US-TST 2014

Let $A B C$ be an acute triangle, and let $X$ be a variable interior point on the minor $\operatorname{arc} B C$ of its circumcircle. Let $P$ and $Q$ be the feet of the perpendiculars from $X$ to lines $C A$ and $C B$, respectively. Let $R$ be the intersection of line $P Q$ and the perpendicular from $B$ to $A C$. Let $\ell$ be the line through $P$ parallel to $X R$. Prove that as $X$ varies along minor arc $B C$, the line $\ell$ always passes through a fixed point.

Let $a_{1}, a_{2}, a_{3}, \ldots$ be a sequence of integers, with the property that every consecutive group of $a_{i}$ 's averages to a perfect square. More precisely, for all positive integers $n$ and $k$, the quantity $$\frac{a_{n}+a_{n+1}+\cdots+a_{n+k-1}}{k}$$ is always the square of an integer. Prove that the sequence must be constant (all $a_{i}$ are equal to the same perfect square).

Let $n$ be an even positive integer, and let $G$ be an $n$-vertex (simple) graph with exactly $\frac{n^{2}}{4}$ edges. An unordered pair of distinct vertices $\{x, y\}$ is said to be amicable if they have a common neighbor (there is a vertex $z$ such that $x z$ and $y z$ are both edges). Prove that $G$ has at least $2\binom{n / 2}{2}$ pairs of vertices which are amicable.

Let $n$ be a positive even integer, and let $c_{1}, c_{2}, \ldots, c_{n-1}$ be real numbers satisfying $$\sum_{i=1}^{n-1}\left|c_{i}-1\right|<1$$ Prove that $$2 x^{n}-c_{n-1} x^{n-1}+c_{n-2} x^{n-2}-\cdots-c_{1} x^{1}+2$$ has no real roots.

Let $A B C D$ be a cyclic quadrilateral, and let $E, F, G$, and $H$ be the midpoints of $A B, B C, C D$, and $D A$ respectively. Let $W, X, Y$ and $Z$ be the orthocenters of triangles $A H E, B E F, C F G$ and $D G H$, respectively. Prove that the quadrilaterals $A B C D$ and $W X Y Z$ have the same area.

For a prime $p$, a subset $S$ of residues modulo $p$ is called a sum-free multiplicative subgroup of $\mathbb{F}_{p}$ if - there is a nonzero residue $\alpha$ modulo $p$ such that $S=\left\{1, \alpha^{1}, \alpha^{2}, \ldots\right\}$ (all considered $\bmod p$ ), and - there are no $a, b, c \in S$ (not necessarily distinct) such that $a+b \equiv c(\bmod p)$. Prove that for every integer $N$, there is a prime $p$ and a sum-free multiplicative subgroup $S$ of $\mathbb{F}_{p}$ such that $|S| \geq N$.