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EGMO 2014, problem 2. Let \(D\) and \(E\) be points in the interiors of sides \(AB\) and \(AC\), respectively, of a triangle \(ABC\), such that \(DB=BC=CE\). Let the lines \(CD\) and \(BE\) meet at \(F\). Prove that the incentre \(I\) of triangle \(ABC\), the orthocentre \(H\) of triangle \(DEF\) and the midpoint \(M\) of the arc \(BAC\) of the circumcircle of triangle \(ABC\) are collinear.

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 \(9n^2\).

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.