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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.