Given a triangle ABC, let P and Q be the points on the 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 and S are concyclic (in other words these four points lie on a circle).
Find all integers $latex (n \ge 3 ) $ such that among any n positive real numbers $latex ( a_1 , a_2 , ... , a_n ) $ with $latex \displaystyle {\text(\max)(a_1 , a_2 , ... , a_n) \le n) (\min)(a_1 , a_2 , ... , a_n)} $ there exist three that are the side lengths of an acute triangle.
Let a, b, c be positive real numbers. Prove that $latex \displaystyle {(\frac{a^3 + 3 b^3}{5a + b} + \frac{b^3 + 3c^3}{5b +c} + \frac{c^3 + 3a^3}{5c + a} \ge \frac{2}{3} (a^2 + b^2 + c^2))} $.
Let $latex (\alpha)$ be an irrational number with $latex (0 < \alpha < 1)$, and draw a circle in the plane whose circumference has length 1. Given any integer $latex (n \ge 3 )$, define a sequence of points $latex (P_1 , P_2 , ... , P_n )$ as follows. First select any point $latex (P_1)$ on the circle, and for $latex ( 2 \le k \le n ) $ define $latex (P_k)$ as the point on the circle for which the length of the arc $latex (P_{k-1} P_k)$ is $latex (\alpha)$, when travelling counterclockwise around the circle from $latex (P_{k-1} )$ to $latex (P_k)$. Suppose that $latex (P_a)$ and $latex (P_b)$ are the nearest adjacent points on either side of $latex (P_n)$. Prove that $latex (a+b \le n)$.
For distinct positive integers a, b < 2012, define f(a, b) to be the number of integers k with (1le 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.
Let P be a point in the plane of triangle ABC, and $latex (\gamma)$ be a line passing through P. Let A', B', C' be the points where reflections of the lines PA, PB, PC with respect to $latex (\gamma)$ intersect lines BC, AC, AB, respectively. Prove that A', B' and C' are collinear.
