Subcontests
(3)Sum of the numbers is divisible by k
Find all integers n≥1 such that there exists a permutation (a1,a2,...,an) of (1,2,...,n) such that a1+a2+...+ak is divisible by k for k=1,2,...,n Circumcenters coincide
The incircle of triangle ABC is tangent to BC,CA,AB at D,E,F respectively. Consider the triangle formed by the line joining the midpoints of AE,AF, the line joining the midpoints of BF,BD, and the line joining the midpoints of CD,CE. Prove that the circumcenter of this triangle coincides with the circumcenter of triangle ABC.