MathDB

4

Part of 2016 IMC

Problems(2)

IMC 2016, Problem 4

Source: IMC 2016

7/27/2016
Let n≥kn\ge k be positive integers, and let F\mathcal{F} be a family of finite sets with the following properties: (i) F\mathcal{F} contains at least (nk)+1\binom{n}{k}+1 distinct sets containing exactly kk elements; (ii) for any two sets A,B∈FA, B\in \mathcal{F}, their union A∪BA\cup B also belongs to F\mathcal{F}. Prove that F\mathcal{F} contains at least three sets with at least nn elements.
(Proposed by Fedor Petrov, St. Petersburg State University)
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IMC 2016, Problem 9

Source: IMC 2016

7/28/2016
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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