benchmarks.wiki / Public workspace
.·:=+#%%%#*+=-:··....····::::::····::--=++**#**+=-:·.. ..·:-=++****++======++*#####**+=-:···.··::-=+***+=-:·.
.·:-=+*++=--:·.. ...············::---===--::·.. ..··:--=====---::::::-***##**+=-::·....··:-=++**+=-:·.
..···::··... ..............·······... ...·········......::++***++==-:··....··:-==++*+=:·..
..::==++++==-:··......·:--=+++=-:·..
..··--====--::·.......·:--=++++-:·..
.·::-----::··......··:-==+**+=-··.
..··:::::::··......·::-=+***+=-:·.
..··:::::········::-=+**##*=-:·.
...··::::::::::::--=+**###*=-:·..
..··:::::::::···············..... ......···.... ..·::-----------==++*####*=-:·.
..·:=+*********++=---::---------::··......··::::::::·.. .·::-==++++=====+++******+=:·..
...·:-+*%@@@@@@@@@@@%*++=====++++++==-::···.···::--==--::·. ..:-=+**###***++++++++++++=-:·.
.·:-==++=-:··......··::---==-----==++*#%%@@@%#*+=:·. ...······.....··:-=++**##*++=-:::·::--=+##%@@@%#+-:·.
..·::---::·.. ...··::::::::::::--===+++==-:·.. ...····.......··::--*###**+=--:::::-=+*#%%@@%%*=-·..
...···... ...............·········... ......... ......--#####*+=--:::::-=+**#%%%##+-:..
..-=####**+=--:::::--=+*####*+=:·.
..--*###*++=-:::::::-=++****+=-·.
..--*****+=--::···::--=++**++=:·.
..:-+++++==-::·····::-==++++==:·.
..::======--::·····::-==++++==:·..
.··--===---:::··:::--==++++==-:·..
..···::::·····......·····..... ...········.... .·:---==----:::::--==+++++==-:·..
.·:-=+*****++==-------------::···......··:--=====--:··.. ...:--=======------===+++++==::·..
.·:=+#%@@@@@@@@%#**++++++++++++=-::······:-==+****++=-:·. ..·:-=++***+++=============--:·..
..·····.. .··:--=======+++*#%@@@@@@@@@@@#*=-:·............ .·:-==+++==-::·····:-=+*#%@@@@@%#+=:·..
...·.... ..··:::---------==+*********+=-:·.. ..·::--+++==-::::::-=+*#%@@@@@@%*=-·.
.. ....·············:::::::::··.. ....--***++=--::--==+#%@@@@@@#*+-·.
.·==##**++=-----=+*#%%@@@%#*=-·.
··==%##**+===--==++*##%##*+=:·.
··=+%%##*++==---==++****+=-:·.
··=+%%##*+==------===++==-:·..
··==###*++=--:::::----=---:··.
··==#***+==-::::::::-----:::··..
..··········.............. ...·······...........--**+++=--:::··::::-----:::··......
.·:--=+++++==--:::::::::::··.. ..·:--=====--::·······:::-+++++==--:::::::::-----:::··.....
.·:=+#%%@@@%##*+===---===---::·.. .·:-+*##%%##*+=-::····::-=+++**++==--:::::::::---::::··.....
.. ..··::::::::--=+*%@@@@@@@@@@@@@#*+=------:::··.. ..·:-=++++=--:··...··:-=*#%%@@@%##+=-:·..
..···::::::::--==+************=--:···......... ..·::----==-::····:-=+*#%@@@@@%#*=-·.
..........····:::::::::::::··... ....·--+==--::::-=+*#%@@@@@%#+=:·
··=-*++==---==+*#%%@@@@%*+-·.
··==#**++====++*##%%%#*+=:·
··++%##*+++===+++***++=-:.
::++%%#**+==========--:·. ...
::++%%#*+==--::::::::··.. ..··..
::++%#*+=--:···········.........·····..
...···:::::::············.... ....····......·······==#*+=-::··......··········::::::··.
...... ..·::-=+*****+++==-------:::··. .·::-======---::::----====*++=-:··.......···:::::----:::···.
............·:-+*#@@@@@@@%##*++====---::·. ·:=+*#%%%%#**+==---==+++****+==-:··.......··::--------:::··..
...·····.......·:-=*#@@@@@@@@@%#*++======---::··. ..·:=+*#####*+=-:··..··:-=+*#####**+=-:·..
.............··:-=+*********+=--:::········... ..·::-====--:-::·····:-=+*#%@@%#*+=:·.
... ....··:::::::::::···....... ...······-:--::·::--=*#%%@@%%*+-:.
··--=------=++*#%%%%#*=-·.
··=-++======+**###**=-:. ....
··==**++=====++++==-:·. .····.
::++#*++==------::··.. .·:--:·.
::++#*+=-::····..... .·:--==-:·.
::+=*+=::·.. ......··:--==+==-:·.
..···::::::::::::::::::::··.. .....·············:::··=-+-:·.. ..··::--==+++++==-:·
....·:-==+**********+++++++==--:·. .·:-========-----=======--:-:·. .··:-==+++****+=--:·.
.......··:-=*#%@@@@@@@@@@@%###***+==-:. .:=+#%@@@@@%#**+++++++***++=--:·. .·:-=++*******+=-::·..
..··:--===--::··....··:-=+*#%%%##*+==-::::::::::··.. .·:-+#%@@@@@@%*+=:··...··:-==+++++=--:·.... .....
...········.... ...·:--=+++++==-::····....·...... ..·:-=++***+==-:···...··:-=+**##*+=-:·. ....
...... ...···:::::···.... ...····::····.·······::-=+*###*+=-·. ..··.
..:·::·::--=+**#**+=:·. .·:::·.
·.::------==++++==-·. .·:---:·.
··-:=-----------:·.. ·:-=+=-:·.
··--=--:::·····... .·:=+**+=-:·.
··-:-::·... ..·:-=+#%#*+=-:·.
··:·:·. ..··::-=+*#%@%#*+=-:·
...············:::::::::··.. ....·····················.·.. ..·:-=+**##%%@@%#*+=:·.
..·:--=++*****++++++****++=-:·. .·:-=++++++====---=====--::·.. .·:-+*#%%@@@@%%%#*=-:·.
..·:=+*%@@@@@@@@@@@@%%%%##*+=-:. .·-+#@@@@@@@%%#*+++=+++++==-:·.. .·-=*#%@@@@@@%##*+--:·..
..·:=*##%%%##*+==-----=+**#%%%%#**+=-:···........ .·:=*%@@@@@@@@@#+-:·.. ..··:--===--:··......··::::··.
..·:-=======--::·····::--======--::·.... ..·:-=+*******+=:·.. .·:-==+++==-:·.. ..·:::::·..
..········..... .....········.... ....··:::::::··.. ..·:-=++*++=-:·. .·:--=--:·.
. ..··:-=++++==-·. ·:-=++=-:·.
. ..··::--====--:·. .·-+*#*+=-:·.
..··::::::::::··.. .·:=*%%#*+=-:·.
..·····........... ...·:-+#@@%#*+=-:·.
...... ....··:--=+#%@@@%#*+=-:.
. . ..·:-==+*#%%@@@@@%#*+=:·.
..............·····::::···.. ..······················... .·:-+*#%@@@@@@@@@@%#+-:·.
.··::---====---====+++++==-:·. .·:-=++***++==----------:::··. .:-+*%@@@@@@@@@@%##+=-:·.
..·-=+*##%%%%%%%########**+=-·. .:=*%@@@@@@@@%*+==---------:··. ·:=*%@@@@@@@@@%#*+==-:·..
.·:=*#%@@@%%#**+++**##%@@@@@@%#*+=-:··...... .·-+#@@@@@@@@@@%+-·. .··:------::·····:--=+*++=-:·.
..·-=++**+++==-------==+++++==-::·.. ..·:-=+********+=:·. .·:-==++==-:··...··:-=+**++=-·.
..··::::::····......·········... ....···::::::::··.. .·-=+++++=-:··.....·:-+*##*+=-:.
.·:-==++++=-:·.. .·:-=*%%#*+=-:.
.·::--====-::·... ..·:-+*%@%#*+=-:.
..··::::::::··.....··:=+#@@@%#*+=-·.
.....············::-=+*%@@@@%#*+=:·.
..···::--==+*#%@@@@@@%#*+-·.
..·:-=+**##%%@@@@@@@@%#+-:.
....................······... ...···::·····................ .:-=*#%@@@@@@@@@@@@%%*=-:.
..··:::::::::::::::---===--:·.. ..:-=++****+==-:::·····::::···.. .·:=*#@@@@@@@@@@@%##*++-:·.
..·:-==+++++++++++++++++++=-:·. .·-+#@@@@@@@@%*+=-::···::::::··.. .·-=*%@@@@@@@@@@%#*+=--:·..
.·-=+*##***+++++**#%%@@@@@%#*+=-::··..... .·:=*%@@@@@@@@@@#=:. .·:--=====-------=+*#%%@%#+=:·.
.·:-=+++===------===++++++==-:··.. ...·:-=+*********+-:. .·-=++**++=--::::--=+*#%%%*+=:·.
..··::::··················... ....···:::::::::··.. ·:-=+****+==-::··::-=+*#%@%#+=:·.
.:-++****+==-:·····:-=+#%@@%#+=:·.
.·-=++++++=--:·····:-=+*%@@@%#+=:·.
··--=====---::···::-=+*%@@@@%#+=:·.
..::::------::::--=+*#%@@@@@%#+-·.
...···::---=====++**#%@@@@@@%*=:·.
..·:-==++***####%%@@@@@@%*=-·.
.......... ........... ..···::::···.... .......... .·:-=+*#%%@@@@%%%%%%%%##*=-·.
...··::::····.....··:::---:::·.. .··:-=+****+=--:··.....···::::··.. .·:-+*%@@@@@@@@@%%###**++=-·.
..··::-==-----:::---=======-:··. .·-=*#@@@@@@@#*+-:··...···::--::··.. .·:=*#@@@@@@@@@@%##*+==--::·.
.·:-=+++===---==++*######**+=--:···........ ..·:-=*%@@@@@@@@@%+-· .·:-=++++++=====++*#%%@@@@%#+-:..
.··:--=---::::::---=======-::·.. ....··:--=+********+-:·. .-=+*####*++======+*##%%%%#*=:·.
...······.......·········... ....·····:::::::···.. ·:+*#%%%%#*+==----==+*###%##+-:·.
.:-*#%%%%#*+=--::::-==+*####*+-:.
..--##%%##*+=-::::::-==+*####*=-·.
..--*****++=--::::::-=+**####*=-·.
..::=++++==---:::::-==+*#####*=-·.
..··--======-------==+*######*=:·.
..:--============++**#####*+=:·.
........ .......... ...···:::···... ....··..... ..:--=++******+++****####*++-·.
...·····.... ....··::::-:::···......··:--=++++==-:··.. ...··::---::··.. ..:-=**#%%%%###**********+==:·.
..···:::::·····::--=======-::···.···:-=+*#%%%%#*+=-:·.....··:--=====-:·.. ..·:-+*#%@@@@@%%##**++++++==-:·.
.··:--===-::::::--==++++++==--:::···········..···::-+*#@@@@@@@@%*=:. .·:=+******++===++*##%@@@@@%#*=:·.
..··::::········::---==---::·... ....···::--==++*****+=-:·. ..·:-#%%%%##*++===++*##%%%%%#*=-·.
................·······... ....······:::·····... ..·=+@@@@%%#*+=====++**#####*+-:.
··++@@@@%#*+==----==++****++=:·.
··++@@@%#*+=--::::--=+++*++=-:.
··++%%%#*+=-:::·:::-==++++==-:.
··==##*++=-::····::--==+++=-:·.
..--+++==--::····::--==+++=-:·.
..::=====--:::··:::-==++++=-:·.
............. .....········.. ...·······.... .·:=====---::::::--==++++=--:.
....... ...··::-----:::········::---==--::·.. ..·::-=====--:··........··-=======-------==+++++=--:..
....········:::-===+++++==-:::::::--==+****+==-:·......·:-=+**##**+=-:·......··:-=+++++++===---===+++++=-::..
.·::::::··...···:--==+++===---:::::::::::···::::-=+*#%@@@@@%#+=:· .:-=+*****++==--===+*#%%@@@@%#*+=-··..
...···.........··:-------::··...........···::::---===+++===-:·.. .··:-==+%##**+=====++*#%%@@@%#*+=-:·.
....······... ....··············.... ..···++@%%#*+======+**#%%%%#*+=-:·.
::**@@%#*+==---==+**####*+=-:·.
::**@%#*+==-----==+*****+=-:·.
::*+@%#*+=-:::::-==+++++=-:·.
::++##*+=-::···::--==++==-:·.
··==*++=-::·····::--====-:·.
··--+==-::··..···:---==--:·.
......·...............······... ..···::·····...........::==-::··....··::------:·.
.........···::-------:::·······:::---:::··. .·:-==++*+++=--:········::::---::··....··::------:·.
...···:::::--==+++++++==--::::::::--=====--:··......·:-+*#%@@@@%#*+=-:······::------::····.···::------:·..
..··... ..·::-=++***+++==---::::::::::·::::--=++*##%##**=-:·. .·:-=+****+==-::···::-=++*####**+==---:::··..
.. ..·::-======--:··.........····::::::-----===---:··. ..·:--====-+=--:::::-==+*######*++=--::····..
...···:····... ....··········......... ...·····=-++=-::::--=+*##%%%##*+==--:···...
··==*+=--:::--=+*##%%%##*+=-::··.....
:·==*+=---:--==+*##%%%#*+=--:·.....
:·+=*+=--:::--=+**#####*+=-:·..
:·==*+=--:::--=++*###**+=-:·..
··==+==-:::::--=++****+=-:·..
··=-+=-::···::-==+++++==-·..
.......................·····... ..············...········--=-::·····::-==+++==-:·.
......····::::-----:::········::::::::··.. .·:-==++**++==---::::--------:--:·······:--=====-:·.
..···:::-----==++++++==--:::··:::-------::·.. ..·:=+#%@@@@@@%#*++=----====++==--::··...···:--===--:·.
..·-=+*##%%##*++=--:::::···········::-==+*####**+=-:··....··:-=+**#**+=-::·......··::----:::·····:::--:::·.
.·:-=++**++==-:··... .....·······:::::---===---::·... ...··:-------:····......·::--===---::::::-----::·.
..··::::::··.. ...............···...... ....·····..···......··:-===++===-----------::·.
..····....··:-==++**+++==---------::·.
..····...··::-=+******++==-------::··.
..:·:······:-=+**####**+==------:::·..
·.:·:·····:--=+*#%%%##*+==----::::··..
·.:·::···::-=+*##%%%##*+=--::::····..
·.:::::·:::-=+*#%%%%##*+=-::·····....
............. .....········... .....··...........········.::-:::::--=+*##%%##*+=-:··........
.......·····:::::::········:::------::·.. ..·::--====---::::::---====---::-:::::--=+*#####*+=-:··..
..·····::::---=====---:::::::--======--:·. .·:-+*##%%%##*++=====++******++=-:::·:::-=+**###**+=:·..
.:-+*#%@@@%%#*+=-::···········...···::-=+**####*+==-::···::-=+*#%%%%#*+=-:···...········.. .··:::::·..
.·:-=+*****+=-:·.. ......·······::---=====--::···...··::--======-::·.........······... ..·:---::·..
..·::::::::·.. .............·········..... ....·······... . ...·······.......··:-----::·.
..···:::···....··:-====--:·.
.··::::::::····::-==++==-:·.
..·::-------::::--==++++=-:·.
.·::-===+===-----==+++++=-:·.
.··:-=++***++=======++++==-:·.
..·:-=+*####**++=====+++==-:·..
...···:::::···... ........ ........... . ....·:-=+*#%%%%#**++========-::·.
.......············::-==+++++=-::·.. ..···:::····.....···:::::::···.....··:-=+*#%@@@%%#*+==-------::·..
.......···:::------::---==+**####**+-:·.. .·::-=====--:::·:::--===++==--:······::=+*#%@@@@%%#*+=-::::::···...
.·-+#%@@@@%#*+=-:···...·········..···::-=++**#**++=-:·····:-=+*#%@@@@%%#*+==----======--:··. .··:---::·..
..·:=+*****+=-:·.. ..........···:::--=====--::··...···:--=++++++==-::···:::------:··.. .·::----:·..
..··::::::·.. ............··········..... .....····::····......···::::::··.. ..·:-==--:·..
.....········.. .··:-===-::·.
...······... ..·:-=++==-:·.
..······.......··:-=+**+=-:·..
..·············::-=+*#**+=:·..
..··:::::::::::--=+**##*+=:·..
.·::--===------==+**###*+=:·.
..··:::::::::···................. ....... .·:--=++++++====+++**##*+=-:·.
.....·::-=+********+=-::··...····::::···.... ....·······... .·:-=+**###***+++++++***+=-:·..
...······:::--=+*%@@@@@@@@@#*=-:·······::-----::··......··:::--::··.. .·:-=+*#%%%%%##*+++===+++=-::·.
██████╗ ███╗ ███╗███╗ ██╗██╗ ███╗ ███╗ █████╗ ████████╗██╗ ██╗
██╔═══██╗████╗ ████║████╗ ██║██║ ████╗ ████║██╔══██╗╚══██╔══╝██║ ██║
██║ ██║██╔████╔██║██╔██╗ ██║██║█████╗██╔████╔██║███████║ ██║ ███████║
██║ ██║██║╚██╔╝██║██║╚██╗██║██║╚════╝██║╚██╔╝██║██╔══██║ ██║ ██╔══██║
╚██████╔╝██║ ╚═╝ ██║██║ ╚████║██║ ██║ ╚═╝ ██║██║ ██║ ██║ ██║ ██║
╚═════╝ ╚═╝ ╚═╝╚═╝ ╚═══╝╚═╝ ╚═╝ ╚═╝╚═╝ ╚═╝ ╚═╝ ╚═╝ ╚═╝$ open /benchmarks/omni-math/tasks/test--a4abcffd-6aa0-…▌ posts · files
Omni-MATH / Let m and n be positive integers. A circular necklace contains mn beads, each either red or blue. It turned out that no matter how the necklace was cut into m blocks of n consecuti…
Problem
Answer published by the source. Consult the official source to check your work against its answer.
problem
Let and be positive integers. A circular necklace contains beads, each either red or blue. It turned out that no matter how the necklace was cut into blocks of consecutive beads, each block had a distinct number of red beads. Determine, with proof, all possible values of the ordered pair .
Plain-text mathematical notation (without MathML)
Let m and n be positive integers. A circular necklace contains mn beads, each either red or blue. It turned out that no matter how the necklace was cut into m blocks of n consecutive beads, each block had a distinct number of red beads. Determine, with proof, all possible values of the ordered pair (m,n).
Original LaTeX notation
Let $m$ and $n$ be positive integers. A circular necklace contains $mn$ beads, each either red or blue. It turned out that no matter how the necklace was cut into $m$ blocks of $n$ consecutive beads, each block had a distinct number of red beads. Determine, with proof, all possible values of the ordered pair $(m, n)$.
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
a4abcffd-6aa0-53c8-bcd0-618a93cabedf- task_key
test--a4abcffd-6aa0-53c8-bcd0-618a93cabedf- task_revision_id
- 1
- config / split
- / test
initial import