MathDB

1

Part of 2012 Tuymaada Olympiad

Problems(3)

Chips on a chessboard

Source: Tuymaada 2012, Problem 1, Day 1, Seniors and Juniors

7/21/2012
Tanya and Serezha take turns putting chips in empty squares of a chessboard. Tanya starts with a chip in an arbitrary square. At every next move, Serezha must put a chip in the column where Tanya put her last chip, while Tanya must put a chip in the row where Serezha put his last chip. The player who cannot make a move loses. Which of the players has a winning strategy?
Proposed by A. Golovanov
symmetrycombinatorics proposedcombinatoricscombinatorics solvedgridrectangle
Diophantine equation

Source: Tuymaada 2012, Problem 5/6, Day 2, Seniors/Juniors

7/18/2012
Solve in positive integers the following equation: 1n2−32n3=1m2{1\over n^2}-{3\over 2n^3}={1\over m^2}
Proposed by A. Golovanov
number theory unsolvednumber theory
Labeling of the vertices of a regular 2012-gon

Source: Tuymaada 2012, Problem 5, Day 2, Juniors

7/21/2012
The vertices of a regular 20122012-gon are labeled A1,A2,…,A2012A_1,A_2,\ldots, A_{2012} in some order. It is known that if k+ℓk+\ell and m+nm+n leave the same remainder when divided by 20122012, then the chords AkAℓA_kA_{\ell} and AmAnA_mA_n have no common points. Vasya walks around the polygon and sees that the first two vertices are labeled A1A_1 and A4A_4. How is the tenth vertex labeled?
Proposed by A. Golovanov
modular arithmeticratioarithmetic sequencecombinatorics proposedcombinatorics