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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.