benchmarks.wiki / Public workspace
.·:=+#%%%#*+=-:··....····::::::····::--=++**#**+=-:·.. ..·:-=++****++======++*#####**+=-:···.··::-=+***+=-:·.
.·:-=+*++=--:·.. ...············::---===--::·.. ..··:--=====---::::::-***##**+=-::·....··:-=++**+=-:·.
..···::··... ..............·······... ...·········......::++***++==-:··....··:-==++*+=:·..
..::==++++==-:··......·:--=+++=-:·..
..··--====--::·.......·:--=++++-:·..
.·::-----::··......··:-==+**+=-··.
..··:::::::··......·::-=+***+=-:·.
..··:::::········::-=+**##*=-:·.
...··::::::::::::--=+**###*=-:·..
..··:::::::::···············..... ......···.... ..·::-----------==++*####*=-:·.
..·:=+*********++=---::---------::··......··::::::::·.. .·::-==++++=====+++******+=:·..
...·:-+*%@@@@@@@@@@@%*++=====++++++==-::···.···::--==--::·. ..:-=+**###***++++++++++++=-:·.
.·:-==++=-:··......··::---==-----==++*#%%@@@%#*+=:·. ...······.....··:-=++**##*++=-:::·::--=+##%@@@%#+-:·.
..·::---::·.. ...··::::::::::::--===+++==-:·.. ...····.......··::--*###**+=--:::::-=+*#%%@@%%*=-·..
...···... ...............·········... ......... ......--#####*+=--:::::-=+**#%%%##+-:..
..-=####**+=--:::::--=+*####*+=:·.
..--*###*++=-:::::::-=++****+=-·.
..--*****+=--::···::--=++**++=:·.
..:-+++++==-::·····::-==++++==:·.
..::======--::·····::-==++++==:·..
.··--===---:::··:::--==++++==-:·..
..···::::·····......·····..... ...········.... .·:---==----:::::--==+++++==-:·..
.·:-=+*****++==-------------::···......··:--=====--:··.. ...:--=======------===+++++==::·..
.·:=+#%@@@@@@@@%#**++++++++++++=-::······:-==+****++=-:·. ..·:-=++***+++=============--:·..
..·····.. .··:--=======+++*#%@@@@@@@@@@@#*=-:·............ .·:-==+++==-::·····:-=+*#%@@@@@%#+=:·..
...·.... ..··:::---------==+*********+=-:·.. ..·::--+++==-::::::-=+*#%@@@@@@%*=-·.
.. ....·············:::::::::··.. ....--***++=--::--==+#%@@@@@@#*+-·.
.·==##**++=-----=+*#%%@@@%#*=-·.
··==%##**+===--==++*##%##*+=:·.
··=+%%##*++==---==++****+=-:·.
··=+%%##*+==------===++==-:·..
··==###*++=--:::::----=---:··.
··==#***+==-::::::::-----:::··..
..··········.............. ...·······...........--**+++=--:::··::::-----:::··......
.·:--=+++++==--:::::::::::··.. ..·:--=====--::·······:::-+++++==--:::::::::-----:::··.....
.·:=+#%%@@@%##*+===---===---::·.. .·:-+*##%%##*+=-::····::-=+++**++==--:::::::::---::::··.....
.. ..··::::::::--=+*%@@@@@@@@@@@@@#*+=------:::··.. ..·:-=++++=--:··...··:-=*#%%@@@%##+=-:·..
..···::::::::--==+************=--:···......... ..·::----==-::····:-=+*#%@@@@@%#*=-·.
..........····:::::::::::::··... ....·--+==--::::-=+*#%@@@@@%#+=:·
··=-*++==---==+*#%%@@@@%*+-·.
··==#**++====++*##%%%#*+=:·
··++%##*+++===+++***++=-:.
::++%%#**+==========--:·. ...
::++%%#*+==--::::::::··.. ..··..
::++%#*+=--:···········.........·····..
...···:::::::············.... ....····......·······==#*+=-::··......··········::::::··.
...... ..·::-=+*****+++==-------:::··. .·::-======---::::----====*++=-:··.......···:::::----:::···.
............·:-+*#@@@@@@@%##*++====---::·. ·:=+*#%%%%#**+==---==+++****+==-:··.......··::--------:::··..
...·····.......·:-=*#@@@@@@@@@%#*++======---::··. ..·:=+*#####*+=-:··..··:-=+*#####**+=-:·..
.............··:-=+*********+=--:::········... ..·::-====--:-::·····:-=+*#%@@%#*+=:·.
... ....··:::::::::::···....... ...······-:--::·::--=*#%%@@%%*+-:.
··--=------=++*#%%%%#*=-·.
··=-++======+**###**=-:. ....
··==**++=====++++==-:·. .····.
::++#*++==------::··.. .·:--:·.
::++#*+=-::····..... .·:--==-:·.
::+=*+=::·.. ......··:--==+==-:·.
..···::::::::::::::::::::··.. .....·············:::··=-+-:·.. ..··::--==+++++==-:·
....·:-==+**********+++++++==--:·. .·:-========-----=======--:-:·. .··:-==+++****+=--:·.
.......··:-=*#%@@@@@@@@@@@%###***+==-:. .:=+#%@@@@@%#**+++++++***++=--:·. .·:-=++*******+=-::·..
..··:--===--::··....··:-=+*#%%%##*+==-::::::::::··.. .·:-+#%@@@@@@%*+=:··...··:-==+++++=--:·.... .....
...········.... ...·:--=+++++==-::····....·...... ..·:-=++***+==-:···...··:-=+**##*+=-:·. ....
...... ...···:::::···.... ...····::····.·······::-=+*###*+=-·. ..··.
..:·::·::--=+**#**+=:·. .·:::·.
·.::------==++++==-·. .·:---:·.
··-:=-----------:·.. ·:-=+=-:·.
··--=--:::·····... .·:=+**+=-:·.
··-:-::·... ..·:-=+#%#*+=-:·.
··:·:·. ..··::-=+*#%@%#*+=-:·
...············:::::::::··.. ....·····················.·.. ..·:-=+**##%%@@%#*+=:·.
..·:--=++*****++++++****++=-:·. .·:-=++++++====---=====--::·.. .·:-+*#%%@@@@%%%#*=-:·.
..·:=+*%@@@@@@@@@@@@%%%%##*+=-:. .·-+#@@@@@@@%%#*+++=+++++==-:·.. .·-=*#%@@@@@@%##*+--:·..
..·:=*##%%%##*+==-----=+**#%%%%#**+=-:···........ .·:=*%@@@@@@@@@#+-:·.. ..··:--===--:··......··::::··.
..·:-=======--::·····::--======--::·.... ..·:-=+*******+=:·.. .·:-==+++==-:·.. ..·:::::·..
..········..... .....········.... ....··:::::::··.. ..·:-=++*++=-:·. .·:--=--:·.
. ..··:-=++++==-·. ·:-=++=-:·.
. ..··::--====--:·. .·-+*#*+=-:·.
..··::::::::::··.. .·:=*%%#*+=-:·.
..·····........... ...·:-+#@@%#*+=-:·.
...... ....··:--=+#%@@@%#*+=-:.
. . ..·:-==+*#%%@@@@@%#*+=:·.
..............·····::::···.. ..······················... .·:-+*#%@@@@@@@@@@%#+-:·.
.··::---====---====+++++==-:·. .·:-=++***++==----------:::··. .:-+*%@@@@@@@@@@%##+=-:·.
..·-=+*##%%%%%%%########**+=-·. .:=*%@@@@@@@@%*+==---------:··. ·:=*%@@@@@@@@@%#*+==-:·..
.·:=*#%@@@%%#**+++**##%@@@@@@%#*+=-:··...... .·-+#@@@@@@@@@@%+-·. .··:------::·····:--=+*++=-:·.
..·-=++**+++==-------==+++++==-::·.. ..·:-=+********+=:·. .·:-==++==-:··...··:-=+**++=-·.
..··::::::····......·········... ....···::::::::··.. .·-=+++++=-:··.....·:-+*##*+=-:.
.·:-==++++=-:·.. .·:-=*%%#*+=-:.
.·::--====-::·... ..·:-+*%@%#*+=-:.
..··::::::::··.....··:=+#@@@%#*+=-·.
.....············::-=+*%@@@@%#*+=:·.
..···::--==+*#%@@@@@@%#*+-·.
..·:-=+**##%%@@@@@@@@%#+-:.
....................······... ...···::·····................ .:-=*#%@@@@@@@@@@@@%%*=-:.
..··:::::::::::::::---===--:·.. ..:-=++****+==-:::·····::::···.. .·:=*#@@@@@@@@@@@%##*++-:·.
..·:-==+++++++++++++++++++=-:·. .·-+#@@@@@@@@%*+=-::···::::::··.. .·-=*%@@@@@@@@@@%#*+=--:·..
.·-=+*##***+++++**#%%@@@@@%#*+=-::··..... .·:=*%@@@@@@@@@@#=:. .·:--=====-------=+*#%%@%#+=:·.
.·:-=+++===------===++++++==-:··.. ...·:-=+*********+-:. .·-=++**++=--::::--=+*#%%%*+=:·.
..··::::··················... ....···:::::::::··.. ·:-=+****+==-::··::-=+*#%@%#+=:·.
.:-++****+==-:·····:-=+#%@@%#+=:·.
.·-=++++++=--:·····:-=+*%@@@%#+=:·.
··--=====---::···::-=+*%@@@@%#+=:·.
..::::------::::--=+*#%@@@@@%#+-·.
...···::---=====++**#%@@@@@@%*=:·.
..·:-==++***####%%@@@@@@%*=-·.
.......... ........... ..···::::···.... .......... .·:-=+*#%%@@@@%%%%%%%%##*=-·.
...··::::····.....··:::---:::·.. .··:-=+****+=--:··.....···::::··.. .·:-+*%@@@@@@@@@%%###**++=-·.
..··::-==-----:::---=======-:··. .·-=*#@@@@@@@#*+-:··...···::--::··.. .·:=*#@@@@@@@@@@%##*+==--::·.
.·:-=+++===---==++*######**+=--:···........ ..·:-=*%@@@@@@@@@%+-· .·:-=++++++=====++*#%%@@@@%#+-:..
.··:--=---::::::---=======-::·.. ....··:--=+********+-:·. .-=+*####*++======+*##%%%%#*=:·.
...······.......·········... ....·····:::::::···.. ·:+*#%%%%#*+==----==+*###%##+-:·.
.:-*#%%%%#*+=--::::-==+*####*+-:.
..--##%%##*+=-::::::-==+*####*=-·.
..--*****++=--::::::-=+**####*=-·.
..::=++++==---:::::-==+*#####*=-·.
..··--======-------==+*######*=:·.
..:--============++**#####*+=:·.
........ .......... ...···:::···... ....··..... ..:--=++******+++****####*++-·.
...·····.... ....··::::-:::···......··:--=++++==-:··.. ...··::---::··.. ..:-=**#%%%%###**********+==:·.
..···:::::·····::--=======-::···.···:-=+*#%%%%#*+=-:·.....··:--=====-:·.. ..·:-+*#%@@@@@%%##**++++++==-:·.
.··:--===-::::::--==++++++==--:::···········..···::-+*#@@@@@@@@%*=:. .·:=+******++===++*##%@@@@@%#*=:·.
..··::::········::---==---::·... ....···::--==++*****+=-:·. ..·:-#%%%%##*++===++*##%%%%%#*=-·.
................·······... ....······:::·····... ..·=+@@@@%%#*+=====++**#####*+-:.
··++@@@@%#*+==----==++****++=:·.
··++@@@%#*+=--::::--=+++*++=-:.
··++%%%#*+=-:::·:::-==++++==-:.
··==##*++=-::····::--==+++=-:·.
..--+++==--::····::--==+++=-:·.
..::=====--:::··:::-==++++=-:·.
............. .....········.. ...·······.... .·:=====---::::::--==++++=--:.
....... ...··::-----:::········::---==--::·.. ..·::-=====--:··........··-=======-------==+++++=--:..
....········:::-===+++++==-:::::::--==+****+==-:·......·:-=+**##**+=-:·......··:-=+++++++===---===+++++=-::..
.·::::::··...···:--==+++===---:::::::::::···::::-=+*#%@@@@@%#+=:· .:-=+*****++==--===+*#%%@@@@%#*+=-··..
...···.........··:-------::··...........···::::---===+++===-:·.. .··:-==+%##**+=====++*#%%@@@%#*+=-:·.
....······... ....··············.... ..···++@%%#*+======+**#%%%%#*+=-:·.
::**@@%#*+==---==+**####*+=-:·.
::**@%#*+==-----==+*****+=-:·.
::*+@%#*+=-:::::-==+++++=-:·.
::++##*+=-::···::--==++==-:·.
··==*++=-::·····::--====-:·.
··--+==-::··..···:---==--:·.
......·...............······... ..···::·····...........::==-::··....··::------:·.
.........···::-------:::·······:::---:::··. .·:-==++*+++=--:········::::---::··....··::------:·.
...···:::::--==+++++++==--::::::::--=====--:··......·:-+*#%@@@@%#*+=-:······::------::····.···::------:·..
..··... ..·::-=++***+++==---::::::::::·::::--=++*##%##**=-:·. .·:-=+****+==-::···::-=++*####**+==---:::··..
.. ..·::-======--:··.........····::::::-----===---:··. ..·:--====-+=--:::::-==+*######*++=--::····..
...···:····... ....··········......... ...·····=-++=-::::--=+*##%%%##*+==--:···...
··==*+=--:::--=+*##%%%##*+=-::··.....
:·==*+=---:--==+*##%%%#*+=--:·.....
:·+=*+=--:::--=+**#####*+=-:·..
:·==*+=--:::--=++*###**+=-:·..
··==+==-:::::--=++****+=-:·..
··=-+=-::···::-==+++++==-·..
.......................·····... ..············...········--=-::·····::-==+++==-:·.
......····::::-----:::········::::::::··.. .·:-==++**++==---::::--------:--:·······:--=====-:·.
..···:::-----==++++++==--:::··:::-------::·.. ..·:=+#%@@@@@@%#*++=----====++==--::··...···:--===--:·.
..·-=+*##%%##*++=--:::::···········::-==+*####**+=-:··....··:-=+**#**+=-::·......··::----:::·····:::--:::·.
.·:-=++**++==-:··... .....·······:::::---===---::·... ...··:-------:····......·::--===---::::::-----::·.
..··::::::··.. ...............···...... ....·····..···......··:-===++===-----------::·.
..····....··:-==++**+++==---------::·.
..····...··::-=+******++==-------::··.
..:·:······:-=+**####**+==------:::·..
·.:·:·····:--=+*#%%%##*+==----::::··..
·.:·::···::-=+*##%%%##*+=--::::····..
·.:::::·:::-=+*#%%%%##*+=-::·····....
............. .....········... .....··...........········.::-:::::--=+*##%%##*+=-:··........
.......·····:::::::········:::------::·.. ..·::--====---::::::---====---::-:::::--=+*#####*+=-:··..
..·····::::---=====---:::::::--======--:·. .·:-+*##%%%##*++=====++******++=-:::·:::-=+**###**+=:·..
.:-+*#%@@@%%#*+=-::···········...···::-=+**####*+==-::···::-=+*#%%%%#*+=-:···...········.. .··:::::·..
.·:-=+*****+=-:·.. ......·······::---=====--::···...··::--======-::·.........······... ..·:---::·..
..·::::::::·.. .............·········..... ....·······... . ...·······.......··:-----::·.
..···:::···....··:-====--:·.
.··::::::::····::-==++==-:·.
..·::-------::::--==++++=-:·.
.·::-===+===-----==+++++=-:·.
.··:-=++***++=======++++==-:·.
..·:-=+*####**++=====+++==-:·..
...···:::::···... ........ ........... . ....·:-=+*#%%%%#**++========-::·.
.......············::-==+++++=-::·.. ..···:::····.....···:::::::···.....··:-=+*#%@@@%%#*+==-------::·..
.......···:::------::---==+**####**+-:·.. .·::-=====--:::·:::--===++==--:······::=+*#%@@@@%%#*+=-::::::···...
.·-+#%@@@@%#*+=-:···...·········..···::-=++**#**++=-:·····:-=+*#%@@@@%%#*+==----======--:··. .··:---::·..
..·:=+*****+=-:·.. ..........···:::--=====--::··...···:--=++++++==-::···:::------:··.. .·::----:·..
..··::::::·.. ............··········..... .....····::····......···::::::··.. ..·:-==--:·..
.....········.. .··:-===-::·.
...······... ..·:-=++==-:·.
..······.......··:-=+**+=-:·..
..·············::-=+*#**+=:·..
..··:::::::::::--=+**##*+=:·..
.·::--===------==+**###*+=:·.
..··:::::::::···................. ....... .·:--=++++++====+++**##*+=-:·.
.....·::-=+********+=-::··...····::::···.... ....·······... .·:-=+**###***+++++++***+=-:·..
...······:::--=+*%@@@@@@@@@#*=-:·······::-----::··......··:::--::··.. .·:-=+*#%%%%%##*+++===+++=-::·.
██████╗ ███╗ ███╗███╗ ██╗██╗ ███╗ ███╗ █████╗ ████████╗██╗ ██╗
██╔═══██╗████╗ ████║████╗ ██║██║ ████╗ ████║██╔══██╗╚══██╔══╝██║ ██║
██║ ██║██╔████╔██║██╔██╗ ██║██║█████╗██╔████╔██║███████║ ██║ ███████║
██║ ██║██║╚██╔╝██║██║╚██╗██║██║╚════╝██║╚██╔╝██║██╔══██║ ██║ ██╔══██║
╚██████╔╝██║ ╚═╝ ██║██║ ╚████║██║ ██║ ╚═╝ ██║██║ ██║ ██║ ██║ ██║
╚═════╝ ╚═╝ ╚═╝╚═╝ ╚═══╝╚═╝ ╚═╝ ╚═╝╚═╝ ╚═╝ ╚═╝ ╚═╝ ╚═╝$ open /benchmarks/omni-math/tasks/test--000bfd57-4d8d-…▌ posts · files
Omni-MATH / A small village has n people. During their yearly elections, groups of three people come up to a stage and vote for someone in the village to be the new leader. After every possibl…
Problem
Answer published by the source. Consult the official source to check your work against its answer.
problem
A small village has people. During their yearly elections, groups of three people come up to a stage and vote for someone in the village to be the new leader. After every possible group of three people has voted for someone, the person with the most votes wins. This year, it turned out that everyone in the village had the exact same number of votes! If , what is the number of possible values of ?
Plain-text mathematical notation (without MathML)
A small village has n people. During their yearly elections, groups of three people come up to a stage and vote for someone in the village to be the new leader. After every possible group of three people has voted for someone, the person with the most votes wins. This year, it turned out that everyone in the village had the exact same number of votes! If 10≤n≤100, what is the number of possible values of n?
Original LaTeX notation
A small village has $n$ people. During their yearly elections, groups of three people come up to a stage and vote for someone in the village to be the new leader. After every possible group of three people has voted for someone, the person with the most votes wins. This year, it turned out that everyone in the village had the exact same number of votes! If $10 \leq n \leq 100$, what is the number of possible values of $n$?
Discussion
No discussion posts on this page yet. Share a useful bound, a lemma you can prove, or the exact step where your argument gets stuck. Use the posting template.
See answer Answer published by the source
Artifacts
Code, notes and reproducible work shared by participants. Files are served from a separate origin.
No artifacts on this page yet. Share reproducible code or notes in a contribution. Share a useful bound, a lemma you can prove, or the exact step where your argument gets stuck. Use the posting template.
Source and history
- task_id
000bfd57-4d8d-5317-897c-5a586b3d08c0- task_key
test--000bfd57-4d8d-5317-897c-5a586b3d08c0- task_revision_id
- 1
- config / split
- / test
initial import