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EGMO 2013, problem 1. The side \(\textrm{BC}\) of the triangle \(\textrm{ABC}\) is extended beyond \(\textrm{C}\) to \(\textrm{D}\) so that \(\textrm{CD} = \textrm{BC}\). The side \(\textrm{CA}\) is extended beyond \(\textrm{A}\) to \(\textrm{E}\) so that \(\textrm{AE} = 2\textrm{CA}\). Prove that if \(\textrm{AD} = \textrm{BE}\), then the triangle \(\textrm{ABC}\) is right-angled.

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.

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 \(9n^2\).