Subcontests
(5)Inequality on a sequence [Poland 2017, P6]
Three sequences (a0,a1,…,an), (b0,b1,…,bn), (c0,c1,…,c2n) of non-negative real numbers are given such that for all 0≤i,j≤n we have aibj≤(ci+j)2. Prove that
i=0∑nai⋅j=0∑nbj≤(k=0∑2nck)2. Poland 2017 P4
Prove that the set of positive integers Z+ can be represented as a sum of five pairwise distinct subsets with the following property: each 5-tuple of numbers of form (n,2n,3n,4n,5n), where n∈Z+, contains exactly one number from each of these five subsets. Poland 2017 P2
A sequence (a1,a2,…,ak) consisting of pairwise distinct squares of an n×n chessboard is called a cycle if k≥4 and squares ai and ai+1 have a common side for all i=1,2,…,k, where ak+1=a1. Subset X of this chessboard's squares is mischievous if each cycle on it contains at least one square in X.Determine all real numbers C with the following property: for each integer n≥2, on an n×n chessboard there exists a mischievous subset consisting of at most Cn2 squares.