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Part of 2006 Tuymaada Olympiad
Problems(2)
seven primes
Source: tuymaada 2006 - problem 1
7/17/2006
Seven different odd primes are given. Is it possible that for any two of them, the difference of their eight powers to be divisible by all the remaining ones ?Proposed by F. Petrov, K. Sukhov
number theory unsolvednumber theory
boxing championship
Source: tuymaada 2006 - problem 5
7/17/2006
There are 100 boxers, each of them having different strengths, who participate in a tournament. Any of them fights each other only once. Several boxers form a plot. In one of their matches, they hide in their glove a horse shoe. If in a fight, only one of the boxers has a horse shoe hidden, he wins the fight; otherwise, the stronger boxer wins. It is known that there are three boxers who obtained (strictly) more wins than the strongest three boxers. What is the minimum number of plotters ?Proposed by N. Kalinin
combinatorics solvedcombinatorics