TSTST 2017

Let $A B C$ be a triangle with circumcircle $\Gamma$, circumcenter $O$, and orthocenter $H$. Assume that $A B \neq A C$ and $\angle A \neq 90^{\circ}$. Let $M$ and $N$ be the midpoints of $\overline{A B}$ and $\overline{A C}$, respectively, and let $E$ and $F$ be the feet of the altitudes from $B$ and $C$ in $\triangle A B C$, respectively. Let $P$ be the intersection point of line $M N$ with the tangent line to $\Gamma$ at $A$. Let $Q$ be the intersection point, other than $A$, of $\Gamma$ with the circumcircle of $\triangle A E F$. Let $R$ be the intersection point of lines $A Q$ and $E F$. Prove that $\overline{P R} \perp \overline{O H}$.

Ana and Banana are playing a game. First Ana picks a word, which is defined to be a nonempty sequence of capital English letters. Then Banana picks a nonnegative integer $k$ and challenges Ana to supply a word with exactly $k$ subsequences which are equal to Ana's word. Ana wins if she is able to supply such a word, otherwise she loses. For example, if Ana picks the word "TST", and Banana chooses $k=4$, then Ana can supply the word "TSTST" which has 4 subsequences which are equal to Ana's word. Which words can Ana pick so that she can win no matter what value of $k$ Banana chooses?

Consider solutions to the equation $$x^{2}-c x+1=\frac{f(x)}{g(x)}$$ where $f$ and $g$ are nonzero polynomials with nonnegative real coefficients. For each $c>0$, determine the minimum possible degree of $f$, or show that no such $f, g$ exist.

Find all nonnegative integer solutions to $$2^{a}+3^{b}+5^{c}=n!$$

Let $A B C$ be a triangle with incenter $I$. Let $D$ be a point on side $B C$ and let $\omega_{B}$ and $\omega_{C}$ be the incircles of $\triangle A B D$ and $\triangle A C D$, respectively. Suppose that $\omega_{B}$ and $\omega_{C}$ are tangent to segment $B C$ at points $E$ and $F$, respectively. Let $P$ be the intersection of segment $A D$ with the line joining the centers of $\omega_{B}$ and $\omega_{C}$. Let $X$ be the intersection point of lines $B I$ and $C P$ and let $Y$ be the intersection point of lines $C I$ and $B P$. Prove that lines $E X$ and $F Y$ meet on the incircle of $\triangle A B C$.

A sequence of positive integers $\left(a_{n}\right)_{n \geq 1}$ is of Fibonacci type if it satisfies the recursive relation $a_{n+2}=a_{n+1}+a_{n}$ for all $n \geq 1$. Is it possible to partition the set of positive integers into an infinite number of Fibonacci type sequences?