TSTST 2016

Let $A=A(x, y)$ and $B=B(x, y)$ be two-variable polynomials with real coefficients. Suppose that $A(x, y) / B(x, y)$ is a polynomial in $x$ for infinitely many values of $y$, and a polynomial in $y$ for infinitely many values of $x$. Prove that $B$ divides $A$, meaning there exists a third polynomial $C$ with real coefficients such that $A=B \cdot C$.

Let $A B C$ be a scalene triangle with orthocenter $H$ and circumcenter $O$ and denote by $M, N$ the midpoints of $\overline{A H}, \overline{B C}$. Suppose the circle $\gamma$ with diameter $\overline{A H}$ meets the circumcircle of $A B C$ at $G \neq A$, and meets line $\overline{A N}$ at $Q \neq A$. The tangent to $\gamma$ at $G$ meets line $O M$ at $P$. Show that the circumcircles of $\triangle G N Q$ and $\triangle M B C$ intersect on $\overline{P N}$.

Decide whether or not there exists a nonconstant polynomial $Q(x)$ with integer coefficients with the following property: for every positive integer $n>2$, the numbers $$Q(0), Q(1), Q(2), \ldots, Q(n-1)$$ produce at most $0.499 n$ distinct residues when taken modulo $n$.

Prove that if $n$ and $k$ are positive integers satisfying $\varphi^{k}(n)=1$, then $n \leq 3^{k}$. (Here $\varphi^{k}$ denotes $k$ applications of the Euler phi function.)

In the coordinate plane are finitely many walls, which are disjoint line segments, none of which are parallel to either axis. A bulldozer starts at an arbitrary point and moves in the $+x$ direction. Every time it hits a wall, it turns at a right angle to its path, away from the wall, and continues moving. (Thus the bulldozer always moves parallel to the axes.) Prove that it is impossible for the bulldozer to hit both sides of every wall.

Let $A B C$ be a triangle with incenter $I$, and whose incircle is tangent to $\overline{B C}, \overline{C A}$, $\overline{A B}$ at $D, E, F$, respectively. Let $K$ be the foot of the altitude from $D$ to $\overline{E F}$. Suppose that the circumcircle of $\triangle A I B$ meets the incircle at two distinct points $C_{1}$ and $C_{2}$, while the circumcircle of $\triangle A I C$ meets the incircle at two distinct points $B_{1}$ and $B_{2}$. Prove that the radical axis of the circumcircles of $\triangle B B_{1} B_{2}$ and $\triangle C C_{1} C_{2}$ passes through the midpoint $M$ of $\overline{D K}$.