Subcontests
(5)finite set
Let ∣U∣,σ(U) and π(U) denote the number of elements, the sum, and the product, respectively, of a finite set U of positive integers. (If U is the empty set, ∣U∣=0,σ(U)=0,π(U)=1.) Let S be a finite set of positive integers. As usual, let (kn) denote k!(n−k)!n!. Prove that U⊆S∑(−1)∣U∣(∣S∣m−σ(U))=π(S) for all integers m≥σ(S). Positive sequence
Let a1,a2,a3,… be a sequence of positive real numbers satisfying ∑j=1naj≥n for all n≥1. Prove that, for all n≥1, j=1∑naj2>41(1+21+⋯+n1). Colored 99-gon
The sides of a 99-gon are initially colored so that consecutive sides are red, blue, red, blue, …, red, blue, yellow. We make a sequence of modifications in the coloring, changing the color of one side at a time to one of the three given colors (red, blue, yellow), under the constraint that no two adjacent sides may be the same color. By making a sequence of such modifications, is it possible to arrive at the coloring in which consecutive sides
are red, blue, red, blue, red, blue, …, red, yellow, blue? Classic algebra problem
Let k1<k2<k3<⋯ be positive integers, no two consecutive, and let sm=k1+k2+⋯+km for m=1,2,3,…. Prove that, for each positive integer n, the interval [sn,sn+1) contains at least one perfect square.