Question bank USAJMO 2012

In triangle $ABC$, let $P\in AB$, $Q\in AC$ satisfy $AP=AQ$. Let $S,R$ be distinct points on $BC$, with $S$ between $B$ and $R$, such that $\angle BPS=\angle PRS$ and $\angle CQR=\angle QSR$. Prove that $P,Q,R,S$ lie on one circle.

Proposed by Sungyoon Kim, Inseok Seo (credit in the supplied notes).

Find all integers $n \ge 3$ such that among any $n$ positive real numbers
$a_1$, $a_2$, …, $a_n$ with

\[\max(a_1,a_2,\dots,a_n)
\le n \cdot \min(a_1,a_2,\dots,a_n),\]

there exist three that are the side lengths of an acute triangle.

Proposed by Titu Andreescu (credit in the supplied notes).

For positive real numbers $a,b,c$, prove

$$\frac{a^3+3b^3}{5a+b}+\frac{b^3+3c^3}{5b+c}+\frac{c^3+3a^3}{5c+a}\ge\frac23(a^2+b^2+c^2).$$

Proposed by Titu Andreescu (credit in the supplied notes).

Let $0\lt \alpha\lt 1$ be irrational. On a circle of circumference 1 choose $P_1$, and define $P_k$ by moving counterclockwise an arc length $\alpha$ from $P_{k-1}$, for $2\le k\le n$, where $n\ge3$. If $P_a,P_b$ are the adjacent points immediately on either side of $P_n$ among these $n$ points, prove $a+b\le n$.

Proposed by Sam Vandervelde (credit in the supplied notes).

For distinct positive integers $a,b\lt 2012$, let $f(a,b)$ count the integers $1\le k\lt 2012$ for which the remainder of $ak$ modulo 2012 exceeds that of $bk$. Determine the minimum $S$ over all such pairs $(a,b)$.

Proposed by Warut Suksompong (credit in the supplied notes).

Let $P$ be a point in the plane of $\triangle ABC$,
and $\gamma$ a line through $P$.
Let $A'$, $B'$, $C'$ be the points where the
reflections of lines $PA$, $PB$, $PC$ with respect to $\gamma$
intersect lines $BC$, $CA$, $AB$ respectively.
Prove that $A'$, $B'$, $C'$ are collinear.

Proposed by Titu Andreescu, Cosmin Pohoata (credit in the supplied notes).