Subcontests
(6)ak mod 2012 > bk mod 2012
For distinct positive integers a,b<2012, define f(a,b) to be the number of integers k with 1≤k<2012 such that the remainder when ak divided by 2012 is greater than that of bk divided by 2012. Let S be the minimum value of f(a,b), where a and b range over all pairs of distinct positive integers less than 2012. Determine S. Prove Collinearity
Let P be a point in the plane of △ABC, and γ a line passing through P. Let A′,B′,C′ be the points where the reflections of lines PA,PB,PC with respect to γ intersect lines BC,AC,AB respectively. Prove that A′,B′,C′ are collinear. Sequence of Points on a Circle
Let α be an irrational number with 0<α<1, and draw a circle in the plane whose circumference has length 1. Given any integer n≥3, define a sequence of points P1,P2,…,Pn as follows. First select any point P1 on the circle, and for 2≤k≤n define Pk as the point on the circle for which the length of arc Pk−1Pk is α, when travelling counterclockwise around the circle from Pk−1 to Pk. Suppose that Pa and Pb are the nearest adjacent points on either side of Pn. Prove that a+b≤n. USAJMO problem 2: Side lengths of an acute triangle
Find all integers n≥3 such that among any n positive real numbers a_1, a_2, \hdots, a_n with \text{max}(a_1,a_2,\hdots,a_n) \leq n \cdot \text{min}(a_1,a_2,\hdots,a_n), there exist three that are the side lengths of an acute triangle. USAJMO problem 1: Prove 4 points are concyclic
Given a triangle ABC, let P and Q be points on segments AB and AC, respectively, such that AP=AQ. Let S and R be distinct points on segment BC such that S lies between B and R, ∠BPS=∠PRS, and ∠CQR=∠QSR. Prove that P,Q,R,S are concyclic (in other words, these four points lie on a circle).