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2

Part of 2013 Middle European Mathematical Olympiad

Problems(2)

4n tokens on 4n*4n square

Source: Middle European Mathematical Olympiad 2013 I-2

5/17/2014
Let nn be a positive integer. On a board consisting of 4n×4n4n \times 4n squares, exactly 4n4n tokens are placed so that each row and each column contains one token. In a step, a token is moved horizontally of vertically to a neighbouring square. Several tokens may occupy the same square at the same time. The tokens are to be moved to occupy all the squares of one of the two diagonals. Determine the smallest number k(n)k(n) such that for any initial situation, we can do it in at most k(n)k(n) steps.
combinatorics proposedcombinatorics
Four variables

Source: Middle European Mathematical Olympiad 2013 T-2

5/17/2014
Let x,y,z,w x, y, z, w be nonzero real numbers such that x+y≠0 x+y \ne 0, z+w≠0 z+w \ne 0 , and xy+zw≥0 xy+zw \ge 0 . Prove that (x+yz+w+z+wx+y)−1+12≥(xz+zx)−1+(yw+wy)−1 \left( \frac{x+y}{z+w} + \frac{z+w}{x+y} \right) ^{-1} + \frac{1}{2} \ge \left( \frac{x}{z} + \frac{z}{x} \right) ^{-1} + \left( \frac{y}{w} + \frac{w}{y} \right) ^{-1}
inequalities proposedinequalities