Subcontests
(7)r_1+r_2+r_3+...+r_{p-1}=\frac{p^2(p-1)}{2} where r_k = remainder k^p : p^2
Suppose that p>2 is a prime number. For each integer k=1,2,...,p−1, denote rk as the remainder of the division kp by p2. Prove that r1+r2+r3+...+rp−1=2p2(p−1) angle chasing with an orthodiagonal ABCD
Let ABCD be a cyclic quadrilateral wih both diagonals perpendicular to each other and intersecting at point O. Let E,F,G,H be the orthogonal projections of O on sides AB,BC,CD,DA respectively.
a. Prove that ∠EFG+∠GHE=180o
b. Prove that OE bisects angle ∠FEH . Indonesian MO 2016 Last Problem
Determine with proof, the number of permutations a1,a2,a3,...,a2016 of 1,2,3,...,2016 such that the value of ∣ai−i∣ is fixed for all i=1,2,3,...,2016, and its value is an integer multiple of 3.