# International Mathematics Competition for University Students 2021

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IMC 2022
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## IMC2021: Day 1, Problem 2

Problem 2. Let $\displaystyle n$ and $\displaystyle k$ be fixed positive integers, and let $\displaystyle a$ be an arbitrary non-negative integer. Choose a random $\displaystyle k$-element subset $\displaystyle X$ of $\displaystyle \{1,2,\ldots,k+a\}$ uniformly (i.e., all $\displaystyle k$-element subsets are chosen with the same probability) and, independently of $\displaystyle X$, choose a random $\displaystyle n$-element subset $\displaystyle Y$ of $\displaystyle \{1,\ldots,k+n+a\}$ uniformly.

Prove that the probability

$\displaystyle \mathsf{P}\Big(\min(Y)>\max(X)\Big)$

does not depend on $\displaystyle a$.

Fedor Petrov, St. Petersburg State University

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