Viewing posts for the category EGMO

As the countdown to EGMO keeps going it is time to get introduced to one treasure Italy has to offer: food! We focus only on the less-known specialities from Florence and Tuscany.

We are now roughly one month away from EGMO, and we are looking forward to welcoming you in Florence. We decided to collect some basic information about Italy and Florence to help you better plan your visit and pack your luggages.

**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.