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China 2010 quiz2 problem 2

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September 8, 2010
polynomialinequalitiescombinatorics

Problem Statement

Let M={1,2,⋯ ,n}M=\{1,2,\cdots,n\}, each element of MM is colored in either red, blue or yellow. Set A={(x,y,z)∈M×M×M∣x+y+z≡0mod  nA=\{(x,y,z)\in M\times M\times M|x+y+z\equiv 0\mod n, x,y,zx,y,z are of same color},\}, B={(x,y,z)∈M×M×M∣x+y+z≡0mod  n,B=\{(x,y,z)\in M\times M\times M|x+y+z\equiv 0\mod n, x,y,zx,y,z are of pairwise distinct color}.\}. Prove that 2∣A∣≥∣B∣2|A|\geq |B|.
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