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Thinking Out Loud – EGMO 2013 Problem 2

EGMO 2013, problem 2. Determine all integers \(m\) for which the \(m\times m\) square can be dissected into five rectangles, the side lengths of which are the integers \(1, 2, 3, \dotsc, 10\) in some order.

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Thinking Out Loud – EGMO 2013 Problem 3

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 \(9n^2\).

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Thinking Out Loud – EGMO 2013 Problem 4

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.

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