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EGMO 2013, problem 3. Let \(n\) be a positive integer.

1. Prove that there exists a set \(S\) of \(6n\) pairwise different positive integers, such that the least common multiple of any two elements of \(S\) is no larger than \(32n^2\).
2. Prove that every set \(T\) of \(6n\) pairwise different positive integers contains two elements the least common multiple of which is larger than \(9n^2\).

EGMO 2013, problem 4. Find all positive integers \(a\) and \(b\) for which there are three consecutive integers at which the polynomial \[P(n)=\frac{n^5+a}{b}\] takes integer values.

EGMO 2015, Problem 5. Let \(m, n\) be positive integers with \(m > 1\). Anastasia partitions the integers \(1, 2, \dots , 2m\) into \(m\) pairs. Boris then chooses one integer from each pair and finds the sum of these chosen integers. Prove that Anastasia can select the pairs so that Boris cannot make his sum equal to \(n.\)

by/di Alessandra Caraceni