̵̪̩̭͉̥͉̤̫̽͂i̴̭̐̎ȏ̵̧̺̦̯̥̘̎ͅň̷̛̟̜̪͔̇͝ ̷̪̈́̓̀̅̈́̕í̷̭̮̲s̷͖̳̑̆̈́̉ ̷̲̩͉̝̂͂̾̾̊͜t̷͇͙͐̀̀h̵͈͋̀̈́̒̃e̵̺͇͖͈̪̹̿̐ ̴̯̞̉̈͝s̷̙̜͛̍̃͘̚a̸̗͛͘m̵̧̯̉e̵̛̱̣̟͗̊̀͑̆̕ ̵͕̙̆a̴̡̖̣̭̺̟͚̔͌̋s̷̘̬̣̼̦̫̙̅̕ ̶̢̧̫̺͚̤̦̀͌̄̌̚t̵͓̫͕̦̄̂̇̿ȟ̵̠͔͖͛̋̇̍̓͝e̴̢̪͕͍̻̩͉̥̅̐̃͝ ̴̨̺̩̈̆̊̎̆̀͒͜ṡ̸̢̨̝̟͈͓̯̯u̷̡̨̝̭̽̀́̀̅̑̂͝ͅm̵͉̊̋͂͌̀̓̕ ̶̼̘̘͆͒́͐ȯ̶͎̤̭̿̅͑̀̂͝͠f̶̛͍̼͔̼̙̭̿͗́̈́̚͝ ̵̹͔͚̬̙̈́t̵͍̩̻͆̈́̔̎̃̾h̶̟̞̞͗̈́͐͗ę̵̥̫̙̟̗͌̌̒̍̇̚͘͝ͅ ̴̛͖̍̈̈́͋̓́̏n̴͉̤̬̠̗̏͜͝ͅu̸͔̠̙̫̲̔̍̀̓̍̊̓m̸̡͍̜̹̹̫̗͊̀͂͠e̶͇̜̙̜͒̅̄r̵̨͉̠̝̤͓̤͐̾̍̔̾̓͑â̷͓͍̟͗̆t̴̗͐͂͂̅͒̈́o̷͍̱̼͙̍͊͠r̷̨̳͔̍̾̈͝ ̴̳̩̱̳̰̳̪̗̍̈́͋̀̅̿a̷̛̛̖̱̰̙̝͙͍̫̓̑͆͆̓ñ̴̖̚͝d̵̮̠̔̾̽̈̃̈́̀͘ͅ ̶̼̝̬͍̥̒͂d̴̩̳͙̜͐e̵̼͓̜͚̺̊̈́͘ͅṇ̶̀͒́o̴̫̦͈̿̓̉̎̍̍̈́̄m̶̲̤̾̍͒̉͠ỉ̴̹́̾n̴͓̼̗̹̣̈́̆̚a̴̠̟̠̍͐ṭ̵̨̧̢͎͖͍̌̃͊̈͑̀o̷̗̹͙̭͉̰̻̓͗͛r̶̛͈͉͆͐͐̀̕ ̸̢͔͍̦͖̟̮̓͋̿̓͝ỏ̷̹̟̼͍͚͍̳̗͝f̸̨̛͚̭͆͋̈́̔͑̕̕ ̶̳͊͛̽͌͂͝t̸̟͈̎̔̔͊͝h̶̖͈͔̗̖̹͑̉̍̿͑̾́͘e̵͎̪̯̐͠ ̴͉̪̌̕̕o̶͇̬͋̌̾̄̀t̷̢̞̜̖́̄h̷̨̗̩̘̦͓̟̀̇̄͋͝͝e̶̤̺̖̤̕͘͝r̴͙͉̿̑̉̽͂́͒̾ ̶̪͖͔̎̂f̵̧͇̜͍̙̂͗̇͒̀̀͗͠ȑ̸̡̹̩͖̳͍̒̾̾̊̕ä̴̧͓̙̠̰ĉ̵̱̬̔͌̾̐͛͐͝t̷̞̲̹͎̲̲͐̑͂̽į̸̛͚͔̮̰̯̟̔̐̓̈o̶̟̦̻̱̫̒̀ṋ̵̡̢͚͈̗̦̎̂̃̿̒̚͠.̶̦̌̉̕ ̵̨̋̃̍͒̃̚͝T̶̪͍̤̀h̸̢͙̖̠͓͑̄̇͠e̵̡͎̞͈͓̝͆̔͌̓̈̈́͂ ̵̮̖͖̗̽̃̕͝s̴̻̘͇͎͚̰̤̓̈́̌̐̈́͋̇̚ư̷̡̥̪̮͚͕̈́̅͑̚̚̕͝m̴̛̜̖̤̭̭͖̙̂̃́̕ ̶͇̳͖͉͇̅o̶̬̳̩̊͑̏̀̓͝f̷̛̲̣͈̘͎̼͋͝ͅ ̶̨̛̛̰̼̣̏̄̅̍̋̚ã̵͉͔͙͔̥̜̋̀̇̈ĺ̷̨͎̪͔̂̂͠ļ̸̛̜̮̬͍̱̱̖̎͑̓ ̴̨̛̰̻͔̠͘ṫ̵̥̳̭́h̴͓͗̌̈͠͝ḙ̷͉̪̱̘̺̊̈́̑̆͜ͅ ̷̙̯̹̤͕̃e̴͈̪̽͗ļ̷͍̊́̐̍̉ė̴͕̉́͋m̴̨̱͕͎̰̓̅e̴̢̲̦̺͖̳̐̐̎̚͜n̴̝̖̖̙̣̈́͐͐̀̿͘͝t̷̠̫̖̙̅̄ͅs̶͇̞̗̙̈ ̷̢͖̲̺̟̻̀̏̈̿̏͒̑̚o̴̠͈̠̬̳̦̓͒̈͗̍̆f̴̱̒͌̾͗̈͑̉̈ ̶̙̏̆̓̕<math>̷͈̪̬͕͈͓̑̊̀̆̈̀̊S̴̡̗̬͠</math>̵͔͚̂̂͝ ̶̲̭̭̲͆̌̿c̴̯̤̯̙͂a̴͔͈̼̯̽͛̔̕n̵͎͈̯̘̜̅̒͋́͋͗̂ ̵̻̹̰͐̽̒̈̌̽̈̕b̶̟̜̏̒̉͐̀e̶̬̩͌̈́ ̸̰̺̝͙͐̀̎̅̓͛̓ẹ̸̱͊͛͊x̵̦̹̠̺̌p̷̡̲̍r̶̨̦̱̠̙͚̻͖̾̊e̶̟̮̭̜̝͕̰̓̾͂s̷̹͔̮̔̓̄̀̾͆s̴̘̜̭͍͍͙̮̓̿͝͠ĕ̷͔̯͂̇̀̕d̸̰̦̑̈ ̴̜̣̺͍̠͇̈̈́̾̑̕͝ỉ̴̛̹̟̝͚̗̤͙͍̈̒n̵̡̦̹̲̱̙̝̼̑͂̍̍̕͠ ̷̢̡̰̗̻̮͎̥͐̽ţ̴̯̫̬͔̒̈́̂̽̓̚h̴̺͔̩͒͛̊̌̓̈́͋ͅę̵̧̣̺̺̞̾̽ ̷̨̪̯̘̼̣͓̞̽̈́̚f̴̙̹̙͖̮͖͉̹̾̉͆̂́̓ơ̵̧̰͍̹̠͉̟͉̈̍͝r̴͇̰̪͗͠m̶̨̫̥͎̈́̂͂̈͠ ̶̡͈͚̼͑̃̑̄́̏̀<math>̷̹̙̮̬̤̞͑̏\̷̼̪̬̜͑͆̆̍̆̔̄̚͜f̵̢̢̱̹̟̉̿̇͠ͅŕ̵̨̡̳͓ͅa̷̧̨̪̦̙͒̇͋̑͋̊̊c̸̣̣̙̙͠{̶̗̥̖̌́͂̈́̽̚p̸̣͇̟̿̊̊͊̓͗̚͜}̴̯̮̗̎͊͗̕͜{̷̙͍̭͕̄̒́̄̈̾̄q̷̥̈͆̌}̴̛͙͇̪̦͇͊̈́̔̽́̑,̴̢͇̝̦̬̠̮̞̈́̑̏͐͌͝</math>̴̡̢̯͎̲͊̂̄͂̔̿ ̶̛͈̜w̵͚̹̳̠̪͔͖̐́͌̀̈̕ͅh̷͔̺̻̱̜͕̀̉͒̽̋͘e̴̢̖͚̦͍͇͚͔̎r̸̖͇͕̗̘̈́̈́̔̌̀̇͑͘ę̵̞̟̖̎̆͒̾̒ ̴͈̠̮̩̆͋̔̅͆͆͘̚<math≯̢̰̗̜̀͜ͅp̶̯̗̍͘</math≯͓͙͚͉̲̞́͘ ̴̛̣̪̀̔ẩ̶̛̻̅́̄n̵̢̦͙̜̫̺̼͊́̽̅̋͠d̸̻̦̗͓̖̞̓́́̚͜ ̷̖̤͇̪̰̫̱̀̋͗́<math>̷̬͓͈̙͖̘̰̉̑̐͝q̷̠̮̘̬͈̥̠̋̾̾̇̈́</math>̶̰̿̾͋́ ̴̩̼̯̈͜ä̸̝́͛̾̏̊r̷̭̍͂̍͑͝ẽ̷̬̱̫̰̈́͘̚͜ ̷̞̝͎͕͉͖͐̊r̴̨͚̟͓͇̔̑̋̌̔̾̂̋e̴̥̱͓͎̹̋̑̔͐͒l̷̡̙͔͈̠̐a̷̮̿̎̇̕t̶̡̠͎̭̝̄̎͛̌͂̈͘į̷̻͍̤̫̮̄̾̄̀̎̾̓̈ͅv̴̟͈̯̓͘͜e̷̛͊̐͊̉̓͆͜l̴̟̜̞͋̑̈́̑̇͗̍̕ý̷̜͚̈́̀̀̕ ̷̱̺̤̟̇p̵̪̗͕̉̈́̀́̓̑̓͠ṙ̸̳̮̫͇̘̜̟̭̒̋͝ī̸̧̲͖̱̗̗̟̊̑̒̀́̀m̷̤̪͍̩͍̖̱̺̑̿e̸̯͝ ̴̧̥̩̗̦̼͋̎͊p̶̩̹̣͕̫̪̞͎̂̋͐̎ö̵̢̮̩͈̟̦̫́͗̂͌̓ş̵̩̻̻̺͇͖̲͗͂̀̆̅͘ḯ̸̡̛͖̹̰͙̫̦̘̐͆̏̑͊͠t̸̡̨̥̙̳̘̱͑̓̎̔̚i̸̢̫̬̫̦͓͌͜v̵̧̢͙̐͒̂̂̋̀͒e̴̢͈͋̈́̋́̈́͝ ̵̠̳͕͚̫̭̻̀̎i̴̜̹̻̬̳̩̼͐̄̒̏ͅn̶̻̞̲͍̥͉̝̳̓ẗ̸̡̛̮̭̜̝̙͊͐̂̃͆͒é̸͖̦͐̏͗̕͝g̴̛̮̮͕̻̣͈̎̀̀̇̽͆͝e̷̲̜̩̯͛̏̀̍̒́̚r̵͕̖̀̋̍̚͜͝s̸̘͓̮̬̓̈́̊̃͜.̵̢̨̛̹͈̖͖̤̂̈́́̚̕̕ ̸̡̛͕̾̚F̶͍̩̮̦̯͔̤̂ȉ̸̛̘͕̦̙̜͗̔̓̈́̀ṅ̴̼̅d̸̛͓̗͍͎́̑̋͠ ̸͚̲̳̹̰͉̗̉<math>̵̨̧̥̟́̑̕̚p̸̛̱̽̓̉̂̕̚͝+̸̠̳̟͗̏͋͝q̸̢̢̮̪̯̫̾͠.̷̭̲͚͈̿́̚͜͜</math>̷̺̲̻̲͇̂͒͊́
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Let <math>S</math> be the set of all positive rational numbers <math>r</math> such that when the two numbers <math>r</math> and <math>55r</math> are written as fractions in lowest terms, the sum of the numerator and denominator of one fraction is the same as the sum of the numerator and denominator of the other fraction. The sum of all the elements of <math>S</math> can be expressed in the form <math>\frac{p}{q},</math> where <math>p</math> and <math>q</math> are relatively prime positive integers. Find <math>p+q.</math>
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̷̡̹̪͎͚̥̠̌͒͆͗̽̾
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̶̦̯̙̱́̐̈̄́͜[̷̢̪̟̻͔̭̰͜͝[̸̛̝̳͓̗̳̞̀̇̑̂̎͂2̸͙̹͎̰̃́́̌͑̋͘̚0̶͓̒̉͂̋̓2̵͍̤̖͙̳̘̒3̵̥͖̭̹̩͔̌̐̀́͝ ̷̯̟͈͗̐́͑̀À̵̭̗͕̲̙̖̋̑̈͗I̵̼̮̰̾́M̶͙͓̰̟̘̬͑͗̆E̸͉̓́̿̚͝ ̸̘̠̏̂̐̆̔̇̚Ĭ̸̭̟͕̤̪I̸̞͊͋̀͂̽̾͜͝͝ ̸̲̫̭̭̈́̆͘̚͝ͅP̶̩̠̘̞̻͖̖̔̈͆̀̑̓͠r̷̗̬̠̺͖̟͎̅̄͋ͅo̴̝͐̈́b̷͔̟̺̰͆̅͊̈́̔̈́͝͝ļ̷͎͇̌̿̊̈̑ͅe̶̬̦̻̥͊̇̕ḿ̸̧̦̘̣͖̝̓͒̽͆͜s̷̛̤̜̜̓̌̇͆̚̕͜͠/̸͇̘̘̓̔̆́̓͆̎P̶̨̪̳̺̟̠̑̚ͅr̶͚̯̥͈̣̉̈́̽̔͠o̶̬͖̠̥̞͋̿͆̽͗̈́̚b̵̨̫͎̜͈̹͙̟̓̆̌̿͝l̵͔̹̯̰̥̰͆͒ͅe̶̺̮̳͑ͅm̷̗͊̉̄ ̵̡͙͚̦̺̺̃̋5̴̪͙̞̈̆̚͝͝|̶͕̞̱̭̀̉͊̎̿S̷̨̧̡̛̳̻̼̼̖o̴̹͉͖͈͑̏̔̎́̿̓͝ḽ̵̛u̷̫͌̌͊̒̋̐̚͠ẗ̶̫̟̜i̵̼̩͓͇̪̜̗͑̈́͛ǫ̶͓̝̩̭͈͍̋̑̉́̀̏̆n̶̛̖̹̜͖̞͙̟͋͂̃̕]̷̢̧̨̉͒̍̾]̵͙͑̆͊̀̚͘̚ͅ
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[[2023 AIME II Problems/Problem 5|Solution]]
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̷̨̗̹̯͇͎͖͇̑̂̔͠͝
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̴̡͉̼̯̐̿≠̧͙̼̯̗̯͇̖̍͂̔́͒͌̒=̶̖͍̠͌̓̒̌̄̐͆͠P̵̬͍͖̮̍̽͜r̷̡͕̜͕̙̻̺̠͝o̵̳͍̭̣̻̻̾̑̊̋͂̕͝ḇ̸̢̘͔͈̼̈͑l̷̲̠̜̫͇̅͋̇̌̓͛ë̸͉͚̪͚́̋m̶̡͖͈̮̦̝̉͑͌͐͘ ̷̢̦͎̺̮͚́6̵̨̧̦͍͍͉̭̀̈̆̋̑=̴̟̝̣͎̳̰̯̆̈̿̓͋͑͜͠=̴̧̠͓͈̩̿̑̆̀̊͘ͅ
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==Problem 6==
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̷͇̟̥͉̱̼̭́͌͗̋́̂͘ͅ
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̶̢̻̹̩̋̀̍̆͆̆ͅC̸̺͐͆̄o̵̜͈̖͔͋̒̀̔̄͝n̴̪̻̪͊s̴͕̲̟̊̈́̓͆͘͝ͅi̸̠̰̫̗̻͌̽d̴̦̩̞̥̳̰̀͌̎e̴̛̫̩̥͊̾̀̅͐̚r̴̢̺̱̺̙͙̉ ̷̙̖͖̓t̴̢̯͇͕̐̾̿h̴̢̡̠̖͚͖̽̈́̄͜é̷̜͖̤́ ̴̩͚̺̈̃͆͐L̶͔͇͖͓͓̥̊͌̃͝-̶̤̇̊͘s̶̡̮̦̜̞̄̀͗̓̅h̶̭̰͉͇̫̋̈́̍̃̌ă̶̳̣̺̗̩̟̔̃͝p̵̹̲̯̪̞͚̤̾̾͆̓͌̿̏ȇ̷͇͍̣̘̼͗̏̋͘͝͝d̴̲̪̐͐̒͒̾̏͠ ̸̧̨̢̹̱̰̗͌̈̈͆̀̽͘ṛ̴̢͎͔̱̦̩͗̄̆̉̇̊̒̏ͅe̴̛͔̖̋͒ǵ̸̝̰͘i̵̢̛̖͚̖̥͋͊̿̇́̕ơ̶̢̞̘͉̱̝̺̌͋̊͐̂͘͝ṋ̸͇̕͜ ̵̧̛̭͖͍̪̦̮̻̍͑͑̍̀͠͝f̸̤̦̺̋̔̋̋̆̽͗o̸͚̠̫͉̫̮͉͔͑̈̔̎̊r̸̢̧͕͖͉̣̫̽́̉̓̄̉m̶̛̞̟̎̉̈́͠ę̵͙̪̹͚̙̘͈̔̓͠d̴̺̮̼̓͂͒̋͘ ̶̛̺̠͉̱͍́̎̀̓b̴̡͈̜͆̐̈̕ͅy̵̟͙̍̀ ̸̢̧̮̞̌̓̈́͆̾̈́̾̐͜t̴̳̠̖̖̜̒̽̌ͅh̶̡̨̘͐͜r̶͙͛̿͆̆͌̂́̂ȩ̴͙̱̘̓̊͊͝ĕ̶̗̝̜̣̱̲͕͙́͆̃́ ̶͎̮̭̎̐̉̇́͝ǘ̶̲̠͈̦̂̍̓͗̉́͝ń̶̯̰̬͇̝̆̅̏i̴̧̗͓̔̒ṫ̸̡͓̥͕̞͍̈͗͌ ̷͓̭̿̄̅s̵̰̭̹̫͕̮̄̈́q̵̡̘̪͗̋̒͐͗̌̓̔u̵̝̰̜̻̦͇͗̾ͅͅạ̶̬̙̑̎̓̃͜ŕ̷̢̛̲͈͎̯̟͉̊̆̿̈͝ĕ̸͍̺͈̈́́̂͆̊͗͝s̷̺̤̲̠̱̙̅ ̶̨̨̮̳͖̳̆̆͜j̵̡̠̜̻̳̓̔ơ̷̢̡̗̫͉̩̼͋͋̋̔̈́̔̓i̴̩̝̪̊̽͗͘͘n̴̨̡̳̺͙̟̉̒͊̎̐͗̀̾ė̵̛̜̫̞̫̀̅̇̎̓d̴͓̖̗̩̾̅͘ ̵̛͓̬̳͇̓͒͝ͅa̸̞̬̮̰̯͒̊̓̕t̴̺̰͙̺̹̆́͊͌̊́̎̃ ̶̢͍̙̗̰̎̀͆͒̏̈́͠t̴̮͒̊̚h̵̗̱̠͍̞͊͂̿͌́̿̚͜ẽ̴̛͙̭̳̳̀̏͘i̷̟̾̈́̕̚͠r̴̜̣̿͘͘͝ ̴̡̠̘͈̬̣̏͋̐s̸̭̼̝͈̘̜̀͗͌ĭ̶̙̰̩̼̼̊̓̃̏̕d̵͖̻̼̃̏͒̄̃̾͝͝e̶͓̩͘͝s̴͎̼̍͌̿̚,̷̡̭͇̿̎̌͛̀̽͘ ̵̳̪͓͛̀̉͌̍͠a̴̡̨̖̯͇̱͆s̷̼̥̯͎̽̋͛̊̃̐̓ ̸̧͙̱̓̏͊̕̚͘͝s̷̡̛͙͌͋̄̚͘h̸͙̘̝͌͆͋̄̈̑o̸̜̮̔̈w̴̭̘̼̃̂̊͗̈ͅn̸̗͚̮̫̣̥͔̚̚ ̴͚̙̮̘͉̳͙̈͊ḃ̵̰̏̊͘̕̕̚e̵̡̟̰͉̳̮̩̽͑ľ̴͓̞͕̬͂̾̎͋̋̕ö̷̜́͌̍̃̽̕̚͜w̷̢̛̥̘̥̺̬͓̞̾͝.̷̨̻̭̱̍̓̉́̄͘͝ ̷̹̱̼̰̆ͅͅT̴̬̘̋̂͊̚w̵͎̞͂̃̾̐̎̋̌͜o̴͇̍͌̃͆̈́͆̚͠ ̴̟͍͛̍̓͊̕͝p̷̞̫̫̓̊̃͊̏̈́o̶̞̭̜͈͐̀̈́̒͜i̴̛̤̲̮̺̻̤͂́̀͛ͅͅn̷̤̖͕̥͐̈́͊̕͠͝t̷̻̰̹͘s̸̛͚̗͇̺̻͙̅͂̔̇̿ͅ ̷̩͆̿̇̆<math>̴̡͍͎͉̦̫̥̂͐͝ͅA̸̡͖͖̳̜̰̰̘̎̊̉͑͘</math≯̞͔̤̭̰̏̏̊ ̵̣͉̩̻͉̫́́̍a̸̡͔̭̿ṉ̸͓̜̊̿̃͂̈́͘d̷̨̯̞̬̮̲̉ͅ ̴̡̜̀͜<math≯̬̠̼̓Ḃ̷̨̯̗</math>̷̡̝̖̘͓̱͓̼̽͝͠ ̷̱̳̘̣̙͉͋̆̇̚ȃ̵̢̡̻͉̰̟̩͌̈̚͝r̵͔͚̩͋̀͋̕͘͝e̴̡̢̼̲̝̅͐͂ ̵̧̡̼͇͖̇́̆c̸̡̪̫̭̿́͗̋͗́͠h̷̫̼̠͛̅̾͐̉̕̕ó̷͓̙̲̂̿͒̆̚ͅs̸̨̱̃̍̂́̾͜ę̴͓̜̦̟͗͛̚n̴̡̹̥̗͖̬͆͠ͅ ̸̧̺̝̲̼̉ͅi̶̢̡̫̯͙͍̓ͅņ̵̖̩̲̳̅̋̊̎͑̚d̷̢̩̰̦̖̣̝͈̐͋̓̊ȩ̴̛͍̱̓̋̒́͐́͠p̶̮̲̖̱̦̄̇́̈̓͘ę̸̦̝̝͇͍͔͖́̈́̑͂͌n̵̫̖̬̙͉̺̯̼͆͛̑́̐̑̃́d̸̥̼͕̫̺̝̬́͐͌̽̍̏ͅe̷̼̍͌n̷̨̞̣̱̟̎͛͠ͅt̸͖͕͔͂̅̊͊͝l̵̼̙̬̤͊̌̄y̸̧̯̮̗̩͋͘ ̶̝̝̪̂̏̓ͅả̸̡͇̘̩͙͔̍̓͜͝n̴̫̭͚̠͂̏͌̕ͅd̵͔̮̺̅̍̾ͅ ̴̛͎̤̙̈͋́́̑ư̸͓̋͂̌̌̓̍̾ǹ̸̡̨̖̬͕̩̈́̀̑́̒̒̈ͅi̷̡̼̝͆͌͜f̷̢̹̠̍̒̾̇͠ò̴̠͈͎̜͔r̸̡̼̊̒̔͒̈́́͜͜ḿ̸̠̺̮̄̊l̸̢̧̗̤̣͈̪̾͂͂̚y̴̧̥̳̤͆̓̐̓̏̆̒̄ͅ ̷̛̫̮̤̮̦̯͌̉̆͘͘a̷͉͕͐̋̎ț̵̞̼̂̿̄̆̿̀̒̕ ̷̹̠̖͔̣̎̒́̀̓̈́̚r̶̡̢͈͕͍̜̞̠̍̈́͋̋͆̒͊â̸̯̹͓͚̦͓n̶̛͉̏̑͂͊̀̏̚d̴̛̜͇̟̾̒̂͆̌̉͠ơ̵̛͇̺͍̘̐̈́̄̈́͗̐m̷͎̘̞͚̼͐̇́̋͌̊̚ ̶͙̝͍͛͘f̶̨̳͔͑͛̾͒͋̔̇̾ȓ̴̜̤͔̜͉̭͠ȍ̵̭̼͍̠̍m̶̛̹̦̄̎̀͛͗̊͘ ̷̙̞̳̟͖̫̀́͑͗́͛͒ͅi̴̛̦͉̟͔͚͋̾͋͂͗͗́n̴̻̺͇̮̹̾͜s̵̬̅͊͒͆͐͋̌̚i̴̺̝̬̮͛̊͗̊̿d̴͎̠̩̽͋ḛ̵̡̝̯̽̾͋̊ ̶̨͙̭̏̍t̵̙̚͝h̸͕̲͐̒̈̂ë̴̡̡̳͇́̐̿̆ͅ ̵̠̦̀r̴͎͇͉̺̱̊̈̔͌̓ę̵̦̬̹̟͇̮̒͒͆͘g̸̡̣̼̯͌́̄͛͜i̴͕̲̦̼̮̝̺̽̂̚ô̸̳͔͂̃͘ǹ̴̹̩̤̊̂́͘.̷̰̹̼͇̩͕̋͑̀͜ ̴̧͖̝͖̯̝̖̋̇̑ͅT̵͇̎h̵̨̛̤̝͙̻̠͈̆̈̏̋̃͐͘ͅḛ̴̛͕̜͗͆͌̿̏̚ ̸̨̗̼̭̱͚̘͓͑́̕p̸̢̹̰̫̀̊́̉̋̈́̕͘r̶͙̫̘͒̍̔͗̔͆̍ǫ̵̱̔̊̈́͛͆̆͝b̴̯̔͗̇̀̓̽̐́a̴̞͙͉̺̗̯͓̓̒̒́b̸̩͙̖̀͝i̸̩͗̇̀l̷͖̻͓̳̙̪̍̊͐́͜i̶̢̛͕͕͇̻̭̇̂̏͑͘t̶̢̥̞̗͖̤͉̊͗͐͝͠ẙ̶̨̯̮̦͊̐̽̏͝ ̸̺̰̒̿̚t̴͙̫̹́h̶̡̟͎͓̥̳͔̔́͐̀̚ą̴̣̖̳̟̟͒t̶̨͔̫͊̒̔̒̆͘̚͝ ̷̣̠̥̯̬̗́ͅt̷͙͎̫̙̂͌̅͆̀̒̕h̵̠͔̉̋̓̌̚̚͠e̷͇̻͇͔̘͕̐ ̸̯̰̍͆͝m̸̬͍̼̮͖͍̈́̿̃ï̷̧͔̫d̷̬̻͎̉̐̀̃͛̕p̸̨̢̡̪̘̌͐̈́̆͜ͅͅo̴̭͖͊͒̒͋̀̕͝i̵̤̱̮͔̣̯̎̊̔̇̐̀̕͝ͅn̴̯̞̠̣̙͉͔̟̾́t̵͚̻̊͌ ̴̮͚͉̘͠͠ȍ̶͒͗͗̅́͛́ͅf̴̨̧͉̲͆͒͝ ̷̛̛̪̮̽̓̅͘<math≯̢͔̩̦̪͚̌̓\̷̢̯̠̪̻̟͂͒́̈̀̚͠ö̴̳̪̙́͋ṽ̴̡̛͎͓̘̝̤̞͊ë̶̡͚́̀r̶͍͈̙͈̓̈́̀̄ľ̷̫̫̰͎̮̯͔͊̅̈́͒i̷̪͉͓̇͐̈́̔̒̏͜n̶͕͓̘͗ȩ̴̛͇̣͛̈́̊{̸̩͐̀̓̋͐͌̀͝Á̸̧̳̼͇̲̙̃͐̉̒͌B̶͕̤̉̉̃̌̚}̶̜̃̐̉</math≯͓̯͇̣͋͛̈̋̀̈́͝͠ ̸̹̲͉̎̄ͅa̶̞̤͊̎͋̽͋͠ͅl̵̗͖̟̘̮͓̲̓s̷̮̆̀ơ̷̤̍͌͂̾͝ ̴͔̫̹̠̯̈́̽̕͘l̴̫̙̀̓͗̑̌̀̍i̷̮͓͙͕͒̇̋̂̈́̈́͝ę̵̧͙̙̘͎̭̱͋s̸͈̱̆ ̶̱̝̲͖̼̪͇͈͋̏͆̃̏͝͝ï̴̱̇͋̈̚̚͠n̸̬͎̩̣̠͇̋̽̑̏́͝s̸̨̤͚̘̜̳̬̈́̕͠i̶͔̐͆̈́̓̑̐ď̴̪̗̳̑͑̈́̽̊͘è̴̢͍̙̗͚͕́͘͝ͅ ̸̛͕͈͈̯͎̗̱̯t̴̡̹̟͚͉̥̭̩̓̃͛̐̔̇͝h̸̢̛͕̣̖̹̙̯̐͂̏͌͘ͅi̸̟͖͎̳̻͎̥̋̊̒̍͝s̴̢̨͇̳̤̫̜̍͊ ̷̛̠̒̅̋́̐̾̂L̵̨͕̹̞̫̩̃͑-̷̧͔̤̯̼͈̩̐̌ṥ̸̰̙͚͍̮͌ḣ̷͓̺̞̦͝͠a̴͖͙̜̦͚̪̮͌͒p̴̡͎̪̙͇̊ē̵̱͕͉̜̤̗̈́͝d̶̘̗͍͎͒͛́̿͗̆́͜ ̴̰̀͌̒̓̇̍̊͝r̸͓͍͑̐̅͐̓ȩ̴̼̖͖̓͜g̷̡̥̠̜̏̉ḯ̸̹̑̓̆͌̇͆o̸̧͈̼͉̺͈̼̭̎n̵͈̯͑̓̂͋̋̏͛͠ ̸̢̘̻̌̃c̸̯̪̦̱͕̿͐̚͠a̸̮̜̕͝n̶̜͉̅̈̽͝ ̷̛̯̯͂̄̀̑͂b̶͚͈̘̭͍̄̒̀̆͆̎̒͘e̴̝̥̍͒̔̚ ̸̜̟͎̂̌͒ė̷̯̻͚̩̱x̷̡̳͕̪̗̹͊́͗ͅp̷̢̯̰̥̼̚r̷̨̭̺͇̹̲͂͆ͅͅe̴̩̭̱̪̩̍̉͠s̶̟̝̫̎͂ś̴̹̭̳͕̮̈̾͠͝e̴̹̼̖̳̝͛̈́̆ḍ̴̥̲̅̀̔̏́͠ ̸̨̧̺̥̣̺̲̀a̷̧̮͉̹̜̥̥̼̋͂ş̵͌ ̸̥̤͈͎̻̫̊̋̔̈́<math>̷̨̙̎̅̾̂͝\̷̛̙̹͇̳̪̓f̶̹͛̄̓̕͜r̵̛̖̣̠̩̩̪̉̆̉͝á̴̧̻̘͓͔̐͐̆͐̾̕͝ç̴̫͓̝̻̰̪̯̏̀̃͠͝{̸̢͕̣̳͉͎̹̀̊ͅm̵̳̩̈́̆̇̾̐̒}̸̡̧̛͇̝̘̮͂̊͆͊̔{̶̟̞͘n̸̨̘̅}̴̜̣͊͌͗͛́͆,̷̡͈̠͈͈̯́̾̌̌̈͊͘͝ͅ</math>̵̨̧̠͚̳͓̣̰̿͛ ̸͖̻͙͓̎̿͜͝ͅw̴͕͔̼͈̗̩̌̿͐̇̆̚̚͜ḧ̷̲̺͉̼͕͙͓̝͆̓͋̄͘ę̶̜͎̳̱̥̼͎̔̈́̂̅͐ṟ̴̻̲̀̄̄͛͜ę̸͉̳̯̍̚ ̴͖̲̪̯̻͍̈́́͒͌͊̀̃̕<math≯̬̝̮̟̦̏̈́̌̚m̵̡̽̈́̈́̑́̈́</math>̶̡̳̖͖̼͚͖̹̅̈́̿̿ ̸̢̭̠̙̗̲͎̄̇̔̃̌a̴͖̜͇̯̬̣͋̒ń̷̳̳̞͇̻͈̙̘͐d̵̗̫̼̤͎͂́ ̷̗̈̂̌̚<math>̷̝̠̜̝͗̑͠n̶͓̳̳͓͚͍̯͂͗̓́̚͝͝</math>̷̨̪̜̞̲̙̹͗̒̾̈́̚͝ ̷̪͕̳̣̫̦͒ȧ̴͇͍̗͉̞͙̻̱̒͊r̸̺͓͂͌̊̏͠ȅ̴̙͛̓́̀͋̐ͅ ̶̞̲̏̿̑̈́̈́̇r̵̢̻̠̥̠̰̤̈́͆̀̃ę̶̧̰̬̰̖́̅l̵͖͌́̒͑á̴̦̣̬͛t̷̥̮̫̅̀̾́i̶̛͍͓̭͇͋̄̎̕v̶̼͖̜͚̫̠͓̖͋̓̕ȩ̷̰̰͇͙̠̓̿̒̂̚͠͝l̵̛̟͎̱̃̊͛͋y̵̛̗̹̙̪̳̝̬̆̿͛͒ ̴̩̰̪̮̬̪͛̃̔̈́͋͘p̸̢͕͚͓̀r̸̳̤͔̣̲͕̠̃̀͠i̸̪̔̔̓͂̚m̵̨̬̮̗̻̟̰̅̑́̾̆̾̏e̴̫͛̒̃͊͘͝ ̷̪͂͝p̶̺̹̱̯̣̤͗̿̔͜͝ͅơ̵̤͛͂̔͝s̴͖̖͕̽̑̄͑͂̀͊̎ǐ̷̛̜͓̫̹͙̖̣̹̇͆̅͋͝͠t̶̙͆į̵͈̺̜͙̗͙̜̀̈̉̈͌̽̀͘v̵̻͈̣͚̥̆̀e̶̠̘͓̦̒̾́̔ ̸͕͖̦̩̘̓̍i̴͈͊͆͐͂͛͒́̌ṇ̸̡̛̮̞͔̩͎͕̋̓̍͋̃̂̃t̵̢̤͇͇͉͖́͝e̸͍̫̔͊́̓͆͑̌̚g̶̨͈̫̰͖̻̲͕̎͂͌̈́͘ë̶̩͎̺̗̪̘̰̼́̄͊͐̊̈́̅r̶͈̝͇̘͉̋ş̴͎̰̪̮͔̋̐̍ͅ.̵̨̭͌́̌͒̂̀̕͘ ̵͇͖̬̌̐̆̈̓͝͝F̴͔̦̬̼̫̤̬̀͊̃̈́͘ḯ̷͓̼̯̲̯̮̘͆n̸̨̡͖̯̣͇͖͂̎̾̋̅̓̅d̸̬͍͚͒̃̂̑̿̀́ ̴̝̯͓̺̩͚̓̏̈́͌̽̔̽̓͜<math>̶͚͖̱̍͂͘͘͝ͅm̴͕̋̀̈́̕͝͝+̸̢̫̻͕̜̉̆̈́̓̑͌̚͝ǹ̷̨̼̪̻̀̔͊́̾̕.̴̦͓͔́͐̍ͅ</math>̷̧͖̞͚̿͑̐̀
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Consider the L-shaped region formed by three unit squares joined at their sides, as shown below. Two points <math>A</math> and <math>B</math> are chosen independently and uniformly at random from inside the region. The probability that the midpoint of <math>\overline{AB}</math> also lies inside this L-shaped region can be expressed as <math>\frac{m}{n},</math> where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n.</math>
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̶̧̞̗̝̝̫̰̏<̶̹̂̏͒̆͝ä̵̛̳̿́͑̾̀s̶̪̔ỳ̷̞͑̚̚͝>̷̡̬̱̖͈͖̟̈́̄̀
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<asy>
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̸̳̩͉͚̫̙̬̽̈́̽̕ȗ̷̗̟̟̰̑̎͊͛n̶̢̟̟̯̬͋̇̐̌͐̔͆į̴̳͔͎̲̘̲̏̈́̉̓͗͒͠ţ̷͈͚͔͠ͅs̵̢̨̱̖̜͚̽̈́͜ͅi̸̮͔̥͚̟͓̾z̷̨̠̅̐́è̵͖̅̅͝(̷̡̂̌͋ͅ2̴͚̇̅͂c̷̪͊͐m̸̳̖̠͓̜̯̝̊̈͒͌͗͛͐)̵̫̬̠͖͖̿̾̓;̵̦͎̬̣̫̓́̇̏̾̓͘̕
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unitsize(2cm);
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̴̡̫̪̹̹͖̏͛d̴̜͈̩̖̪̲̼̮͐̀͋͐̀̔̋r̵̢̙̱͒a̸̫͇͕͗̎͗̊̉̀̿ẁ̵͔̟̼̖͚̬̠͖́̍̃̎͠(̴̡̼̝̥̗̳̦̝̉̂͊̃͐̿(̷̡͈̻̹̹̔͋͊͘͠0̶̨̨̛̣̦͎̆̿͑̄̔,̶̟̭̟̼̀͋̀͜ͅ0̸̨̧̹̟̥̩̹̓̐̂̂̄̉ͅ)̴̭͉̪̼͚̪͉́̿͘-̴̳̦͕͇̬͚̟̎̔̐̾͌͘-̵̢̻͓̳̮̠͉͂̎̽̒(̶̠̳͍̖͠ͅ2̷͔̥̺̥̼̞͙̠͂,̵̼̹̱͉̮̮̤̦͑̐͒̀̍̍͝0̸̡̒)̵̝͕̲̮̓̉̈́͌̍̚͜-̴̧͉̜̈̽̓͛-̵̞͈̟͙͖̈́̂̏̆̿͌̌̓(̶̟̠͂͛̓͋̂̊̈͆2̶̭̜̻̾̏̓,̷̠̯͂͐͂̑ͅ1̵̡̺̩̠̈́͆͋̀̈́̕)̶̢̛̗͚̖̋̀̓̐͗͘-̸̭̗̬͍̮͔͙͑͐̑-̴̛̛̩̪͊̇̀̕͝(̷̡̥̏1̴͍̰̩̠̣́̈́͗͂̉̆͜,̴̡̻̱̩̉̀̆̍́̿͜1̷͔̥̥̞̔͜)̵̡̤̞̘̯̪̳̔-̷͍͔̻̦̭̯͂̅̎͜͜-̸̙̫̤͑̃̽͌ͅ(̷̰̰̃1̸̡͔̞͈͊̉͗̍̏,̷̣͖̦̙͔̭̜̓͘2̴̡͈̤̳͍̤̲͂̅͌̈́̍̕͠)̸̧̻͕̫̝̯̓̚͠-̷̛̮̱̳̓̊͒̌̌́-̶͚̫̹̮̔̾̀̂̎̀̀(̴̦͖͌̆̀͠0̸̛̠͉̯̈́̔̾͛̃,̶̡̨̥̦͈͍̹̀̓͆́̕͠2̸̡̨̗̗̿͜ͅ)̴̨̺̳̘̱̹͂̕̕͝͠͝-̵̨͕̣͕̤͓̣͒̿͛̆̚̕͝-̴͕͙̺̗̂̎̀͝͠c̷̡̛̛̹͉͓͔̣͚͌̃̓́̿͜ý̴̛͖͚̫̳̹̪̮̽̀̈́̾̒c̷̮͎̤̳̟͖̩̲̊̆͝ḷ̵̖̼̤̯͚̍̉̉͒è̸̢̤̞̟̖)̵̛̦͓̐̊͒̃̎͌̈́;̵͚͑̎͛͑͋̽̐̕
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draw((0,0)--(2,0)--(2,1)--(1,1)--(1,2)--(0,2)--cycle);
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̴̪͌͆͐̋͛d̴̝͛̄̎̆́̄͗͋r̶̹̽͒a̸̭̜̩̟̭͛̔͗͝ẉ̵͍̫̮̺̇̆͒͘͘(̷̣͓̅̅̈́́͌͗̽(̸̛͈̻̥̥̫̑̓͋̏́̑̓0̸̡̧̣̱̖̪̅͛̾̾͂͜͜,̷̧̠̮͇̻̓1̴̨̼̱̤͖̈́̌̇̅)̴̻͙̌̇̏̄͝-̸̲̗͇̌ͅ-̶͓͕̺̝̯̌̔͌̿͜(̸̧͚̬̺͕͙̟̓͆͠͠͝1̷̝̰̯̳̖̓̈́,̴̨̛̫̜̭͎͔͙͚͋1̸̳̯̆̈́͐̂̕͘̚̕)̸̢̱͔̪̠͚̰̉̈́́͛͜-̵͖͍̭̻̒͝-̴̳̤͕͂͊͝ͅ(̸̢̫̦͓͕̞̘̣̀͛͒̐1̶̭̬̱̲̥̉̓̈́̓̐̄̑̒,̸̙̰̠̠́̊̀͝͝0̵̛̽̈́̊̐̉̌ͅ)̶̞̅̚,̸͈͖̝̳͕̈́̇̽́̐d̵̖̟͂̀̓̚͝͠a̵̬͎͉̰̘̲͊̊̄̓͆̕͠ş̸̪͉͗̏̀͋̓ḥ̴̓̃̒̉͜͝e̸͔̰̟̟̔̀͜ͅͅd̶̢̯͖͎̜̮̑̅)̶̯̕;̸͔̍̌͑̄̅̂͑͂
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draw((0,1)--(1,1)--(1,0),dashed);
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̴̼̫̆̕<̷̥̪̹̠̣̀̈́̈́̃/̵̹̪̩̪̞͉̟̆̈́͒a̷͚̼̺̔͋͜s̴̢̳̘̫͑̏̓̆y̷̺̣̺̍͒̒͐̈́͐͝>̴̙̄͊̕
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</asy>
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̵̭͓͋̀̋̿
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̸͙̰͙̥̈[̷̘̹͈́̈́̾̆̅͝[̷͍̹̯̮̳̭̣̊̾͛̇͜2̸̩͎͉͉̉̊͜ͅͅ0̴̡̬̯̞̫̗͆̓2̴̧̲͑̇̓3̸̧̼̫͓̜̪́̇͆̍͋͘ ̷̨̱̖͍̓̃̕Á̵̦̻͛͋͝Î̶̢̤͇́́M̴̡̱̹̩̌͌͌É̶̛̯̼͍̟̑͆̅̚ ̵͖͙̖͖̦̏̓̇̊͛ͅI̶̢͙͚͇̠̓̈́̋͆̒̋̌͘ͅI̴̭̬͊͛̈́̑͘͝ ̴̨̱͕̞̭̈́̆̃̈́P̸͖͈̥̠͔̲̀͋͌̀̽̉̀͜͝r̵̢̘̻͈̪̞̮̽͌̃̉͛̏̾̀ơ̶̢̢̧͕̮͔̮͐̒̊̈́̕b̴͙̰̺̣̖̜̘̈́̿̾̿͜ļ̴̰̝̗̯͖̯̘͂e̵̥̖͎͍͕̟̻̣͛͒̀̓͂̃̌m̴̼̤̖̝̳̌̓͘͝s̷̭͉̹̈/̷̢̣̖̰̣̤̃̃P̷̙̤̪̘̙͇̋͆͘͜r̵̨̥̲̼̲͖̲̖͗̉̆ǫ̴̢̙̥̟͉͉̳̋̂͝b̶̧̠͔̙͆̒̔͌̈́̉͘l̴͙̳̱͍̦̼̞͗ë̶̜́̔͜͝m̴̧͖̖̟̦͎͂̈́̾̕ͅ ̸̨̓6̵̨͕̯̝̜͚̥̋̇͂̌̑̒̌̚|̷̹̜̝͇̤̮͔̅̔́̏̔̓͘S̴̝̜̘̺̲̘̫̈́̆̌̃̓͝͠o̴̜̺͎̊̔̐̿̎͝l̵̗̬̞̹̀̀͑͐̿͘͠͝ȗ̷̗̘̫̉͜ț̵̛͍̝̃͒̀͝ͅį̴̨̛̠̞͕̀̀̅͊͛̉̚o̷͖̰͈͕̣̦͑̈̒̈́̊̌n̸̛̬̐̓͋̅͝]̴̩̜͒͆̂͊̕͝]̵̟̈́
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[[2023 AIME II Problems/Problem 6|Solution]]
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̷̜̱͔̪̠͊̈̒̈̋́͝
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̵̩̀̂͝=̶̧̘̺̩̒̎͌́=̷̳̬̲̘͒͛̓͗́̍P̴̡̦̪̯̥̈̀͑́̒̒͆͝ȓ̴̰͆̀̄̆̌̒̒o̷̺̯̬͇̐̉̄b̶̮̫̫͝l̵͎̼̼͛̄̆̀͒̒͌͝ͅȩ̴̧̛̛̻̣̗͚̭͎̏̃̇̃̔m̴̼͔̬͕͆̂̆̓͆̅ ̸̥̝̻͓̔̏́̈͜͜7̴̪͖̽=̷̧͇̬̲͖͓͑̐͊̌̈́̈̚=̵̯͕̀̀͋͂͋͗̕
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==Problem 7==
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̶̛͙̣̽͐̈̔̕
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̶̧̰̪̹̳̟̤̀̔̈ͅE̷̛͔̫̣̟̽̂̃̇͝͝ä̸̜́̍͒͑̔̓̕͘c̴̖̪̆̾̉͜h̶̤̞̯̫͇̊̔̓ ̸̘͎͚͚̙͇͈͈̏v̵̳͔͐̕͜ę̴̧̜͓̫̞͉͍̑̌̈́r̶̺̫̦̠͔̉̌̅͜t̴̰̄́̾̃e̶̱̲͎͕̘͓̳͒̄̎̑̒͝͠x̷̭̝͈̒̾͆͂̆̐́̈ͅ ̷̢̮̥̃͊̓̅o̵̢̢̧̭̻͇̲͗̋͑̊͝f̵̧̘͖͕̱̠̟̻͑ ̷͓̬̣͉̱͇̟̐ǎ̴̺͔̠̝̐̅͘ ̷͚͓̑̍̉̋̔̚r̸̲͍̙̠͇̯͋̅̇ê̷̞̣͚̰̞̱̰̙͑͗͛̀̾̀͝g̶̦̮̹͖̻̭͙̎̌͊̉̏͠u̶͖̩̟͍̯͎̰̓̒͊͆͝l̷̢͈͒͂́̍͂͠a̷̧͔̣͉̖͐̅r̸̭̥̯͖̰̈ ̶̛̤̪͈̥̌̑͜͜d̷̙̫̻͊ͅo̷̰͚̩͙̜̮̤͊d̸͇̬͎͍̮͚͂̀͆̔̌͝ę̴͖͔͐̎͋͗̑͘̚͝c̵̡̜̘̽̿̾a̴̢̩̫͎̠̩̅͑̎̊͠͝ͅg̶̛̤͔̣̩̲̻̔́̓̀̽͘͠ͅo̵͕̻͚͇͛̋̄͑n̵̜̯̝̪̱̊̕ ̶̢̞̉͜͠͝(̷̨͂̇̽͗̑̇͝<math>̶̛͔̰͚͍͙̬̗̋̄̍͋̀͐̓1̶̢̟̰͉͕͖̻̣̔2̴̭̲̰͛̊̏̓̄͊͂͝</math>̵̝̱̥̩̻̊͗-̸̢͚̜͓͚́̌̾͂͛͐ǧ̵̲̦̫̖̤̼̀̏͌̿́̿̈o̷̺̲͒̒͘̚͝͠n̵̫͙͔̓̉)̴̬̾̀̽̄̈ ̸̩̭͚͇̩͚̤̗̇̀̉̂́͑̌i̶̧͈̪͇̺̱̼͚̕ş̸̙̖́͒͒ͅͅ ̴̢͖̫̞̗͖̙̣̃̇̑͑̌̔͠͝ţ̸̡̘̻̦̤̣̆̋̃̚ͅǒ̶̡̩̠̲̻͔̗ͅ ̵̢̦͖̘̟̾̉̓̊̕b̵̡̲̲͍̲̘͍̋̆͛é̶̹̤̲̗̌ ̷͔̲̤̜̜̿͂͘c̷̨̛͙͇̲̠͎̗̈̐̆̂͠o̴̡̬̜̰̹͇͛̿̎́́l̴̜̱̱̻͚͔̋̓̓͂̔̆̀̕ò̴͓̹̤̤̟̋̓́͋͐r̷̡͉̫͑̔̈́͒e̶̢͓̤̺̻̰̪͗̚͠d̵͉͚͘ ̷̛̮͉̞̫͗͑e̵̢̲̮̫͋͗̍͂i̷̡̢̢̭͍͙̰̊̀͊̊̈́̄͘t̴̺̺͖̝̺̟̦͆̕h̸̝̱͉͙̓̅͜ẽ̵̻̖̀̿͑̈́̀r̵̰͕̹͔̗͂͐̎ ̴̻̑̓́̆̊̎͠r̶͎̜̀e̴̡̨͙͍̩͓̍̒̂̆̈́͘̚ḏ̵̢̛̘̍͒ͅ ̵̛̼̖̠͖͎͖̞͎͊͗̆̂̕ǒ̶̱̆̑̋̔̈́r̴͕͉̘̻̈́̍̇͆͠ ̶̧̨͙̥̠̐b̵̙̰̮̤̉̀͊̿͂̈́͠l̶̤̫͉̓̓̃͆̑͝ͅu̵̟̜̱͉͓̜͕̓̏̇̂̄̂͝ȇ̶̢̫͔͚̝̈́̽̒̀,̴̩͕͖̦̖͊̊̔͂́͜ ̵̢̞̪͖͍̕a̸̞̱͈̻̟̝͂̍ͅņ̶̛͎̣̱͒̑̐͠d̸̢̼̦͖̲̳͗̆ ̶̛͎̟̘̩̳̩͉̽̈́͒̎̾͠͝ṫ̶̨̡̢̺̲̣̭̠̑͗̄̉̄͠h̷̦̤̮͔̾ů̶̺̯̳͙̜̘̲̯̀̈́̃̅̃͝s̴͎͚̱̏̅ ̵̨̰̹͚̋͠t̶̨͇͉̲̲̋h̷͚̝͚̲̤̪̬͑̈́͐̈͝͠è̷͉͕̣̩͇̜̅͊͠r̴̰͖̖̣̩̬͉͇̿ẹ̶̡̯̉͆̀̓͘ ̷̘̕a̸̹̙̰̟̅̊̚r̸͍̪͝è̸̖̱̝̀̚ ̴̱̥̿<math>̴͊͑ͅ2̶̞̦̃^̶̡̡͈̤̬̺̽̐͆̿͐͝͝{̷̻̭̽͗̑͂͊1̸̥̊2̸̦͔̪̲͚̦̬͚̇̄̈́͐͗̀}̵̡̢̛͓̹͎̬̇́͒̏͗̚͝</math>̷̜͇̲͕̊̏̋̅͂͘ ̷̡̛̬͉͔̠̓̂͋̕p̸̛̲͂̽̋͊̀̅ŏ̷̡̧̲̮̂̊̀̃̏͝s̵̗̦͉̮͗̒̾̀̃̚s̷̩͓̦͛͜i̵͕̲̻̤͚͕̔̾͜b̷̨̟̈́̏̀̃͂͋͜ļ̶̧͇̥̀̀̌̌̅͘͝ę̴͓̞̈́̔̕ ̴̢̦̻̣̦̫̟͊c̴͉͚̪̥̤̻̙̊o̴̰̞̺͔̍͂l̵̹͈̲̰̟̔͆͊͒̾̎͗͑ò̴̻ř̶̨̯̗̣̮̪̣͋͆͋i̶̦̣̘̎͛͒̌̀n̵̨̬̼̬͖̖̆̅̾͋g̵̡̧͓̻͊s̷̹̜̭͉̄̇͌̃̈́͜.̵̛̺̖͎͕͚̎͋̋̍̐̎ ̶̹̣̺̾̀Ḟ̸̘̫̬͓͐i̷̱̮̱͓̙̊n̵̗͈̖͚̦̙͓̂d̷̼̤͖͔̲̓̍͑̅̚͠͝ ̷̟̈̎t̵̡͚̹̮̜̬̘̔͠h̴̨̦͔̗̰̩̘̻̿́̀̇͛̐ę̴̜̼͙̣̮̇̒̑̌̊͝ ̷͕͌́n̸̨̾́͐̈́́̀̕u̵̲̇̂̚m̷̦͓͙͇̊̌̀̉̚b̶̺͕̈̊̎͊̚͝͝ȩ̷̦̘̬̯̺̀̽̀̆̀͛͛̒r̸̫͍̈́ ̸̢̹͓̹̝͗͝o̸̡̯̞͖͍̣̠͊̔͌̀̈̈͠͝f̴̫̜̬̞̮͉̳̝͛ ̸͎̗̣̩͉͌͒̀̾́͆̋͠t̶͉͎͎͙̜̞͇͌̆̈́͌͜h̷̜̜̝̪̀̍̾̀̋ë̷̝̼̤̗̒̇́̿̈͋̉s̸̢̤̗̭͇͙͙͊̀̿͌̅̾̒̆ẹ̵̡̙̦̯̈́̈́̋̌ͅ ̶̭͖̱̄̆͆c̸͚͚̯̠̻̪̄̈́̔ŏ̷̢̧̧̤̟̬̗̱̈̓̆̐̉̆l̸̲̝̅̊̓͋ǫ̷̝̣̙̻͔̱͈̔̈́͑́͂̏r̵̜̦̠̳͕͈̲̽͑̄̋̕ǐ̸̛͚͛̂̄͆͝͝ņ̶̛͙̗͓̗̉́͘g̸̢̟̩͓̜͙̿̓̈́͠s̴̛̗͛ ̵̘̰͑͘̚ẁ̴̱̻̞͇̮̭̏i̸͖̻̟̹̻̬̓̄͑̉t̶̲̎͐̋h̷̡̛͕̮̹̼̗̐͑̊͋̃͗ ̷͚̠̃͜t̴̨̮͚̥͐̂h̷̛̩̙̟́̒͠͝͝ḛ̸̥͚̘͖͈̩̭̾̉̕͠ ̸͓̜͕̥͐̌̎͂ṕ̸̤̭͈̝̹̤̫̪͛͝ŕ̴̛̮̣̘͉̰͗̀̈̄̓͝o̸͈̅́̒p̶͖̮̑ẽ̷̹̹̪̼̰̏̈́͠r̸̡̧̙̬͋̐ͅͅt̸̡̥̣̯̰̙̺͊̋̂̒͆̄͐y̶̡̭͍̮̑͗́̂̽̚͘ ̸͍̇̆͐̑͒̽ẗ̶̢́͌͌͋͗̈́̀͠ḧ̸̛̗́̆̑̆̊͘à̸̲͓̜͒́̇͗͠͠t̴̠͔͈̥͖̫̠͕̆͊̿͛̈́ ̴͕̤̻̯̖͚̹̓͑̀n̶̨̧̜͎̝͎̠̺̍̀̏̎̚o̸̲̬̕ ̶̢͔̳̗̭̟̊͒̈́̾̚ͅf̶̻͖͖̝̼͇̐͌̃́̾͛̈̂o̴̡̘͔̪̹̠̩͙͑̈́̃̓̇̋͘ū̴̙̰̺̦͍̿̕̚r̵̙̩̼͔͙̥̳̖̽͐̊̐̎̎̎͘ ̴͖̈́͌̀͜v̷̼̲̫̫̦̱̲͆͜ę̷̢̧̦̬̤̣́́͘̚͜r̴̢̧̪̥̰̿̕t̴͕͗̌̒̅̈́́͛̈́i̴̹̝̹̜͖̋̕c̸̞̉́̈́̅̚͘̚e̷̛͍͇̳͉̫̓͊͊̀̾͝s̶̡̤̠͙̩͖̓͗̌̊̀͂̋ ̵͓̏̍c̵̨̧̖̟͈̦̟̘̓͛́̀͂̇̕̚o̶͍̜̦͈̠̹̣̿͂͒̄́͜͝͠l̴̡͙̻͇͎̯͉̽̌̃͊͆͐ơ̴̭̊̾̇̎̌̾̒r̴̫͖͔͔̞͐̃̋̚͠é̵͍̹̿͂̀̉́d̵̢̦͓̑͐̓͌̃̌͝ ̸̦̲͙̤̱͉̭̿̋͑͝t̸͈͖̃͌ͅh̴̖̅̄e̵̡̳͙͉͔̪̤̓̎͋͗̈́̒͒̋ ̸̲̇̎̆́͘s̶̛̛̳͖̎̈̍̃͊a̴̱͖͎͑̆̈̀͘m̸̹̀͋̀̅̉e̸̠̯͋͆̀̀͗͆̆̀ ̵̧̥̯̎͊̃͛͘c̷̞̝͕̞͉͙̻̱̅̎̅̓̈́͝o̴̯̱͂̂̌̀͛̃͘̕l̵̯̩̲̔̅̇͑̅̑̆͑o̴̰̺̱̳̳͐́̃̉̈́̕͠͝r̵̢̹̤͈̍̑̉̇̀́͑ ̶̗͎̅̒̇̀̕͠a̸̻͌͌̆̎͆͝r̴̡̙͕̉̏͒̌͂̑̎͜ȩ̴̢̮̺͔̩̲̺̊̆͑͌̒̅̎ ̵̭͕̮͕̙̄̊̄̃̏̈ͅͅt̷̢̢̤̼̱̫̏́̂́͆͠h̸̡͇͖͍́͑̈́́e̷̢̲̗̺̰̽͂͐̌̿͐͑ ̴͉̫̬͊̓͋̆̐̇̓͛f̴̩̖͜͠ó̸̯͓́͊ủ̶̬̠̲̗͉̜͑̉͛̎̓ͅͅr̷̘̱̰̘̭͕̼̥̒̄̅̕͘ ̶̢͖̘̠̣̊͗̓v̶̼̦̳̠͎͍͐͋̿̓̓̇̚ȅ̸̢̬̖̤̓̄̉r̴͔̦͌̅́͜͜t̷̢͎̭̦̞̤̑͌̀̑́͌i̶͙͇̯̞̖̪̜̼͐̀c̸̲̓̕e̷̲͔͎̕s̵͖̜̥̥͎̅̑̒̓̚͘ ̷̠͍̤̒̈́̾̕o̷̧̮͛̇̿̕f̵̛͚̞̗͔̍̒̄ ̴̧͔̱̲̩͚̪̙͆̈̊̐͌a̵͔̞͇͑̐͒̑̏̾̚ ̶̼̆̋̈́̇̊͘͘͝r̷͚̜͉̯̰͈̹̊̾̿͠é̴̡̺̩̀̉̈́͋̽́ͅč̶̢͕̘̤͉̀͐͆͂͐t̴̢̯̩̘́͐͑̾̌̒̈́́͜a̷̡̦̗̘̼͐͌̕̚ͅn̷̖̯͈̮̜̣̈́g̵̱̪̈́l̵̛͕͉̒̈́̌͑̚ͅe̴̫̦̼̲͆̎͒̂̂̓͊͘.̴͈̥̯̇̄̄̎́̅̄
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Each vertex of a regular dodecagon (<math>12</math>-gon) is to be colored either red or blue, and thus there are <math>2^{12}</math> possible colorings. Find the number of these colorings with the property that no four vertices colored the same color are the four vertices of a rectangle.
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̵̻͙̹̱̠̀̓̈́͜
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̶͇̜̜̑̇̒̆͊̔̎[̵̨̘̳̭͎͉͈̃ͅ[̷̢͙̤̮̙͉͉̂2̵̛͉̥̀̍́̉̓̿͜͜0̷͙̙̪͈̰̩̌̀͛̅͂̐͋̏2̷̢̫͖̭̹̾͋̿͜3̶̛̼̟̒ ̷̣̦̻̯̥̝̬̀͛̂̐A̵̪͙͕̩̱̻̺͋̉I̴̼͕͎͜͝M̸̧͕̫̝̞̥͔̞̍̈́̃̋͐Ē̴̙̔̑̅̀̕ ̷̨̲̈́͊̐̽I̷̼̼̠͔̭͍͕̱͛̋Į̴͍̖̞̊̑ ̷̘̻̠̪̟̦͓̏̄̈́̕P̴̜̭̜̏̾̐̓ṙ̵̛̼̈̿o̶̰̳͕͖̿̎́̀̾͌̋͘b̸̠̙̎͝l̸̛͈̜̱̬͉̺̟̈̏̋̉̒̾͝ͅḙ̷̝͖̯̏m̷̢̠̼̒́s̷̖͚͚̹͒̍̈́͛̂̄̕/̷̠̻̮̳͒̈̅͌̋ͅͅP̶̛̥̦̀̃͑̏̈̕r̷̜̦̥̼͂̏́̔͋̎͜͝ǫ̸̜̈̔̈́b̶̨̯̟͖̃̄̾l̵̹͔̣̣̟̙̤̀ȩ̷͍͍͎̤̐̓m̴̳͓͉͓͖̤̍̽͋͒̄͘̚ ̴̻̭͈̺̽̓̄̔7̵̪͚̟̞̌̇͊̎͗̄͠|̷̢͕͔̩̹͋̎́S̶̯͆̈̿o̷̡̲̔͂͐̎̒̓l̴̡͕̜̙͚̩̋̏̇ủ̷̱̬͖̜̺̥͑̔̊̃̎̂ͅt̵̰̘̭̦͒̚į̸̝̻̯̦̩̌o̷̠̖̼͖͙͕̦͉͝ņ̵̧̗̙̫̖̱͙̈̊̽͒̔͌͝]̶̨̬͎̦̤̈̆̿̈̇͘͜ͅͅ]̷̨̛͈̰̭͚̙̒̀̾
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[[2023 AIME II Problems/Problem 7|Solution]]
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̴̣̙̫̖͇̞̑͛
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̵̨̢̮̮̭̩̎͛͝=̴̢̥̺̞͆̓͘͠=̷̡͖͔̮͉̐̆͐̇̆P̷̩̹̻̍̅͛ͅr̸̯͖͆̾͒ȍ̴̼̱̍͐͘b̶̛͙̠͈͕̿̈́̃̈̏ͅl̶̙͚͚͆̐͐̓͝è̷̢̞̼̤̖̬̳͋́m̵̤͍͕̫͉͌͑ͅ ̵̱̠̼̦̓8̸̰̘̝̠̏≠̝͈̬̙̰̥̣͉́͂̔͝=̵̞̣̱̻̟̞̾͌̏͜͝
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==Problem 8==
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̴͕͔̦̹̗̫̐͊͊ͅ
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̷̡̧̢̛̜̳̮̰͓̂̈́̈́Ḷ̷̡̧̡͖̯̉͐̐̿̈́e̴͚̹̬̋̍̊͜ţ̵̗̰͍͑̋́̍̀̀̓͘ ̶̞͖͓̳͍̎͗͒́̅̓̆͠<math≯͉̰͆\̸̛̞̣̱̰̈́̅ȏ̸̢͔͓͎͔m̴̲̞͚̹̔̉̃e̸͕̙͖̎̓́g̸̫̼̭̻͙͓͌̋̑a̶͍̙̼̞̅̈́̔̂͘͠ ̶͉̰̝͐̿͛̀≠̹̼̯̒̂̚͜ ̸̢̯̲̠̰̤̀͆̓̂͌̉ͅͅ\̴̺̀̅̽̌̕͘ç̵͓͇͎̹̋̍̌̈́͒̌͗͝ǫ̵̯̞͎̥̙̯̭͆͌͐̒̍͆̚s̵̘̠͛͌̈́̆͛̇̍̚\̷̰̼̂́͆͋́͜f̷̛͇̮̤͎̽͂́͑̽̐͝r̴̲̹͈̺̚a̴̹̖͐̑c̶͔͚͚̟̗̓̔͋̅͠{̷̪͈̬͔͙̞̹͒͜2̸̡͍̤̘̟̓͆́̆͐\̴̮̪̫̜͆̋͗p̶̨͇̮͋̆i̸̛̱̒̀̌͋̇͝}̷̛̲͇̝̙͙̃̎͌̀͗{̸̧͈͎̝͆͐̋͛̅́7̸̱̻̟͇͉̺̅͐̄ͅ}̶̡͚͍̈̈͌͊͐͊́ ̵̨͉̳͕͆̈͌̋̈̾̏ͅ+̶̬̞̝̭͋͊̇͝ͅ ̴̗͔̯̥̦̪̱̔͋̄͐̊ḭ̷͗̏̈ ̵͇͈̟̖̼̜̲͒̚\̷̤̝͉͖̆̈́̽͑̏͘͝͝c̸̯̙̆̈́̃̅͌̆̿d̶̩͇̪̭̞̣͔͑̀̒o̴͇̝̩̪̱̾̆͒ţ̴̼̯̲̤̤̀͑͊͋̍̈́̊̄͜ ̸̹̓\̴̣̟̜̺͎̞̼̫̐̕s̷̹̹̠̲͂͘i̵̪̺̗̥̳̅͐͊͊͝ṇ̶̨̧̝͖͎̱̣̔͊̏\̸̤̯͊̌̾͑̓͝f̶̧̝̗͇̅̂̽͐͆r̶̘͍͎̝̖̞̼͌͌͂̾̔ą̵͚̥̦̽̏̎͜c̵̬̺̲̯̬̩̫̈́̃́̃͌͘{̴̛̪̰͙̱̞̰̜̈́̑̔̈́̋̌͠ͅ2̸̧̨̤͒́̾\̴̛̟̞̫͚̼̪̱̿͒̒p̶̗͕̀ĩ̸̡̮̘̏̓̽}̸̹̞̘̳̃̑̀{̷̨̼͒̿̆̏̆7̴̢̼̼͉͙̩̮͐́̃̀͛̎͝}̶̖̻͗̊,̷̨̛̭̠͍̬͇̻͖̾̋̓͛̀̓</math>̷̼͊̓̄̃̍̀ͅ ̸̡̦̠͋́͒͂ͅw̴̢̦̯̩̙͓̅̐̂̈́̃h̸̞̫͍̗̆̽̽̍̈̈́̚ͅę̷̝̝͕̖̜̓r̵̗͛͂̈̑̃̆̒e̵̛̟̼͉͇̻̺̠̥̾̈́̉̂̕ ̸̢̫̠̾́̈͋̐͊̊<math>̷̘̿į̶̝͎̓͊͋̽̀ ̸͕̠̺̜̝͗̾̀̇̕≠̺̘̉ͅ ̷͖̉͂̋͊͑́̔\̸̜̺͐́͘s̸̨͈͕̲̃͛̃q̸̜̤͖̙̹̦̮͋̍̓͘͝r̶̨̩͎̗̍̍͒̄t̴̪̘͝{̷̨̲͔̖̌͑̋-̸̧͓̎̇́1̴̯̱̟̤͈͓͐̄̚͜}̵̹͓͉͈̥̩͚̪͌̊͌́̄.̶̩͕̲̺̼̃</math>̷̛͎̗̤̭̃̿͑͆̋̚ ̶̣̬͚̺͈͂̽́̊́̇͘F̸͔͇͆̓̿̀͂̔̚͠ì̴̹͕̻̯͍̪̫͉̀͒̿̓̓̀n̸̳̘͓͇̱̗͈̖̏͑́͘͠d̴̙̍͋̊̉͒̾ ̴̛͇͆̽̿͆̿̏͘t̴̥̰͆͆͐͊̕h̶̡̡̞͚͇̩̮̜̆̆̀͌̾ȩ̴̪̜̺̩̘́ͅ ̴̞͊̌̈̃͝v̷͕̫̞̯̱̯̅͐͘͜͜a̴̧̿l̶̢̼̦͓̤̘̓͒̚u̵̘͗̿̅̔̔͌e̸̜̪̝̖͂̇͠ ̸̭͇̳͈̣̀o̶̠̤̅̉͂̈̌̉̚f̵͕̗͈̰̘͇̹̿͊͌̿̚̕͠͝ ̷̧̡͍̗̦̺̮̃͒̌̄͑t̴͕̋̿͒ẖ̸̛̞̄̾̈́́͌̚ę̸͙͕̞̱̖̰̫̽ ̵̛͚̹̲͂͂̚p̷̨̡̹̦̪̱͛̈́͝r̷̟͇̻͚̝̒ơ̶̰͉͐͗̐͌̃͝d̴͎̝̮͙̗̰̥̼͊̈́̈́u̷͙̯͈͎̟̣̎̎̄͌̅͝c̸͙̯͆͂t̷̗̓̈́̋̉̉͑̈́̈́ͅ ̴͈͙̹͖̮̰͇̓<math>̵͉̺̮̥̀͊̿</math≯͕̠͖̦̣͛͐͝\̵͕͓̬͋̄̈̈͜p̷̗̳̞͆̃̔̀̈́̎̍̋r̵̩͓̙̬̬̱͖͑̈o̶̯̬̩͙̱̝͊̑͑̓̕d̴̮̜̟̩̻̺̽͊̌_̴̳̹̲͘͝{̴̣̰̪̣͈́̃̑͘ḱ̶̛̩͇̒̄̊̚͝=̷̡̧̖͎̺̖̤̾͆̄̒0̷̢̲̈́̏̏̐}̵̬̼̙̂̓̃̌̀̈́̈^̵̨̦̘̜͙͙̘̞̃̈̒͐͋6̸̠̼͎͓̗̲̌̍͂͝ ̴̡̢̞͙̤̉̄̑̚ͅ\̵̡͉͙̤̀̄̑̏͂́̌͑ͅl̸͔̺̰̖̄̅̂̓̾͒̚̕ē̶̖͈̫̭̰̪̓͆f̴̗̺̦̟̭̩̗̂́͗̈́̿͠ẗ̸͚̭̝̍(̶͚̫̹̲͔́́̔\̵̢̖̟͓͉͉͙̙̃́ö̵̖͔͂́̅̚m̷̨̢̠͛͊̀̀ẹ̸̭̼̝̰̥̒͋́̅͝g̷̜̽̽́̎͝a̷͓̥̲̐^̴̢̻͕̻̱̗̺̓̽̃͘{̸̣͙̜̺͚̪̌͜3̴̘͙͚̪͇͓̼͓̂͌̍͌͒k̶̡̰̿̑͑̈͘͘}̵͇͖̯͍̖̭̲̤͒̀͂͐͘͠ ̸̛͍͎͇̂̅̓̀͝͝+̵͈̱̲̬̗̱̊́͛ ̸̡͚͕̂\̴̡̱͆͌̄̎͝ȯ̵̲̹̹̙m̸̜̝̠̮̣̦̜͐̔̃̒̐ͅȩ̴̧̞̪͈͝ǧ̷̱̟̍̾̉͌̀̊̃ä̷̻̥̹́̊̾͛́͛^̴̹̣̈́̌̂̇̀̚͝k̴̢̮̪̖̘͔̲̊̀̔ ̶̝̰̱̄̇+̸̛̳͊̆͐̋̑͝ ̵̮̄͊̊͑̓̕͝͝1̷̢͔̖̮̮̄̌͒̄\̶̧͖̬͙͖̫̺͈̄̀́r̷̢͔̹̫̪̫̼̟͗̌̃́͐̽́͝i̵͌̌ͅg̵̩̥̮̉̍͊̎̎̈́h̶̘̪̦̠̲͊̏͗̃ṱ̵̛̞̣͈̯̗̓͐)̶̠̳͈̄̈́̋̓̋̈́̎̿.̶̧̡̱͔̣̽͑̓̈́͗̊̚<math>̵̡̢̻͕͇̘͍͍̾̎̑̉́͗</math>̴͙͙̜̩͊͌̎͐̑͝ͅ
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Let <math>\omega = \cos\frac{2\pi}{7} + i \cdot \sin\frac{2\pi}{7},</math> where <math>i = \sqrt{-1}.</math> Find the value of the product <cmath>\prod_{k=0}^6 \left(\omega^{3k} + \omega^k + 1\right).</cmath>
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̸̧̻̫̮̼̬̑͆
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̸̝͛̉̔͛[̵̥̤̹͙̣̍ͅ[̵͉̜͎̺͎͉͔̹̈́̓̆̑͒̕2̶͓͚̝̺̘̈̇̉͂0̵̺͗̂͗̒2̸̧̡͈̫͇͖͚͐̊͆̏̎̀͛͝3̷̩̩̞̄͊̋̊̽̀ ̷͎̙̝͎̼̤̬̓͒̚͜͝Ạ̸̡͆̐͜Ỉ̴̩̑M̵̢͕̳̩͕̘̖͕̓̎Ẻ̸̡͍̦̦̖̭̳ ̵͈̗̲͚͑͐̄͝͝Ì̴̧͉I̵͇̎̏͑̽ ̸̝̞̰̻̼̝̿͜P̴̗͚̾r̷̢̳̞̽̆͂̅̋͒̈ō̸͓̞̩̪͇͈̂̉͗̈͐̀ͅb̴̮̩̙̩̠͉̏ľ̵͈̤̫͍̠̪ȩ̷͇̯̱͖̬̓͜ͅm̷̡̛͈̺̠̤̝̔̊s̷̰̰̗̣̖̫̏̈́̈́/̷̞̓͛P̷̧̮̤̖͆̑́̇́̓͘̚r̴̢̫̗̱̙̓͊̐ô̸̠̦̟͚͎͋̅͑͌ͅb̴̭͇͉̗̀͌͒̑̂̈͑̚l̴͔̞̯͊̂̔͐͘͠ě̶͉̤̭̺͓̈̀͒͘̚͜m̵̪̯͚̹̻͚̀̓͒ ̵̛͔͍̔̀̔͂̇͠8̷̫̩̱̺̞͇̲̯͒|̸̛̬̙͊̈́̀̍̋̍S̷̛͚̻̫̙̙̱͈̣͑͆̾̾̈͝o̸̟̤͓̺̤͆̄͋̐͋l̷̡͛̃̂̋́̎̕͘ȕ̷͚̱͓͎̺̫̐̾́̈́̀̚t̴̡̧̘͕̫̥͐i̴̢̦̘͍͇̺͇̎o̶̬̬͎̖̞̓̃̋́̃͂̽̈́n̸̻̱̝̥̯̩̒̈́̇̐]̷̡̤̞̰̊͋]̶̛͎̎́̃̀̄͐͗
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[[2023 AIME II Problems/Problem 8|Solution]]
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̵͕̇
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̶̡̘̣̫̾̓=̷̩̥̬͉̣̊̋̑͠=̵̫͒͗̊̓̚P̴̛̼̘͉̘̪̳̊̍̓̀̽r̴̡̛̤͖̩͙̮͇͈̾́o̴͎̣̬͇̲͎͇̠̊b̷̝̤̭̬̥͕͈̄͒͝l̶̛̹̩̰̯̲̬̘̿̀͂̈́̉̂͜͝e̸̛̩͈̭̤̯̩͎̽͋̊́̈͊̈m̶̡̗̭͑͐ ̶̢̟̻̘͕̱̿͑̾̔̚͝9̵̛͈͕̠̤͖̝̹͓́̀̍̕≠̙̩̻̭͖̳̦͌̆̅̄≠̣͖͎̤̯͔̭͌̂̍͑͋
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==Problem 9==
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̶̨̧̧̨̩̬̗͓̈́͌͗̌͒͑͠
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̷̫̘͈̯͇͎̲̋͒͌̐̾C̵͍̟̬̼̪̘̝̎͒̑i̵̢̻̻̺͊̚͜ͅŗ̷̯̞̹̱̒͒c̷͚̃̍̄̾̓̕ĺ̸͍̓͆ẽ̴̫̦͓̙͉̐̀̍̚͝ș̵̡͇̠̃̌͑́̕͝ ̴͓̀̊͗<math>̷̛̗͉̖̆͗͆͝\̸̗̜̜̥̆́̆̍̄̚͝o̶̼͚͇̓͋̉̿m̵̨̡͈̟͉̲͙̈͛͒́̾̌̐͋e̸̳̠͓̍̌̀͊̕g̷̢͚̻̜̜̓̏̽͑a̴͎̠̩͗̓̄_̷͖̉ͅ1̴̡̗͓͎͖̟̈̍</math>̶̟̩̮̫͎͚̀̐̽̌̅̌͛̂ ̸̮̟̃̈́͋̾̓̂ā̵̢̖̙͉͚̘̈͗͛̒̚͜ͅn̸̙̻̳͛͗͌̄̄́̎d̶͕͕̝̖̬̞̱͊͂͐͜ ̸̡͇̱̬̳͙̀̀̊̉͌̚<math≯̰̱̠̇̉͘͘͝͠͠\̸͕̥̭͖͓͓͕̟̍̐̋͠ơ̷̧̬̜͙̊̎̾̄̽̒͝ḿ̸̧͇̮̗̞̟͉͓̿͒͘e̵̛̗͈̊̈̿͒̕͜͜ģ̶͍̳̮̖̌̅͋̂́͜͠͝͠a̸̢͛̀̐͑ͅ_̵̳̻͕̫̈̀͂͂̎̚͘2̵̜̮͇̠̫͚̫̋̅̂̑͑ͅ</math>̶͉̝̙͈̓̓̊̿͐̃͂͠ ̴̹͉̞̱͚̌̾́̇̃̂͋̈́í̸̩͓́̑̂͂͘͘͝ṉ̷̟̖̪̒ͅt̴͇̼̗̦̠͗̐̅̐ẽ̶̯̭̹͕͜r̵̜̺̼̟̜̔̀̑s̵̤͕̅̑͛̅͝ȩ̴͈̯̜͕̄́͗͠ċ̵̢͕̦̰̓̆̍̈̈̀͠t̴̢̛͈̬̥̹̬̣̽̋͝͠ͅ ̸̧͙͚͓̙͋̊͌̿́̎a̴̺̦̅̑̔̓t̸̢̨̟̞͈̩͔̿̃̊̂̃̂̅͠ ̸͖̗̃̇̈́t̷̘̯͐ẃ̸̦͕͈̬͆̋̈́͝o̶̖̳̳̖̯̩̓́͋̊̄̎ ̸̝̭̬̰̊̃̔̇͜p̷̧͍̟͖̥̗̈́̀͛̓ò̸͕͕̤̯̻̑̾̈́͝i̴̺̣͆ṋ̵̘͈̱̝̰̭̄̋̍͜ẗ̸̨̠́s̶̹̺͊͠ ̴̢̺͉͈̟̹͓̮͊̃<math≯̮̰̘́̐̓̀͐P̴͎̦̽̄̔͘</math>̵̯̮̓͊̊̽͌̋ͅ ̸̩̮̳̰̮̥̟͈̍̓̚ḁ̶̾n̵̨̧͎̓͊̒̇̅̒̒̽d̸̹̿ ̵̻̒̈́<math>̵̛̮͉͙̜̓̄͋͊͒̋̌Q̷̠̫̫̠̫͎̥͗̅̚,̶̭̺̳͎͙̐̇͊͌̓̕͜</math>̷̨̩̩̰̣̋ ̷͔̣̠̒̓̍̇̈́͌̾͑a̴̭̣̫̣̪̜͗̀̍̏̌̕͝͝n̵̯͈̠̥̗̘͔͒͆̓̈́̂͝ḓ̵̛̯̃̋̕̕ ̴̨͐́̂ṭ̴̭͋̆͒̀̾̈h̷̢̩̻̜̳̟̮̀̆̑̐e̸̡̱̲̣̩͐͌̚͝i̷̧̡̜̼̬̼͑͝r̷̝̙̦͇̿ ̸̨̧̹̰͍͖͎͛ç̸̠͍͙̪̬̉̃͛̚ö̴̤̺́̎̌̔̈́̍͌ͅm̷̲̳̰̭̅̀̋͒͠m̶̝̠̮̖̳̩͒̓̾̈́̃̕͝͠ơ̶̡̢̪̤̳̇̂́͒̈́̏̀n̷̟̗͙̥̗̗͕̓̍͛̆́̄̕ ̵̱̰̤̥̤͚̃̏̽̎̃̕͝t̸̗̅̽̍a̷̛̭͙͍̾́̏̊̀́̕n̴̦͈͚̱͇͆̈́͂͝g̷͕̱̠̼̤͒͛͒̔͒͝ė̶͉͍̭̱̣͊̌́̎n̸͔̚t̴͓̞͠ ̵̜̰̪̯͐̏̋͌̑̚͝ͅl̵̞̓̓̽i̵̡̦͎͚͌͒͗̕͝͠ņ̷̡͉͕͕̖̎͝e̷̻̜̥͑ ̶̛̱̫͓̰̖͍̆̾̚ć̸̯̩̖̲̞̘̻̩l̶̼̂̇̈́̀͝ȏ̸͙̉̽̅͒̿̚͝s̷̻̞͓̰̰̻͊̽̅͐̑̐͜ẻ̶̛̠͍̰͐̄̆͝r̸͚̎ ̷̜͗̄̈́̀̿̓͜t̵͉͆̈̊̑͝͝ọ̷̙̞̭̜̦̒͜ͅ ̶̛̘̣̻͙͗̏́́͝<math>̷̳͓̥̭̭̳̥̪͗̆͒̍͛̊P̸̤̝̼̲̹̻̒</math>̶̟̯͍̩͍̬̩͋͛̈́̊̕̕͘͜͝ ̸͉̳͉̟̜͒̈̿͛̕͠͝i̴͕̖̟̿̈̀̔̉n̵̢̨̺͍̫̠͕̹͌͝t̵̡͚͚͖̲̅̎̍̍e̸̳̫͉̋̐̿̉͝r̷̢̢͔͙̤͍͎̾́́̇̈́͝s̵̢̛͈̎̎̐̍͂͑̅ë̷̢̱̬̝̳͉̞́̋͛c̵̪̻̥̖̟͙͙͂͗͌̇͑̚t̷̠̗͔̯͖̲̱͒͗̽͆̏̔s̶̛͓̞̩̆͋̉̈́͗̕ ̸̛̛̬̈͐͆̉̇͝<math>̶̨͍͉̱̞͎̭̓̉̃̏͘̕̚\̴̝̯͍͔̰́̈́̍̀͠ò̷̢͉̞̝̘̺̜̀͂̋̌̀̓m̸̦̮̩͕̉́̒̇͑ë̵͎̦̳̘̤̦́̐̂̈́̒̚͜͜g̴̢͇̬̘̰̻̎̋̿̽ͅa̴͉̣͊̎̎̏̒̽_̵̡̈́̂̐̄͝1̵̨̧̗̠͔͔̐̋͌̐̔͘̕͝ͅͅ</math>̴̧̱̭̑́ ̵̠̻͑́̍̇͜ă̷͔͆͠ņ̴̬̙̺͎͆̈̈́̽̈́ͅḓ̶̢̤̙̪̹̰̪̔̿͂̀͗͗ ̵̺̬̀̅̃̋̕<math≯̘͙̪̫̮̠̰̅̄͊\̴̜̲̥̣̦͑̓̀̆́̂ő̴̭̰̟͎́̊͛m̸͕̘͇̤̋̆̈ȩ̴̡̈̓ĝ̸̡̤͖̞̠̻͕͑̚á̴͖̬̼͙̹̟͍͓̆̆͆̚_̵̨̳̲̺̫̹̠͎̈́̄2̶̯̉͐͊</math>̶̹̦̯̮͇̥̺̒̎̀̓ ̸̡̧͚̝͈̐̎͘͘ā̵̡̝̤̱t̷̛̠͔͋̈́̌̂͠ ̷̛̬̝͉͕̙͇̱͌̄̔̐ṕ̶̹͚̰̺̩̫̺́̓͌̅͆̒̕o̵̜̝̳̮̰͂͒̿́̕͜î̵͚̋̅̓̐ṅ̴͔̦̣͕͋ͅt̶̨̢̫͎͓̯̍̒͜͝s̸͔͔͓̦̽̏̔̔͠͝ ̷̙̈́<math>̶̻͚̬̰̝̽̈́̍͌̿́̌͜À̵̢̛͔͓̗̞̇̂̐̽</math>̶̭̰͙̫̜̄̈̂̌̀̈̂ ̸̤̱̱̔̀͘a̶̡̼̣͕̫͂͑̇̾͠n̵̗͙̻̄̅͂̈̀d̴̪̗̹͋̀͝ ̶̢͙̠̔̈́̅͂̎̅͘<math>̵̩̏́͒̑̈́̚͠B̷͉̙̩͓̰̟̻̦̅̈́͒̇̈̈́͒͠,̵̝̦̫̯̳̅̉͘͜</math>̵̣̮̝͍̫͖͐̏͛̽͌ ̶̰͖̜̤̘̈͂̈́̾ͅr̴͈̤̓͆̂̊͝ḗ̵̛̮̺̦͈̹̄s̴̳͊̀̐͘ṗ̶̛͍̻̻͇̻̯͕̅ḙ̵̛̛͂̈́̃̌̓̇č̸̨̀͘ẗ̸̗̳̦̝i̷̧̨̱̮̪̳͋̓̇͌̀̈́͘͜͝v̷̜͇̓̑e̵̡̢̟̲̠̥̦̋̐̏͗͊͜l̸̮͇̗̺̰̔͊̎̃̇͝y̴̞͍̰̘̍̓̋́̾͝.̶̧͖̞̠͖̞̈́̌͛̅̈́ ̸̡̗̲͕̽̏͋̆́͠T̸͉̘͔͙̳̠͌͛ḩ̷̨̛͙̳͔̺͊͐͒͜ͅe̵̫͕̯͔̍͋͑͠ ̸̣͖̙̍̈́̀̾̿̏͗l̵̩̫̮̺̜̏͛̾̈́̇̕͘i̷̢̡̱̳̤̤͂̃͒ǹ̷̻̻̥͈͚̅̂̾̾̈̕ȩ̷̩̰͈̰̥̘͌̉̃̾̌ ̶̭̬̿́̀̋̑̄p̷̠̎̍a̴̘͓̺̮̹̔̇͠͝r̶̠̭͖̜̞̠̠̃̀̇̽ͅa̷͙̖͕͚͂͑̐͊l̴̺̒l̷̨̳̬̗͍̭̣̊̂̅͗͌ę̷̘̫̙̤̒̈l̵̢̟͍̦̼̘̓ ̴̢̟̜͔̞͚͑̊̃̒̀͘͘͠t̵͇̱͓̹͈̤̑̏͠o̵̢̺̼̙͓̾̈́͠ ̴̨͔͕̣̹̒<math>̶̧̛͓͐̈́͗̆́̀A̶̙͔̭͗͌̽̃͂̊͠ͅB̴̡̭͚͚̑̎͊͌̑́̐͒</math>̶̧͍̘͉͙̞̙̓̿͋́̒̍̓ ̸͚͍̗͔͓̊̇͜t̵̨̡̰͍̝̲̰́̈̇͐̇́̓͠ḥ̷͖̝͙̳͔͙̃͆á̶̜̦͇̦̖͉̫͔t̷̟̻̲̭̱̹̹̐̉̈̃̈́̋ ̸͇̭̱̲̠̹́͂͝ṕ̸̨͕͉̼̳͔̈́̔̄̀̐̿̿a̴̡̛̙͊͌̈́̿͆̀ͅs̸̛͓̘̜̯͎̏̌̇͗͐̈̍š̴͎̹̣̺͍̞̹e̵̡͉̝̖̺̘̭̥͂͊̆s̴̱͓̠̱̞͇̤̃̑́̀͝ ̵͙̥̻̺̙̖̰̠͗̒̍̈́̿͒͐t̵̗̱̮͎͛̈̀̃͆̋̚͝h̵̨̛̙̻̗͖̙̮̅́̋̽̕͘͠r̶̭͇̈́̾ơ̵͉̋̌ǘ̷̡͍̗̊̍̀̈́̓̚g̶̜͍͚͔͆ḩ̵̖̠̗̦̃ ̶̠̰͚̼̗̥̪̺̎͒̒̍͛<math>̶͎̊P̴̼̰̟̣̓̆́̆̕ͅ</math>̶̟̥̦̼̐̀ ̷̢̺̤͇͖̬̱̽̓̅i̵̢͈͌͆͌͂̆̊́n̴͍̹͙͔͉̖͚̍̎̉͜t̴͔͑̑̈́e̷̢͈͚̹̱̼͋͂͋r̶̨̛̓̄́̎̋̓š̷̭̻̳̾̌͌̒̊̐e̸̢͎͚͔̾͂͝c̷̛̯̖̜̬̓̋̽t̸̬̭̠̜̖̃̆̑̄̓s̴̜͍͔̬̅ ̵̥̠̑͛̅́<math>̶͈̮͂͑̊́\̸͍̱̟̲̈́̆͛̀̓̒͝ő̶̩͖̮̰̙̼̻́̀͂̏͐ḿ̴͉̃̇e̵͎̞̐̓̋̏ġ̴̦͚̉̀̇͂̿͂a̶̝̠̠̬̫͙̐̉̚ͅ_̴̧͖͍͙̟͐̔̈́̃͜1̶̛̝̰͋́͐͌̍͠</math>̵̡̟̼͕͇̈́͂̍͗̐ ̸͎͛͂ą̴̜̞͆͂̏̾̒n̷̜̜͕̦̹̉̈́̒̈́̍̏̋ͅd̷̦̖͕̰̝̠̐̀̀ ̵̧͕̩͒̃̔̅͘<math>̶̡̺̔͜͝\̶̢̮̩̳͊̍o̷̠̻̳̦̰̾̇m̴̧̲̻̘̱̻̞̝͝ĕ̶̬͈͇͇͎̳̂̏̀̒̀͐͠g̴͉̖̳͓̰̝̥̋͠ͅả̷̧̩͖̮̜̻͘͝_̵̫̮̥͙̯̎̀2̴̤͈̅̌̀̌͝</math>̴̛̝̼̰̺̦͍̞̱̍̉͝ ̷͉̗͎̈́̌̒͒̉́̆͊f̴̠̝̥̬̿͋͌́̅͑͆ȍ̵͔̳̭̞̼̉͂̃̆̐̾̈́ř̵̢̢̯̮̯̗͚̞́̉̎̓̎̐ ̵̨̧͔͔̝̅̀͂̍̊̂͘t̶̛̤̃͐̇̿̓ḣ̷̡̰̟̋̈̀̄́͘e̸̛̞̺͝ ̶̮̟͓̫̉́ş̵͙͙͙͖̆́͐͝e̷̗̻͈͂̈́͆̈́͗̂̍͠ć̵̲̫̠̩̲̹̤͙͒̉o̴̢̢̞̫̹͍̤̥̽̊̏͠͠ṇ̶̙̱͍́́͌̐̽͒͝͝d̸̙̺͎̫͊̄̋͘͝͝ ̸̘̤̃̿̈́̓̆̏͆t̷̡̨͙͕̱͖̥͒͆͠i̶͇̬̮̒͜m̸̠̫͎͂͗̂ȩ̸͍̖͌̉͘͜ ̷̧̠̙̗͚̫̭̰͋͑̿͗̾͐͝à̵̰̘̪t̵͉̖̣̫̲̋͠ ̵͍͋̉p̴̖̝̂̾̔̍͒̅͠͝o̵̢̧͇̟̣͙̱̓͝͝i̸̧̭̪̼͎͕̒͂̉͊̾̑́̀n̴̡̦͉̹̫̘̈́͌t̶̰̟̜͒̈́̆̒̊̿̌ͅͅş̸̠̖̒͠ ̷̪̦̦͖̺̿͛͘<math>̵̨̢͉̘̜̜̝̻̑̆̽͑̌̒̚X̵̗̫̱͍͔͎̯̾͠</math>̴̨́͌̄͌ ̷̥̙̭̈̅a̷̪̼̬̥̣͒̊̀͐n̷̡̼͚̻͉̅ͅd̸͇̩̹̫̹̆̊̈́̉̑ ̶̢̻̬̯͑͑ͅ<math>̵̭͎͚̜̋Y̶̢̢̡̩̘͉̳̔̿̿̆̓͠͠,̷͈͔̖̔̇̐͊͝</math>̴̨̩̱̃̅ ̸̨͓̪̻̬̅̍͊͗ŕ̷̨̺̱̥̼̱̤̜͆̓̔̓͗̽e̴̛̞̽̒͋̋̈͠s̸̻͚̫̩̠̓̒͂͊̓̄͐̏͜p̶̮͎̂̂͠e̶̝̰͔̍́̉c̶͍͆͑͛̏̃̆t̶̛̺̄͛i̴̢̢͔̳̥͉̻͑ͅv̴͓̣͇̋̑̔̒͆̕e̶̛̟̤͇̱̹̣̥͂̐͛͆͠ͅl̷̡͙͓͍̻̃̋̄̊̀̐͠y̵͎͔̪̣̘͙̜̋ͅ.̵͖̈́̾͐̑̌͐̓ ̷̣̦̇̎̽̎̏̚S̶̢̹̥̻̬̝͉̰̔͛͝ṵ̵̧̨͔̣̥͛̾͂̚p̶̢̟͔͎̼̐͛͆̕ͅp̸̡̛̝͉̂̀̅̎̊o̴̜̼͎͌͋͠s̶̡̱̭̣͗̒͋͊̀̉͝e̷̡͍̎̓̇͗̌͑͂͜ ̶̡͕̱͔̈̌̓̑͊̓̾<math>̷̬̲̜̆́̎̒̎P̷̛̳̥̠̤̺̳̭͌X̷̨̡̢̬̗͇̩̀̑=̷̻̦̞̟̯͑̍͗̑̈1̶̨͖̯̣̱̉̈́͒̈́͛̽͘̕͜0̵̳̤͕͖͖̜̎̎,̸̢̧͍̰͔͓̅͊̂̄͝</math>̵͕̰̿̈́̍̆̔̎͝ ̸̭͖̙̜̯͚͂̈́<math>̴̯̝̬̙͔̓͒̉̈́̽̐̿P̴̱̪̹̲͈͆̀̈́̅̒͝ͅY̶̞̠̰͌͒=̶̟̖͇̙͔́̑̚̕1̶͖̘͎͖̜͊̐̃͐4̵̱̔͋̆̀̌̐̚͘,̸̡̢̧̨̞͍̽̑</math≯̞̬̗̑̾͗̋̒ ̷̞̗̯͖̦͔̓ā̸̙̜͙̲̯̿n̶̢̗͚̪͙̱͛d̴̞̮̮̿̑̈́̋́̃̆͠ ̶̫̼̓̎̒̒͑̉̋̋<math>̶̼̝͎̯̪̹̈́̈́̊̄͝P̵͈̙̹̯̞̺̑̎ͅQ̸̨̗͖͗̋͛̓͠≠̫͌̈́̔5̴͔͎̣̌̌̅͝͝.̵̨͍̭̟̫́̾͆͠͝͝</math>̷̡͖̰̘̼̆̒͌͠ ̴̯̠͎̓͘͜͝T̸̙̑̚ͅh̶͒̎̚͜͠ê̴̺͕̻͕̘̏̉͑̆ņ̵̖̺̟͍͉̣̳̄̎̇́̔̚ ̶̳͋̓̽̿͑͠͝t̵̨̫͔͙͉̝̄͊͊̿̋͌̐͜͠ḫ̸̡̛̦̥̘̂͠è̸̤̫͚̫͍̘̻̭ ̵͙̬̚a̴̧͍̻̝̰͉̔̒̎r̴̡̼̙̥̩̗͗͊͋̽͂ë̸̛͍̻̤́a̵̻̩͍̣̻̞̙̽́̋̀͒͒̍͜ ̵͔̣̠́͂̈ỏ̶̩̅f̸̼̼̪̉͊͑͗̎͘ ̵̖̌̇̃̏̈̾͝͝t̵̩̳̘͋̍̈́̌͝r̵̡̧̞̘͉̻͚̽̀̂̅̈́͐a̴͍͉̜͈̱̝̮̗͂̓̒͊̕p̴̢̭̲̭̈e̶̡̗̜͈͔̖̿̈͌̆͝͝z̷̘̮̣͛͂͊́̓͛̔̄ö̷̘̫̺̬̖́̏ͅi̵̳̘͓̙̬͖̳̹̅̌d̸͇̖̲̪͈̩͕̑̈̓̐̈̽͑͝ ̶̪͑͑̔̋͠͝<math>̶̠̣̻̭͓̱̜͔̀̾̃͑͌͝Ẋ̵̼̰̈́̑̅̿̚̚Ā̸̱̪̭̺͙͉̼̀B̷̦̥̱́́̿́̃̓̚Y̷̰̫̚</math>̵̯̋̏̌̎̔̏ ̷̥̿̅̊̎͑͌̕͝ͅì̸̤̲̘̪̾͝͠s̵͇̀͐͋͆͊ ̴͇͙̘̝̥͑͐̔͐̏͋͊͘<math>̵͈̭̯͇̣̖̱̺͗̇̀̑m̴͔̭̫̣̜̝̅̿̚\̸̡̮̥͓̮̍̒̀̇̃͒s̶̥̩̣͇̰̹͖̄̈́̀͒̀̀̈̐q̵̻̐̽̊͆ͅr̸̢͉̣̼̈́͐̒̈́͘t̸͎̳̫̒̄̀͂͆͂{̷̞̹͉̎ñ̸̨̛̩̱̀̊̆̐̏͠ͅ}̸̙͐͋̊̀͝,̴̯͎̏̂</math>̷̰͔̼̟̫̜̾̃͛̓ ̵̝͍͉̆͂͘ẅ̷̤̰̮̺́̾͂ͅh̷̯͓̗̩͒̅̅̑̾̾͝e̶̢̫̩͕̻̪̭͒́̐̍̾͋ͅr̸͓͘ĕ̴̢̞͈̜ ̴̧̞̠̼̪̻̰̝̈́̓<math>̵͖͖̪̝͗̈̓͑̾m̶̢̼̺̘̱͐͒͊̿̓͒͝</math>̶̫̦͕̳̯͕͔͊͂ ̴̻̘͕̹̟̓̅̕͝a̸̩̖̣̘̓̔̓̈́̇̋͋̊n̷̛͉̳͎͚̆͒̀̒̉͐͆ͅd̸͙̜͌̾̈́̿̌͆ ̵̗̱͇͆͋̇͗̿̽̾̀<math>̵̛̩̌̎n̶̳̬̏̍̑̓͝</math>̴̧̠̼̱̹͓̠̀̑͂̃̄͂͜͝ ̷͉̖̝̺̖͕͋͊̓̄̓̈͘͝a̵̦̣̣̻͎͂̃r̴̢̢̮̹̥̼͖̗͂ȩ̷̨̢͉͔̺̼̓͛̄͛̍̐̌̕ͅ ̷̢̨̨͎͔̞̩̐̈́̒̀́͐͜p̸̧͓̱̥̠̂͂̋o̵͕̺͖̲͕͆̉ͅs̶̢̧̨͇̖̜̜̓̈́̔i̵̱̳̠͖͐̌͐̊t̵̢̯͑i̸̛͙̊͛́v̶̡͖̰͗̀̒̆̚e̸̡̮̤̝͚̋͜ ̵̞͍͉̊͑́͌̅͆̎̀ͅí̷̧̀́̆͘͜n̴̨̛͓̋͛̀̈̓̕̚͜t̷̡̧̞̙͚̻̰͆̽͜ĕ̸̬͒ĝ̸̛̲͚͕̈́̔̈́͛͘̚e̷͙̣͓̮͆̔̉̃̑ṙ̶͔͍̘͓̲͎͎ş̷̙̝̲͕̤̋̿̇̍̑̔̕͝ ̸̦̲̮̽̇͆̎̄a̸͈͉͉̿̄́͂̚͝n̶̠̓͛́͋̋̕d̵͖̓́ ̴̨̛̞͙̝̌̓̃̽̒̔͝ͅ<math>̶̢͒ͅn̸̬̘̤̙̼̈̏͜ͅ</math>̵̛͈̹̟̌͋̐̔ ̵̥̈́͌͌̀̋̌ì̶̮̖͕͎̘̮̼̐̂̆ͅs̷͔̣̞̐̃̏̿͂̎̽̂ ̵̧̗̼̳͚̻͚̃̎̄̂̉̍̇͜ņ̶͈̊̑ͅͅö̴̺̠̙̼̜́͗̒̚t̴͙̓ͅ ̴̡̻̐̅͂͒̇̀͆͋͜d̷̬͙̜̲͚̘̪͂̓̑͘͝ͅi̵̡͙̽͆͒͗v̷̢̠̅́̍̈̃̉̕͜i̶̢̜̽s̸̠͍̟̞̠̓̑̎̿̏̂̈́͠į̶͓̫̺̺̺̳̈́ͅb̵͍̙͗̇͋̃̄͠͝͝l̴̢͔̜̈́́̆̀̓̂ẹ̵̣̊̄ ̶͙͕͈̎͊̐́b̷̡̦̬̫̗͎̞̫̑̐͌̕ỵ̷́̈́̓͊͗̕ ̸͇̺̟̜͈̻̤̑t̵̨̢͇̺̼͍̰̬͊͂̂͌̆̕͝ḥ̸͔͕̪̭̥̇͌͌e̴̯̍̽̌̉͋̈́̍͝ ̷̛̺̻̜͔͓̺̰̦͒̈̀̀́ś̷͕̹͙͓̪́͒̓̆̕q̵̡̛̂̓̉̏́u̵̧̨͈̰̓a̶̧̱̬͖͓̞͐̊̏̈́͘͝r̷̖̠̣̎̋̉̌̌e̶̝̰̦̦̻̳̼̾ ̶̨̮̳̯̪̥̲̂ŏ̷͔̬̀͑͜f̶̻͈̠͓̘̰̄͗̑̌̋͝͠ ̸͍͙̪̰̟̞͒̋̈͊͊̐å̵͔̺ǹ̷̢͓́͋̀̚͝ŷ̸̟̫̹̳͚̤̀͂͐̚͠ ̵͉̘̰̋̅̀̎͐̚p̴̢̼̯̱͍̓͆̾̏̉͊͗r̸̠͋i̴̡̛̺̣͒ḿ̷̢̠̝̤̦̓͒͂͘e̷̯̠͍͈̟͎͔̹̔̇̈́.̴͚̮͖̕ ̶͖̬͉͚̓̏͌̓͗́̕̕F̵͚̄̐̌̓̾i̷̲̣̇́͐̏͝n̶̳̈́̑̈́̈̔̽ḏ̷̅͋͂͊̉̕͝͝ ̸̡͎͙̲̤̏͛́͗̋͋͒<math>̷̦̬͍̜̮̙̔͐̀̉m̵̧̡̱͍̜̦̂̆͗̂̀͆͝+̵̡͈͈̥̍͋̈́̓̕n̵̖̱̪̬̑.̶̨̅́͑̏</math≯̛̞̈́̾̋́̍̕
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Circles <math>\omega_1</math> and <math>\omega_2</math> intersect at two points <math>P</math> and <math>Q,</math> and their common tangent line closer to <math>P</math> intersects <math>\omega_1</math> and <math>\omega_2</math> at points <math>A</math> and <math>B,</math> respectively. The line parallel to <math>AB</math> that passes through <math>P</math> intersects <math>\omega_1</math> and <math>\omega_2</math> for the second time at points <math>X</math> and <math>Y,</math> respectively. Suppose <math>PX=10,</math> <math>PY=14,</math> and <math>PQ=5.</math> Then the area of trapezoid <math>XABY</math> is <math>m\sqrt{n},</math> where <math>m</math> and <math>n</math> are positive integers and <math>n</math> is not divisible by the square of any prime. Find <math>m+n.</math>
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̴̧̬̘͗͛̑́̎̆͌
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̵̪̍́͠[̶͖̺͔͈̩̣̭̦̒́[̸̜̠̜͇͕̼̥̞̈́̕2̸̧̛̼̲͖͓͈͘0̷̢̙̙̮̽́̆̔̊̓ͅ2̵̣̭̌̈́̑͠3̸̪̠̹͈̥͛̃̄̎̉̀ ̶̛̜̮̌͜A̵̡͔̲͙̒͑̈́̀́͛̾̀Ị̶͉͕͇̭̕M̵̝̹̭̉̎̃̾̕͜͝E̴̥̘̪͆̈́̓͒̅̃͝ ̷̝͓͌͆̎̆̌͝͝Ĭ̴͓̫͔̞͝Ḯ̶̠̭͈͉̪̩̒̃̌͋ ̴̹̻̑͌̾̾̅̕͝͠P̶̢̧͕̰̹̫̺͓͆̅̍̎̐̚r̷̛̪̀̈͘͠o̸̢͒̃̉b̸̡̢̢̙̝̼͕̎͊ḷ̵̎̎͝e̸̼̼̫͐̈̋m̸̥͆̏͑͘s̷̨̥͇͒́͑͘͜/̴̺̯͚͛͂̈́̿͘̚P̴̡̼̜̯̝̱͆r̶̖̼̟̱̱̀͆͐̈́͆̿̕͝ͅô̵̟̦̻̳̮̱̦b̶̹̤̞̻̲͐͂̀̎̑̀̔l̶͕̗̩͍̥̤͋́͋͘ĕ̸̥͕͎͖͒̒͌̄̕m̷͎̜͉̈́̏̿̈́͛̚͘͝ͅͅ ̸̢̮̭̲͓͕̩̑̈͂͌̀͌͑͜͝9̶͚̥̳̠̻̙̋͆̀̔̋͠|̷̡̪̺̞̬̉̊̈́̓̂͝S̶̬̏ͅo̵̞̼͗̋̅̃͂͘l̴̟̣͓̰̹̎̈̄̏̽͌̕͜͝ű̴̱̈̓͒̈́̋͑̓ț̶̢̮͕̀͌͜͜i̶̠̟̗͓̣̥̾̾̓̏̄̓̐́o̵͚̚n̵̢̢͕̥̤̒͠]̵̛͔̑̐͋̑͘]̴̭̜̗̟͖̎̀̾̓
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[[2023 AIME II Problems/Problem 9|Solution]]
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̸̛͎̍͂̉̒͑
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̷̼̼͑̀͌͛ͅ=̵̻̫̲͕̙̥͈͕̌̊̾̑͐=̴̣̻̦͕̑̆̃̈́P̴̙͗̄͛̀̈͜r̶͇̬̱̭̝͔͇͎̍̿̎͘ǫ̵̛̦́̑̈̄͠͝b̸͖̜̥̩̱̙̗̈͐̂̏̌̌͌ͅl̶̡̩̲̥̩̫͌̌̀̌̅͊̀e̵̖̮͚͔̱͖̊̈̿̾́̄͜m̷͙̗͈͗̿̉͌͂͝ ̵̺̖͔̺̩̝͙̼͗̂̈́͑1̷̘͓͙͚̭̀͌̔̂̍̽͌0̴̪̗̟͙̐͌͒͑̀͜͝≠̤͔͓̺̾͛=̵͔̤̣͈͖͉̯̤̿͛̍̈́̆̚
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==Problem 10==
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̴͎̹͍̩̰̃̉̀̀͘
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̷̝͖̲̦̦̦̦͇̄͘L̴̜͙̬͍͇͒͌ͅͅe̶̦̠̜̅̐͘t̸̳͍̱̠̺̙̫̋̄̂͜ ̴͉̩̻̯̥̏̿̋͋̅͂̾̾͜<math≯̹͉̎̈́N̴̹̪͓̖͋̀̎</math>̶̱̠͇̹̖̖̠͎̆́͋̈́ ̷̥̠̘͆͊b̷͔̭̎̍͐̈͝e̶̺͖̟͚͈͉̾ ̴̧͚̘̳̐͋͝t̴͚̳͈̪̳̠̤̺̎͌̑̈h̴̩͙̺̺̻̼̾̄̎̍̚͜e̵͙͓̮̣͉͖̬͋̏͐͂̎̓͜͠ ̴̹͎̥̊̈́̂̒̈́̃̎̓n̷̤͍͓͋̌̎̐̑̿͘͠u̷̧̝͎̅͒͂̔̃̾̃͝m̶͈̥͉̗̪̭͚͉̓̋̉̈̒b̷̢̛͇̜͇̰̀̔̀͐͘̚͜͜e̸̯͉̘̮̘͕͉̣̾̒̍͊̋̕r̴̝̪̰̼͗̏̽̇̎͜ ̷͔̦̬͂͐ō̸͔͉̝̹͑͛̌̽f̵͕̉ ̷̙̈́̄̾̓͗̿͒ẘ̷̠̟͚̚ä̶̱͈̼͚̹̺̭̉̾y̸̨̤͉͌̏̎s̶̖̼̼̪͇͇̻̮̀̆ ̸̘̭̮͎͓͕̝͈̈́͛̐̈̕͘͝ẗ̷̗̘̰̫́̌͝ȍ̶͓̠͠ ̸̛̘̬̜͌̈́͐͝p̶̡̘̠͍̩͒̒̏͜ͅl̶̪̖̦̯͖̪̦̀ả̵̼̱̀́͒̋̐͆c̶̡͖̻̎̒͆̅͘ę̴̠͉̬̲̥͚̻̀̀̃͗ ̷̬̼̺̈́͜ţ̸̧̙͈͖̺͑̊̀̑̊̆̕ͅh̷̙̥̭͂̇͂e̸̻̭͋̽ ̷͈̳̞͍̞́͐̿̈́͜͝ì̵̪͚̼̻̹̰̀̃ͅn̵̯͙̻͂͒̕̕t̷̙̯̗͍̪̎e̵̙̝̺͜͝ǵ̵̢̳̻͕̜̪͛͠ë̸̲͓͔͇̭̘́̊̀̃̓̕̕͝r̵͚͕͉͉͚̜͋̿͂̔͆s̴̢̛̮̝̫̤̙̤̻̑̆ ̸̢̼̪̑̿̀̉͌<math>̷͚̟͍̟͙͋̓͝1̷̢̭͓̣̼̟̞̲̏̎̎̊̔̚</math>̵̛̖̟̤̗̐͗̊͘̕ ̸̹̱͙̘͗̏͒̐͋̇͋̓t̸̢̙͇̉̂͆̚ĥ̸̘̺̽̈́̍̍̚͝r̷͎̎͆̀̈̈́ō̵̠̖̫̖̮̭͙̞̇̌̆ǘ̴̪͓̌̔͝ģ̶̱̪͔͕̜͉̽͒͗̂͘͝ḩ̵̺̙̗̠̹̞̓͒͗̍͝ ̸̝́͗̿̐͂̀<math≯̜͙̪̀̈̉1̶̦͖̙̞͕̜̈́̿̈́̽̂̋2̸͚̪̖̱̪̱̫̓̍̂̓</math>̷̢̱̲̙̘͉͖̈́̅̎̀͑̚͠ ̷̘͎͍̤̖̺͎̬̂̌̎̂i̵͇͋͝͠n̶̡̢̝̯̣͚̹̠̿̉͐̓̒ ̵̡̡͎͚̬͓̟̎̑̏͜t̷̥͎̰̮̕͜h̶̼̀̈́́̚ę̵̛͙̖̖͇̟̥̅̔̐͆ ̶̢̥̦̻̺̯̈͗͋͑͌̈́̐͗͜<math>̴̧̪͓̲̗̟͂̔͌͊͆̓̈̄1̴̼̜̳̯͇̙͉̓̌̏̍̚2̴̦͓̺̒̔̾̍͐̚͘</math>̵̬͚̭̒̋̌̉́͜͝͝ ̵̱̰̤͉͖͆͂̅̀̽̈́͘ç̶̰͈̩̥͍̎͋̓e̸̢͚̠̞͈͍̣͛̂̄͛̽̀̚ͅḻ̸̱̯̰̾̿͐̾̋̐̈́l̴̨̩̬͈̓̇͘ş̴̨̟͉̮̖̻̫̀̽̿͂̊̚̚͝ ̷͓̃̑͑́̚ô̵͈̲̝͍͇̄͌ͅf̷͙͉͗̈́̅͛ ̵̧͚͙͔̈́͛ͅa̷̘̲̮̓̔̂̿͜͝ ̴͚̣͍̘̖̻͋ͅ<math>̶̨̮̖̪̦̌͑̈́̊͘̚ͅ2̴͉͎͓͎̦̅̅̕̚͝ͅ ̸̘̫̳̳͇̹̖̳̎̆͠\̶̣̜̖̣̯͚͒͜t̸̩͈͕̉̄̏̓ỉ̷͈̝̤͇͔͎̀̈́̅̽̆͜͠ḿ̶̨̗̳̬͎͇̬͍ę̶̦̏ś̸̢̢̭̱̈́̄͐͐͘͝ ̸̦͆̆̓͐̄͘6̷̡̩̳̠̪̳͋̔̓̾̋̀̀</math>̴̨̝͙̂̅ ̵̞̥̱͐́̽̾̌g̶̪͂̀͋͑͠r̶̳͍̾̿̒͛i̶̦̭̽͛̏̋̄̌͠͠d̷͓̥̓̇̕͘ ̸͉̘̤̯͕̯̊̓̏̉̈́͛́ś̵͎͔̟̱̳͔͔̟̏̀́͘ö̴̙́̍͋̎͋̐̀ ̷̤̱̾̉̾̏́́t̵̤͉͍̣̑͋̄h̵͈̜͊̐̀̀̚ă̶͖̘̺̫̠̗̰͇t̵̩͙͚̯̅̇̒͗̓̌ ̴̳͉͉̪̈́̊̐͋̂f̷̫̲͈̈̾̃̇̈͂͜ò̸̗̼̮̠̭̰̱̯̋r̵̮̠͖͗ ̴̧͓̟͐̇̈͜á̶͎͖̎͗̓̽́̍ñ̷͜ỵ̴̧̣͕̭̼̙̈ ̶̤̫̤͂̌́̑̆t̶̟̼̟̰͍̳̲̀̑̎͛̽͊͜͝w̸̹̳̎̐͋̉͝ơ̸̪͍̞͙͉̽͑̽̚ ̸̪̖̱̦̪̟̃͐͝c̴̛̹̀̚ẻ̵̩̗͌̓͌̓̈́͝l̴̞̙̩̬̝͖̠͊l̶̢̺̎͗̊̑̄̈́̆̕s̵̟͚͐͗͗̐͒͑͊͝ ̵͍͈̪̠̀̀̕s̶̡̛̗͇̟͕̆̇̇̂̿̉h̸̛̥͍̟͚͐͗̚͝͠a̶̼̙͊͑̍̍͂͋͝ȑ̸̛̭̲̯̮̔̃͋͑͜͠i̴͖͈͍̥͋̅ǹ̴̲̺̭̣̀g̴̛̻̤̗̖͇̻̔͊̏̍̔͝ ̴̝̜̗̍́̈́́̈́̎͠ͅą̵̳̲̋̌͆͆͘ ̷̖̜̜̝̮͎̭̓͑̽̽̃͘s̸̢͍̠͓͉̋̐̒̌͘ḯ̷̺̯̜͍̼̔͐͌̈́̉̕d̸̢̧̳̥̙̠͂̍͜͠ͅē̸̡̗͕,̴̱̖̲̾̒ ̸̹͙͓͖̈́͑̅̋̔̎t̴̩͂̿͌͆̕h̷̟̹͓̞͇͙̯̏͐̐̆ȩ̷̹̮͖̫̮̪̓̋̄̑̚͝ͅ ̵̢͔̦͒͌̿͛͂d̶̝̯̥̩̤̠̪͖̾͊̋̂̕i̶̼̾̍̂̓͝f̴̛̻͈͋̂͌̿̀̋f̸̨͚͖̙̪̓̂̎̈́ě̶̠̬̣̥̰͈͆̑ͅȓ̴̡̤̐e̴̮̯̅̓͠n̸̨̝̺̆̊̇̇̓͋c̴͈̦͇͓͙͓̔̉̋̎̋̏͝͠ḛ̵̼̘̺̮̹̎͜ͅ ̷̦͓̖̼̈b̴̬͍̱̰̘̟̔̍ë̷̢̙́͂̈́̿t̵̪̠̦̉̀̆̅̓̀̅̄w̴̭͕̑͐̉̎̈̆̾͝e̸̯͂͝e̵̠͇͈͒̀̈́̐n̶͇̂̅͐̐̋͝ ̷̧̙̝̫͉͚̰̤̾͒̉͛̄̓̃̓t̴̩͔̙̮̦̞̣̄͌̆̃̉͌̆ͅh̴̦̻͛͋͛̋͜͠͠ė̵̤͘ͅ ̴͍̝̊͆͆n̶̡͇̟̺̺̝̩̾̅̉̆̈́̕ų̴̫̠̟̙̗͆̃̓̔̀̕͝ͅm̸̛̯̘̩̱̲͑̽̑͋̐̐̽͜b̵̨̢̨̥͖̭͊́̆̾͗͘͠ẽ̴̘̻̭̹̜̟͔̆̅̿ͅr̶͔͉̙̹̠̰͚̆̓̑̄̾ṡ̴͙͆̌̽̾̈́ ̵̨̘̮̓̏̈͛͐͑ì̴̞̱̲̒̈́͘n̷͔͕͌̎͘ ̵̛͎̰̯̻̭͂͋̀͆ţ̸͉͉̽͋̓̈̃̓͜͠͠h̸̡͖̩̗̥̱̃͝o̶̠͓͙̲͖̲̓̄͘s̴̛̰̼͓̯̈͂͛̉͝ě̷͉̭̝̭̱̖̍̈́͌̈́̂̚ ̷͓̟̻͂͠c̷͕̺̗̚ē̴͓͈l̵̡̦͙͈̜̏̅̌͑ĺ̸̢͍̬̋̈̄̈́̆s̵̥̗̖̭̤̣̅̂͑͒̍͝͠ͅ ̵͈́i̴̙̹̔̀̂͋s̷̡̺͑ ̷̦̝̥͗͘n̵͉̕͝o̷͎̍̓ẗ̴̨̛͎̣̼͇́͌̎͋̆̇ͅ ̸̖̩͓͎̠̪̗̀́̄̈́͊̐͜͝d̸̝̑̑̇͌̈͌͜ĭ̷͖͖̪̭͍͍͎̈͌̈́͝v̷̞̗̫͕͝i̷̪̖̿̔̊͠s̶͖̠̲͉̱̓i̸̛̦͕̅̾́̿̽͐͠b̸̬̰̤͕̼̏̍̚̕l̸̡̩͈̐̑̈́͋e̸̛͓̱̙̮̒̃̓̕̚ͅͅ ̶̨̡͙̙͚̣̤͒̏̒̍͋͒́̉b̸̯̤̝̐̋̅̕y̶̛̼̮͎͈̙̺̤͋̎͂̈́̾̕ ̶̯̥̻̀<math>̴̧̮̭̭̻̱́̐̈́̈́3̷͓̖̗̓.̶̙̤̹̓̀̄͗̀̃͠</math>̴̟̗́̈̏̆͝ ̶̬͚͕͋̊̒Ö̸̭͚́̎̍̒̑̚n̶͚͈͙̝̭̤̮͌͛́̌̏̄́e̷̳̪̜̽̓̉̈̚ͅ ̴̦͚͖̦̉w̷̡̨͔͉̦͇͋á̸̢̛̬͇̗͖̺̑̔y̷̛͍̺̱̩͐͛̇̅̐̾͘ ̸̫̓̈́̓̓̀t̵͈͔̐̍̑̓̌̏̚o̵̬̲̤͉̹̲̻̠͂̈̾̈̕͘ ̶̦̭͎̬̍̏̇̊̇͛͝d̵̡̼̺̠͖̈̍͋̊̀̈́͗ͅơ̶͇̻̩͈̝ ̵͇̪̫͙͕̩̆̋̎͒͗t̸̛̮͙̜̼͈̗̂͐̓ḩ̷̨̱̜͚̝̉̅̈̑̅̍̾̒i̷̛̞̾̀̉́̅̚s̸̪͕͈̀͗͐̄͒̔ ̴̥̲̤͆̈̏̅̿͜î̷̛͚̲̲̞͕̜͕͖́͊̾̍̆͠s̶̩̘̣̞̳̪̪͙̀̃̔̆͐͐ ̸̜̋̒̐̂̈́̈͌̌s̷̬͋̓̃h̵̛͚͗̇̃͝ǒ̴̡̭͚̟̭̜̅̅͛͆̔͊̀w̵̢̖̫̭̰̩͍̒̓̔̐͘n̷̨͈̙͙͙̬̳̳̊͛͊ ̶̫͔̋b̷̨͉̹̲͓̃e̷̡͉̠̳̱̜̕ͅl̸̳̇͋͐̐͂͝͠o̵̗̖̣̼͇͈̩̔w̴̛̫̩̣̍̌̉̏̆͑.̵͈̮̠̥͇͙̏̓̽̾͛͘͜ ̷̫̖͛̒̿̏̏̚͘F̸̖͎̟̬̯̩͍̒͗i̴̲̞͎̲̞͆̎ṇ̴̛͍̱̟̩̂̓̀́͊͝ͅd̴͕̎̉͋̊́͂ ̸̗̰̮͆̂̂̔͆̔͑̚ţ̶̣̭̘̮̀͐͊͂̈̀͘͜h̵͖̹̳̬̤̬͈̊̿͐ę̵̫͇̗͓͐̇̍͝͝ ̸̧̢͍̟̽̀͗ņ̶̨̢̲̜̖̓̔ư̴̤̂̾͒̅̈́m̴̢̮̦͉̰̪͐̐͗̆̌͋͘͜͠b̵͚̈̅͗̇̍͜ę̴̢̪͇̬̝̱̞̔͋̎̀͒̾̐r̵̻̭͈̦̮̾̈́͒͛͐̚ ̶̡̧̬̦͕̱̻̗͗o̵͇̘͎̖̬̠͗̂̔̑̌f̵̛̩͈̀͒̿̕̚͝ ̵͚̘̩͂͂̌̈̏͂̅̚p̷̹̖̫̣͎̳͕͋͂͐͋̚ö̴̗́̈́͒̒̀ṣ̸̰̲̳̙̠̏̐͑͗̌̕͝i̵̭̫̬̩̥̍͛͋͛̓͋̏̇t̴̜̲͌͐̄̌̕i̴̘̓̀v̷̢̹́͗̉e̸̝͍̳̝̥͌͆͑̎̕͝͝͝ ̴̦̼͖̯̞̩̪̱̈́̀̾͆͛̒̋͋ḯ̸̡͇̭̥̻͖̌̂̒̉̎̀͝n̶͉̈́̈̎t̷̺̎̔̽͜͝e̴͎̦͓͇̲̩̲͂̌̽͝g̸̤͆̊̈́ę̸͔͎͓̒̅͌̒̽͛̈́͝r̷͕̬̣̓̃ ̷̺̣̪̹͋̿̊̈́̕d̴̤̅͆͐͂̑͆̕͠i̷̛͙͎͈͕̐́̽͐̈́̿͑͜v̶̠̈́ĭ̸̺̦̬̩̣̹͖̀͘ş̵͓̲̩̦̱̩̫͐͒͌̏̽͘͝o̵̢̎̈́́̃̕͝ŗ̴̩͉̣̤̣͍͌̏̓̈́̕s̶͕̖͖̞͋͒̓ ̷̰̭̓̍̚o̸̢̳̯̪̮̓̂̿͛̚͜f̴̱̯͈͇͋̈́̋͆̄͠ ̴̛̞̜͚̲͎͍̫͛̃͂͊̕̚<math≯͔͉̄͂͘ͅŇ̴̨̛̖͙͖̆̋.̶͍̾͑ͅ</math>̷̢̯̣̝̊̇̏̕͝
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Let <math>N</math> be the number of ways to place the integers <math>1</math> through <math>12</math> in the <math>12</math> cells of a <math>2 \times 6</math> grid so that for any two cells sharing a side, the difference between the numbers in those cells is not divisible by <math>3.</math> One way to do this is shown below. Find the number of positive integer divisors of <math>N.</math>
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̶̛̰̤͋̊̈̅͑̕͝<math>̶̺͙͙͎̂͂̀̊̏̓̈́̆</math>̶̺͖̗̑͆̂̓͘͘ͅ\̵̧̼̭̣͕̼̞͕̂͒͂̓͌b̵͓̪̊͒͗̽͠͝͠e̷̝̞̽͐̈̓̕͠͝g̷͉̕ị̴̛̌͑n̵̢̛͔̰͖̖̹̉̈́͝{̸̧̠̘̀̓̿́̐a̴͓̦̺̮̥̬͒̍ȑ̶͇͚̮̤̤̗͖̤́̔̅̊̓̽r̵̫̟͚̯͎̳͙͋͒͗̓̎́a̴̧͍̤̦̝̠̖͛́̍͒́y̶̼̞͇̳̑}̵̧̥̦͚͆̑̊͋̅͒̏͆{̸̫̞͌̽͗|̶͎͍̥̮̘̮͙̣͗̑̓̇͝c̴̛̤̱̆́͑|̶̟͒͊͛̓͂̄c̶̛͔̳̎͆̆̅͜ͅ|̷̢͔̋̕̕c̵̳̳̍̈́͆|̷̦͓̼̳͍̆̑͂̇̅c̵̜͎̳̖͋̈́̎́͘|̴̩̎͗̈̈́͘͠c̷̦̳͇̦͈͉̺͇̔̆̓͒̎|̶͍̝̀̇͐͛̒̇͂ç̷͍̬̀͗|̸̗̩̼̬͚͕͔̒̚͝}̴͇̄̉͗ ̴̱͕̿̐͒̂\̵̦̀̑̓̿͐̂ḥ̵̣͉̘̬̥̂̀̚l̷̮̰̟͇̟͙̖͋͆͂͜͝i̴̡̻͇̱͇̤͌̈́̃́̀̀̕͝n̴̯̥͖̭̤̉ẹ̵̜̭̬̝̺̯̉̈́̈́̄
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<cmath>\begin{array}{|c|c|c|c|c|c|} \hline
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̸̢̛̬̰̟̳͔͂̾̀͒̉̕ͅ\̸̭̫͚͕͕̟̝̦͋̃̈́̈́̉͊̿̏,̴̙̠̥̖͕̻̖̄̋̅̃̑̕1̷̬̺̼̰̃̊\̷̡̣̙̗̙̍͐̽̍͒̏̍,̵͖͎̠̱̳͔͐́̄͜ ̷͉̉͑̇͝&̶̢̖̼͊́ ̵̼̓̐͠\̴̡̛̩̪͝,̷͓͗̊3̶̢̨̮̫͇̍͆́̀͊̏͛͝\̶̭̪̟̮̀̃̈́͂͛̚͘͝,̴̢̛̱̞̇͂̋̐̐̓̊ ̸̖̘̝̲̑̽͘ͅ&̵̩̞̪̪̾̂ ̷̢̋̒̅͐\̶̢̝͓̬̹̙̯̲̅,̸̨̭̤̤̜̰͐͌5̸̛͇͛̏\̵̤̥͔͖͈͊̊̒̈͌̀̍͘,̵̬͕̘͚͙͕̓̒̄̃͋̓͘͜͠ͅ ̵͕̲̯͕̏̍̎́͝͝&̶̛͈͉̲̬͝ ̷̟͚̔\̷̥̳̙̺̂͆̈́͑̋̿͆͜,̸̧̜̥̞́7̴̢̙̖̹͋̊̒́̽\̵̧̯͖̗͌̉̾͛̏̐̈̽͜,̶͕͚̙̈͘ ̵̜̳͕̬̭̭̋̾̋̿́͌͠͝ͅ&̶̫͖̪̮͂̍̾̚͠ ̶͖̩̥͇̀͗́̿̓͝͠\̷̻̬̜̮̜̫̈̋͝,̶̢̰̘̞͓̭̞̫̏̊̊̍̏̋͋̚9̸͔̤̐̓̆͌͘\̶̛̻̜͉̮̳̜̌͗̅̈́̆͝͠ͅ,̶̡͖̬͖̜͎̎͒̅͛͐̈́̈́̊ ̷̤̙͕̥̃͑̿͋&̸̲̫͉̠͓̇̅͛̅̚̕͜ ̴̺̦̘̘̯̲̓͘͜1̸͉͐̔1̷̡̀͋͋̎̽ ̴̠̦̊̆̐̋\̸̧͍͎̭̙͐̈́̍̊̏͑̀\̸͚̹͈̝͐͜͜ ̶̢̧̼͕̈́̃ͅ\̷̞̙͔̱͓̠̑͘h̶͔̣̾ļ̷̡̯̣͚̻̣͙͒͒͂̏̑͌į̴̝̣̿̋n̸̮̯͚͇͎̹̔e̷̢̳̲̩̩̥͓̊
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\,1\, & \,3\, & \,5\, & \,7\, & \,9\, & 11 \\ \hline
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̷̛͍͔̯͉̩̺̫́̃̆̎̊̄\̶̢̛͓͙̟͍̭͍̌̓̅́̌ͅ,̵̠͎̒̎́̈2̵̻͈̖̮̺͋̑̆\̵̗̬̌͑,̸͚̺̺̖̺͙̯̃͂̊̑̕͠ ̵̜̌&̷̬͙̤̫͛̏̏ͅ ̶̙̯̳͋̈́̊̎\̸͖̌̔,̸̲̞̒̿͌̽4̷̝̪͉̯͓̆̉́̂̕ͅ\̴̛̙̜̖͚͈͕̦͙̊͌̀͊͗̃̎,̶̧̦͕̲̯͉̪̑͐̉̔̉̋ ̵̹̱̱̳̮͍̦͋&̶͔̊̀̽̃̈́̿̋̕ ̵̱̗̺̺͙̦̭̂̑͑̂̚͜͝͝\̵̭͎̓́͊̊̽,̷̛̱̰̪̰̝̹̼̓̔̑̉̇6̸͚̺̺̭̻͉̗͗̂̄\̴̹̰̥̟̓̒̇͜͠,̸͚̰̱̼̼̺͝ ̵̦̮̼͐̈́́͂́̕͠͝&̴̻͔̙̞͖̀͊̂͆̂̏̐͛ ̴̳̼̫̜̂̄̔̓̽̅͝\̷̡͙̗̹̳͓̄̆̿̔̂̇͝ͅ,̴̧̰̗͈̂͌ͅ8̶͚͚̏͛\̷̼͙̠̓̌̄͊̉̇͠,̴̪̀͊ ̴̠̗̜̇͝&̴̨̞̘͍͎͈̳̘͊̄̊́̿̔ ̴̢̳̹͈̤̤̊͑1̵̠̤̼͍̀̈́0̸̯̰̼̼́̏̑̉̎̕͜͜ ̶͇͍̝̻͇̉͊͂̎̂͜&̴̲̦̊̓̊͑̑͘ ̶̫̪̙͖͔͉͕̽1̷̖̤̽̈́2̷̭̗̯̙̘́̅̅̀͌̀̈̈́ ̴͉͎̣̼͕̦̆̍̊͝͝\̴̡̬͗\̷̻̩̫͔̝̙̪͉̊͊̆ ̷̭̼̬̟͎̭́̅̇\̵̨̛̳̦͊̌͋̓͝h̴͍̗̔̀͛̌͜͝l̴̢̧̘̺̰̰̺̥̀̿̀į̷̡̪͓͔̲̭̲̓̅̊́̕n̴̨̪̻͍͋̎͋͆̕͘͝e̷͇͑
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\,2\, & \,4\, & \,6\, & \,8\, & 10 & 12 \\ \hline
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̶͇͔͉̥̮̜̅\̶̝̪̽͆͒̈́̓̍͝ë̶̢͎̟̱̘͙͕̇̈́̑́̽ͅn̶̲͕̩̘̪̬̎̓̋̏̎̑̚͝d̴̲̖̩͉͉͚̄͐̀͛͊̐̎͜{̴̨̖̮̤̭̝͖̱̊̀́͛̔̀͝ą̵̖͉͓͙̪͎̍ŗ̷̳̗̖̤̰̙͐́͑͠ŗ̸͇̭̩̎͋a̸͍̘͚͓͂͛̊̂̔y̸̧̟͙̗̳̘̙̙̽}̵̝̊̊̃̅̕͝<math>̴̢͖̘̪̀͊̓</math≯̢̤̱̙̗͕͆͑̆
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\end{array}</cmath>
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̷̡̡̼̩͔̣̺̪̇̉͐̅̅̚͝
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̷̛̲͖͒̅̆̊ͅ[̷̠̭̙͚͎̅́ͅ[̵̜̹̱̉͑͗̈͊2̶̨̝͔̃0̶̙͇̙̯̤͇͉͐͛́̀̑2̸͈̺͗3̵̼͉͔̻͛̿̈́͒̓̄ͅ ̸̩̯̗̰̏Á̴̧̙̣̘̤̥̰́͌͑͂͆̅͜͝Ȉ̵̭̦͔̻̯̬̊̔͗͘͜Ṃ̸͚͍̠̦͕̫̝͂̀͑̅Ę̷͍͓̠̼͈̿̒̇̿̏̓̾ ̴̨̢̼͎̽̓̍̓̋̀̆ͅI̸̞͇͉͉͎̎̑͋͊̿̓͘Ǐ̸̧̨̗̼͎̦̙̀̂̀ͅ ̷̟͂P̴͖̳̺͉͑̅̄̉͘͠r̶̛͚̥͓̒̌́́̅̈̚ő̵̮̝͙̭̦̼͇͌̆́͜b̸̡͕̲̓̊̑̽͌l̸͖͈͇̆ę̴̳͇̭̩͈͛̿̋̈́m̸̠̠̳̜̩͙̟̆́͑̓ś̴̠͔̫̎̉/̵͍̫̼̦̯̃͗͊̾P̴̯͇̉r̶̨̤̗̠͎͐̊͗̃̽̎ö̵̹̫́́̒̈́͝͝b̴̖̻̋̈́͂͂́l̶̘͓̻͖̮̐͂̀͐̾ͅȅ̷̹̠͕̫̖̾m̸̛̻͕̓̀̒̀͛̑ ̸̝̭͓͚͔̾̽͘͘1̴̢̛̣͉̣̮̰̈̓̒̽0̸̣̘̹̃́̑̽̈ͅͅ|̷̱̱̜͌̃̒̄͘S̸̢͚̙͇̅́̃͘͝ơ̴̺̑͛̇͝l̸̢̧̥͉̏̇͛́̌͐͘u̷̻̬̱͖͚̱̇̉́̓͑͆͗t̵͍̊͑̒̐͝ȋ̷̺͙ö̸̬̩̂͋̇͛͂̀n̶̠̻̜̭̏]̷̻͍̯̭̜͕̖͌]̶͎̥̱̺̟̇
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[[2023 AIME II Problems/Problem 10|Solution]]
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̸̲̥̞̈̈̏̑̊͊̚
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̴̗̻͎͙̣͋≠̝̘̀̅̀̈́̀=̶̡͖̹̖̰͕̯̉̾̐̚̕Ṕ̵̮̱̼̬̝̔̎ř̸̜͉̣͗̈́ȯ̷̢̮̼̗̌b̶̡͇̱̲̠̲̝̃̃̀͛͜l̶̨̩̪̼͙͙͔̠͊ẹ̵̲̤̃̆͌ḿ̴̭͛̐͘͝͠ ̴̛͕̺̼͇̟͔͂̓͊̆̌́̿ͅ1̸̡̢̘̼͔̙̟̙̽͛̂̌̕͠1̵͖̱̪̽̒̎͋̀̈́̄̒=̶͙̩͓̠͍͓̼̊̀͐͗=̶̧̗̹̞̻͌̎͋͛̓̀̐͌ͅ
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==Problem 11==
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̵̧̲͖̽̂̈́͆̇̃̄
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̸̩̥̖̀̓̓͊F̴̞̲̮̗̩͍͜͝ḭ̴̅́͊͌͘ǹ̵͔̦̰d̵͎̖̤͗̀̌̍͘͝ ̵͍̫͙̞͖̪͎̗͌͒t̷̨͖̤̣̪̜̳͋͛ͅḫ̴̡̢̯̰̻̖̗̅͆ȩ̷̧̬̺̗̲̦̀̈́́̍̕ ̶̛͔̩̬͛̾͌ņ̸̟̙̠̹͚̊͌̓͝u̸̧̦͇̺̺͂̀̎̓̈́͘m̸̪̣͇͈̝̘̹̈́̅͜͝b̵̛͕͔̼̤͓̽͠ͅȅ̸̟͊̉͌̎́͋r̴̢̧̧̥̫̩̮̩͌͂ ̴͕͈̫̽͗͛̿ô̶̖̲̍̑͐́̒̉̇f̷̤̕ ̴̠͗̇̅͛̏̏͝c̴̛̜̓̋̾̓̈͘͠ͅo̴̮͍̫̻̪̓ͅl̸̬͖̮̗͉̍̄l̸̻̼̪̮͒̆e̸̘̟͗c̵̛̪̘̻̀̀͗̕t̴̪͈̲͕͗̄͛̏̚ì̵̳̙͊̃̄̽̌ó̶̠͇̱̞͚͐n̵̛͓͙̺͚̗̈̂s̷̨̢̖͈̥͉̣̋͑̍̈́̋ͅ ̶̙̔̏̏̉̆̈́o̵̧̬͇̦̤̦͓͎̅f̴̹̤̹̘͊̌͠͝ ̶̲̗̙̔̊͊̒̐͋<math>̶͎̭͈̤̩͂̉̊1̶͖̤͓̰͎̩̟͙̊͋́6̸̖̺̗͇̺̞͐̊̏̉̅̈̌</math>̷̢̦͓̝̼͍̮̘͋̽̓̇̍̂͊ ̷̨͍͙͎̙̱̣̦͐̓̄̽̔͒̚d̵̺͇̞̆̅i̶̥̾̆̈́s̶̘̤̘̠̳̉̿́̓̓͘͝t̵̢̪̞̣̮̪͐̀͗ḭ̴̩͙̠̣̱̩̑́ņ̸̢̤̖̭̞̝́̐͠c̵̭̼͓̝̲̱̟̿̍̓̄͜͝t̷͇͇̍̊͒͜ ̶̡̠̮̥͖̬̆͂́͋̍́͝s̵̹̖̒̔̍̈́͐̓͝͝ü̷̬̈̆ḇ̸̟̊́͗̈́̓̐̀s̴̥͇̮̕ě̶̛̗̲̤̀̇̑͘t̸̨̼̗̭̪͔̏͌̈͛s̵̱̻̜̓̔̃̓̍͑̈͝ ̵͖̘͖͕͉͙̑͐o̷̡̢̼͎̤̻̰͙̿̀̈̌̓͘͝͠f̸̡̻̤̗̘̫͔̕͠ ̶̡͔̑<math>̶̗̣͔͕͍͐̃̌͋̓̃̍̎ͅ\̸͚͍̩̹̐͑̐̋̌̃͒͘{̸̨̹̭͛̏͜1̸̡̢͕͇̳̃̎̇̅̍͒͝,̸̱͚̃̓̐2̵͈͕̌̎̽̚͝,̶͕̭͋3̶̫̝̘̠̼̫͉͓̌̎̿̒,̶͓̓̃͒̓̆̃ͅ4̸͙̫̖̊́̽̕,̸͚͍͙̮͇̹̣͋̐̃̚͠5̶͉̲̺͆̀̋\̷̨̫̼̪̺̆̂͠}̸̻̘̖͉̱̥̽ͅ</math≯̛̗̀͊͆͑̋͂͘ ̶̡̪̳̫̖͒w̵̧̲̼̦̲̥͊̑̄̈͗̀i̴͓̐t̸̹̄̀̑̇̑̉̀̍ẖ̴̭͉̗̰̓͑̒ͅ ̶̢̮͎̗̜͌̅͐̋̄ͅț̵͉̫͎͆̓h̸̨̪̦͖̯͂́̅̔̽͆̀̌e̶̡̲̪͔̠̦͋̀̋̈́͘̕ͅ ̷͔̭̟̥͍͓̠̭̀̿̿̇͗͑͐͘p̴̭̱̖̦͙̬͙͑̈́ŕ̸̛̝͓̫̀͒͛̆̅̋ŏ̷̢̥̣̰͙̬́̾̍̄̏̚͜͠ͅp̸̡̦̱̮͕̣̞̂e̵͍̭̍̒r̶̨̤͕͍̤͂̀ṱ̸̙͌̾̂͘ÿ̵̢̨̧̠̥̹̱́̒̅͆̒͆̏̕ ̷̡͙̣̹̞̆͐̽̆̄̈t̷̗͊̀̽̑h̸̗̻̽ǻ̵̪̩̳̪͉̗̠̩͐̐͐͠t̴͔̝̠̩̫̺͐͒̆̎̅͝ͅ ̷͙̯̓̓̓͑f̸̪̞̹͇̈́̌͂̆̌̚ȯ̴̬̿͊̌͑r̸̛̜̎̂́͂̒̕ ̴̡̧͖̤̮̼̱̍͆͜a̸̧̙̠̦̐̎͂̓̕̚͠n̴̩̉ŷ̵̡̢̨̳̤̜̌̐͋̈̀͂̚ ̶̪͎͚̲̜͈͖̔̀̅̋̑͝͝ţ̵̧̢̠̜̀̆͊w̷̯͚̺̾̅͑̌͋̈͘͜ȍ̸̟̠́͝ ̶̫̙͍̬͍̞͗̔͐̇̈́͜͠s̷̯͚̠̳̘̈́̓̑͒̍ͅu̷͈̠̘̗̠̟̔͒b̸̧̥̬̲̭̈́̄̓̍̆s̵̻̗͔͚̗̻͎̝͂e̵̢͓̲̖̣̐̇̂̎t̷̡̠͙͈̺́̿̈́̄͊̿͗s̷̤̻͓͚͖̳̠̲͗̉̓ ̵̤̬̓͘<math>̵̧̳̯͉͕̫̌̄̋͋̓X̸̛̼̮͈͕͔̭̩͒̊̇̇͐̀̕ͅ</math>̵͎̅̿̋̋̄̋̎ ̷̨͈͉̀̓̽a̸͉̎ṉ̸̈d̴̯̑̂͆̔̊͂̍̑ ̸̧̰̰̼͖̲̈́̄̈́̀́͂̐̕͜<math>̶̠̺͇̭͌̂Ÿ̵̤̖͖̳̲́͐</math≯̡̖̩̩͋̉ ̷̨̘̜͍̥̳͆̒̿̄͐̊͘̕͜͜ĭ̴̞̰͍̌̈̔̈̂͋͊ṋ̴͎̣̘̜̼̱̱͂̒͋͘ ̴̢͚̞̫̳͖̗̓͌̋̈̚̚t̵͍͖̘̎̃͝͠h̵̝͗̀̾̄̐ȩ̵͎̠̤̀̏́̌͠ ̸͉̝̱̻̾̅͂̌̇͊͝c̴̺͈̍̈o̷͉̼̚ͅͅl̵͉͛͑̔̑̄̍̓l̵͎̺̼̺̙̔̂̇͆̈́̈́e̴̛̪̬̦̝̗̯̓̏c̴̯͍̤̞̣͙̜̒̀̄̒͗͊͝t̶̡̡̮͎̼͊̍̀i̷̺͆͂͐̈́̍õ̷̳̇͌̾̉̕n̷̹̦̮͇̗̮̈́͗͂̐̓͘̕͠,̴̢͚̋͆̓̃̇̀ ̷̨̞͚͓̼̇̈́̏̒̈́̂̚<math>̷̢̨̘̠̥̰͙̫̎̃͌͗͘͠͝X̴͚̖̘͒̃̔̑̏̋͜͝ ̶̩̈́\̵̢͎̥̦͉͍̉c̴̡̟̭̽̐̈͒̄ã̵̧̝̏͒́͑̚ṕ̵̟̆͌ ̴̨͔̔͂͌̓͘͠Y̵̻̮̝̺͛͋̌̃͌̇͜ ̸̧̛̲̲̬̺̯̰̃ͅ\̶͓͓̫̖̯̜͊̊̆n̵͍͈̹̣͇̝̎̀͝ͅo̵̗̜̎t̸̖̣͎̭̀̏̀͌̕͝ͅ=̴̨̛̉͂̀͑̏̆͘ ̷͍̦̥͕̈́͗̕\̸̡̦͍̲̳̏̓͆̆́̓̈͠e̷̡̡̢̲̙̒̃̒̓͛̿͝m̸̩̥̱̘̺̳̣͂̑̐̈́͛͑̎p̷̫̟̬̳̤͎̂̀̉̋͊̇t̷̨̞͋͝ͅý̶̝͇̆̍s̶̮̍͛è̴̩͕͛̏͜t̷̝̀́͗̔̈͗͘.̷̻̪͖̂́̕</math>̵͖͙͓̬̌̒
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Find the number of collections of <math>16</math> distinct subsets of <math>\{1,2,3,4,5\}</math> with the property that for any two subsets <math>X</math> and <math>Y</math> in the collection, <math>X \cap Y \not= \emptyset.</math>
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̴̧̡̛̝͖͔͇̂̇͛̈́̾́͝
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̶̢̺̫̦̟̼̱̱̔[̴̘̯̓̔̍͝[̸̱̼͗͊̕2̶̤̮͑̐̍̏͗̀͒͠0̶̩̦̙̜͕̼̂̆̌̏̋ͅ2̶̦͉̰̒͌͐͝3̴͍̈́ ̵̡͈̪̺̹̺͙̑̄̓A̶̤͈̅̋͛͝Ī̶̧̲̯̣̫̗̲̓̓̀͋M̶̭̊̒̉́́E̸̬̩̟͖̖͑̈́͂̊ ̶̤̣͓̣͆̈̇̾̚Ȋ̴̙̞̻̯̖̌̎I̶̢͇͕̜̮̞͍̣͗̅̈́͗͠ ̷̛̘̟͎̓͗͛Ṕ̶̧̮̤̬̙̼̠̔̀͌̆̿͂͝ŕ̷̻͖̝͛̈́̔̚̕ŏ̶̡̐͗͒̇̊̕͝b̶̛͈̑̑͒̌̍͛̏ͅl̶̛̞̈́̚ȩ̴̙̩͙̦̗̺̋̀̿̌͜m̴͇̭͙̥̙̤̞̖̽ş̸̨̻̫̝̞͉͑̀̑̚͝/̴̢̣̬̭̼̻̫̭́̀̓́̏͗̏Ṕ̷̡̪̻̮͖͊́̉͘͝r̴̬̮̖̦̲̐̈̈͜ͅō̸̢̱̙̫̙͖̭̠͝b̸̳͎̩͈͗͜l̵͇̞̟͕͎̺͂͗̎̍ȇ̵̛̗̋̈́͌̈́̆m̷̨̯̣͎̘̤̘̈́̄̌̎̍̅̍̃ ̶̹͔̟̖̙̂͐̎̾͗́1̷͎̦̣̼̪̹̳̃̃̃̈́̌̿͒̎1̶̪̅|̸̳̬̬̫̃̀̒S̵̡̡̻͕̙̒ͅo̷̢̹̯͕̓̾͒͌l̴̳̉͛͐͗͌u̶͇̫̾ͅt̸̨͋͐̾̂͝í̴̞̗̱̼͕͔̩́ǫ̵͎̂n̸̹̩̯͎̮͓̎͆]̵̨̛̻̮̟͉̈̔͌͑͒̚ͅ]̶̢͖̺̻̟̲̗̹̔̄͂͘
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[[2023 AIME II Problems/Problem 11|Solution]]
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̶̯̣̯͎̂̇̍͝
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̴̹͎̬͆̏͗̔̃̚=̶̡̭̼̉̀̃̍͂́͛̃≠͎̺̑̀͆͛͆P̸̖͕̹̰̘̂r̸̨̦̥̳͕̬̓̒̄̎̕ô̷̡͎̣̳̟̥͕̬͛̓̆͆͌̆͝b̸̢̯̹̘̻͍͗l̶̡͍͈͑͑̎̈́̑ę̵͈̼̩̟̺̊̑̽͊͘m̵͔̠̠̣̩̳̻̂͊̇̈́͜ ̶̛̟͗̃͊͒͒́͂1̵̰̝̹̘̦̐̈́̄2̸̙̻̱̞͗͐̋̌͐͠͠ͅ=̷̛̣̤̈́̓̇̀̕=̵̛̘̤͕̙̘̥͎͊̏̿̋͜
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==Problem 12==
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̴̛͍̪͑́̈́̃͛
+
 
−
̶̨̙̘͋̽ͅI̵̧̯̱̝̅̅̐̂̇͝ṋ̷̽͋ ̷̨̱̺͈͑̄̀͐͝<math≯̯̥̘͑̔̍̍̕͝\̸̘͔̇͌̈͑̽͘͜͝t̷̨̧̧̻̫̖̲͛̓͌̒r̴̢̨̨̩̪͕̠͇͂̚͝ì̵̛̤̻̯̉͠ȧ̵̦͓͇̝̣͕͑͜ͅn̵͚̜̭̚͜͠ǧ̵͎͇̫̌l̸̬͎̘͐͜͝ͅe̴̹͚̬̘̬͋͗͐ ̴̻̝̥̜̫̫̲̎̔̈̋̑Ā̸̻B̸̛̯̯̓͒͛́͠C̸̨̹͈̈́̓̔̅̔̑</math>̴̨̪̦̩̱̰̘̬͘͝ ̶̟̦̋w̷̟͔̫̬̑̅͆͗̆̆̈́i̸̧̧̛̜͂̒̋̓̈́́͠ͅẗ̷͍́́̽̅̾̊͆͘ḩ̸̧͉̳̳͚̭̀ ̵̢̛̣̯̰̿͌s̷̬̝͊̆͌̑į̴̠̠̜̯̳̣̌̒̃̈́͛͠ͅd̶̻̝̺͓̭͈̼̥̉̑̚ẽ̸͔̺̤͂ ̸͎̟̏͌͊̌͂l̷̖͛͒͑̎̍̕ę̶̼̺͓̹͍̒n̸̹͆͊̍͊̈́g̸̬̣͎̤͖̪̲͋͗͆̚͝t̵̥̩̲̼̥̗͖̊͋̑̀̓̉͠h̷̨̟̻̮̱͎̲̃̃͐̎͛s̵̢̬̥̖̪̈͜ ̵̜̤̌̃̌̇̑͑̓̕<math≯͓̐͛͌͊́̚Ą̸͙̮͈̱͈̃̓͋̃̓B̵̺̗̜̺̜̹̬̄͑͛̾̈̃̂ ̶̢̼͋̃̀̽̍̄͝≠̱͈̱̙̖̂͝ ̸̯̦͇̰̟̟̪̩̏̓̽̇̄̇̓̄1̷͉̼͇͒̀̊̿̋̚͝3̴̡̥̲͚̬̳̄͒͒͌̿͠,̸͓̅͗̓̽͗́͘͠</math>̶̧̛̹̟͚͎̈́̿̒̔͗ ̶͍̜͙̫̈́͘<math>̵̢̧̻̯̖̬͖̾̚B̵̘̤̪͚̯̱̋̈́͜C̶̝̿̃͗̈̈́̽͘ ̶̡̩͉͚̦͉̼̦̈́͆̐̓=̵̦̓͋̾̀̒͐ ̸̧̛̦̫̣̮̬͖̒̑͐̄͝1̶̢͉̭̣͓̖̅̓̓̄̇͑̉̓4̴̛̠͎̰͕̩̹̼͍̔̋͋̓͐,̴̬̙̲͒̾̽̈̇̑̚</math>̵̪̥̤̫͕͆̆͠ ̶̼̥̳̿͆̈̏́̊͂̕a̴̡̹̝̰̠̯̋̃̈́̏̾̾̚n̸̥̘͔̣͚͇̻͌̈́̀͋̑̔́͐d̸͉͕̔̽͆̚ ̸̜̿̇̓̔̈́͝<math>̷͍̫̞̯͓̖͋̂Ç̶̧̠̱̟͓̰̳̈́A̷̜̓̎ ̶͚̯̞̟̳̺͂=̷̙̟̠̣̞̪͎̓́́̊̓ ̷̹̙̥͇̟͋̏̾̋̽̎̚1̴̨̦̪̦̩̻̎̐́̕͜ͅ5̸̝̣̲̌́̀̽̓̏̌,̴̲̭͍͎͙̠͌̿̑͛̀͝</math>̶̢̛̪͉͖̲̙͌͐̾͜ ̷̲͎͚̀̀͗̚l̴̖̱̗̿ė̸͍̞̖̼̉̅͂͠t̶̺̰͇̏̆̀ͅ ̸̢̠́̓̽̿̂̔͜<math>̷̰͆̅̉Ḿ̴̼̯̪͝</math≯̘͙̼͛̓́̈́̈́͘̚ͅ ̵̠͎͍̣̆̊̀͑̿̕͝b̷̢͇̦̰̳́é̵̪͍̮͙̱̾̏̂ ̴̪̮̠͈̼̯̙͆́͋͛͛̈́t̸̨̠͚̥̃̓́̈́̓̽̈̓ḩ̵̣̿͊͜ĕ̴̥͍͋͗̓̃̃̌ ̵͖̟̱̦̻̳̞̽̀̎̆̆̀̋͠ͅm̷̥̏̂̏̀͑̓̃̈́i̷̛͎̯͉̝̒̉̀d̵͖̩̩͔͓̫́͆̇̎̆͜p̶̨̞̞̒̎̔̆͂̈́̕o̸̡̳̙̼̅̀͠ȉ̵̢̛̛͖̹̥̱͈͙͑̀̄̄ń̷͇̜͚̂̋t̶̮̱̗̞̜̓ ̴̧̨̧̘̉̋͐͂͒͒̀o̶̩͍̣͚̫̝̓́͜͜f̷̣̜̪̘̱̌́̓͑̊ ̴̢̡̛̤̩̄̑͐́̀͋̎<math>̶̹̈̈̑̊͘\̸̡̗͓͔̟͚̏͗͗ǫ̶̡͙͕͎͚͉̍́̒̔̑v̷͇̲̤̞̀é̷̘̯͐̅͋r̵͈͊͂ͅl̴̡̩̊i̵̧͓͕̤͎̾̂̐n̵̹̞͉͍͊̊̇̾ͅe̵̛̝̥̝̘̙̾̃̈́́̕͜͝{̵̛̞̤̤͗̿͋̕͝B̴̧̅̐C̶̜̱̹̯̦̱̫̒}̴͍̓̑͒.̶̡̛̪͙̙̦̘̐̿͊̀̚͠͠</math>̴̫̗̝̒͒̓̅̾͆͠͠ ̴̟̞̥̘͖͐̀͜ͅL̵͙͖̯̱͉̪̹̱̈̊͐e̸̡̛̖͕͙̯̪̼̓̀t̸̳̱̱̹̖̀̈́̀͒̃̓ ̴̨̗̎͆<math>̴̮͌̄̊͒̃̈́P̴̢̖̠̦̜̖͒͌͜͝</math≯̥̩̺͔͗̃̂͂͛͠ ̶̢̨̫̬̫͕̻̙̄̓̿̕͠b̷̳̰̯̹̼͖́̂̇̿͛́͆e̸͉̋̉̊̅ ̵̧̧̯͇̩̥̭̝̏t̴̨̮͔̥͕̀h̸̛̥̫̟̟̼̺̱̔̈́̀́̌̋è̶̪ ̸̩̝͉̋̔p̵̛̱͚̬͌͂̚ǫ̵͍̘͙̥̓͊ͅḭ̸̭̤͖̳̘̹͋͂̏̓͝͝n̴͈̰͙̺̈́t̴̳̿̂̓̐ ̸̝̲̑̀̆o̸̭̪͇̤̱̤͚̊̇͋̽̊̚ń̶̛̪̤̻̩̺͇͈̝̄̑͐̈́ ̴̝̞̐̋̐̉t̷̫̓h̴̺͓͎͇̲̼͙͝ͅe̵͚̤̣͍̤̾ ̷̩̝̜̖̫̾̋͐c̵͓̹̣̼͔͖̣̱̏̊̀̾i̸͈̤͓͙̱̩̝̻̍̈́́͒̉̚ŗ̷̗̲͈͉͛̑̍č̸̢̧̯̩͈̤̭̤̉͌́̈́u̷̧̧̱̘̦͇͓͉͛̿̏́̈m̷̛̼͍̣̫̯̾͛͐͘͝͝c̸̖̦̦̺̑͆́̒̏i̶̡͙̺̻͆͐̍͋̒͛̃̇r̶͎͖̩̅̅̀͝c̸͖̀̉̂̎͌̏͘͝l̸̳̯̘͍͚͚̏͊̀͋̉̓̈́͝e̶̝̽̂̕ ̴͕̩̥̮̺̱͍̓͌͑ơ̸͉̯̞̞̭̳̿̀̚f̶̞̒̍̌͊ ̸̹̻̯̻̣̔́̊̽̇͆͑̕͜<math>̴̮̯̥̝̥̞̀́͝\̵̼̳͈͕̖̀t̴̩̲͚̆́̇͝ŗ̸̝̰͇͉͔͐i̴͎̥̗̙͖̹̬͗̏͐́ầ̶̩̮̹̻̾̃̾̈̕n̵̬̜͒͗̓͘͝g̷͎͌̀͗ĺ̶̼̦̘̜͓͊̄̾̇e̴̛̼̳̯̚ͅ ̴̛̣͖̹̝͉̾̆̐̂͘͠Ą̷̛̰͙̪̱͇́̎B̸̡͔̘̳͖͎̟̺̈́̋͆̿̑͋͘C̷̩͓͉̖̍̚</math>̶̛͕̌͛́̀ ̶̨̢̢̨̙̱̲͇͒̎̈́̋͌͘s̶̢̤̙̞̭͈̆͆̇̚u̶͚̾̐̏͝c̶͕̬̦̱͠͠ḧ̶̻͈̲̬̫͙̞́̔̀͊͋͘͝ ̵̧̺͍͕͕̅̽̍̃t̵̢̀͛̌̌̆h̸̢̡̩̳̼̮͍̅̈́͒̀̆͌̾͘a̶̜͓̮̒̐t̵͓̭̤̱̦͌̇̎ ̷̲̙̲̖͈̖̻̱̍͊<math>̷̡͍̮͔̐͑M̷͉̓́̽̊̀͒̌͝</math>̵̳̘̙̝͖͔͒ ̴̛̟͋i̶͉͙͕̫͔͋́́́̕͘͝ͅs̴̘̟̯͉̜̀̐̉ ̶̢͖͕͊̅ọ̶̢͇̱͍͖̀̏͋̓̈͝͝n̸̬̝̩͉̮̫͋̓̾̽̚ ̴̰̇͛̾̓<math>̵̨̨͈͓̰̣͆͆̉͂͘\̴̛͍̔̆̓̋̽̕ͅo̷̫͉̽̍͛̑̈́̐̃̀v̵̢̢̟̞͚͉̓̎̑̕e̷̛̻̣͐r̷̯̼̼͇̣̳̘̺͗̓l̷̠̥͎͋͆͂̊͑̀̑͝i̵̧̳̻̗̓̉̇́̃̊̚n̷̨͎͋é̷̛̟̘̪̞̱̼̖͌̌̚ͅ{̸̢̞̞̺̫͉͇̅͑̀̊̅̐A̸̩̝̒̊P̷̜̱̦̗̹̄͋̿̽̒}̷̯̩̰̞̳͋̀.̷̨͓͓̳͓͖͘ͅ</math>̷̨̫͍̿̉̎͜͠ͅ ̴̟͇͓̥͛́̑̓T̶̞͎̞̘̭͇̰̩́͐h̴͇̖̣͇̼͔̳̏̔͊͑è̴̛̼̆̈̾̿̈͘r̵̹̼͚̝̅͐̽ę̸̛̦̮̖͉͕̈̅͛̈́͆̚͝ ̸̡͛͆̇͝è̷̻̬̭̟͕̎̽x̴̙̫̦͗͊̒̇͗̕ḭ̶̛͓̪̀͒̆̔̈̕s̸̡͝ṫ̸̪͎̜͚͋̈͌ŝ̶̝͆̌̿͗̈́̓ ̴̹̟̺̣̹̂̍͂͗̊̈́a̵̖̻̗̽̚ ̶̜̳̺͈̓͑u̸̢̩̤̦̅̕n̶̜͍̝̒̿̃̔͑̾̆̕i̴̯͊́q̶̧̤̇̏̏͛ȕ̶̙ȩ̴̧̹̰̩̮̏͌͜ ̸͚̭͆͜͜͝p̵̢̙͇̰̋̕ọ̷͉̙̰̱̯͑́̐̾͆͘̕ͅi̶̪̞͎̣͍̩͙̺̽̅̔̅͘͠ṇ̷̢̲͉̜͔̔t̵͕̗̟̺͈̞̟̒̊̽͑̈͒̕ ̷͔͐͐͆̋͜͝͝ͅ<math>̷̗̥͉̝͙̑̈́̆̒͌Q̴͖͙̼͕̈</math>̶͕̤̓̆ ̷̠̘̙̥͚̘̄o̷̜͚̹͊̈́́n̵͗̋͆͜ ̶̛͔͍̳̹̐͌̌̌͑́ͅs̷̨̳̀͗̈́ě̸̺ģ̵̠̈m̴̤̓͜ḛ̶͇̝̀̄̇n̶̨͓̳̗̗͓̽́̀̊̋̚͜t̷̲̳͙̹͕̪̣̙͛̀͐̄̏̔̚ ̴̘̠̙͍̲͖̅͜<math>̵̢͚̫̥̹̲͆͜ͅ\̶̮̹̄̽̐o̷̦̍͠v̶͍̍͌̂̌̅̆͘ȇ̵͙̯̼̝̍̓ṛ̷̲̜͓̣͐̾́̎͐͘͘l̶̨̥̱͍͎̈̾́͊̃̂͝͝ỉ̷̫̟̲̮͍͔̊̏̔͑͆̚n̵͕̱̭̳͈͚͊͊̀̆̋̑͘͝ͅe̸͔̫̙̱̜̅̍̂̍͑̓̋̕{̴͖͗́͗́͠Å̵̪͚̞̲͕͗̀M̵̨͉̟̓̏͌́̀͐͝}̸̫̾̀͊̊͝</math>̷̢͉̮̥̼̃̈̽̓ ̴͍̂͐͂͐͘s̵̨̡͚̳̫̿̅̆̽̂͝u̸̞͛ĉ̴̻̼̣͖͎̝̿̐̆͂͘h̵͓̖̲̀́̅̚ ̶̧̡̧͔̬̘̞̈́̂͆̐t̵̳̹͙̝̝͚́͆̚h̵̗̗̫̅͒̎̎͒́̌̕a̷̢͎̟̯̲͂̾ͅț̸͒̓̀̆͒ ̴̝̯̊̌̅͝͝<math>̶̢͇̝̲̩̞͍͆̋͘̚\̸̜̮͚̣̂͋͌́̓̉͘a̵̰͔̍́͛͝͝n̶̪̩̙̲̬̑̈́̓̈́̔͝ͅg̵̢̜̏̋͜l̴̲̎̽̋̎̀̉ĕ̸̢̨͓̮͚͕̼͑ ̸̬͙̓̇̎P̷̛̩͐̈̀͊̀̎͝B̴̩̤̭̻̄̀͜Q̴̧͇̯̞̻̲͑͐͐ ̷̛̥̤̳͖̺͖̦̀=̶̧̰̩̓̑̌ ̶̗̋͛́͑\̵̡̳̘͕̱̝̹̲͌ą̶̗̼͚̓ͅn̴̨̧̖̭̉̏̾̅́̄g̵̢̭͚̒͂̆̏l̴̦̬͚̍͘e̶͕̭̟͓͂̑̎̈́̌ ̶̧̛͚̼̲̭̱̩̅̾̈́͂͘͠P̷̧͇̠̤̒̏̀̿͝Ç̴̹̯̰͙̍̅̾͗͂̄͆͝Q̵̗̠͇̖͚̑̍͑̚.̴̧̟̪̲̃͆̾͌̿̆̇̓</math>̵̢͍̟̦̅̋̄̍̇ ̵̪̈́́̉͂̓͘T̶̛̪̪͋͒̀̿̆͝h̵͕̦̅̿̀̍̽͗͐͝ẻ̶͖̬̣̘͊̍n̶̢̤̼̞̫͖͎̙͗͒̂͌͂̋ ̵̟̻̞͎̈́̎̔̂̉͘<math>̴̻͋̔̈̊̕A̵̗̭͕̻̿̽̉̀͗Q̵̘̄̓̒̽̅</math>̷̡̙̗͋́́̎̄͑ ̴̟̖̞̌͊c̸̜̳̲̳̭͇͚͌̈͝ạ̷͙̣͕̔͆n̶̡̲̘̼̩̗̭͛̒̉̽ͅ ̸̨̡̡̯̖͕͑̾͆̇͘͜͠b̷̟̀̑͜ë̶̼̣̰͎̖́̅̂̌̈̏ ̶̢̙̗̹͙͙̪̭̾̑ẅ̸̗̥ṛ̸̺͇̥̔́͛̀̉̅i̷͚̲̖̗̜̻̣͊̀̔̀͋ṭ̵̫̜́͆̽̽̉͠t̴̛̰̪͙̲̆͂́͘̕͜é̶̳̪̀͆̅̊͒n̶̫̳̑͝ ̵̨͔̣́̄̕ä̵͔̣̺̰̣̜͈̔͑̋͐̈̕s̷̲̤̪͉̉̑͂̿ ̵̬̪̑̂<math>̶̤̅̃\̶̛̝̗̥̬͖̙͛͒̿f̴̜̮̲͒̀ͅr̵̢̬̳͒̑̇̑ą̷̥̠̞͉͕̮̉͐c̸̘͓̥͈̄͋́{̵̨͘m̶̧̧̆̑}̵̝͍̭͊͊̂̄̀̀̈́͜͠ͅ{̸̡̫͚̮̯̀͌̊̃͜\̸̥̠́̕͝ŝ̶͚̟͚̜̝͚q̷͎̫̞̞̓r̴̹̈́͑̉̈́̿t̵̠͈̥̬́͐̀̿̊{̴̨̦̽̐̓ñ̴̢̧͖̼̽͐̍͆̏̌̀}̸̢̺̘̻̤̐̔̆̔̓}̴͔̦̱̗̖̐̅̇͜͠,̴̤͖̠͉̯̟̀̿̾́͝</math>̶͉̒͜ ̶̡̬̲͎͉̩͋͊̈͑̆͗̒w̸̝̬̳̰̤̼̐͗͊̆͂̌̋̐ḣ̷̙̩̠̿̓͐è̵̢̩͈̘̍̅̀͋̈́͜͜r̸̺̠̘͑͛͒͠e̵̮̼̓̆͐͗̊ ̶̧̧̮͖͙̝̃̈́̃̆̃̇̂͝<math>̴͈͖̤͙͐̎̎̈́̀͝͝m̶͚̱̜͛̀͑͋̂̍̐</math>̶̡̜̃͐͊̋̿̚̚ ̷̢͍͖̉͌͗̓̍̾a̸̲͋͐̾̈́̆͝͝n̶̡̝̞̭͓̳̈́̃͌͝d̸̨̤̓̔̈́ ̶̟̮̰̭͙̟̬̓̐<math>̵̯̿̏̎̄͐̎̈́̌ń̴̟̲̣͔̮̮̖̘̅͊</math>̴̡̳̿̈͌̔̓͆͗͘͜ ̵̪̙̈̆̿͜å̵̹̘̝̘̘͙̹r̸̠̒̋̓̎́͆è̵͓̝̞̦́͋͘ ̴̪̹͔̹̓̀r̵͚̄͛e̵̫̩̘̣̭͈͖̊̐̕l̸̲̐ȃ̴̡͍͉̺̼̺̓͠t̵͇̭͒̂̆̽̔̀͝ͅi̶̘̯͒̏̐͛̀͠͝v̷̛̼̙̟̲̉̓̂͘͜e̷͈̬̣͆l̶̡̙̯̱͇̳̟̈́̈̓͘y̸̥͎͇̘̦͕̥͠ͅ ̸̛̩͔̩̪͔̥͊p̸̪̹̼͙̗̯͕̏́̓̇̿͂̄r̵̞͔͓̥͂̅́̈́͘͜͝i̸̛̮̠̟̽͋̏̂͘ͅm̷̟̓̂̅͌̑̕͝ë̸̪̝̞͇́̈́͝ ̶̢̓͘p̶̨̾o̸̬̲̺̲͐͘s̴̮̥̻͓̥̎̈́͂͗̉̑̓͝i̷̢̦̗̗͚̗͋̂͆̃̀̓͛̌t̴̘̫̥̬̭̣̏́̐i̸̫̲͔̫͋͂v̸̘̐̓͛̉̋̇ȇ̸̩̣̖̼̱͚̐̎́̒̆ ̵̢̦͔̝̮̱̀͊̋̅́͗͜i̴͎̖̱̍͂̄͌͛n̸̡̯̱̙̍̍͆̇t̴̝͌͗͝͝͝ḙ̷̛̝̒̓́͂͒̕͜͝g̴̢̜̤͈̯̮̹̬̿͆́͋e̶̛̙̮̳̣̗̱͔͛͌̑͘ř̸͙̜̰͛́̂͋ş̷̨͇͖̼͔̫͑̈́̊͊͋̐͊̎.̶̰̣̗̇̓̃͊͑̀̔ ̷̖̗͚̮̤̗̹̝̏̀̇͂̏̈́̍̐F̴̖̞̫͓̳͚̔̏͜ḭ̶͙̝̮͍͊͆̑͆̚ͅn̸̡̤̪̣͚͍̠̚d̵̲̆̉̆̀̾͝ ̵̯̰̬̞̓̇̒̿̅͋͐̚<math≯͖̩̻͈̜̾̾̕̕̚ͅḿ̷̥̺̣̳͂͊̔̊͂+̵̛̮̪̱̞̊̑̓́̽̋ń̶͇̜̤͈̯͓͔̹̾͗͒͌͘.̸̠̩̮̟̈̐͋̐̌̑̈́ͅ</math≯̺͎̲̓͒̕͝
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In <math>\triangle ABC</math> with side lengths <math>AB = 13,</math> <math>BC = 14,</math> and <math>CA = 15,</math> let <math>M</math> be the midpoint of <math>\overline{BC}.</math> Let <math>P</math> be the point on the circumcircle of <math>\triangle ABC</math> such that <math>M</math> is on <math>\overline{AP}.</math> There exists a unique point <math>Q</math> on segment <math>\overline{AM}</math> such that <math>\angle PBQ = \angle PCQ.</math> Then <math>AQ</math> can be written as <math>\frac{m}{\sqrt{n}},</math> where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n.</math>
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̴̪̄̐̉͛̈͝
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̴̧̢̫̩̲̱̐̓́[̸̡͉̓͂̾͋͒͂̓[̷̨̣̩̝̦̜̃̾̑͒2̵̨̨̡̜͓̫͖̂͛͛͜0̶͎̘͙͂̒̂̆̕2̷̜͕̬͈̀͂͒̑͑̐̊͠3̵̢̨͉̔͑͘͘͘ ̴̭̘̟͇̟̮̼̾A̵̡̟͉̪̪͊̃̄̈́̒͝I̶̛̗̓̋̇M̷̧̧̼̣̥̉̀͒̇Ḙ̶̥͉͍̥̭̪͛̈̈́̆͜͠ ̷͙̪͈͓̚Ị̶̣͉͌̌͌Ḯ̶̡̡̖̪̘͆̈́̉͐͂ ̷̻͍̫̫̞̤͖̋̓̿̏͆̔͑̕͜P̷͎̞̩͔̻̟̣̖͋r̵̡̗̎ǫ̶̟̌̽́b̷̡̗͍̬͓̥̬̰͒̑̅̋̇̍̔l̴͙̝͌ë̴̝͛̈̔̾̕̚ṁ̷̨̛͚͛̊͝s̷̢̪̹̹̈́͒/̵̘̹̠̙̞͕͚̔̾̆̄́͠P̷̖͓̺̩̠̘̗̪͑ȓ̸̡̟̭̲͙͚̘͐̆̿̈́̎̑ͅǫ̷̫͉̲̾b̵̝̈́̔̽l̶̖̹͎̮̽̿̉̄̔̒̒͝e̶̩͈͙͈̼̦͂̿̂̾̒̅̽̚͜m̸̡̰͕̤͖̠̌̾̾̊͛ͅ ̶̧̡̣͉̝̲̹͉̓̌͆̑1̶̢̧͇̩̀̕2̸̳̭̙̟̬̳̝͑̾́̑͜|̷̖̮̲̪͓̈́̓͋͊̔̎͘S̵̹̺͉͉̺̻̹̥̒̄̍̈́̇͐̌ǒ̶͎̪̪̓̽̅͂͐͛̚ḷ̷̟̬͎̻̦́̇̄̽̅u̶̡̢̪̼͊̀̓́t̴͓͚̏̌̅͑i̶̛̛͔͈̙̹͊̀̏o̶̧̪̭̤͇̰̰̊̑͑̈̈́̾̕͘n̴̢̥͉͛̈́̑̎̏̌̏͜͜͠]̴̖̎]̵͕͍̖͔̖̠͉̄̋̔̊͋
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[[2023 AIME II Problems/Problem 12|Solution]]
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̴͓͚̩̗̣̞̹̯͆̀̂
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̴͚̹̯̂̐̄̔̀̋̂͘͜≠̡̩͉̰̮̰̟͇̎̔͑̂́͠=̷͙̫̾̀̔̌̏̑P̷̧̧̥̺̂͛̋́r̷̞̉̉o̷̠̗̊͂̅̈̇̓͂͝ḃ̷̻͔͚͙̮̆̿̐̓̀̏ḻ̸̣́ͅe̸̯͎͔̦͓͚̍̒ḿ̵̡̞̦̰ ̶̛̤̓̀͑̔1̸̰͎̞̂̅3̶̘͓͕̣͗͗=̵̰͉͇͔̻͒̓̍̋̋͘͠͝=̵͚̗̏͌̏̕
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==Problem 13==
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̵̢͈̃
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̴̺̺͕̳̱̝͉̒̽̕̚L̸̤͇̪͊ė̵̛̛̳̖͚̌͂̎̄͒ẗ̷͈̜̘͎̭̲́͜ ̸̺͍͔̋̏̌̍̔͛̉͜͠<math≯̠̳͕͓͓̩̀̆̐̉̓́̐̾A̴̢̧͔͚̙̟͗̅̉̈́̊ͅ</math>̵̯̻̭͓̳͗̋̌̂͌̌̂͝ ̸̱̬͎͉̭̐̀͐͑̚b̵̨̛͍͕̜̩̐̍̕̕͜ͅê̸̹͍̩͓͇͇͛̏̚̕ ̴̹̹̪̾̕a̴͈̦͖͌͗̓͛̒͋̚͠n̵̢͚̼̳̻̳͊́ ̴̬̝͇̣͖̈́̌͛͠á̴̳̼̺͓͇̑c̶̫̥͇͚̀̋u̸̳͈̠͘t̴̯̔̑̐̏̀͋͋ͅe̴͚̯̙̍ ̵̺͖̽̾͌͐͝a̸̛̞̗̦͚̪͉͋̈́͂̔͜n̴̢̠̩̞̚͝g̸̹̰͔̺͑̆̅͌̓͑l̸̲̳̝̩̤̹͍̓̐̄́̂̓̚͜͠ȩ̷̜̯͙̭̘͍̩̏͝ ̶͙͐͌̃͂̌̕s̷͔͕̖̼̩̬̒́͑̚͠ǘ̷̖̇͂͋͌̕͜c̸̘̀̉͂h̵͕̬͍̼̭̒̇̉͋̾͗ ̴̲̅̂̍̎̾̀͝t̷͈́͐̌̈͝h̸̨͉̠̺͖͎͊̉͆͛͗ͅȃ̴̲̚t̴̛̜͚͕̀͂͆̀̏͝ ̸̛̹̱͚̮͑̉̽̀͑̽̒<math>̴̻̣̫͙̈́͋͝\̴̛̮̹͛̓̆͗͝t̷̘̝̠͐̀́̓͊̍͗͘a̵̢̱͇̟̲͖͚̼̕n̸̢̪͈̯͇̞̄͆̈́͆ͅ ̶̧̗̲͚̭̊̀̇͐̔̉͒͠A̸̡̡̛̱̗̘̙͑͐̋̊̏ͅ ̶̖̲͉͖̀̋̍͂͛͜=̵̛͇̣̌̀͒̄̆͊̚ ̸͔̲͇͖̙̘̥̦̔́̕2̷̢̻̣̹̤̿͜ ̷̰̈́̈͒\̵̤̤͉͒͗c̴̨̞̳͍̫̬͓̝̔́̚͠o̶̢̱̲̮̐̕s̸͓͓͇̫̘̀͑̋̎͐ ̴̨̭͙͉̬͈̎̄̑͛͛̆͠͝Ä̵̘̠́̊̓́̑͆̇͐ͅ.̴͉͚͕̝͚̖̿̓͂̏͝</math>̵̧̼̰̮̪͇̈̌̋͋̚ ̷̜̙͈̬̝́́͠F̸̖̜̘̜͕͍̼̄͐͂̐͌̔̍i̷̡̢̯̲̰̘̣̔̄̆͜ṇ̴̘̗̝͈̏̈͌͑͑d̸͎̐̀̓̓̑̐͝͝ ̵͔͛͒͑̈́ţ̶͙̜̲̹̱̊̆ͅh̸̛̫͈̟͌̀͘̕͜ȅ̸̡̺̝̞̫̯̝̫̌̐̃͒̃̆͂ ̷̱͇̼̮͚́͐̉͂̈͜ͅn̷̩̞̦͎̳̣͖͋̅͒̚͝ư̵͓͈͍͎͙͖̲̊̆̔͋m̴̧̜̞͙͓̯͙͍͑͆̓͘b̶̯̅́̓̓̀̚͜ę̶̝̙̦̲̙̎̈́̓͆̈r̶̢͈̭̺̬͎̂̈́͊̽͝ ̵̗͖́̈̒̈́̆ǫ̵̰̩̠̖̙͇̀͝f̵͇͂̉͊̔̐̆ ̷̡͎̗̤̞̮̩͚͊p̸̡̥͈̩̉̽͝o̶̬̽͂̒̿͠s̵͚̍̚i̴̱̳̫̯̲͊͐͐́̿͝t̴͍̮̼̋̇̇̿̑̚͘i̶̗̳̰̱̻̗̫͛͝v̶͖͔̈̀̉͊͑͌͝ȩ̴͖͎̠̽̌̓͂͜ ̵̱͙͉̜̈́͒̽͜͠i̸̧̞̦̥̻̪̰̍̇͋͑͐̈́n̴̯̤̱̥̊͋͆̈́̓̒t̷̳̯͙̹̖̆̑̔̎̎̄̚͝ͅḙ̸̢́͑́̇̅g̵̯̹̥͑̋̂̂è̴͍r̸̰̭͌̍͑̎͝s̶̛̼͎̝͍͒̾̾̽̓͜͠ͅ ̸̬͙̱̙̂<math>̵̻̜͚͖̒̊͐̀̄͊ṉ̷̭̰͐͋̓̿͝</math≯͓̯̜͖̜͕̌͆̍͠ ̶̫̠̹̩͍͈̓̆l̶̨̮͕̺̉̅̌̑̽͜͠e̸̛͖̺͕͖̖͑̏̋́̕s̴̨̛̳͔̩̟̳̾̊͐̃͛̓̕s̵̹̏͂̆̚͠ ̷̻̱̬͊͒̽̒͝t̵̘͑͌̅ḫ̶̫̘̒̀̈̓͊͂ḁ̸̡̡̗̫͍͇̜̽̔n̸̝̺̤͉̙̜̔̅͌ ̵̥͕̞̯́͆̃̽̕̕͠͠ò̸̲̝͈̫̆̒͗͒̽̚ͅr̶̛͎͈̟̄͒̈́͛̓͝ ̷̨̳̑̐̈́͒̍̀͒͠e̵͕̱̖͈͚̳̾̈́͆̒̀q̵͖̦̜̳̱̔̏͛̈́͊̽͛u̵͍͖̬̬̰͈͈̘͛̀̇̄̇̓̇â̴̳̻͎͇͖͌̎̏ļ̴̲̬̺͍̠͍̀́̈́̽́́͛͜ ̸̫̘̣͉̝͗̄͛̒̚̕t̴̗̫̠͍̼̗̂ò̸̲̹̈́͑͒̌̓̊̕ͅ ̴̪̅͛̈̌͝<math>̶̙̠̙́̔̍̆̂̋͂̚1̵̢̠̖͈̝̇̾̊̄͐̌̊0̵̢̜̎͒̓0̷͎̦̟͎̀͂̉͊̇0̴̡͉̝̿̈́̐̐͝</math>̵͙̣̙͔͂̾ ̷̬̉̋̓s̴̮̜͔̅̃ū̴͈̻̍c̴̹̦̍̃͂̋̉͝ḫ̶̨̡̺̤͉̫̿͋ ̵̳̳͍̺̙͕͒̍͝͝ṯ̸̱̤̞̮̎͂̈́͐̀̂͘͝h̶̡̯͇̞͗̎̒̄a̷͈̗͗t̵̟͖̀͌͛̆͐͊̚̚ ̶̤̭̹͍̺͂̂͊͝<math≯̪͑͑͒̐͊̏͗͘\̵̝͚̩̭͚̀ṡ̵̳͇̞̈̆̔̾é̶̼̰̦͉͖̯̫̊͗͛͊̽ͅc̸̦̗͐^̷̲̜͍͌̈́̚̚n̸͚͇͕͛͐͐ ̸̬͇̦́͊̒͒̊̍͌͠A̸̟̻͔̻͑̓̋͆̈ͅ ̸̢̯͈̞͈̝̣̓̓͊̓̓͘͜͠͝+̸̨̙̟̺̺͈̯̦͛̈́̉ ̸̧̛̮̙͂̾̐͝͝\̶̠̖̣̊̽͗͆̑̚̕͘ṫ̸̗̘̽̍̈́̑̀ă̵̠͆n̵̙̍̽͒̿͛̏̔͝^̶̡̈́̂́̑̕ņ̶̱͉̜̿̀̽͋̇̅͆̃ ̵̘̯͚́͊͑A̵͇̲̺̤̜͆̀̋͒́͘͝͠</math>̷̨̨̺̳̝̯̓ ̷̻̞̮͍͋́͐͜i̶̪͔͖͈͐̅͜͠s̶̛̤̮̫̉̒̿͑͌͑͒ ̸̖̓̓a̵̖͈͖̟̤̹͇͍̓̉̈̏̉̀͠ ̶̭̯̳̳͋p̸̢̟͋͋̆̀o̵̟̫͔͇͍̬͎̞͆͐́̋̓s̴͍̩̏̓̽̚į̴͚̩̳͓̿ṭ̸̻̹̗͎̇̒͜ḭ̸͇̣̲̒͒̂ͅv̷̝̦́̌͊̀͝e̵͎̜̻̩̱̣̓͜ ̸̠̰̻͕͓̞̟̅̆̈̎́̋͘͠ī̷̛̩͕̗̪̮̅͗̕ñ̸̠̻́̇̅͘t̷̠̬̜̀͠e̷̩̹̝͙̾͆͌̀̑͜͝g̴̼̲͍͇̘̋̔é̵̺̯̭͊ŕ̷͉̞̤͉̎̓̅̽̔͆͋ ̷͔̞̪̟̗̼̄̚͝ẅ̸̨̟̗̰̹́͋h̶̭̾͒̍͊o̶̜͊͐ṡ̷͈̯̫́͘ě̷̩͒͛̒ ̸̧̢̧͓̬͊u̵͓̻͂̇n̴̢̠̞̹̼̲͖͇̈́̏̇̕ì̷͜t̶̙̦̟̺̣͉̪̙̍̀̓s̵̢̋̈́ ̷͕̤̮̺͎̩͇̿d̴̗̩̹̣̹͋̂́̃͋̏͌͝i̴̢͍̯̝͈͈̞̾̍̽̇͝g̷͎͐̃i̶̡̡͚̜̊͌̌͜ţ̵̙͉̓̋͠ ̶̨̤̗͕͍̥̺̾̋̈́̈́͒̉̕ĭ̷̻̇̿s̸̭̟͍͙͙͚̟͒̂̎̂ ̴̝̹̏̈͒̽<math>̷͍̹̚9̷̪̖̩̯͇̏.̶̛̗̪̝̖̋̊̾́̆̀̐</math>̶̛̤͉̀̈͐͋̉̃͌
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Let <math>A</math> be an acute angle such that <math>\tan A = 2 \cos A.</math> Find the number of positive integers <math>n</math> less than or equal to <math>1000</math> such that <math>\sec^n A + \tan^n A</math> is a positive integer whose units digit is <math>9.</math>
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̵̢̛͖̜͇͍̟̫̋̑̈́͘ͅ
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̸̡̣̰͚̐̂́͌[̴̼̰̟͕̓̈́̌͐ͅ[̸̗̭̬͙̓͒2̷̛̳̈́̃́̕0̵̢̛͇͓͂2̸̖̺͇̮̝̳͇̈́3̵̰̳̗̞̤͈̇̑ ̴̝̜͎͚͇̎̔̍͂͐͑̕A̴̮̯̟̩̗̻͒̌͊̒͘I̷̢͖̙̮̍̏͝M̵̱̘͕̤͎͈͋̈́͐͗͝Ę̴̻̱͛̍͐̇̋ ̵̬̻̤̫̂̾̒̋̅̾̍ͅI̷̠̼̰̐͋̈́́̈́̂̿̌I̸̬̔̾̔̑́́́ ̷̛̹͇̼̞̩͗̐̅̍̓̿͘͜P̷̬͍̲̟͉͖̬͂͜ŗ̴̻̘̱̞̄͌̑̒́̇͝͝o̴̜͊̔̏̾͜b̷̘̟̫̟̦̼͑l̶̜̺̂̊ḛ̵͇̥̟̬͓̝̑̆̈́̃͋̒̐m̷̟͖̤̦̳̐̈́̔̄̇͝ŝ̵̡͇̬̲̔̅͂̔/̵͍̳͚̙̪͉͐̓̆̃̄̕P̷̡̡̰̞̲̎̃͐͌́͆r̸̤͙͚̰̖̤̬͋́̈̿̌̐̕̚ǭ̴͓́̑̒b̸̩͔͍̂̍̒͒͌͠͠͝l̵̡̺̮̩̥̱̫͛̊̈́̉̔̒̚̕͜ȩ̴̰̣̯̻̩̹̉̈͑ͅm̴̳͉͇̤͈͈͛͝ ̷̯̬̬̐̔͗̇̍͑̕1̸̜̰͉͚̗̰͑̾̾͒̒͘̕͝3̵̨̡̧̞͓͔̝̀͂̋͌̈̍͠͠|̵̰͇̪̖̽̌̓͜͜͝S̵̬̯͓̟̞̫̈̏͂̕ȯ̴͖̌̿l̵͖̾u̸̘͂̍̆͂͋̇̒̚t̷͙͚̱͎͆̆̿̏͘͘̕͝i̴̜̊͒̎ǒ̴͔̫͉̼͝n̶̛̪̾͝͠]̷̩̙̆̅̚]̸̜̐̉̽͌̿͝
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[[2023 AIME II Problems/Problem 13|Solution]]
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̵̡̭̻̗̻̘̗̥̽
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̴̗͓̰̰̤̳͓̀͘≠̬̙͍̭̣̰̀͐̊͜≠̮͈̠͇̳͉̙͌̅̌͝͝Ṕ̸͇̖̳̲͙̣͈̗͑̐͛͆ṟ̴͎̭̊̑o̵̧̐̐b̶̺̫̩̫̜̻̊̃̾͑̈̋͜͝ͅl̶̪̑ě̶̢̺͘͝͠m̴̬͕̱̙̠̟͈͛ ̷͇̳͉͍̉͒̉̽̇́͠͝1̴͖͔͐͊̌̽4̶̺̺̪͑̃̇̈́̊̈́̋̚=̴͚̉̂͑̊̈́=̶̛̝̹̰̹̖͇̊̏̒̿́̚͜
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==Problem 14==
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̶̜̱̰̭̊͋
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̶̭̖̮͎̇͛̅̑̔͌̍͜A̷͓͋̄͊̾͆͛̓̈́ ̴̻̪̮̙͙̭̣̓̀͐͊̅c̴̞̖͘ṳ̷̢̧̧̯̟̗͕̈́̚b̷̻̱̯̳̳̋́͊͋̃́̒̿ȅ̴̟͕̜̒͊̋-̶̲̺͊̈́̈́͐̈́͝s̸͔̤̯̲̈͊̚̚͠h̶͈̘̝̩̹͙͘á̶̩̩̞͈͙͓̾̉p̵̡̨̢̪͙̤̾̌͜͝ę̷̢̝͖̯͔̹̫͐͂̈́̏d̶̛͖͉̩̂͌̈́̑̋̓͠ͅ ̵̗͚͖͉̐́̔̀̓͑̒c̵̬͙͚̘̩̹̊̔ͅo̸̭͎͇̘̺̔́͒n̴̨͈̗̲̣̪͈̈́̄t̴̡̃̍̄a̴̧̱̞͗ḯ̸̝̯͕̆̚͝n̵̥̍̇e̸̢̺̗̘̔͘ͅr̷̟̬̼̠̗̞̻̊̎̽̉͑͜͝ ̵̛͈̱̬̥̈́̅͛́̅̆h̶̠̲͆ả̸̢͈̬̦̗̂̉̌̒̕̚͠s̶̟͔̮̤̺̹̘̣̑ ̶̢͔̗̂̈̽̕͝v̵̧̻̜͖̉͐͑̕͝͠e̷͔̪̒́͂͆͂r̶̭̼͇̘͈̿̑͐̾̆́̓̕͜ͅť̷̗̙̙̹̒̓͌̂́̏͝ͅḯ̴̻͂̓̊̄̈́̋̂c̷̡̳̑e̴͕̭̙̣͆͆̾̒̍̅ŝ̷̮͓͇̋̽̎̕ ̶̧̥̮̄ͅ<math>̶͕͍̹̟͎̀̄̐́̊͂̑̇ͅA̴͈̲͙͒̒,̸͚͓̖͇̳̣̇͗</math>̵̞̜̻̫̺̯̰͊̓̒̍͆͛̚͘ ̸̖̤̄̓̎̓͋̕ͅ<math>̷͇̪̻̳̭͘B̸̞͓͈̬̖̘͓͈͊͗,̴̡̅́̿͗͑̈́͝</math>̷̧͖͑̊͗̆̕ ̸̧̹͕͚͉̮̅̃͝͝͝<math≯̫̣̾̍C̷͈̉͊,̵̞̪͕̻͛͠</math>̷̳̼̘̝̰̽̆́̓͝͠ ̷̨̥̝͍̽̒ä̸͓́n̴̨̧̝̯͌̈́̾͑̉̓̇͝d̴̫̙͕͉̒̀͌̓̔̒́̃ ̸̗̣̗̫̎͌́̂͂̿̆͘<math>̵̦̳̘̹͍͍̠͆͆̆ͅḒ̸̭̖̜̦̠͛͒͜,̶̢̮̘͆͂͜ͅͅ</math>̴̹̳͈̟͐̌̋̔̈́̚ ̴̢̦͎̔̔̅̕ẃ̷̨̭̮͌͐̈́͑̓͝h̴̢̰̦͖͇͊͗̂e̸̪̙̝̪̠̬͇͔͂͝ŕ̶̳̤̀ȅ̴̯̬͎̪̆͐̇̏͑̐͝ ̵̡̠͚̩̽̐<math>̴̛̱̦͉\̴̧̣̭̬̌̃͋̊̑́̇ö̴̱̻̺̀̽̉̄͝v̵̛̘̪̲͚͙̗̘͔͋͛͑͌̈́͠e̶͔͗̐̃́̈ŗ̵͍̟̦̘̪͓̂̍́ͅĺ̵̦͚̬̣̗̎́̽̇i̴͙̲̋̿̂̈́̅n̶̛̮͙̔̊͋͆̚͝e̴̜͐͒͊͌͘{̵̧̬͋͋̐̉̽͗̽͜͝A̶̮̭̹̮̩̦̣͗̃́̈́̀̌̌B̴̝̉̅̐̀̑͛͗̅}̵̦͎͍͖͇͈̋͝</math>̷̢̞̬͉̫̤͚̱̃́ ̶̼͎̪͉̰͑̊̓̐ạ̶̢̡͖̫̐̿̀͘͜n̸̢̘̼̳̩̰̏d̵̝̑͂̏͗̌́̕͝ ̵̲̏͂<math>̷̡̗̮̝͔̇͂͠ͅ\̷̤̭̝͙̙̰̀̈́̆͠ơ̵̜̠͎͙̥̘ͅv̴̬̯̀ę̶̙͌̐̈́͌͂̕͝ṙ̷͔̹̥̩̞̥̣̀l̷̡̨͖̆̎ḯ̸̫͓͎͖̲ņ̵̛̤̣̬̬̪̥̜̎͊̅̊̀e̵̢̢̢̘̻̳͆̍̅ͅ{̴͙͉̳̤͂͛C̴̝̭̮͊̀ͅD̸̛͈̓͛̿̽͐̚͠}̶̡̛̳̘͙̈́́͌̾̏̕</math>̷̖̯͊ ̶̹͉̦̤͍̔͗̈̀͜ã̷̢̮͍̙̥̥̘͘͜r̸̟̭̐̿̓̈́̀͘è̴͕̐̂̍͋͂ ̸̜͊͆͗p̵͔̻̏̐̆̆a̴̘̝̦̐̔͆̊́ͅr̵̡̦̥̯̗̗̯̙̋̋̇̎͆̒a̴̯̍͑͑̍͛͛l̸̞̇l̸̢͔̱̻̪̞̏͊̆̏́͒͠é̷̤̭̣́́̎͌͌͝͝l̷̨̠̙̫̞͍͚̻̀̿̅̎̇͗̆ ̴̰̳͚̲͔̖̏̓̌͊͌͐ë̸̟̥́͌d̷̨̡̨͚̘͓̙͗̋͑g̸̼͕͎̻͐̏e̴̡̤̹̘͆̐͒͜s̶̥͍͔̼̮̾̿͑͌̊̉͘ ̵̮̤͚̫̦̥͇̌o̷͎̦̤̦͌͛̾f̶̮̏̇̿̔̒͆̍̄ ̴̧̡̩͚̟͓̥̩͒̈́͗̃͘ṫ̷͈͉͔͇̔̓̅̍ḩ̵̦̜̠͓̊̅͜e̴̹̼̠̅̊́ ̸̰̜̱̽̓c̴̢̧̢̲͖̖̪̱͗͒u̴̞̫̗̗̭̳̳̓̏͒̿̌ͅb̴̪̙͙̋ȅ̴̛͖̙̱̘͆̔̈́,̸͉͑͒̔̍ ̵̡͈̗̭̉̽̎͌̃͑̎̈â̸̡̿̅̀͒̕n̵̦͔̪͉̠̟̄̌͗ḋ̸͚̑̌ͅ ̵͕͇̺͌̀͒̓͘ͅ<math≯̖͝\̶̨̭̙͇̣̱͖̿͜o̵̰̬̠̎̀̇̿͊̽v̴͓̰̩̩̈́̈́̋͛ẻ̵̡̝̻̫͋̋̋̈͘͠r̴̡̪͕̪̅̋͋̌͜͠l̸̮̦̙͎̀i̵̢͈̥̲̱͔͎̒ǹ̸͖͓͎͉̗͌͝e̷̩̹̮̯̣̊̆̅̃͑{̵͕͉̝̦͕̞͊͛̿̾̃͜͝͠ͅĄ̴̦͕͙͚̘͕͓̅̈́̃̇͛C̵̦̳̀̍̆̓͝}̶̝̩͔̦̓̎̔̏̀͘͜ͅ</math>̵̭̩͍̦̐̿̒̃̾̀͘ ̴͈̇͜á̴̡̠̤̬n̴͕͋d̶̤̤̥̿͑͝ͅ ̷̥͔̳̌͊̈͒̑<math≯̻͒̇̐̑\̶̛̥̭̮̬̋̈́͊͊̐̌̕ṓ̶͎͋̒̚v̶̛͍̰̗̝͑̐̃̈́̚̕͜͝ē̷͇̖̼͉̰̻̘̬͗̊̊r̵̨͔͑̓̐͗̌̽̇̕l̴̼̝̖̯͊́͝ỉ̵̲͙̰͇̔̂̚n̵͇̳͍̠͘e̶̲̜̩͂̆̑͌̀̊̔̏{̵̛͇̗́͌B̶͉̮̂̽̾Ḑ̸̠̿̈́̌̂͊}̴͍̂͘͝</math>̴̜̘͙͓̖̯̰̰̔̑͊̍͒̊ ̷̡̫̞͎̃̉̾̕͝a̸̲̦̓̆̄͑̉̈̓͗ŗ̴͍͈̫͈̺͇̰̃̈́͘è̷̢͚̻͜ ̷̢̪͈͙̰̣͖̕͜ḑ̴̰̜̙̥̠͍͑̅̀̋̉͝͝i̶̢̙̞̤̪̍ḁ̶̘̀͒̅̆̀g̶̡̧͙̼͇̥̈́̀̄̀̍̆͂͝ơ̵̲̖̩̈́̌̽̒̈̅͜n̸̼͎͑̉̀̀̚ą̵̨̞̦̟͔̗̅͆͠ľ̵̪̥̱̪̬̅̉̈̃͋̊s̷̺̞̳̭̔ ̴̧̹͔̤͔̳̙̂ŏ̶̘̮͕̲̮̳͚́̽̂͆͝f̷͉̮̂ ̴͇̲͔̝̠̓͆̇͘f̶̱͉̟̎͒͝ǎ̴͚̝͐c̴͎̠̃̌e̵͕̼̣̺͑̽̄͠š̷̗̳̜͚͉̜͍̈̈́̀̌͘͘͝ ̸̬̂ͅo̶͕̟̠͂̊̄̀̈́̑͜f̴̼̪͓̥̍̏̐̋̔̐͝ ̸̥̼̬͖̭̃̌̕͠͝ẗ̴̢͙̩̠̬͖͙̀̓͛͛͜͝h̴̨͙͉̯͍̪̒̏́̈́e̸̱̟̪̖͍͐ ̴̣̻̘͖͉̈́͗̀̀̚͝c̶̛̻̺̗̥̦͆̓̚ͅứ̴̭̥̺̦̂ḇ̴̗͆̌̆ȩ̵͕̩̖̭̩͇̠̐̇̐͂̈́̿,̶̹͕̻̣̯̜͆͗ ̵̗̯̟͇̭̩͛͘a̷̧̪̰͇̽̿s̵̯̗͐͐̈́̽ ̸̜̯͇̥̜͈͑͐̈́́͘͜s̸͕͑̑ḩ̷̢͉͔͔̪̪̰́̎̀͂̓ȯ̷͍̹͕͇̟̳̗͚̈́̆̉̒w̵̧̌̒̍͑̌n̶̥̱͚̩͎̣͋̉͊͆̚͝͝.̷̺̤͙̥̯̻͎̔̓̈́̔̒͊͜ ̶̩͍̮̯͌̍͆Ṿ̶̧̛̱̜̰̘̮̯̑̒͐̏͘͝ę̴̲̻̹̮̩͇̓̒̋̐̔͋́͌ŕ̶̦̳̭̲̯͉̗̎t̶͔̣̰̥̫̐̀͂̌̀͌̚͝e̷̮͐̀̅͛̌́͒x̶̡͓̩̩̯̅̊̔̚͝ ̷̜̦̀̾̐̔̚<math>̴̧̨̻̼̳̉͒̀͝Ả̵̛͎͈͛̄̎̂͜͝</math>̶̙͉̩͗ ̸̧̙́̆̓́̎̽͘͝o̷͓̦͂͒̋̉͒̏͜͝͝f̸͔̙̙̭̩͔̦͂̿ ̴̰͍͒́̎ṭ̶̨̨͓̟̗̮̍̉̎̒h̴͓̯̗̻̫̩̙͐͝ȇ̶̛̠̮͆̌̏͊͘ ̴̭̀̓́͗̑͋̄͝c̶͔͆ư̸̇̋̀̒̎̕͝ͅb̴̰͖͎̒̏͆̓͘͠ẽ̷̟̣̮̼̩̝̉̔̑̔͝ ̴̮̣͔̪̤̦͕̋̔̐̎̾́͝ͅi̶͚̙͈̰̬̇̔̊̓̎̕͜͝͝š̷̹̝̈̎͐͆ ̴̧̛̦̲̜̘̔̍͑̑̌̃̕s̸̙͂͂̔̚ȅ̴̠̳̳̦̺̘̳͉̅͋̀̀̓͊t̵̰̟̖̭͔̻͖͂́̓̀ ̸̨̟͍̯̦̂͊̋o̵͉̼̞͙̐̒͑̓̕͜͝͝n̶̺̬̯̠̲͉͇̂́̇̓́͘͝ ̸̨̡̥̠̯̓͂̈́̎͐̊͘͜ͅa̶̧̢̤̲̻͙̾̾̀̓̊̓͘ ̴̧̨̩̬̣̓̅͋̏̋̉̀̓h̶̛͕͇̗͖͚̫̮̘̒̈́̏̑̊o̶͍͊͘r̶͓̝̦͓̅̿̑̀̉͂̕͝į̷͇̻̬̳̞̬̰̇z̶̛̭̞̱͊̉̇͗̇o̴̲͑̂͛͗͆̉ñ̵̯͔̊t̶̳̜͖̣̓͑̃̍̕̕ã̵̤̮͖͉͙͉͉͙l̴̛͚̮̻͉̙̇͝ ̶̛͓̯̘̗͎͖́͗̈́̅͆̿̀p̵̨̗̹̪̓̏͑̏͌̓l̸̞̱̱̹̈́̐̅͑́͝͝á̴̟͉̑͒̍̂̿̊̆n̵̹̳̑̿̆͋̋͑͆̕ͅe̵̬͈̓ ̵͕͈̠̐̏̃̿͝<math>̵̛̙͉̖̐̒̉͂̆̍͝\̶̢͔̫̭̺̹͒͊m̴̻͇̟̈́̈́͆͛̈́̈́̚͝ͅä̶̗̟͖̘̰̳̑̓̍t̵̰͓̪̗̙̰̰̒ĥ̴͚̑̑͌͑̋̏̾c̷̤̀ǎ̶͔̦͈̳̈́́͆̑ĺ̵̡̛͙̙̫̙̈́̿̀̚͜͜{̵̨̳̠͍̻̥̞̰͌̑̋̽̍̆̕P̴̬͎̝̯̞̖̜̒̓̈́̋̈́̉̚͝}̸̬̖̺̟̞̒̈́̂̚͠͠</math≯̧̫̖̂̔͑̚͠ ̸͕͇̼̩̤̐͌͜͝s̷̪͖̣͍̥̠̙̑͑̄̀̂͝͝o̸̥͇͍̺̱͓̓̇̅̀̃́͌͜ ̴̮̬̤̬̣̭̹̜̐t̵̘̠̬̳̹͎͓̿̈́̄̌̓̏͐̐ͅh̴̙̍͗̔̊̕̚a̴̛̙̐̄̍̏͝t̵̯̾͋̇͒̏̌̆͝ ̸̩̞̳͖̤̑̿͗͐ͅt̷̤̜̱̝̰̿̄͊ͅh̶͓͈̰͙͕̳̅ͅe̶̢̿̈́̈́̔̄͌̚ͅ ̵͇̖̬̔p̶̨̯̯̺̖̖̲̟̌̐̐͛̽̀́̓ļ̵̢̤̺̮͖̭̇̉̉͛̆͌̔͝a̶͚̠̤̩̤̲̤̹̾͌̔̈͝ṅ̵̼̣͙͙͚͎͊͐́͆̏́̀ͅͅe̶͔̹͉͎͝ ̵̝͚̻͋o̸̬͇͛͋͛̚̚f̴̡̜̪̦̻̯̋͗̈́̀ ̵̛͚̩͍̬͒̑̋͂̓̃̚ţ̵̧̘̼̜͙̫̙̈́͛́̇͊̀͐̊h̶̭̥͍͙̗̭̲̓͛̑̎̊ͅể̷̖̖͒́̈́̋ ̸͓̲͒̇̔̀̃͗̕r̵̨̛̗̥̙̣̊̌̓͜ė̶̱̺̝͇̝̟ć̴̢͙̩̜͎̹̥̮̔t̴̲̝́͑̈̕͜ạ̴̈́̈́́ņ̵͇̮̩̻̥̄̽̅̚g̵̨͉͎̖̐̏͑̎̂̒͝ͅl̷͈̫̪̑ḙ̸̡̧̡̩͓̖̠̒̍̀̌̈́͐ ̶̢͙̮͂͋͋͒̊̓͜ͅ<math>̶͖̚A̵̪̎͑̐͝͝B̷̡̬̯̦̬̪̃͂̈̍̓̔͜͠͝D̴͕̞̼̮̥͆̀̊͋ͅC̶̡͈͖̊̚</math>̴̡̛͓̬͔̱͎̖͛̈́̈̄̈́͝ ̵͉̥̳̅̄́͊̀̈͋̕i̷̮͑͌͊́̌͘͝s̸̤̺͙͍̹̾ ̸̩̜̥̫̻͎͊͂̃́͊̄̂p̴̨̢͇̬̲̫̥̈́̄͝͝ë̶̢́̎̋̑͘ŕ̷̛̗̗̻̬̿̀̀p̵̨̝̝̿͗͒͛̉͘͘é̸͈ň̴̥̝d̷̢̛͕̰̻̺̤̍͌̉̅̈́̓ͅǐ̷̻͛c̵̡̳̤̘͚̳̥̕͜ư̷̲̝̹̗̠͂̊̈́̕l̵͇͗̿̌͂̐̐͘a̴̖̳̥̣͒̈̿r̶̼͎̰̼̟̭̘̲̔̃́͂͐̉̐̚ ̴̖̮̹̋̎͋ṫ̷̼͔͙̩ő̴͕̀̋̓ ̸̛͉͖͍͙̓̎̂̋̏͒͠<math>̷̜̠͔͔͚͓̙̍̾̅\̶̨̂̓̒̓̇͘m̷̖̟͙̲͖͔̪̂̈́̅͆͆a̸̡͖̝̖͈͙̋͛ţ̶̢͎͇̪͔̳͉̎̀̆h̸̡͚͔̎̀̊̔c̵͈̩͈͕̣̟̗̻͒̈́̇͝a̶̼̪̺͎̫͗̈͛l̷̨͝{̷̨̨̭̙̰̰͓͗P̴̥̥̜̼͚̻͂̎}̸̢͚̝͂̂̇̉̒ͅ,̶̖̞̞̜̝̗̟̬̍̈́̂̃͑̎̍̕</math>̷̘̫̤̺̓̈́̈́͝ ̸̝͚̰̈̎̽̅̏͌͝v̷̫͑̍̐̈́̈́̍̓̊e̷̝̜̬̱̯͊̐͐̾̾̇r̴̭̲͎̹͇͑̈́̊̌ť̸̡̤͙͒̋é̵̖̼̙̓̀̎̄͝x̴̧͚͓̬̺͌͋́̾̊͆̋͒ ̶͈͓͈͔̭̤̉<math>̷̡̤̤̫́͌̏̅̇̌B̷̰͍̪͂</math>̶̡̿̊͂̈̌̉́̿ ̷̡̡̭̜͕̀̌i̸̢̡̞̠̠͉̙͙̾̾̂̀͘ş̵͎̳̞̪̮͔̑ ̷̨̗͈̭̯̳̲͊̏́̈́̈́̚<math>̵͖̓̒̄̿2̶͈̬̬̰̈́͛̀̑</math>̶̛̳̍̀̀̈́͊̎͠ ̶̯͛̄̌̍̓̀̓̀ͅm̷̞͕̞̓̊̎̕͝e̴̹͍̓̄̋͆̾͘t̸̪̰̼͛͑̊̇̇̒̓é̶̘̳̦̀͛̌̔͝r̴̛̛̗̱̟̩̯̳̞̅̾̑͘̕̚s̶̯̆́̍͒̎͊͝ ̷̻̓͐͗͆̈͑͘ą̷͎̙̖̩͈̳͓̐́̔̅̉͌͂̌b̷̠̮̒͠ọ̷̖͗̒̀̿̏̂̒v̶̨͍̮͙̫͓̜̀̃͌̐̂̎̊͐e̵̩̭͎̒̀̄ ̵̤̜̰̏̂͒<math>̷̨̺̲̖̻̞̙̹̍\̵͎̺̣͍͕̉m̵̛̗̖͕͕̩̺̈́̎̀̀̈͛͘ͅą̷̞̯̯͖̍̉͜͜t̶̡̡̛̛͉̻̩͖̑̂͋͛̋ͅh̴̛͎̥̗̲̘̪̺̾̉̔̕̚c̶̝̈͗̐a̶̢͇͕̪͋̕l̵̗̤͚͎͈̽̈́̎͒̾̈̄͝{̵͇̑̈́́̀̑̕̕P̴͎̩̭̠̤̦̣̔̄̿̒͘̕͝͝}̷̧̟̯̓̂͗̍͂̀͘͠,̴̪͇̞̥̙̗̦͌̿̆̔͝</math>̴̧͉̣̣̟̗̃͒̄̉̚͠ ̴̥̀̃͂͂̓́̒v̵̲͔̞̰͖̲̟͑̿̐͆̆͠ͅė̴͇̫͔̯͇͕r̵̩̭̙͚̱̯͒̆͌͝t̵̲͔̂̔̽̌̈́̾͋ȩ̷̧̏̓̄͋x̵̲͉̣͈͍̜͈̎̒̄̌̎̇ ̷̝̮͂<math>̴̲͉̗̼́͊̕C̶̛̮̲̹̠̪̳̦͐́</math>̴̡̥̤͚̤̥͍͉͗ ̶̧̯̤̹͉̈́̄̍̍͑̉͗̚i̶̳͓͖̬͑͑s̶̪͚͍̖̔̚̚ ̶̝̟̳̠̮͋̆<math>̷̡̇͛̈́̉̌̕8̵̡͉͓͚̙̜̅̈́̍͐</math>̷͖͓͌͑̈́͛̋̾̈͘ ̴͚̗̮͒͌͑̈͌̃͜m̷̭̋̓̏e̷͈͚͕͐̽̅̚t̶͉̻̳͖̺͔̰̓̈́ͅę̴̼̬̬̥̹̃̄̿̀͜r̶̺͉̘͌͆̅̓̋͆̉̀͜s̶̛̱̙̱̺͙̞̤̓͜ ̴̧̯̻̗̤̭̦͓̑ä̷̧͓͚̥ḃ̸͍͔͙̥́̀o̶̪͎̹̼̘̔̑̾̌v̷̟͌̈ẽ̸͖̲̻̠̰͕̞̈́̆̓́ ̴̮̭̹̜̖̤̳̂͗͒̀<math>̵̛̛̯̰̳̹̞̼́̈̀̔̐ͅͅ\̵̡̭͚͆̑́m̷̝͚̝̙̗̮̪̩̌̕͝a̵̛̠̻̹͉͈͖̿͗̽̽͘͝ţ̶̨͙͕͈̥̈́͊̈́͝ͅh̴͇͓̘̫͗͆́͋͘͜ͅc̴͚͆̊́a̸͚̮͉̺̅̀̽̾̈́͝l̸̗̫̕ͅ{̴̝͎̝̫̤͗͌͌͒P̸̢̫̥̙͓̺̆͛̄̆̕͝ͅ}̸̳̪͇̙͈̣̀̍͋̔̽̍̕,̶̟̀̈́̔̀̿̅͠͝</math>̷̨̹͍̔̿̍͌̐͝ ̴̹̣͈͓̪̗́a̸͙̜͆̅͒̒͒́̕n̴̢̙͇̰͉͗͂͑͑d̵̢̧̛̠̹̖̱̝͒̊̌̄̐̀̚ ̴̰͎̳̥̘̺͙̩́͗̈́̊̈̕͝v̷̢̞̭͂̐͊̈͘e̸̢͚̪̟͚̗̓͊̏͝͠ṙ̷̛̙̮̩̀͂̾͜͜͠t̴̠̽̈̽͒̽̎̕e̶̛̫͙̦̝̺̎̋̈́̿͐̂̀͜x̸̢̟̻͙̱͈̙̀ ̵̨͎̗̣̲͖̹̊̿̌̀̀<math≯̤̖̜̤̲̺̀̓̈́͒̄D̸̝̮̍̏́̎͝</math>̵̡̨̡̬͓̳̼͊̌̒̆̚͝ ̵̼̗̠̤̘̉͜͝ͅȋ̷̩̺̘̬́͆̈́̏s̶̖̤͚̠͈̔̎̿͆̉̕͝ͅ ̴̛͚͚̲͎͐̃̈<math>̶͖̬͇̘͕̮̀͜1̶̡͚̪̱̱̖̂̀̍̄̿̿͂0̷̫̖͈̖̘̜́̾̾͋̽̀ͅ</math>̴̖͉̪͒͊ ̸̛͕̭̝̤͐m̵̩̰̘̣͕͎̩͓̂͘͝ę̸̳̮̭̱͐͜͝͠ţ̷͎͖͖̞͙͐e̴̳͍͇̳̺̣̒̎̅̈̽̓̈̈́r̴̥̝̄͂͝ṡ̷̺͕̖̜̣̕ ̸̭͙̞͚̠̭͑͜a̴̢̡͎̽̂̒̕b̵̫͙͇̳͐͊̔̀̋o̷̢̒̿̿͗͋̕̕v̴̧͔̰͔̉͑̔̿̈́͑̚͝ḛ̵̼̳͉̼̩͈͝ ̵̳͍́̋̆̀̎͗͒̈<math>̷̡̧̼̬̞̱̓̄̉̐̚\̶͚̝̝́͗̽͂͋͊͝͝ṃ̴̹̠̼̏͂̏̂̂̆̀͘ą̶̭̩̩͇̤̼̬̈́̀̌̕ţ̷̲̰̞̦̦͇̖̂̃̎͂̇h̸̯̑͑c̶͇̥̙͍͕͖̮̐̆͐̕a̶̳͙̫̙̍̿̆͌̑͛̀ͅͅl̸̨̩̔͂̎{̷͎͂̏̄̕P̶̟͔̯̊̓ͅ}̶̤͙̳̟͚̾̑̍.̴̘̜͌̊̓̌̏͜</math>̷̮̏͂́̽͜ ̸͙̗̫͈̖̜̈́Ť̴̡͇̟̬̮̼̦̻̈́̏h̴͖͕͛̄e̸̱͚͋̓̓̉̍̌͘͠ ̴̘̰̱͂̏̍̓̐̍̓c̴̛͓̭̫̃̔̄́̐̒ȕ̷͇͈͍̤̖̮͒b̸̧̰̼̱̒̓̎̈́͗̒ẹ̵̡͓̎͊̒̏̑͗͝ ̵͓̹͔͕̩̌̽c̴̢͕̟̲̬̭̮̿̂̋̋ͅo̵̢̮̪̣̩̠̅n̷͖͕̺̖̹̳̏̇͋̐̋̆̚̕t̸̩͚́͐̔̅̊̕ả̸͔͉͕̦̹̈́̈̌̐͝ĩ̷̪̆̿̀̌͝͝n̵̢͕̬͎̙͒͂̕͘s̵͕̈́̚ ̷̧̡̠̘̱̩̼̌̐̓̂̄̑̿̃͜ẃ̸̡̺̼̭̀͌͋̂̍̍a̴̞̝͈̮̺̜̤̿ţ̸̢̮̙͚͝è̷̙̬͇̎͗r̵̨̰̠̰͇̠̤̺͗͂ ̵̛̲̀̆̆͋͝w̶͚̪͈̼͔̩͈͑͑̕͝h̸̢̘̝͎̮̼̞͊́̿ỏ̸̧̨͓͜s̶̙̖̦̬̻͚͇͗̏̇̈͌̑͛e̶̡͖̪̘͉͚͗ ̷̨͇̻̻͙̲̬̇̒̈s̷̤̮̞̰̾̌́̃ͅu̶̢̝̼̪͚̱͎̇͊́̇r̷̪͔̿̃̃̊͐f̷̙̝̰̝̬̩̟͚͊̍̂̐́͒̚a̶̻̖̜͔̩͛̽̈́̔̊̚͘͜͜͝ç̴̟̫̪͖̍̓̊̊̚ͅë̶̗̘͈̭̺̅̊̋͒̀͝ ̴̢̞̗̪͎̇̌̔̍̒̈́͒͛ỉ̸̘̺̺s̴̢̢̝͓̹͖̞͍͋͒ ̴͇͈̇̀̉̓͛̃͘͠p̶͕̻̼͈͕͓͒̈́̉̉a̴͓͕̮̫͈͙̜̓̈̂̑͌͝ͅr̴̡̹͇̄̆̒̕̕͝ǎ̷̗͚̦͛ļ̴̭͖͓͔͊̔̀̒̽ͅl̵̨̮̗̓̈́̇̑ȩ̷̛̬̞̹̐ḻ̴̛̖̥̟͈̥͍̹̊̂̀̔̋̋̚ ̵̢̪̲͎̭͕͍̞̃̆͗̀͋͠ṱ̵̫̪̜͙̻̰̄ó̶̩̞̤̗̮̉̓͌̈̈̚ ̶͕̲͚͆̐̂͛̓̕<math>̶̖̗͉̜̖͙͚̈̇͒̐̉̀̋͘\̵̧̠̰̱͈̈̈́̍m̵̜̥͊͌à̵̡̛̛̟̝͒̑̈̓t̷̡̻̠͓̙̀̃ͅh̴̝́̄͘c̷͎͎͇͇̬̗̀͐ǎ̵̳̝̤͉̘̊͘ļ̵̛̑{̴͕̘͂P̸͉͎̻͓̈́͛̏̀͘͝}̷̹͔̤̦̠̩͖̏</math>̵̢̛̞̲̒͗͐̌̔̒̕ ̷̛̗̥͈̻̗̯̒̇̈́̏̐͠͝a̵̹͌̽̓t̶͍͒̋͒̅̚̚ ̵͇̟͓͔̤͈̆͊̚͝ͅȁ̷̻ ̷̖̻͕̮̤̰̀͐̓́̍̆ͅh̴̹͍̍̇͐́͘͝ḛ̶͓̳̠̳͛̀̔͗͜ͅi̸̦͋͊g̵̗͕̟̦̪̔͒̈́̀̽́̀͘͜h̵̫̫͗́́͘̚͠ṱ̷̣͇̙̭̄͜ͅ ̷͇̪̈̉̊̍̔̈̊͠o̸̦͍̒̉͜f̶̏̋ͅ ̶̮̈́̅̋̑́<math>̴̜̥̇̒͒̀̊͋͂7̶̜̀̏̀̓</math≯̖̫̹͓͍̌̔̆͛͐ ̴̩̭͒̇̓̄̐͝͝ṃ̵̢̡̱̎͘̚e̵̝̻͍̜͕͐̐t̴̳̜̖̥͎͇̮̄̀ĕ̸̱͔͚̌̒̃́r̴͙͚̹̠͖̩̐̋̑̓͊̈́͗̕s̵̟̼̹͗̃̄̀̄ ̶̫͓̤̖̉̀͘a̵͓̒̐͒ḇ̴̿̐̆̀͝ỡ̴̦̄̒̍̕͝v̷̢̭͚͖̲̉̌͜ͅe̴̤͖̱̳͑̂̌͘̕ ̸̡̧̝̯͚̐<math>̷̢̭͖̩̱͙̼͑̒͜\̴̛̱̙̱̰̈́͆̊͂̽́m̶͉͔̖͍̄̋͂͝a̵͍̝̟̝̤͠t̵̛̬͊h̵̙̉̑̍c̷̟͈͆̏͂̋̇̕͘ą̷̣̤͔̜̑́ļ̶̦̻̾̈́͊̉̀́͘{̸̠̜̒P̵̟̟̈̑̓̍͂͠}̶͔͔̪̈́͋̇̾̔̕͠.̸̭͔͊̏</math>̷̜͙͎̤͍͋̈́͆́̀̅͑ ̴̡̹̯̗̫̦̾̀́̀̓͆̄͛T̵̨͚̜̣̲̣̟́͂̽h̵̟͍͓͂̈́̈́̍͋͘ȩ̵̪̓́̂̈́͆͂̅ ̶͓̘͈̉̚v̶̫̌ͅọ̷̬̋̉̓̋͐ļ̴̗̣̙́̾͂͒̈́̕͝u̵̡̨͔͙͊ͅm̷̢͇̼̪̟̪̄͜e̷̯̟͈̞͉͖̪͛̃̄̎̎̾̚ ̵̧͉̊o̵̫̣̪͖͗͑̒͜͝f̵̝̲̲̑̀ ̷̝̠͖͍̤͚̞̯̉́́̅͠w̷̨̬̼̪̙͉̲̒͂̏́͠ͅà̷̖͉̜̊͊̀̚t̷̩͖͖͉̿̓̅̽̚͝e̷͈̣̜̲͖͛̈́̏͘r̷̟͕̪͖͔̫̓̈́ ̵͖͍́̐i̸̛̲̜̥͒̓͆͐s̷̹̯̩̫͊̓̄̆̀̐͠ ̵̛̤̣͇͙̺̐̎͛̊̕<math>̵͓̐̽̈́̍͐͛͗͝\̴͓́̎͜f̵̯̮̰͈̻̮̊̌̌r̷̯̮̜͙̍̅̑a̴̱̦̅̓͂͛̍c̸̩͇̺̭̾̈́͒̂́͝{̷̯͂m̴̡̙̻͗̈̈́̄̈́}̶͔̪̲͍̗̺̀̌͗͊͂̍͠ͅ{̶̢͓̻̹͋̈̇̀̏͌̄̚n̵͖̭̗̠̙̘̎̒͠}̷͓̆</math≯̗͍̜̉̈͗͊͊̕ ̵͇̣̻̙̱̯͌̿͐͠c̸̡̛̭̻̠̞̟̀́̚u̵̯͙͈̥͌̔͋̔b̸̨̛̭͉̭͈̫̬͈͒̓̽͗͗͊ĭ̸̤̔̈͑c̶̖̲̠̬͕̬̳̐́ ̴͉͕͙̱̬̠̘̅̑̀̃m̸̺̞͊ę̸̰̻͖͍͖͋̆͐ṫ̸͇͙̙͇̟̈́͐e̴͎̙͙̹͑̋̓̂̈́͒ŗ̶͖̺̲͈̟̥͛̽̈͝s̷̺͇̩̺̻̻̐͘,̶̪͔̗̟̖̭̥̞̓̀̊ ̷̗̤͎̳̦͍͑́͠w̵̧̱̾̓̐͛h̶̨̛̩̯͔̖̟͂͐͐̍͒͝ę̸͉̝̱͇͉̥̯͋̾̿̊̑̌r̷̨̞̃̔͐͑͌̆͐̒ę̴̗̜̦͈̹̩̆̎̄̈ ̸̞̪̞̖̥̟̦̉̐̑̐͐̽̕<math≯̬͔̿̔̋̃̔́̕̚m̸̨̤̙̼̠̭̲͊̂̐̿</math>̶̧̧͇̝͉̖͖̾͒̒̾͑̆̋̀͜ ̴̢̢̥̜̳͓͎̍̄͗̑̆̈́͜a̸̭̦͂̌͐̽̈́ṋ̷̜̮̩̲́̔̄̒̇͒̇̂͜ͅd̶̞͉̤̲̠̑ͅ ̴̨̠͉͖̖̈̀̒<math>̷̝̹̀̆̕n̸̨͕͎̟͉̟͚̂</math>̶̢̲̌ ̸̝̙̱̬̩̀̆̿̀̉̂͝à̶̬̲̉̆̿̈͝͝ŗ̶̣̟̝̦̱̳̩̾é̴̠͙̗̯̐̽́̽̈́͋ͅ ̴̯̞͒̓͜ṟ̴͇̺͈̞̰̾̇̋̄̂́ĕ̶̡̥̘͇̬̼l̴̤̑̌͂͗̀̄͝ͅà̴̛̹͕̠̄́͒̕͝t̴̲̳̘͇͆͛͜i̶̢̼̒͂v̵̜̼̈̀͗͋̾̏͋ẻ̸̲ĺ̴̜̲̏̏̔̔̚͜͝͠ŷ̷̱̾͒ ̵̼̬́͑̕͜͝͝͠p̶̧̞̹̻͙̻̊̎̔͑͋̓͑ͅr̸̟̯̩̠͚̮͌́̄͝ͅͅḯ̷̡̧̺̬̰̺̒̽̅̂̕m̴̯̂e̴͔͙̎͊͂̄̑̎͝ ̷̛͓̬͉͔̖͚̃̏̋̀̋̔p̷̩̰̼̯̽̃̈̋̕o̴̱̱̥̎̑̎̆͜͠ͅś̸̞̗̥̅̇͌̂͘͠ì̶͔̠̞͔̬̫̺̤̍̾ť̵̡̙̣i̶̡͇͕͇͉̼̭̎̌͛̾͆̈́͜͝v̶̨̞͙̦̪̯͎̼̆̍̓͝͠e̷̞̰̜̱̯̘͙͊̾̿͛̈́̆͝ ̸̢͚̮̞̣̥̝͗̀͊̿̊͆̕ȋ̸̧̾́̃̋ņ̷̛̥̤̠͎̈́t̶͙̝̂͗̐g̵̛̼͎̪͖̀͆̅͌͋̃e̷̡̧̞̰̝͙̘̙͆̈́̉̎͘͝r̶̡̢̘͎̩̻̽͜s̵͙̥̣̝̔̃̓.̶̧͎̳͔̲̫͇̻͂ ̴̣̮̩̒̈́̚F̸͓̫̃̍͑̚i̴̲̳̪̞̭͗̆̊̎́̕̕ǹ̸̬͈̝͎͈͜ͅd̶̗̒ ̷̢̡̛̗͈̲̼̹̏̎̚͠<math>̷̟̞̤̥̯̩̈́̓̓̉m̸̼͕̠̟͑+̸̺̼̩̻̚͝ͅn̸̤͛̃̎̐̓̏͑̆.̴͉͈̦̪͂̒͘͜</math>̶̫͍͍̬͉̥͈̤͌̆͌͊͠͝
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A cube-shaped container has vertices <math>A,</math> <math>B,</math> <math>C,</math> and <math>D,</math> where <math>\overline{AB}</math> and <math>\overline{CD}</math> are parallel edges of the cube, and <math>\overline{AC}</math> and <math>\overline{BD}</math> are diagonals of faces of the cube, as shown. Vertex <math>A</math> of the cube is set on a horizontal plane <math>\mathcal{P}</math> so that the plane of the rectangle <math>ABDC</math> is perpendicular to <math>\mathcal{P},</math> vertex <math>B</math> is <math>2</math> meters above <math>\mathcal{P},</math> vertex <math>C</math> is <math>8</math> meters above <math>\mathcal{P},</math> and vertex <math>D</math> is <math>10</math> meters above <math>\mathcal{P}.</math> The cube contains water whose surface is parallel to <math>\mathcal{P}</math> at a height of <math>7</math> meters above <math>\mathcal{P}.</math> The volume of water is <math>\frac{m}{n}</math> cubic meters, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n.</math>
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̵̢̡̝͈͎̩͈̠̂͗̇̍̊̈≮̡̥̺̬͊̈́̆̀̉͋̒͠a̸̦̺̻̬͒̋͝s̷̡͈̱̳̈́̐̌́͝ÿ̸̧̠̥͙́̃̑͋̌̓̽̆ͅ>̵̧̜̝́̎̓̄̕̚͘
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<asy>
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̶̨̨͔̰͔̳͚̱͌̍̓͘͝/̵̺̮̞͉̫̞̻͊͗͆̽/̸͔̰̘͙̩̥̾̋̆̚̚͜͜͝M̴͈̼̮̥͕̾͂̐̒͌͝a̶͂͜ḑ̶̣̳̳͎̮̳̓̔e̸͓͑͗̊̎̒̍ ̸̨͔͕͙̞́b̷͉͔̼̉̓̃͑͗̄y̷̰͎͔̘̰͊̋̈́̑́͘̚ ̵̛̬̅̎Ḍ̶̲̊͌̈̿̈͗j̷̧̩̞̘̈́͛͑͝m̷͕̋̈́̉̐͘͜͝ạ̴͎̱̾ţ̶̧̧͉̣͙͎̓̋̎͐͆̚͝h̵̤̠̥͂͒̔͗͗̓m̵̢̥̥̗͋̐̾͑̕a̶̪̠͕̱̣̼͈̓͋n̶͓̩͠ ̵̨͚̬͓̼͕̒̉̋̑(̸̞͇̘̼͇̝̝̜̕ō̷̡̖̠̓͂͛r̷̖̫̬̣̲̦̣̣̈̇z̸̢̽̄̍̀͘͠͝)̶̢͖̬̭͖̜̘̇̑̌̚
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//Made by Djmathman (orz)
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̸̡͕̼͙̭̔͆̿̕ş̵̟̘̑̎̀̈́î̸̭̀̾͝z̴̤̪̘̰̩͐́̒͜e̷̤͂̇̓̾̿̋́͝(̵̢̙̥̻̿̎͘2̶̱̱̓͂5̵͎͚̀0̵͎̝̣͔͙̆̈́͜)̵͙̻̍̓̐̌̀͠;̴͇̉͗̇
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size(250);
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̴̥̹̾̋͂͌̓̕d̵͎̠̤̘̫̭̗͗͝ͅe̸̙̣̼̝̭̻̓͋̿f̸̦̖̂̒̅̕͝a̴͉̞͈͚̺̩͂̓̀̿̌̚û̷̞͑̿̕l̵̪̀͛̈́t̶͎̮̪͎̫̩̰̩̐̌̓͐̊̋̀̂p̸̢̺̪̤̂̊͒̓͋ĕ̴̝͎͒͒͐̚n̵̡̝͈̯͙̜̩̠̏(̶͈̯̅l̸̲̙͎̭̽̓̒͒̆͌̌͘i̷̛̝̻͒͛̃̆̌̀̚n̵͈̖͓͇̾̑̃͗͌ͅͅȩ̶̛̞̮̩͇͕͍̟̔̓̆̽ẁ̵̡̛͙͎̪͔̹́̈́͆̉i̵̘͛̃̓͂̃d̸̯̣̤̿̉͌t̷̛̜͍͕͉͈̹̂͑͛͝h̷̭͉͎͎̫̹̟͌̌͜(̷̹̆̀͒͌́0̶͉̥͋͐̕.̷͎͂͆̂̋6̵̨̰̞͖̭̀̇͌̔̚)̷̧͕͔̪̲̜͎̾͆́́̚ͅ)̶͓͚͔̭͙͚̟̓̿͒̀͑̎̏͠;̸̗̄͊
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defaultpen(linewidth(0.6));
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̷͓̀͊̈́̊̚͘͠p̸͍̯̳̮͝͝a̸͈̼͇͙͛į̴̥͙̘̥̻̥̥̒̀̍͝r̸͖̝̺̳̘͕̆̿͝͝ ̴̨̹̳̝̕͜͝Ä̵͕͚̟̹̦͔́̿̍͝͝ͅ ̴̨̮̗̈́̀̀̚=̷̨̝̿̅́͒̊̊̕͝ ̸̛͔͚͖̮͇̄̊̓͘ô̴̲̼̤͉̞͚̓̀̚r̶̛̬̮̦̣̓̉̇́ḯ̶̭̦̘͙͛͗ͅg̶̡̞̦̙̗͒ȋ̶̙͕̹̄͋͒̒̾͠n̴̘̭̹̍,̴̢͔͇̣̰͙̈͛͝ ̴̨̥̟̘̦̍͐̋̆̀͘͝ͅB̸̜̼͗ ̵̢̰̟̰̻̖̈̋̉̐̂́͝͠=̶̪̌̽́̋́͂͗̚ ̷̛̙̺̳͐̏̈̃͑̿̒͜(̷̨̋͂͒̎̄͋̀6̸̛̗̟̪͕̻̤͛͆̈́͑,̷̧̱̘̑͂́̎͝3̸̺͛͒̉̏͘)̵͓͋͆̇,̸̬̋͐ ̴̛͙̼̲̲̙̻̹͑̈́̐͘X̷̛̘̿̿̓̀̓ ̴̜̂͛̍̎̈̈́̎ͅ=̴̛̝̹̖̎̀̓̌̓͊ ̶̡̮̮̾r̶͍̳̫̬̮̜̩͌ǫ̶̞͙̘͎̰̂̔͆̃ͅt̷̨̛͚̫̘̲͑̀͊́̐͘͜a̴̺̗̹͖͗̊͊͗̿͝t̶͎͍̩̬͖̪̪̳͒̈́̂̽͛̐̀é̶̙̘̖̬͈̱̯̓(̴̞̆̈́͊͌͝4̴̳̰̩̅0̸͓̀̉̈ͅ)̵̤̫͛́͂̃͋̈̚*̵̨̪̟͇̟̹͙̌͂̔̚̕B̴͈̱̖̘̰̪̒̈̐̋͝ͅ,̷̡̲̟͉̊̈̿̇ ̴̺̃͆Ỳ̸̬̲͓̫̞̦͎͗̀͝ͅ ̷̰̣͕͉̹̼̟̠́̀≠̭̼̹̝͕̊̐͒̇̍ͅ ̷͖̰̝̗̿̊ͅr̴̼̞͎̦͉̫̰̓́̃̇̂̆̕͝o̵̡̞̻̣̜͉͗̌͋͐̈́̃͗̚t̷̲͛̉ã̷̮̗̻͖̪̗ẗ̵͎͚̞̜̳́̀͆̏̏̑̕e̸̯̥̐̄̐̀͂͘(̴̛̗͎̾̓̆́̀͜͠7̴͙͚̞̟̮̹͓́0̷̣̱̻̺̯̑͒̓͌̌͜)̶͈̱͖̂̓́̎́̇͗͝*̶̹͔̯̣̳͖̣͕̊X̶̯̗͇̎̎,̶̧̹̲̝͈̬̥̙͑̇͠ ̷̯̹̜̘̠͗̈̎̉Č̷̪̬̬̭̞͜ ̷̛̣͓̑̓͑͘͠=̴̛͇̂̑̃̋̅͂̚ ̷̢̢̯̤͗͗̑̍̚͠X̵̡̨̝̫͔͈͗̄̎͐̓̇+̴̛͉̘͚͔̝̏̄́̂̐̏Y̶̡̩͓̞͋,̵̞̱̤͌̇̌ ̴̢̛̙͙̙̼̦̪͍̃̽̚̕̕Z̶͉̲̝͔͍̥͂̏̊͑̓͠ ̸̺̩̘̱̹͍͖̈̿͐͝͝͠=̶̰̼̗̹̮̯̩̣̆͑͐͠ ̸̧͖̘̼͍̦̮̓̒̅̔X̵̯͚̺́̆̈̽̕+̴̦̳̻͑B̷̛͔̞̣͊̈̊͗,̶͓̎̌͗͂̽̚ ̴̡͍̼̱̲̣͆̋͆͑D̸̢͈̱̐͆͛̐ ̶̻̤̝̺͔̱̰̉̋̉≠̢̛̩̭̖̑̅̀̌̑͝ ̸̞̹̓̋̏B̴̡̳̩̥̝̮̻̍+̴͇̙͕̬̦̔̌C̸̛̫̈́́̽̚,̷̢͇̅͛̀ ̵̭̃̂̓́̀W̶̺̙͔̑̔̓ ̷̨̧̻̺̪̈́͆͗͛͐̿≠͚̹̬͍̮̀̾ ̵̹̦̯̳̫̘̑̔̿͗̈́͋B̵̞͖̤͂̈͐͊͑̿̉͘ͅ+̶̢̨͚̹̣͎̂̏̍͒͠Y̴̛͔͑͠;̴̳̟̩̱͕́̾̌͂̚͜
+
pair A = origin, B = (6,3), X = rotate(40)*B, Y = rotate(70)*X, C = X+Y, Z = X+B, D = B+C, W = B+Y;
−
̷̳͔̮̍̐̀͂͗̇p̴̫̺̝̬̝̩̦̲̿̈͒̾a̸̛̺͓̞͕̱͕̰̐̓̽̑ȋ̶̛̪̬̙̀̇̏̉̊͝r̷͕̒̅̊̕ ̷̨̛̛͉̍̔̎͋͆P̵̳͚̬͍̻͒̈̍͂̑1̵̛͇̘̭̙͑́̐̈́̍ ̸̹̈́=̴̛̟̰̟͗͆ ̴̩̎̇̌͋͑̊͠0̷̡̡̛̝̘̘̩̗͖̋̋́͝.̷͚̟̀̓8̸̞̱̏̿͗̓̐͜*̵̫̬͖̭̎͐̇͝C̴̛̗̓͐̈́̂̇̒͝+̷̤̖̤͌͂̕͝0̶̛̝̘̙̣̻̲̺̀.̸̨̟̲̗͈̠̈̋̊̽̐͝͠2̶̡̳̬̤̹̲͑̎̂̋ͅ*̸̨̠̭̗͇̪́͛̃̽̿Ý̸͕̠̼̇,̶͙̠̤͇̳̘̥͉̔ ̵͇̄̉̔̉̀̌͠P̸̨̣̫̗͉̩̺̲͊͊͐̽̓̋2̴̢͕̲̫͕̘͔̏̏̿̽̒̕͜ ̸̘̜͖̲̿̏̍͘͜͝=̴̧̧̭̩͖͈̙̥̀̏̚ ̴̧̧̪̫͔̖̻̥̇̾͒̏͑͂͠2̴̣̤̋̃̂/̴͕̘̙̣̣̈́̔̌̋̒͜3̸͇̅̃̍̂̍͠*̵̛̛̪͖͓͍̩̈́̌̈́̇͠͠C̷͖̽̇͝+̷̨̠̩̤̿̊̓1̷͎̂̌͑̃̾͒̕/̷̡̬̦̙̈́3̸̻̱͇̩̳̽̅̇̉̍̕͝*̷̝̞̕͘͝X̷̮̾̾̈́,̴̢̩̙̤̘̃́̍̔̾̆͘͜ ̵̙̩̠͓̀̕P̶̧̛̟̣͆̀̄̇̇̇͠3̶̜̙̺͔̥͂̊͆̿̕͝ ̷͉̹̹̩̺͛̽͆́=̶̥̺͎̻͍̗̏̀̈́́̋͝ ̸͓͔͚̗̞̠̜̺̆̀̓̏̚͠0̴͙͍͎̹̞̯̿͆́͠.̵̪̘͗̓̔̓͠2̵̤̫̣͔̰̉͆͐͝*̵͙͙͎͕̌̈͌̎̂̉̑D̶̲̫̱̩̩̿̊̉̒̓͜ͅ+̴̮̼̼͎̣̟̈̐͌0̷̡̠͓͈͚̅̃͛.̷̛̫͑͌̈́̊̀͌͒8̴̨̨̳̈*̸̣̩͂̆̃̓̽̇̿̕Z̶̢̳̝̩̞͖͛͗̒̍,̸̡̨̥̬͍̲̂̕͜͝ ̴̳̼̙͈̃̉P̶̖͕̦͙̀͒͌̈́4̶̪̙̦̗̉ ̷̪̣̺̫͍̝̈͊̅͌̕̚͝≠͇͕̱͈̋͑̒̀͒̅͌ ̶̟́͊0̵͎̻̦͕̀̋͑͒̈́̂.̵͈̮͇͌6̷͕̠͔̮̮͍̿3̸̧̹͕̻̱̯̌̓́*̵̲̱͇́͌̈́̇͊̕͠͝D̵̼͙̠͙̒͛͗̆͂̑͆̔ͅͅ+̵͖̹̊͋0̷̡̡̩̤͓͍̯̀̐̊̏̃͋͠͠.̷̂͒̾͂̅͜3̶̧͚̫̤̒̈7̸͎̉̄̒͊̐̍̀*̵̡̠͔̣́ͅẄ̷̢̛͉͍̣́̔̓͋͂;̶̝̣̣̓
+
pair P1 = 0.8*C+0.2*Y, P2 = 2/3*C+1/3*X, P3 = 0.2*D+0.8*Z, P4 = 0.63*D+0.37*W;
−
̴̢̠̫͙̲͑́͐̉͆̀̅͊p̶̡̧̠͇̩̹͎͑͂͐̅̚ͅa̷̭̖̙͕̘̓̋̈́͘͝͝i̵̯̯͇̹̠̒̅̋̏̊r̴͕͖̈́̀ ̸̟͔̣̙͔̖͒̌́̈́̽ͅĘ̵͕̠̯̠̤̞̠̈̃͋͛̾̒͝ ̵̤͔̠̦̘͖̜͓͗́̉̀̍=̴͔̤̟̮̠̇̑̇͘̚͝ ̴̬̳̰͌̿̽̽̚(̷͕͚̹̟̺͝-̵̼̦̳̝̟̪̓̓̾̇̀̐͜͝2̵̼͐̉̽0̶̩̘̫̭̣͒́̇,̷̭̪͖͆̑6̶̨̭̱̑̀)̸̃̎̇̊͜͜,̴̄̇͜͝ ̵̡͖͔̩̤̐̆͊̍̃͒͝F̸̠͚͖͛̄̕ ̸̢̛̻̰̦͈̲̻̼̓̓̇̇̚=̷͚̘͛̉ ̴̛͓̣͆͝(̶͉̎-̵̨̨̭̣̖͖̯̋̌͆̈́͐͘̚͝6̵̗̼̩̪̹͕̍̄̏̀̈́͐ͅ,̸̟͎̺̠̺͂͂̊͒̔͛̃-̵̖̟̼̲̫̯̈́̎̓̔̑̑̆5̶͇̥͍͛̍̕̚)̷̼̖̬͚͈́́͠,̵̠̳͇̞̇̊̽̈́̽̈́͜ ̷̲̄͒͝G̷͖͆́̊̉ ̴̣͔̀̐͗͐̍͘͠͝=̶̭̊̏̌̀̎̄̅̚ ̵̧̺͇̮̞͈̩͒(̵͈͖̺̆͛̾̒̈́͛̓̿͜1̷̭̭̠̂̄̅͋͛͑̕8̴̯͝,̸̡̢͉̣̳̩̠̝̾̽̂͗͑͠-̵̧͕̜̖̖̔͌̎̇̔̕͝2̷̦̹̟̤͙̪̘͂̓͆͋͗͆̾)̴͖̪͎̓̅͋͆͗͛͆,̷̨͗͋ ̵̢̛̦͚̘͉̱̠̻̎̋̓̃̚Ḩ̶̛̫̜͍͔̼̲̠͆̈́͑ ̷̺͇̹̓̍̍͊͋̏̕=̷͇̐͗͐̓ ̷͇̔̍͆͐̆̐͒̚(̵͎̃̎̂̄͗̓͠9̸̡̡͙͆͜,̸̲͇͊8̸̯̀̽͒͂͛̽̐)̶̣͕͈̫̱̗͚̿;̶̨̮̝͙̗̺͕͙͗̀͂͘
+
pair E = (-20,6), F = (-6,-5), G = (18,-2), H = (9,8);
−
̵̞͉̮̗͇͑̓̑͊͌̂ͅf̷̢͚̼̪̻̅̿͑̽̅͋͒̾i̵̫͖͔̬̰̬̳͈͐͆͌͛̀̀̑ḻ̴̢̙̫͕͎̟̓̽̈́̉̓l̸͉̮͕̠̲̩̃̒̿͜ḋ̴̳͕͉̬̞̌̄́̽̾ř̴̳̼̱̻̔̊̌ͅȁ̷̞͔̲̻̣̲̩̞̿̑̑̆̈ŵ̴̜̭̈́͆̽̉̒(̶͈͐̄͒́͝͝E̷͍̓̇̉̕-̴̣͍̟̝̃͒̒̓̂͛̈-̷̱̬̙͈̰̝̺̯̄͊͑̇̈́͠F̵̰̻͌͑ͅ-̸̨̛͈̫̭̘͚̥̰̔̋̕-̸̛̲̟̤̼̞̖͔̉͋̒̀͌G̷͙̼̉͗̐́͝-̵̳͔̺̠̮̞̞͍̈́̓͋̎̕-̵̮̥͓͔̹̔͂H̵̨̖͎̰̗̥͑͋̀̿͑̈́͝-̸̹̟͕̠̲̐̈́̚-̶͎̽̅͒c̶̺̺̼̐̋́̎͋͘͘͠y̷̟̾͌͑̃̿̿c̵͚͚͖̰̗̞̅̓̑͛̈ĺ̶̨̟̳̹̠͓̯̩̆̐e̴̞̙̫̹͗͌̔̈́̄̿͗̕,̶̥̺͓̖͉̔̃̄͗́̆̕͠ŕ̶̖̤͉̗̬͛ĝ̶̯͙̖̄̾͠b̸̪̹͓̣̪͇̠̣̑̃͑̀̊̄̓(̵͈́0̸͙͈̬̀͗̒̚͠.̷̰̩̿9̶̹͈͔̰̼͊͑̉̿8̷̛͙̩̯̖̲̤́̂͆͠͝,̶͔͍̘̫̻̀͛͐0̷̨͈̫̦̱̠̬͑̏́̃͘.̷̡̢͎͎͔͙̋͌͊̑͋̓̚9̷̝̗͕̣͋̈́̈́8̷̗̝̰̜̕͜,̴͎̜̣͔̆́͌͋̎͆͑͑0̸̛͚͌́̆̔.̵̼̰͓̠̠͖͚̲͆2̸͈̿ͅ)̵̖͈̆̔͛̀̈́̉̔͘)̶̨͔̼͚̑̀;̴̹̩̩̻̖͉̤̩̈̔̀̑́̚
+
filldraw(E--F--G--H--cycle,rgb(0.98,0.98,0.2));
−
̸̧̬͊̍͒́͊̓͘f̵̨͙̯̫̼̺͒̈́̍į̷̙̱̼̥̜̭̝̋̊̇͝l̴͍͇̫͉̘̏̓̓̂̕l̴̥͂̋̽͆(̷̛̜̰̹̟̟̇̃͆͘A̷͓͈͔̪̠̻̗̐̓͠-̴̧̬͇̀̅́͝-̵̲̩̟͉͐̏̚Ý̷̡̡̺̼̬̥̼͝-̴̧͕̫̬̬̹̇̑̄͗̆̏̌-̶̢̜͙͔̜̜̀̕P̶͕̺̂͋͝1̴̬͈͍͈̠͊̚̚͝-̵̺̤̱̝̺͒-̴̰̮̜̚P̶̧̛̛̥͚̘͚̼̭͚̕̚͝4̶̛͎͈̗͔͖̈́͑̈́͛͘-̴̝̩̻̪͑͜-̷̢͕̖̹͈̖̣͛̅̓ͅP̶̬̳͙͗̀3̸̢̡̞̹̟̦̞̿̀͛̓̋̇͝-̴͚͕̘̳̀̃̊̍̌̐͜-̴̨͉͎͓̤̩̑͂̉̾͑̄͝͠ͅZ̵̨̘͖̼͖̭̈́-̵̹͈̰̦̮̋̍̚͘͝-̷̦̱̗̣͕̱̀̔̀̕B̴̨̡̮̀̈̌-̴͙̹͉͚̙̼̑̂̃ͅ-̶͎̠̟̪͈̺̤͍̈́̓̃̿͌̊͘͝c̶̡̢̛̤̻͔̳̮̺̊̈́̉̐̍̋ȳ̷̰̩͉͇̔̉͐͒ç̶͎̼͉̊̊̑̈́̌̚͘ͅl̶̢̛͕̥̲̾̇̽͋͛̔ͅe̶͎̭̜̣͐͊͝,̷̧̲͇͙͈̞̑r̷͙̟̪̣̞̜̈́̎̇͂͛̂̚͜g̵͍̈́b̶̧̛̜̩̗̬̎͑̄̓̔́̚(̴͍͓̰̳͐0̴̭́́.̶̟̠̘̐̌͋̈́͐̿̑3̸̧̡̡̨̟͖̉̓̈́̏͆͝͠5̶̖͙̄͗̈́̿̽͘,̷̝͍̎̋̑͜0̵̗̙̣̰̞̇͊ͅ.̷̨̳͈͓͈̪̥̅̕7̵̥̺̟̩̆̆̽̀̈̈́͂,̶̨̧̤͙̯̹̣̣̏̓0̷̰̯̥̞̃́̒́͊̐͘.̴͖͋͝9̸̨̝̠̻̼̾͊̈́ͅ)̴͓̜͖̖͈̃̎̊̈́͐̚)̴͔̠̮̤̞̏̃͂͑́̓̀̕;̴̗̗̙̦́͆̎̉̌
+
fill(A--Y--P1--P4--P3--Z--B--cycle,rgb(0.35,0.7,0.9));
−
̴̩̪̥̬͉́d̴̡̢̛̲̠̺͔̰͂̓̑̈́̐͘ṙ̵̬̗͒͋́̀͒̀à̴̛̦̙͈͛͊̾͘͝ẅ̴̨̻̼̻̱̮̲͖́͗͂͑̚(̷̟̎Ą̶̫͍̫͑͂͜-̸̧̪͉̯̲͇͌̐̄͋-̶̪͙̮̜̣̯̲͖̈́̂͊̆̆̈́̐͝B̷͔̪͍̣̟̝̫̉́͌͒͆͝-̸̤̯͖̣͋̔͒-̴̩̫̯̘̱̣̂͛͘͜͝͝Z̴̨̛͈̫̻̀̔̍̓͒̚-̵̳̟̯̻̼́̿̍͒͊̀͘ͅ-̷͉̮̘̊̎̂͊̾̓X̸̞͖̝͈̐͂̒̐͋̀̈́̏-̴̢̪̩̙̠̍͗̾-̵͎̳̘̝̈̕Ȃ̷̛̰͙̺̈̈́́͋-̷̧̻̣̙͆͆̐̉̕̕͠-̴͍̒͑͘Y̵̫̪̗̯͔͛̿̓͜-̴̖̱̘͂͑̿̎͐͘͝͠-̶̳̾̈̑̂́̕C̸̨͈̦̰̻͕̮̖̅̿̃̊͝-̸̛̤͚̜̆̾̋͊̌͑ͅ-̶̢̛̛͋͊̽͑́Ẍ̷̧̠̞̥̮^̴̛̲̱͒̒̆́̍̓^̶͓̜͚̫͈̘͚̍͊̇͗ͅC̸̡͓̫̱̒̍̚̚͝-̶̰̆̏-̸͙͗͗̏̂̅̽̇Ḍ̶̪̦̥̲͊̏̈́͠-̸̧̰͎̰̞̽̍̍̈̊-̸̰̤̼͚́̈̓̚͜͠ͅͅZ̵̨̦̗̫̜̝̎̉̐́̏̚)̶̲̲͉̓͑̑̕͠;̸̡̙̫̬̻̙̝̼́̈́̈́̄̓̆
+
draw(A--B--Z--X--A--Y--C--X^^C--D--Z);
−
̶͎͔̯͎̗̑̀̎̇̇̍d̴̘̏͛̒̚͝r̵͙̰̖̦̖͍̃̓́̏̔̌̕͜a̴̡̞͉̎͛̅́̾͊́͜ẃ̴̖͒̂̑͒(̷̢͎̓̉͑̊̃̚͝Ṕ̴͎̞̺̝̗̂̍1̶̨̩̪̫͊̓-̶̺͇̣̖̞̤̗͒̕-̷̩̀̅͝P̸̬͉̰͆̋2̶͇͚͍̠̮̹̇̕-̵̰̦̯̗̦͙̱͆̍̒-̴̧̛̰͚͌̕P̵̛̰͖͉̗͕3̵͙̮͉͖̆̏̀̐̽́-̴̧͉̺̲̜̦̲̗̈́͑̓̚-̴̖͙̯̙͚̪̏ͅP̶̻̱̜̮̈́͛̍͊͝4̷̘̥̝̙̪̯͖̯̌̂̓̔́͐̃-̶̡͔̳̲̯͓͓̃͌̊̚-̴̧̧̟̳͇̦͑͜č̵̠ÿ̴͎̝̻̱͉̮̌c̷̢̬̤̼͉̰̹͊̄͊͌̈́̊l̶̡̛̤̪̠͖̥̓͂̃͑̋͒̔e̶̞̖̱̠̮̲͗̈́͂͑^̶͙͍̘̺̏̆̓̄͒̀͊͠^̴̖̖̐̾͗̂D̸̢͍̱̪͈͗̿̑͐̒̚̚-̶̜̣̼͈̘̟̀͐͆͋̒͜-̷̳̰̽̇͐̔͗͠͝P̶̥̜̈́͌̊͑̾4̵̛̠̳̼͖̮̩́̂͗̆̇̍͗)̸̛̖̬̹͚̪͂͋̿̌̚;̷̟̗̑̌͝
+
draw(P1--P2--P3--P4--cycle^^D--P4);
−
̶̦̫̻̦̆̋̕ď̶̲̣̟̺̞͎̻͋͐͋̇̚o̵̢̮̭͎̬̰̣͇͠t̵̳̩̥̺̰̪̭͋͂̒͋(̸̗͒͠"̷̼̲̺͉̼̯̫̍͋̅͌͑̓͝<math>̷̫̓͆̂̒̇͠A̸̞͎͚̅̓̐͘</math>̶̧̦̮̜͖̺̩͐̌́̏̐̀͒ͅ"̴̡̻̹̟͐̚,̸̧̈́̂̑̽̀A̵̰͕̎̀̈́̀͋̑̍ͅ,̷̱̠̩̪͉͚́͘S̸̫̄͂̄̆)̸̢̤̣͉̣̙̞͙͋̄̅̃̊;̷̫̠̂́͌͜
+
dot("$A$",A,S);
−
̸̗̞̺͎͕̬͕̟͑̓͝d̴̙̬̦͍̿̒̃̾̇͝ỏ̶̡̞̻̤̭̜͓̱͆̓̊͑͘̕t̸̗̠̫͉̘͖̾̂̂̈́̃̃(̴̨̡̧͕͍͇̀͜ͅ"̶̨̭̱̬̥̓̓͘͘<math>̴̡̨̲̞̹̠͒̈͌̋̂͠ͅB̶̨̛̫̯̮̘̼̎̈́̽̇͌̆</math>̴̤͍͊͑̂̊́̂"̸͔̪͓̣̤̈́̒̿͛̾͂,̶̳̪̘̕͜ͅB̷̢̬̰̯͉̠͌̅̃͒̚,̸̛̰̩͛͂͋̓͝S̶̙̹̻̺̩̤̻̉)̵̢̗͕͈̀̔̀͌̀̅͑͝;̵̼̲̠͂
+
dot("$B$",B,S);
−
̷̯̪̺̈́͒̂̆͘͝d̴̨̛̠̫̦͊̑̐o̸̡̨̺̣̦͔̱͓͂̒͂̒͐̎t̸͖͒͌̈́̽͗̈̓̉ͅ(̴̧͓̺͕̯͇̱̇͗̔͋͜"̴͔̜̠̬̻̞̱͗<math>̶͔͆C̶̺̱̫̄̎̌̎͐̆̽̄</math≯͔͇̥̾̍̈̎̿̅͝ͅ"̴̡̰̲̩͚̭͎̃̔̐̏͘̕,̶̬̆̃͊̂̔̈́̐̕C̷̳͖͈͓͔̦̆,̴̢̯͔̱͈̤̞̌͊̑͛̽͘̚̚N̶̥̑͗͒̊̾͆̽͘)̴̮̦̜̩͓̈́̄;̸̡̰̰̣̖̈̍̐̀́̎͜͠
+
dot("$C$",C,N);
−
̶̡̟͎̋d̸̬̖̓̿̆͜ǫ̸͇̗̬̫̩̤̀́̈́̏͘t̸̹̱͔̺̆̈̏̈́͋̏̓͝(̷̛̘͙̦̞̈̈́͛̀̿ͅ"̷̺̀<math≯̨̢̫͓̦̝̉͛Ḑ̵̨̤̭̿̎̍̔͠͠ͅ</math≯̼̮͕̑"̴̖̼̈́͝,̵̛̻͕͑̈́͛̂̌̃̄D̴̦̣͓͈̼̤̲̘͂̾́̈́̈́̉͝͝,̸̨̙̯͖̞̔̽̆̔̂N̷̪̳̤̭͕̂̄́̇̊)̶̝̼͇̯̔̉̀;̴̯̼̪͎͎̔͛́̂̈́͠͝
+
dot("$D$",D,N);
−
̵̞̒̓͆̈́́͋̀͝ͅl̶̡͖͇̭͕̹͂́͘ã̴̭̬̱̈́̀̀̆̓͝b̵̦̗̰̺̙̥̩͂̈ë̸̮̞̱́l̴͖̳̊̄͊̕͠(̴͈̞̹̘̩̌̓͝"̴̥͔͎̼̼̬̦̃̆̔͆̋͝ͅ<math≯̲̋̊͆̄̂̍̚̚\̴̭̹̟̰̫̗̂̓̃̌̓ḿ̶̩͉͎́͋̌̓̚a̷͈̻̬̅̌t̸̩͚̮̖̳̀̐̉ͅh̴͎͉͈̹͗͊͐c̵̡̗̮͍̫̮͔̕a̸̦̎̎̚l̵̢͍̩͆͒͝ ̷̫̑͋͜P̶̨̨̙̗̭̘̫̈́̏̅̇̃̿̊ͅ</math>̵̛̹̬̼͍͍̬̃̾̒"̴͕̼̼̱̱̩̘͊́̽̒̓̎͜,̴̧͇͙̟̮̿́̌̾̎̾̽͊(̸͔̻̏̊̓̆̀̃̅̚-̶̛̞̪̬̏̅͋̑̈́̍͠1̵̙́̋͗̏̊̔3̸̢͉̰͇̝̘̉̓̃̓̅̈͝,̶͈̿̀̏͝4̷̙̗͕̅̿̀͆̃͑̏͂.̸̫͖̤̰̲̲͕͍̊5̷̺͑̊͝)̴̺̫͂̉͒̂̅̌̕)̵̦͔̙̉̉͋̈̽̂͘̕;̵̦̗̘̰̙̙́
+
label("$\mathcal P$",(-13,4.5));
−
̵̙̺̟͇͓͔̈́̉͊̈́́̈́̈́̃<̷̱̥̰̰̱̫̈͋ͅ/̷̛̝̥̫͓͑̈̔̔̊͝ä̶̜̙̟͋́͌̄͌͘s̸̼̘̞̥͙̥̾̇̂̌͛̇͝y̸̱̥̗͓̳͓̰̍̋̃>̶̢̰̜͚̞͇͇͍̐͂̓͊̐̀̚͝
+
</asy>
−
̵̼̹̝͗͛̈͘
+
 
−
̶̧̟̦͍̖͖̑̈́̈̑[̴̺͇̑̈́̃̈́ͅ[̶̢̞̬̩̘̰̺̙̓2̸̢͓̮̼̽̏̿̈́͝0̸̧̨̞͇̟͐ͅ2̸̪̓3̷̬̩͆͊̄͘ͅ ̸̧̫̘̭̞̥̏̓Ą̵̨̫̮̫̙̯̆̇̀̕I̵̙͐̎̈͑̓M̵̧̘̩̼͊͆É̴͉͈͋͌ ̷̛̝̳̞̦͇̯͉̎̓̑̇͐Ȉ̶̞̹̞̭͍I̸̛͇̓̍͐͘ ̶̡̈̀̈́͝͝͝P̶̬̺͉̫͚̓̏̃͌͜͝ȓ̷̺̖͎̯̜͙̪̓̿̍́o̴͕̳͖̺̝͐b̷̧̨̢̨̲͈̆̋̋͆̇̿́l̵̩̦̍̆e̷͉̫̝̹̋͑̄̈̏́̐͘m̷̨̩͑́̓͋͐s̵͔̳̘̭̳̩̓̇/̵̨̝̬͇̙̞͌͗͋̄P̷̝̱͚͎̰̞̋̽̅ͅr̵̨̼͖̞̹̭͓͇̐̓͒o̶̢͕̝͕͙͐͝b̶̫͉͙̟͆͐̍̚l̴̨̲̱͇̙̄̈̈́̄̍͠͠ḙ̴͈̺̮͇̩͗͊̌̋͌͜ḿ̴̗̼̳͉̆͐ ̷̛̬͓͙̠͉̮̒͒̀͠1̸̡̨̫͇̟̰̲͑̕4̶͇͉̆|̴̯͉̻̪͚͕̿̇̓̀̈̑͘͜S̸̛̛̹̺̞̳̼̆͐̈̎́̆͜ȯ̶̢̺̭͇̪̻̭̍͗̈̏̈́̚͜l̴̨̯̔̊͛̇͆u̵̡̠͍̱̻̥̪̍͐̓͝t̸̞̰͎͈͂̆̊̀i̴̢̖̮̠̠̦̔̇̎̍͜ò̷̜̭̝͖̣̞̅͗ǹ̶̦̥̿̆̏]̴͕̯̼͗́̀̀̑]̸͚͂̕
+
[[2023 AIME II Problems/Problem 14|Solution]]
−
̷̝̾̓
+
 
−
̸̰̮̳͙̲̯̏́̌̅̋≠̮̩̥̹̮͐̇ͅ=̴̨̖̯̟͍̥͒̔͐̽̋̆̕͝P̴̧̳̹͈̲̩̀̊̃r̴̮͉͚̜̝̾̃o̴̧͕̹̺͍͓̘͛̄́̚b̴̨̤͔̣̙͓̃͗̏͒̒͛͠l̶̤̼͍̙͎̲͂̆̓̆̓̇͘ė̷̢̺̠͕̭̭̘͗̅͋̈́͊͜m̶̻͔̬̥̼͊ ̸̯̀͆̀̾͛̆͝1̶̰̬͓̖͒͊͒̂̕͝5̵̛̖̤̪̟̱̘͍̎̀̀́̓̊͝=̴̢̲̦̞̬͔͚͙̿̎̇̎̊͋≠̫͎̪͖̱̀̓͒̉ͅ
+
==Problem 15==
−
̵̟̀̓̑͑̈
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̵̝̩͔͊̈́̇̐̅̍F̷̧͖̭͕̲̯̬͓̓̈́̏̓͌͐ờ̶̩̪̫̩̟̰̤̂̉̇̎͛͝r̶̙͖̅̽́͑͝ ̸͚͙̯͕̘̲͑̈́͝e̴̡̨̫̤̳̪̣͒a̸̻̫͉̪̖͕̦̎̊̈́̃̋̈́̒̕c̶̛̳̝̿̓̔͊̽h̸̥͉͇̦͇̩̒͝ ̶̛̹̿͂̑͛p̶͙̿̈̽͊͠ơ̴̞͍̖̼̆̌̋ͅs̷̨̨̨̫̘̮͙͗̾͗̓̌į̷̺̼̘͕̻̐̽͒̎t̷͍̥̲̏͌͑̐͌͐i̷̧̲̘̯̮͕̦̫̎́̔͠v̸̨̮͇̪͍̟͐ȩ̸̧̢̡̰̠̣̜̌̽͑̔̎̋ ̷̨̯̟̓̆̋ỉ̴̥̮̈̈̈́̄̿̋n̷͈̬̻͕̳̔͒̅ẗ̷̖̭͎͚͆̈́͜͝͠e̴̹̖̾͋̃̀͝͝g̴̡͉͎̑̀ͅe̷̯̊́͐̀͐̌͐̉r̴͚͖̭͓̠̺̿̆̾̓̔ ̸̗̩̩̼̱̬͇͎͂̊́̀̌̾͋<math>̵̻̱̍͗́n̸̘͛̃̒͘̚̚͝</math>̴͇̗̻͗ ̸͍̖͛͂̌͊͌̐̽͂l̴̛̗̟͖̫̻̋͋ĕ̶̥͈̣͙̤͔̫̒ẗ̶̛̝̐̈̕͘͝͝ ̴͎͋́͗͝<math≯̨͓̜͐̋͛̐a̵̧̟̩͎͒̅͝_̶̧̳̯͒̈́͌̆n̶̝͇̬̬̊</math>̴̪̪̔ ̷̡̺̜̘̞̇̈́̿͊̕b̵̧̪͖̝͈̖͙̅̈́e̵͓̤̘̞͈͙͗̀̈́͌͜ ̵̫̦͌̑ẗ̶̯̘̻̝̬́̾̍́̕h̴̨̛̻̖̺̹͊̌͛̓̈́̕͘ḛ̸̜̏ ̸̘̟̖̥̥͖̯͊̓ļ̵̛̟̩̯̥͇̪̤̎̔̀̾́è̷̦̪̒ͅä̶̩̬̭̩͇͎͖́͘ͅs̴̛̰̪̔̍͑̉̎̃͠t̷̗͑̅̿ ̵̡̛̛̘̭̼̝̝̎̏͛́͜ṗ̸̛̦̱̱̗͎̞̙o̴̞̟̹͖̓͒̏͆͘ͅs̷̻̭̙͔̮̅̅͑̄̅̓̃̕i̸̡̱̪̮̠̗̘̽̈́̆̾̀ͅṱ̴̛̜͍̲͆̈͆̉͑̏ḭ̵̆̐̓̍͊͘͠v̴̙͎́̃͜ḙ̸̛̱̻͈̍̓̈́̂̋̈́͠ ̶͔͈̺͑i̸̝̼̦͒̈̌̒n̸̖̺̭̓͊́t̶̡̠̲̯͈̙̏̍̅̇̆̂͒͝e̶̤͆̎̾͌̓̓̚ğ̶̮͇̅ę̸̥̞͓̎̂̉r̵̨̝̝̪̹̟̽̎͝ ̴̜͔͈̲̪͗̄m̷̡̲̝̪͖̥͖̽̿̉̓̈́̂͘u̸͔͓̗̪̝̬͎̔́̅͂̓̋͌l̶̩̻͌͌͂̓̈́̊t̷̛̗̦̬͖̼̝̩̆ì̸̻̙̤̥̦͎͐p̸̼̲̹̗̩̰͍̾̐̕ļ̵͚͎̘̀͐ͅė̴̥̫̀̒͘ ̷̥̿̂̔̆̿ō̴͇̣̘̫̈́̌̅̒̑̓͜f̷̢̙͎͇͈͍̞́̏̔̚͜ ̵̢̢̛̺̩͇̗̑<math>̵̞̼͝ͅ2̵̥͍̇̈͜3̵͙̾̃̀̊</math>̶̯̥̳̔͊͋͗͗̚͘ ̴̤͙̗̲̺̝̔̉̀̏s̵̪̞̭̉̂̔̆̃ǔ̸̩̩͈̈́̑̓͑̈́c̸̡̲̮̪͎͙̑͒̈̓͜h̵̙͚͇̦̿͆̽ ̷̻͈̬̙͕̞̇̐̉t̴͇̞̼̙͈͚̟̯͐h̴̰̫̘̗͍̦̤̲͊͛̋̈́̓̚͝͝ạ̸̹͉̐́̄t̷̞̖̞͍̘̠̟̄ ̵̫̰̘͎̗̜͎̣͌͠<math>̷̢̞͔̺̪̺̀̚ȃ̴̙̺͓͍̒̇_̶̝͖̓́̊̐̀̚n̶͓̠̺̲͕͛̾̾̅͜͝ ̶̧͓̰̜̰̽\̴̛̩͖̜́́̿̚e̵̛̳̜̳̼̓̀̕q̸͇̥̯͙̱͈̑ų̵͎̟̰̹̫̌̆̄̚i̴̢̙̘͖̻̝̒̊̓̅́̄̈͜ṽ̵̫̺̀̂́̚͝ ̵͕̙̫̬͗1̸̰͙̞͈͓͕͠ ̶̨̪̒͌̇̇\̵̫͚͛̓̎̓̂͝p̴̛̩̩̠̗̜͒̾͠͝ͅḿ̴̧̭̻̠̰̎͋̌̆ŏ̴̧̩͎̰̱͓͂̈́͐̈́̔d̵̟̱͖̮̄̐{̶͍̬͕̜̄̀̐2̴͇̯͋̉̕^̷̳̩̑̓ͅn̶͚̗͙̺͈̟͚̆͂̄͜}̴̟̝̗̪̱̿ͅͅ.̵͚͓͇͉͖̤͊</math>̴̧̨̞̭̞̱̯͉̇͌͊̽̊̕͝ ̷̨̛̣̗̭̓̄̓́̚F̶̣̬̅͂͆̉̾͝ī̵͖̜̿̒̀͜n̸̦̜̥̪͕̞̦̆̈͠d̵̠̠͕̘̟̮͖̓͋̉̄̿̅̕͘ ̵̢͍͔̲̈́̊̔̂t̸̩̀̾͒͒̿̿͜͝h̷̖́͒̓̄̆̀͝e̶̩͉̗̽̈͌́͂̇̈ ̵̪͖̞̟̙͖͔͋̾͑̈́͋̐͠ͅn̵̢̝͕͙̤̞͒̆͒͋̄̿͋͝ų̸͓̠̱̭̩͠m̶̫̝̘̩͓̅̄̽b̵̯͕̯̈͒͑͠͝e̴̛͈̯̤̣̗̬̞̟͋͂̂͘̚ř̶̟̱͖̋͆͝ ̴̲̹͔̂̈́͋̉̉̈́̚ò̷̡̪̜̼͕f̸̜͈̟̟̞̠̑̇́̄̆͠ ̵̲̙̜͍͊̊͂̀p̶͖͇͓̯̱͓̐̃̏̾͝o̶͎̦̩͐̀̈́s̵̛̰̔̿̒i̵̬͇͛͠t̵̟͔͚̻̤͔͈͂͜ị̵͙̪͈̓̒̽̈́̈́̔̆̎v̴̧͓̔̐͐̎e̶̥̹͐̾̐̋ ̴̛̣̤̐̑̔̍͝i̶̭̒͗n̸͎͎̮͈͔̥͔̦̂t̷̝͙́͊̊͑̋̿̚͝e̵̦̰͍̫̩̙̓̾̈́͗̒̀͑̽g̴̙͎̗̘̋̅͋́e̵̡̙̳͊̇̿r̴̡̹̺̺͔͛s̷̠̪̙̀ ̴̜͂̔͋̀̔͆͜<math>̷̼͚͌͆͒̈́̈́̊n̴̨̠̪̟͊̂</math>̶̤̻̘͎͇͈̠͊̐͒͜͝ ̴̫̰̗͕̳̪͈͌̌̈́̍́͐͆̈́l̵͓̄̀̇̾̈́̕͝͠ḙ̸̛̤̱̗̪͖̓̕̚̕s̶̼̫̒͂͂̔͘s̷̢̢͇̈̓̽̈́ͅ ̵̨͇͝t̷̬͖̗̘̋͝h̵̰͓̮̪̆̓̅̿̕̕ā̵̼̱̭̬̓̈́n̷̘̻̘̈́̉͝ ̸͎̫̺̎͘ȏ̸̙̝̱̦̱̦̄̅̏̋͛͝ͅr̵̪̺͎̤̠̤͍̀͝ ̴̩͔̓̍́̕͠͝ę̵̛̬̼͇̈̆̃͘͝q̷͙̣̠͎̫̲͓̿͌̊͗̚u̵̳̎̍̀͒̂ą̴̫͈̩͖͎̽̆l̵̠̙̹̞͓̗̇̓̈ ̵̦̫̝̀͛̆̋t̴̡̼̤̰͍̥͊̂̕ͅǫ̷̗̺̪̙͝ͅ ̸̡̛̻͙̟͒̀͆̆͆̇͌͜<math≯̡̹̘̳̖̖͙̇͊́̽͐͐͘1̵̨̞̍̚0̵̱̦̣̰͍̜̠͉̿͂̅0̴̨͕̖͖̠̘̖̻̌̉̅̈̃̅̾̐0̸̢̣͖̜͉̓̏̔̽͌͂͝</math>̶̫̤̭̫̪̺͖̮̽̒̓̄͗͂͂͂ ̵̲̥̒̇̓̉t̸͎̖̺̲̖̙̺̥̄̑̿̆̏͛ḩ̴̥̺̞̄͑ͅa̴̟̖͇̟̺͋t̷͔̲͙͉͎͍̤̣͐͗̌̔͠ ̷̻̤͉̐s̵͔̮̝̻͔͈̫̐̌́͌̒͋̔̾ạ̷̦̻̣̫̱̳͜͝t̴̢̩̠̦͍̤̀̈́i̷̤̞̯̯͐͐̍͝s̵̺͙̫͍͒̀̚͠f̶̣̪̏̓͝ȳ̶̧̤̯̱̤̅̚͠ ̷̯̟̭̙͙̌<math>̴̢̰̣̠̀̈̑̑̿a̵̡̧̰̜͠_̴̮͓̓̈́̓̎͛̿̚͝n̵̢͎͕͇̮̈́͝ ̷̘̫̠͉̪͎̤̋̔́̇̍̇͜=̴̧̧̤̘̝̝̜͊̉ ̴̛̞̣̝̖͓̼̦͂̎̒̀͆a̶̰̥̯̥̗͇͆̒͒̍̕͠͠_̶̠̰̍̚{̸̛̤̐͆n̸̛͖̐͆̔͆̒͘+̷̨̬̩̘̮̞̮̣̔̀̆1̵͖͙̣̤̼̘̇̀͛̔͌͛̇}̸̜̖̻͛́.̵̲̹̹̆̂̋̑͂̏̈́͝</math>̴̮̝̬̻̫̬̣̽̈̇͆̔̄̿́
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For each positive integer <math>n</math> let <math>a_n</math> be the least positive integer multiple of <math>23</math> such that <math>a_n \equiv 1 \pmod{2^n}.</math> Find the number of positive integers <math>n</math> less than or equal to <math>1000</math> that satisfy <math>a_n = a_{n+1}.</math>
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̸͕͎͕̝̬͝
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̷̭͛[̷̙͓̺̭́̊̓̕̕͜͠[̵̧̭̱̗͋̕2̴̧͍̬͓͙̺̌̇̇̕ͅ0̸̙̼̘̤̳̝̻̀̂̎̑̑͐͠2̶̩̲͚́̂͐3̶̢̧̛̘̹̙̫̞̋̐̄͜ ̴̛̮̼̰̰̜̐̎͗A̶̗̮̰̻̤̠̜̦͋̚͘I̷̘̓͐̅̔͊̚M̸̘̄͒E̵͙͔̘̘̫͎̲̐͑͆̾͝ͅ ̶̨̖̠̲̣̭̱̭̈́̈͂͊͝I̵̱̝̻̳̐Į̷̤̯̳͍̰͓̐͐͊͛̿̋ ̸͔̗̲͊̎̅̃̎̊̕P̶̻̭̉͛́̽̔͗ͅr̷̡̝̣͍̝̭͖̋̄̄̚ͅo̴͎̫͕͇͕͕̤̬͊͋̔̈b̵͙̰͓̹̙͎̪̾̾̀l̷̞͒͑̇͊̃̊e̸̗̝̥̐̾̽̊̀͒̚͠m̴̖̙͛̇̿͝s̶̞̺̊̋̍͋̍̇/̴̝̪͋͑̏P̴̨̝̳͓̓́̊̑͒͝r̷̼͔̱̤͕̗͌̎̈́̓͋͛ȯ̷̝̤͉̙̚͝͝b̷̡̹͍̦͉͚̉̈̀͆̾̇ļ̶͉̩͖͗͌͌͂̈́͝è̴̪̭̓͑m̵̡͔͚̰̒͘ ̸̬̠̍̐̅̃1̵̛͉̩̭͙̈́5̵̪̥̖̰̤̟͍̿̌̍̀̈́̕ͅ|̶͙͎̦͈̮̅̈́̊̃̓͂S̵̯̩̺͎͌͛̓̒͘͜o̸̹̰̱̱̹̍̊́̎͜ͅļ̴͔̻̮̤͓́͊̎̓̈́͆u̸̖̅̑̈́͠͠t̵̢͇͖͓̏͛́̾̎͝ḭ̷̗̮͆̊́̈́̓ő̵̦̙̮͎͚̹̙͘͘ṅ̶̥̬̙͉̩̑̓̀̽̈́̕͘]̴̨̛̖̳̥̘͉̈́̋̎]̶̡̯̩̯̫̭̏͋̓̍̋̂͘
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[[2023 AIME II Problems/Problem 15|Solution]]
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̶̜̏͑̀
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̷̻̟͈̬͋̀͌̍̈̊͘=̴̬̝̈̀̿͋͋=̷̟͙͇̯̳̝̽̇͂̀͗͠S̷̰̻̠̦͐̒͆͝͠ȇ̵̜͎̯̦͊̏͋͘͠͝ͅe̸̬͖̊͒̿̀̌͆̓̉ ̶̢̝͎́̀̊̆̍̕a̴͓͔͓̮̘͈̾͐̐̀l̷̳͖̩̣̗̫̇̅̆̕͝s̷̼͒̒̋́̚o̵̯̱̠̝͙̐͂͌̽͆̉̐ͅͅ=̷̫͍̮̐̎̅̅̆̑̀͠=̴̘̙͙̬̪̉̿̍̑̌̔̇͝
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==See also==
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̶̛̩̺̭͇̳͐̉͊͘͘ͅ{̸̧̠̼̪̯̰̤̄̄{̸̖̤̦̓͑A̸̻̪̠͒̑̋I̶̡̞͕̲͂́̍̍͆̚͜Ṃ̵͓̯̰̥̲̲́E̶͕̙̹͓͉͊͌̉̆̽̚͘ͅ ̵͙͓̥̯̆̅̅̽͋ḃ̵̨̜͕̊̈́̔̽̎̐ǫ̶͎̼̤̉͐̓̌̈x̶̯̔̓͒̌|̵̧̨̬͉͙͓͉̿ͅý̵̦͙e̷̡̟̬̼͖̳̬̙͂a̸͈̰̒̇́̆̇ͅr̴͚̞̭̋͐͋=̵̢̪͈̬͋2̸͚̍͌̃͋͘͠0̴̦̩́̓̃͐2̵̧̮̠͚̥͍̋̈̑̈̾̕3̶̛̜̑͐̈́̓|̴̪̑̌̔̉̂͠͝n̴̘̹͍͋̃̓=̷̨̝̯̹̻̺̃I̴̫̳͘Į̷͓͎̥̭̈̉̋̾͘|̶͚̙̺̹͕̜̬͕̊͂̍b̷̻̮̲̝̣̹͓̍̃̔̈́̃͌̚̚e̸̟̲̜̭͗̒͊̋̊̉͝f̷̤̻͕̙͓̮̈́͒̊͝o̸̧͆͊͊͊r̴͍͖̫̥͙͕͇̒̂͗ͅe̵̢̦̪̱̜̔͑ͅ=̷͇̳͉͐͗̀̑̂̉̈̆[̴̞́[̵͍͑͝2̵̥̐0̸͇̭̱͔́͊̂́͊̊͛͝2̵̛̺̐̃̓̃̀͛3̸̯͋̇̅̓̎͠ ̶͓̲̱͎̠̈A̶̗̱̥͎͖̍̍̕I̷͖͕͇̳͕̹͐̋̍͗͗̔̚͘M̶̦̊̃Ê̵̞̤̟̭͉͒͋̌̿ ̵̞̺͂̒̎͛̅Ĭ̵̱̖̤̯̾́̕ ̷̝͙͇̞̙́̑̃̂̍P̸̢̘̖͈̺̤̟̈̊̈́͝ŕ̶̡̭͙̼̱͈̭̆͒̚͜ò̴̱̻̖̤̺̞̟͕͒̌̃̚͠b̵̢̪̙̠̻̎͗̉̑̆͝͝l̶̟̪̣͍͉̫̳̽̏͐e̵̖̘͚͍͓̰͚̽͊͑̚m̸͎̤̔͊͑̉̅̈s̴̢̭͈̫͕̞͙͈͋̆̏̆̇͊͘|̶̛͙͇̙̻͉̗͈̿̈́2̶̮́̿͐̕0̵̟̳̲̞͉̠́̂̽̍́̉̏2̶̦̱̩̟̏3̴̢̟͍͉̺̈́́ ̴̡̛͚͔̙̯͈̻͈̍͂̑͋͛̌͠Ą̵̛̭̖̟̀̓͒̿̓̚͝İ̶͓͍͕̞͇͗̌M̵͍̭̤̣̼͔̙͆͑͝E̴͚̰̺̊͝ ̶̦̜̦͊̊͝͝I̷̦͉̜̣͍̦̋̈͂]̸͇̀̽͆]̶͇̹͎̿̀̊͜|̴̧̼͓̫͍͆͛̀̔̍̇͜a̶̗̜͕̲̬̦̫̱͌̕f̶̡͎̌̊́͒͌͘͘͜ţ̵̨̦̰͂͘̕ȅ̵̮͌̋̈͆̐͝͝ṛ̵̈́̊̀̽̚͠=̷̢̨̝͓̻͓͋͋̀[̵̯̌̆̆͗͒[̸̪͎͖̳̮͖̫̀̓̓̊̕2̵̛̣̜̂̏͜0̴̛̫͎͚̖̹̱̞̫͊̑2̶̹̩̦̀̽̂̂4̴̱̝̯̪͗͛̾ ̴̢͈̦̻̝̺̙̮̎̈́̓͘Ä̶̟̭́̅̐̑̈́Ỉ̶̧̹͍͈̠̳̹̯̇́̿̊M̵̧͔̣͖̌̃̀̄͘Ę̴̯̯̥͓̔ ̴̧͍̞͎̙̗͙̐̈͠I̸̘͕͋͆̓̍͌͐̌̓ ̵̺̯͍͘P̵͇͚͇̫̈́̄r̴̙̜̪͔̅͐̽̓̎͌̾̕ō̴̢̜̜̙̩̝̦͍̈̋̒̀̐̑b̴̳̮̳͉̾͜ḷ̵̿̊͒̈́̆̈é̴̮̻̬͊̌͛͘̚m̶̡͙̠̥̫̮̠̀̊̏̊͗̐̆s̴͔͂͒͂|̷̨͕̲̻̘̖̈́͑̿̾̒̋̌2̶̢̢̱̗̰̙̮͊͌̍0̷̡̢̞̦̖̹͕̙̑̍͆̓͘2̶̛̜̝͈̘̱̃̇̅̕͝4̴̥̲̜̀̏̍̈̄̃̚ ̷̡̰̤͚͕̱̘̟̽͑̚̕̕Ǎ̶̢̛̆̀̄̓͝Í̴̩̟̓͗̓̀M̸̧̧̖͈̦̣̓E̵̩̯̜͍̥͗̎̊̓͐̓̈́ ̸̨̙̫͇͍̓̋̌́͌̋͛͝ͅỈ̶͓͕͎͓͒]̴̻̯̗͎̫̳̤͓͛]̶̩͐̏͐͘͝}̴͆͂̌͗̕͜͝}̵̫̯͎͓̗̼̮̋͐̃͂͌̄́̊
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{{AIME box|year=2023|n=II|before=[[2023 AIME I Problems|2023 AIME I]]|after=[[2024 AIME I Problems|2024 AIME I]]}}
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̸͍̒̐͘*̷̢͙̫̈́ ̶̢̡̯̞̲̥̂̀͌̔̿[̵͎̻͍͊͂͂͜[̴̛̥̥͐̈͐A̸̡̬͒̌m̴̯̮̓̄͛̓̈́̍e̷̼͚̺̰̜̼̹͒̍͐̽́̈̕r̴̻̫̹̠̮̩͊̈́́i̷̫͚͐̂͊͒͐̓̈̚c̸̳̻͈͇̟͂̾͑ä̴̡̡̝̭̝̺́́̊̈́͝n̵̞̳͔͇̹͇̿ ̴̺̫̟͖̘̮̣̃́̀̚I̸̯̲̐͑̐͘n̵̦̮̭̲̦̽͌͂͂v̶̻̼̰̤͑͋͗̆̕̚͜ì̴̡͘͜͜ͅͅț̴̼̀̋̄͌͋̓͠a̴̤̗̿̔̌̍̾͝͝t̷̢͚̲͉͈͒̉̅̏̇̓̀ị̴̢͔̮̲͉̉o̸͇͒̍̚͘͠n̶͖̮̿́̓̎͗͝a̸̪̕͝l̸̨̢̰̩͔̜̇̈́̾̊̒̀̇ ̷̼̺͝͝M̴̡͕̖̗̼̯̎͋̍̾ȁ̶̺̤̠͙̀̋͑̇̃͜t̷̝͙̏̊͝h̶̘̙̣̯̰̟̮͆͑͌̊͆e̸̹̖͈̓̾́͋̚͝m̶̡̫̟͇̾̋̽͂̎͑͗͘ͅä̴̳͎̞̦́̇͒̊͘ͅt̸̹̭̫͠i̶̻̠̖̘͚͛̏̈̀͑̕͠͠ċ̵͇͎͎̟̮̩̩̼̇̉̔ś̷̞̘͉̰̃͑̽̄̑ ̶̨̼̋̊E̶̡̼̫̹̋́̓̈́̀͘ͅx̶̻͚̭̘̔̍̚ͅả̵͖͔͙̫͕̙̆̃̐̈͋̅m̷̜̜̙͇̫̯͔̟̊͗̈́̆̃̈́͝i̴̡͍̗̤̝̱͊̆͑́͜n̴͆̓͋̆̏̉̉ͅa̸͎͔̦̘̔̚̚t̸͙̃i̷̛͎͉̙̬̊̄̅̾o̵̧̙̫͝͝ņ̸͇͓͗͒̍͋͘]̵͎͔͚̼̞͇̤̼͊̏̈́̾̄̕̕̕]̴̛͖̖̿̂̒͝
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* [[American Invitational Mathematics Examination]]
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̴̤̩̘̭̣̍́*̷̡͓͔̌̊̊͌ ̸̛̻̻̼̅̿͋[̷͍̼͗́̋͋͗[̷̘̹̹̇̐͐Ą̵͕̬̞̹̥̪̍̅I̶̞̹̞̦̿M̸̢̯̟͇͓̝̉͠ͅĘ̵̢͙͕̗̠͛ ̵̮̽̏͝P̵͚͔̭̪̭̄ṟ̸̫̹̘̅̊̀̈́͒͆̇͝o̷̤͌̑͌͋̈́̈́ḅ̵̝̩̲̗̪̂̋͝l̴̙̲͖̉̌̐̇̈̆͐̓e̷̛͚̓̄́̐͗m̴͎̭͍͛̃̔̓̕̕s̷̰̥͖̥͚͖̭̋́͆̈́ ̷͖̰̻̦̬̫̤̈́ͅã̸̯̱̤̲̖̌̈́́̾̄̿ń̷̛̳̟̝̩̹͎̃d̶̞̤͑̍͛͆ ̵̝̄̀͂̍̄̂Ṣ̷̡͎̻̝̼̪̾̿̄̾͠o̷̡͍̩͓̘̦̾͆ͅl̷̨̰̩͓͍͂̈́̊͑̍̄u̷̢̲͍̼̭̮͚̓̈t̵̼̰͍͚͈̳͑̈̇͛̌̿͑͜ͅi̶̛͕̙͍̭̒̊͊̂͌́͠ͅơ̸̡͈̣͓̣̣͓̒͒͠n̵̬̘̾́̀͜s̶̤͈̻̠̝̏̈́͝]̵̧̝̖͙͕̤͚͂̀́͂͐̓͑̕]̵̡̜̺̱̭͊̎̍̾̈̔͝
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* [[AIME Problems and Solutions]]
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̸̢͙͇̍̅̆̅͊͑̕͜*̶̝̙̆͝ ̶̩̥̥͎͒̚[̶̡̨̗͙̭̟̖̫̀̀͒̌̇[̸̦̾̓̌̿̔M̵̼̹̊͂͊a̵̯̜͓̖̳̻̥͌̒̊͋̐͌̄͊t̶̥̣̟̺̜̺̯̻͊͒͛̚h̶̹̗̘̠̹̤̬̭̅̒̏̈́́ȩ̶͐̓̾̊̕ṃ̸̢̈́͂̂͂̉a̵̺̭̭̦̾̒͊̉͝ṱ̶̰̣̳̼̇̈́̊͛́̏i̶͇̲̪̗̭̖̭͇͆̓͒̀͛͐c̷͖̺̔̆̉̈̐̊͝ş̴͉͙̠̠̩͐͐̃̿̒͝ͅ ̵̻̖̤̀̍͂̒̑̑͌ċ̵̢̛̣̈̊o̸̦̻̞͔͓̱̯͒̒̔̂̕͜͠m̷͖̪͍͚̈̒p̸̢̻̱̌͆͋͌͂̎͝ê̴̡̫͕̭̥̝̞̫t̸̬̰̞͖͙̣͆̈́͋̒̇̔͋̕ĭ̶̢̧̲̞̹̱̲̰͋̋t̷̢̮͓̠̙̲̘̹͘i̸̡̍̅̂ö̴̳͍́̈́͊̄̐̋̌n̴̲̽͗͆͗ ̶̲̮̯͇̺͕̒̋͊̽̈́͛ȓ̶̢̢̛̬̫̼͙͓̇̊͊̉͠e̶̢͇͖̼̣̅̽̈́̓͜ͅs̵̠̥͍͋̄͋͌̍͊͜͠ȍ̷͓͈̾́̀̑͘͠ũ̶̻̤͚͎̳̯͓̀͒̓̊̑͠r̴̻͗͋̿͂́͘̚c̶̣̑̎̌ȩ̷̖̲̹͚̪̗̬̋s̷̳̰͒]̴̨͍̬̫͇̠̫͌̋̓̀̿́̚]̵̡͈͍̘̙̎̓̓̔̈
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* [[Mathematics competition resources]]
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̸̖̪̰͓͉̏͛̏͛̀̄͝{̷̨̩̜͓̲͒̀̋͒{̵͈̖̭̫͇͈̲͇͑̒̀͗͒͘M̶̢̳̫̝͔̱͇͊̅͘͝Ḁ̶̡̧͎͇͓̤̠̉̃̉̈́̔̚̕Ǎ̷͓̩̩̥̟̹͛̃ ̶̧͈̭̘̠̰͝N̸̗͖͓̰̠̹͑o̸̩͚̫̺̖̲̞͊̔͆ţ̵͔͕͙̙̠̘̹̾͋ḯ̵̮͈̈́̍͗̆̎č̶̗̫̱ȅ̵͍̬̌̓͗͝}̶͍̠̬͓̜̗̖̫͋̿̐́͆͝}̶̳̖̞̗͕̞͙̈́̀͘
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{{MAA Notice}}
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̵̛̬͚̟̏͛̽͊̏
 

Latest revision as of 21:34, 28 January 2024

2023 AIME II (Answer Key)
Printable version | AoPS Contest Collections • PDF

Instructions

  1. This is a 15-question, 3-hour examination. All answers are integers ranging from $000$ to $999$, inclusive. Your score will be the number of correct answers; i.e., there is neither partial credit nor a penalty for wrong answers.
  2. No aids other than scratch paper, graph paper, ruler, compass, and protractor are permitted. In particular, calculators and computers are not permitted.
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15

Problem 1

The numbers of apples growing on each of six apple trees form an arithmetic sequence where the greatest number of apples growing on any of the six trees is double the least number of apples growing on any of the six trees. The total number of apples growing on all six trees is $990.$ Find the greatest number of apples growing on any of the six trees.

Solution

Problem 2

Recall that a palindrome is a number that reads the same forward and backward. Find the greatest integer less than $1000$ that is a palindrome both when written in base ten and when written in base eight, such as $292 = 444_{\text{eight}}.$

Solution

Problem 3

Let $\triangle ABC$ be an isosceles triangle with $\angle A = 90^\circ.$ There exists a point $P$ inside $\triangle ABC$ such that $\angle PAB = \angle PBC = \angle PCA$ and $AP = 10.$ Find the area of $\triangle ABC.$

Solution

Problem 4

Let $x,y,$ and $z$ be real numbers satisfying the system of equations \begin{align*} xy + 4z &= 60 \\ yz + 4x &= 60 \\ zx + 4y &= 60. \end{align*} Let $S$ be the set of possible values of $x.$ Find the sum of the squares of the elements of $S.$

Solution

Problem 5

Let $S$ be the set of all positive rational numbers $r$ such that when the two numbers $r$ and $55r$ are written as fractions in lowest terms, the sum of the numerator and denominator of one fraction is the same as the sum of the numerator and denominator of the other fraction. The sum of all the elements of $S$ can be expressed in the form $\frac{p}{q},$ where $p$ and $q$ are relatively prime positive integers. Find $p+q.$

Solution

Problem 6

Consider the L-shaped region formed by three unit squares joined at their sides, as shown below. Two points $A$ and $B$ are chosen independently and uniformly at random from inside the region. The probability that the midpoint of $\overline{AB}$ also lies inside this L-shaped region can be expressed as $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$ [asy] unitsize(2cm); draw((0,0)--(2,0)--(2,1)--(1,1)--(1,2)--(0,2)--cycle); draw((0,1)--(1,1)--(1,0),dashed); [/asy]

Solution

Problem 7

Each vertex of a regular dodecagon ($12$-gon) is to be colored either red or blue, and thus there are $2^{12}$ possible colorings. Find the number of these colorings with the property that no four vertices colored the same color are the four vertices of a rectangle.

Solution

Problem 8

Let $\omega = \cos\frac{2\pi}{7} + i \cdot \sin\frac{2\pi}{7},$ where $i = \sqrt{-1}.$ Find the value of the product \[\prod_{k=0}^6 \left(\omega^{3k} + \omega^k + 1\right).\]

Solution

Problem 9

Circles $\omega_1$ and $\omega_2$ intersect at two points $P$ and $Q,$ and their common tangent line closer to $P$ intersects $\omega_1$ and $\omega_2$ at points $A$ and $B,$ respectively. The line parallel to $AB$ that passes through $P$ intersects $\omega_1$ and $\omega_2$ for the second time at points $X$ and $Y,$ respectively. Suppose $PX=10,$ $PY=14,$ and $PQ=5.$ Then the area of trapezoid $XABY$ is $m\sqrt{n},$ where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find $m+n.$

Solution

Problem 10

Let $N$ be the number of ways to place the integers $1$ through $12$ in the $12$ cells of a $2 \times 6$ grid so that for any two cells sharing a side, the difference between the numbers in those cells is not divisible by $3.$ One way to do this is shown below. Find the number of positive integer divisors of $N.$ \[\begin{array}{|c|c|c|c|c|c|} \hline \,1\, & \,3\, & \,5\, & \,7\, & \,9\, & 11 \\ \hline \,2\, & \,4\, & \,6\, & \,8\, & 10 & 12 \\ \hline \end{array}\]

Solution

Problem 11

Find the number of collections of $16$ distinct subsets of $\{1,2,3,4,5\}$ with the property that for any two subsets $X$ and $Y$ in the collection, $X \cap Y \not= \emptyset.$

Solution

Problem 12

In $\triangle ABC$ with side lengths $AB = 13,$ $BC = 14,$ and $CA = 15,$ let $M$ be the midpoint of $\overline{BC}.$ Let $P$ be the point on the circumcircle of $\triangle ABC$ such that $M$ is on $\overline{AP}.$ There exists a unique point $Q$ on segment $\overline{AM}$ such that $\angle PBQ = \angle PCQ.$ Then $AQ$ can be written as $\frac{m}{\sqrt{n}},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

Solution

Problem 13

Let $A$ be an acute angle such that $\tan A = 2 \cos A.$ Find the number of positive integers $n$ less than or equal to $1000$ such that $\sec^n A + \tan^n A$ is a positive integer whose units digit is $9.$

Solution

Problem 14

A cube-shaped container has vertices $A,$ $B,$ $C,$ and $D,$ where $\overline{AB}$ and $\overline{CD}$ are parallel edges of the cube, and $\overline{AC}$ and $\overline{BD}$ are diagonals of faces of the cube, as shown. Vertex $A$ of the cube is set on a horizontal plane $\mathcal{P}$ so that the plane of the rectangle $ABDC$ is perpendicular to $\mathcal{P},$ vertex $B$ is $2$ meters above $\mathcal{P},$ vertex $C$ is $8$ meters above $\mathcal{P},$ and vertex $D$ is $10$ meters above $\mathcal{P}.$ The cube contains water whose surface is parallel to $\mathcal{P}$ at a height of $7$ meters above $\mathcal{P}.$ The volume of water is $\frac{m}{n}$ cubic meters, where $m$ and $n$ are relatively prime positive integers. Find $m+n.$ [asy] //Made by Djmathman (orz) size(250); defaultpen(linewidth(0.6)); pair A = origin, B = (6,3), X = rotate(40)*B, Y = rotate(70)*X, C = X+Y, Z = X+B, D = B+C, W = B+Y; pair P1 = 0.8*C+0.2*Y, P2 = 2/3*C+1/3*X, P3 = 0.2*D+0.8*Z, P4 = 0.63*D+0.37*W; pair E = (-20,6), F = (-6,-5), G = (18,-2), H = (9,8); filldraw(E--F--G--H--cycle,rgb(0.98,0.98,0.2)); fill(A--Y--P1--P4--P3--Z--B--cycle,rgb(0.35,0.7,0.9)); draw(A--B--Z--X--A--Y--C--X^^C--D--Z); draw(P1--P2--P3--P4--cycle^^D--P4); dot("$A$",A,S); dot("$B$",B,S); dot("$C$",C,N); dot("$D$",D,N); label("$\mathcal P$",(-13,4.5)); [/asy]

Solution

Problem 15

For each positive integer $n$ let $a_n$ be the least positive integer multiple of $23$ such that $a_n \equiv 1 \pmod{2^n}.$ Find the number of positive integers $n$ less than or equal to $1000$ that satisfy $a_n = a_{n+1}.$

Solution

See also

2023 AIME II (Problems • Answer Key • Resources)
Preceded by
2023 AIME I 		Followed by
2024 AIME I
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

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