TSTST 2022

Let $n$ be a positive integer. Find the smallest positive integer $k$ such that for any set $S$ of $n$ points in the interior of the unit square, there exists a set of $k$ rectangles such that the following hold: - The sides of each rectangle are parallel to the sides of the unit square. - Each point in $S$ is not in the interior of any rectangle. - Each point in the interior of the unit square but not in $S$ is in the interior of at least one of the $k$ rectangles. (The interior of a polygon does not contain its boundary.)

Let $A B C$ be a triangle. Let $\theta$ be a fixed angle for which $$\theta<\frac{1}{2} \min (\angle A, \angle B, \angle C)$$ Points $S_{A}$ and $T_{A}$ lie on segment $B C$ such that $\angle B A S_{A}=\angle T_{A} A C=\theta$. Let $P_{A}$ and $Q_{A}$ be the feet from $B$ and $C$ to $\overline{A S_{A}}$ and $\overline{A T_{A}}$ respectively. Then $\ell_{A}$ is defined as the perpendicular bisector of $\overline{P_{A} Q_{A}}$. Define $\ell_{B}$ and $\ell_{C}$ analogously by repeating this construction two more times (using the same value of $\theta$ ). Prove that $\ell_{A}, \ell_{B}$, and $\ell_{C}$ are concurrent or all parallel.

Determine all positive integers $N$ for which there exists a strictly increasing sequence of positive integers $s_{0}<s_{1}<s_{2}<\cdots$ satisfying the following properties: - the sequence $s_{1}-s_{0}, s_{2}-s_{1}, s_{3}-s_{2}, \ldots$ is periodic; and - $s_{s_{n}}-s_{s_{n-1}} \leq N<s_{1+s_{n}}-s_{s_{n-1}}$ for all positive integers $n$.

A function $f: \mathbb{N} \rightarrow \mathbb{N}$ has the property that for all positive integers $m$ and $n$, exactly one of the $f(n)$ numbers $$f(m+1), f(m+2), \ldots, f(m+f(n))$$ is divisible by $n$. Prove that $f(n)=n$ for infinitely many positive integers $n$.

Let $A_{1}, \ldots, A_{2022}$ be the vertices of a regular 2022 -gon in the plane. Alice and Bob play a game. Alice secretly chooses a line and colors all points in the plane on one side of the line blue, and all points on the other side of the line red. Points on the line are colored blue, so every point in the plane is either red or blue. (Bob cannot see the colors of the points.) In each round, Bob chooses a point in the plane (not necessarily among $A_{1}, \ldots$, $\left.A_{2022}\right)$ and Alice responds truthfully with the color of that point. What is the smallest number $Q$ for which Bob has a strategy to always determine the colors of points $A_{1}, \ldots, A_{2022}$ in $Q$ rounds?

Let $O$ and $H$ be the circumcenter and orthocenter, respectively, of an acute scalene triangle $A B C$. The perpendicular bisector of $\overline{A H}$ intersects $\overline{A B}$ and $\overline{A C}$ at $X_{A}$ and $Y_{A}$ respectively. Let $K_{A}$ denote the intersection of the circumcircles of triangles $O X_{A} Y_{A}$ and $B O C$ other than $O$. Define $K_{B}$ and $K_{C}$ analogously by repeating this construction two more times. Prove that $K_{A}, K_{B}, K_{C}$, and $O$ are concyclic.

Let $A B C D$ be a parallelogram. Point $E$ lies on segment $C D$ such that $$2 \angle A E B=\angle A D B+\angle A C B,$$ and point $F$ lies on segment $B C$ such that $$2 \angle D F A=\angle D C A+\angle D B A .$$ Let $K$ be the circumcenter of triangle $A B D$. Prove that $K E=K F$.

Find all functions $f: \mathbb{N} \rightarrow \mathbb{Z}$ such that $$\left\lfloor\frac{f(m n)}{n}\right\rfloor=f(m)$$ for all positive integers $m, n$.

Let $k>1$ be a fixed positive integer. Prove that if $n$ is a sufficiently large positive integer, there exists a sequence of integers with the following properties: - Each element of the sequence is between 1 and $n$, inclusive. - For any two different contiguous subsequences of the sequence with length between 2 and $k$ inclusive, the multisets of values in those two subsequences is not the same. - The sequence has length at least $0.499 n^{2}$.