MathDB

2007 Kyiv Mathematical Festival

Part of Kyiv Mathematical Festival

Subcontests

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inequalities from Kyiv math festival

Let a,b,c>0a,b,c>0 and abc≥1.abc\ge1. Prove that a) (a+1a+1)(b+1b+1)(c+1c+1)≥278.\left(a+\frac{1}{a+1}\right)\left(b+\frac{1}{b+1}\right) \left(c+\frac{1}{c+1}\right)\ge\frac{27}{8}. b)27(a3+a2+a+1)(b3+b2+b+1)(c3+c2+c+1)≥27(a^{3}+a^{2}+a+1)(b^{3}+b^{2}+b+1)(c^{3}+c^{2}+c+1)\ge ≥64(a2+a+1)(b2+b+1)(c2+c+1).\ge 64(a^{2}+a+1)(b^{2}+b+1)(c^{2}+c+1). [hide="Generalization"]n3(an+…+a+1)(bn+…+b+1)(cn+…+c+1)≥n^{3}(a^{n}+\ldots+a+1)(b^{n}+\ldots+b+1)(c^{n}+\ldots+c+1)\ge ≥(n+1)3(an−1+…+a+1)(bn−1+…+b+1)(cn−1+…+c+1), n≥1.\ge (n+1)^{3}(a^{n-1}+\ldots+a+1)(b^{n-1}+\ldots+b+1)(c^{n-1}+\ldots+c+1),\ n\ge1.
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balancing the stones

a) One has a set of stones with weights 1,2,…,201, 2, \ldots, 20 grams. Find all kk for which it is possible to place kk and the rest 20−k20-k stones from the set respectively on the two pans of a balance so that equilibrium is achieved. b) One has a set of stones with weights 1,2,…,511, 2, \ldots, 51 grams. Find all kk for which it is possible to place kk and the rest 51−k51-k stones from the set respectively on the two pans of a balance so that equilibrium is achieved. c) One has a set of stones with weights 1,2,…,n1, 2, \ldots, n grams (n∈Nn\in\mathbb{N}). Find all nn and kk for which it is possible to place kk and the rest n−kn-k stones from the set respectively on the two pans of a balance so that equilibrium is achieved. a) and b) were proposed at the festival, c) is a generalization

game

The vertices of 100-gon (i.e., polygon with 100 sides) are colored alternately white or black. One of the vertices contains a checker. Two players in turn do two things: move the checker into other vertice along the side of 100-gon and then erase some side. The game ends when it is impossible to move the checker. At the end of the game if the checker is in the white vertice then the first player wins. Otherwise the second player wins. Does any of the players have winning strategy? If yes, then who? Remark. The answer may depend on initial position of the checker.
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