US-TST 2017

In a sports league, each team uses a set of at most $t$ signature colors. A set $S$ of teams is color-identifiable if one can assign each team in $S$ one of their signature colors, such that no team in $S$ is assigned any signature color of a different team in $S$. For all positive integers $n$ and $t$, determine the maximum integer $g(n, t)$ such that: In any sports league with exactly $n$ distinct colors present over all teams, one can always find a color-identifiable set of size at least $g(n, t)$.

Let $A B C$ be an acute scalene triangle with circumcenter $O$, and let $T$ be on line $B C$ such that $\angle T A O=90^{\circ}$. The circle with diameter $\overline{A T}$ intersects the circumcircle of $\triangle B O C$ at two points $A_{1}$ and $A_{2}$, where $O A_{1}<O A_{2}$. Points $B_{1}, B_{2}, C_{1}, C_{2}$ are defined analogously. (a) Prove that $\overline{A A_{1}}, \overline{B B_{1}}, \overline{C C_{1}}$ are concurrent. (b) Prove that $\overline{A A_{2}}, \overline{B B_{2}}, \overline{C C_{2}}$ are concurrent on the Euler line of triangle $A B C$.

Let $P, Q \in \mathbb{R}[x]$ be relatively prime nonconstant polynomials. Show that there can be at most three real numbers $\lambda$ such that $P+\lambda Q$ is the square of a polynomial.

You are cheating at a trivia contest. For each question, you can peek at each of the $n>1$ other contestant's guesses before writing your own. For each question, after all guesses are submitted, the emcee announces the correct answer. A correct guess is worth 0 points. An incorrect guess is worth -2 points for other contestants, but only -1 point for you, because you hacked the scoring system. After announcing the correct answer, the emcee proceeds to read out the next question. Show that if you are leading by $2^{n-1}$ points at any time, then you can surely win first place.

Let $A B C$ be a triangle with altitude $\overline{A E}$. The $A$-excircle touches $\overline{B C}$ at $D$, and intersects the circumcircle at two points $F$ and $G$. Prove that one can select points $V$ and $N$ on lines $D G$ and $D F$ such that quadrilateral $E V A N$ is a rhombus.

Prove that there are infinitely many triples $(a, b, p)$ of integers, with $p$ prime and $0<a \leq b<p$, for which $p^{5}$ divides $(a+b)^{p}-a^{p}-b^{p}$.