EGMO 2017, Problem 1. Let \(ABCD\) be a convex quadrilateral with \(\widehat{DAB} = \widehat{BCD} = 90^\circ\) and \(\widehat{ABC} > \widehat{CDA}\). Let \(Q\) and \(R\) be points on segments \(BC\) and \(CD\), respectively, such that line \(QR\) intersects lines \(AB\) and \(AD\) at points \(P\) and \(S\), respectively. It is given that \(PQ = RS\). Let the midpoint of \(BD\) be \(M\) and the midpoint of \(QR\) be \(N\). Prove that the points \(M, N, A\) and \(C\) lie on a circle.
EGMO 2014, Problem 3. We denote the number of positive divisors of a positive integer \(m\) by \(d(m)\) and the number of distinct prime divisors of \(m\) by \(\omega(m)\). Let \(k\) be a positive integer. Prove that there exist infinitely many positive integers \(n\) such that \(\omega(n) = k\) and \(d(n)\) does not divide \(d(a^2+b^2)\) for any positive integers \(a, b\) satisfying \(a + b = n\).
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.