USAJMO 2012

Given a triangle $ABC$ , let $P$ and $Q$ be points on segments $\overline{AB}$ and $\overline{AC}$ , respectively, such that $AP=AQ$ . Let $S$ and $R$ be distinct points on segment $\overline{BC}$ such that $S$ lies between $B$ and $R$ , $\angle BPS=\angle PRS$ , and $\angle CQR=\angle QSR$ . Prove that $P,Q,R,S$ are concyclic (in other words, these four points lie on a circle).

Let $\alpha$ be an irrational number with $0<\alpha < 1$ , and draw a circle in the plane whose circumference has length $1$ . Given any integer $n\ge 3$ , define a sequence of points $P_1, P_2, \ldots , P_n$ as follows. First select any point $P_1$ on the circle, and for $2\le k\le n$ define $P_k$ as the point on the circle for which the length of arc $P_{k-1}P_k$ is $\alpha$ , when travelling counterclockwise around the circle from $P_{k-1}$ to $P_k$ . Suppose that $P_a$ and $P_b$ are the nearest adjacent points on either side of $P_n$ . Prove that $a+b\le n$ .

For distinct positive integers $a, b<2012$ , define $f(a, b)$ to be the number of integers $k$ with $1\le k<2012$ 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 $\gamma$ a line passing through $P$ . Let $A', B', C'$ be the points where the reflections of lines $PA, PB, PC$ with respect to $\gamma$ intersect lines $BC, AC, AB$ respectively. Prove that $A', B', C'$ are collinear.