Asian Pacific Mathematics Olympiad · 2012
Problems
- Problem 1Let P be a point in the interior of triangle ABC. Let D, E, F be the intersections of AP with BC, BP with CA, and CP with AB, respectively. Prove that the area of triangle ABC is 6 if each of triangles PFA, PDB, and PEC has area 1.Solutions: 1
- Problem 2Into each box of a 2012 by 2012 square grid, insert a real number between 0 and 1 inclusive. Split the grid into two non-empty rectangles of boxes by a line parallel to a side of the grid. Suppose that for every such split at least one resulting rectangle has sum at most 1. Determine the maximum possible sum of all inserted numbers.Solutions: 1
- Problem 3Determine all pairs (p,n), where p is a prime number and n is a positive integer, for which (n^p+1)/(p^n+1) is an integer.Solutions: 1
- Problem 4Let ABC be an acute triangle. Let D be the foot of the perpendicular from A to BC, M the midpoint of BC, and H the orthocenter of ABC. Let E be the intersection of the circumcircle Gamma of ABC with the ray MH, and let F be the other intersection of line ED with Gamma. Prove that BF/CF = AB/AC, where XY denotes the length of segment XY.Solutions: 1
- Problem 5Let n be an integer at least 2. Prove that if real numbers a_1, a_2, ..., a_n satisfy a_1^2+a_2^2+...+a_n^2=n, then the sum over 1≤i<j≤n of 1/(n-a_i a_j) is at most n/2.Solutions: 1