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IMC 2016, Problem 9

Source: IMC 2016

July 28, 2016
IMCIMC 2016college contestsfunctions

Problem Statement

Let kk be a positive integer. For each nonnegative integer nn, let f(n)f(n) be the number of solutions (x1,…,xk)∈Zk(x_1,\ldots,x_k)\in\mathbb{Z}^k of the inequality ∣x1∣+...+∣xk∣≤n|x_1|+...+|x_k|\leq n. Prove that for every n≥1n\ge1, we have f(n−1)f(n+1)≤f(n)2f(n-1)f(n+1)\leq f(n)^2.
(Proposed by Esteban Arreaga, Renan Finder and José Madrid, IMPA, Rio de Janeiro)
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