TSTST 2021

Let $A B C D$ be a quadrilateral inscribed in a circle with center $O$. Points $X$ and $Y$ lie on sides $A B$ and $C D$, respectively. Suppose the circumcircles of $A D X$ and $B C Y$ meet line $X Y$ again at $P$ and $Q$, respectively. Show that $O P=O Q$.

Let $a_{1}<a_{2}<a_{3}<a_{4}<\cdots$ be an infinite sequence of real numbers in the interval $(0,1)$. Show that there exists a number that occurs exactly once in the sequence $$\frac{a_{1}}{1}, \frac{a_{2}}{2}, \frac{a_{3}}{3}, \frac{a_{4}}{4}, \ldots$$

Find all positive integers $k>1$ for which there exists a positive integer $n$ such that $\binom{n}{k}$ is divisible by $n$, and $\binom{n}{m}$ is not divisible by $n$ for $2 \leq m<k$.

Let $a$ and $b$ be positive integers. Suppose that there are infinitely many pairs of positive integers $(m, n)$ for which $m^{2}+a n+b$ and $n^{2}+a m+b$ are both perfect squares. Prove that $a$ divides $2 b$.

Let $T$ be a tree on $n$ vertices with exactly $k$ leaves. Suppose that there exists a subset of at least $\frac{n+k-1}{2}$ vertices of $T$, no two of which are adjacent. Show that the longest path in $T$ contains an even number of edges.

Triangles $A B C$ and $D E F$ share circumcircle $\Omega$ and incircle $\omega$ so that points $A, F$, $B, D, C$, and $E$ occur in this order along $\Omega$. Let $\Delta_{A}$ be the triangle formed by lines $A B, A C$, and $E F$, and define triangles $\Delta_{B}, \Delta_{C}, \ldots, \Delta_{F}$ similarly. Furthermore, let $\Omega_{A}$ and $\omega_{A}$ be the circumcircle and incircle of triangle $\Delta_{A}$, respectively, and define circles $\Omega_{B}, \omega_{B}, \ldots, \Omega_{F}, \omega_{F}$ similarly. (a) Prove that the two common external tangents to circles $\Omega_{A}$ and $\Omega_{D}$ and the two common external tangents to circles $\omega_{A}$ and $\omega_{D}$ are either concurrent or pairwise parallel. (b) Suppose that these four lines meet at point $T_{A}$, and define points $T_{B}$ and $T_{C}$ similarly. Prove that points $T_{A}, T_{B}$, and $T_{C}$ are collinear.

Let $M$ be a finite set of lattice points and $n$ be a positive integer. A mine-avoiding path is a path of lattice points with length $n$, beginning at $(0,0)$ and ending at a point on the line $x+y=n$, that does not contain any point in $M$. Prove that if there exists a mine-avoiding path, then there exist at least $2^{n-|M|}$ mine-avoiding paths.

Let $A B C$ be a scalene triangle. Points $A_{1}, B_{1}$ and $C_{1}$ are chosen on segments $B C$, $C A$, and $A B$, respectively, such that $\triangle A_{1} B_{1} C_{1}$ and $\triangle A B C$ are similar. Let $A_{2}$ be the unique point on line $B_{1} C_{1}$ such that $A A_{2}=A_{1} A_{2}$. Points $B_{2}$ and $C_{2}$ are defined similarly. Prove that $\triangle A_{2} B_{2} C_{2}$ and $\triangle A B C$ are similar.

Let $q=p^{r}$ for a prime number $p$ and positive integer $r$. Let $\zeta=e^{\frac{2 \pi i}{q}}$. Find the least positive integer $n$ such that $$\sum_{\substack{1 \leq k \leq q \\ \operatorname{gcd}(k, p)=1}} \frac{1}{\left(1-\zeta^{k}\right)^{n}}$$ is not an integer. (The sum is over all $1 \leq k \leq q$ with $p$ not dividing $k$.)