Subcontests
(5)Gangnam-Style
Let n be a positive integer. A \emph{pseudo-Gangnam Style} is a dance competition between players A and B. At time 0, both players face to the north. For every k≥1, at time 2k−1, player A can either choose to stay stationary, or turn 90∘ clockwise, and player B is forced to follow him; at time 2k, player B can either choose to stay stationary, or turn 90∘ clockwise, and player A is forced to follow him.After time n, the music stops and the competition is over. If the final position of both players is north or east, A wins. If the final position of both players is south or west, B wins. Determine who has a winning strategy when:(a) n=20132012(b) n=20132013 Special Integers
Find all positive integers a∈{1,2,3,4} such that if b=2a, then there exist infinitely many positive integers n such that 2naa…aa−nbb…bb is a perfect square. Tangent circles
Consider a triangle ABC with height AH and H on BC. Let γ1 and γ2 be the circles with diameter BH,CH respectively, and let their centers be O1 and O2. Points X,Y lie on γ1,γ2 respectively such that AX,AY are tangent to each circle and X,Y,H are all distinct. P is a point such that PO1 is perpendicular to BX and PO2 is perpendicular to CY. Prove that the circumcircles of PXY and AO1O2 are tangent to each other. Mysterious Fixed Sum
The cells of an n×n table are filled with the numbers 1,2,…,n for the first row, n+1,n+2,…,2n for the second, and so on until n2−n,n2−n+1,…,n2 for the n-th row. Peter picks n numbers from this table such that no two of them lie on the same row or column. Peter then calculates the sum S of the numbers he has chosen. Prove that Peter always gets the same number for S, no matter how he chooses his n numbers.