Subcontests
(3)Polish MO Finals 2014, Problem 6
In an acute triangle ABC point D is the point of intersection of altitude ha and side BC, and points M,N are orthogonal projections of point D on sides AB and AC. Lines MN and AD cross the circumcircle of triangle ABC at points P,Q and A,R. Prove that point D is the center of the incircle of PQR. Polish MO Finals 2014, Problem 2
Let k≥2, n≥1, a1,a2,…,ak and b1,b2,…,bn be integers such that 1<a1<a2<⋯<ak<b1<b2<⋯<bn. Prove that if a1+a2+⋯+ak>b1+b2+⋯+bn, then a1⋅a2⋅…⋅ak>b1⋅b2⋅…⋅bn. Polish MO Finals 2014, Problem 1
Let k,n≥1 be relatively prime integers. All positive integers not greater than k+n are written in some order on the blackboard. We can swap two numbers that differ by k or n as many times as we want. Prove that it is possible to obtain the order 1,2,…,k+n−1,k+n.