Let $ABC$ be an isosceles triangle with $BC=CA$, and let $D$ be a point inside side $AB$ such that $AD< DB$. Let $P$ and $Q$ be two points inside sides $BC$ and $CA$, respectively, such that $\angle DPB = \angle DQA = 90^{\circ}$. Let the perpendicular bisector of $PQ$ meet line segment $CQ$ at $E$, and let the circumcircles of triangles $ABC$ and $CPQ$ meet again at point $F$, different from $C$. Suppose that $P$, $E$, $F$ are collinear. Prove that $\angle ACB = 90^{\circ}$.

Let $f:\mathbb{R}^+\to\mathbb{R}^+$ be such that $$f(x+f(y))^2\geq f(x)\left(f(x+f(y))+f(y)\right)$$for all $x,y\in\mathbb{R}^+$. Show that $f$ is unbounded, i.e. for each $M\in\mathbb{R}^+$, there exists $x\in\mathbb{R}^+$ such that $f(x)>M$.

Let $1 \leq n \leq 2021$ be a positive integer. Jack has $2021$ coins arranged in a line where each coin has an $H$ on one side and a $T$ on the other. At the beginning, all coins show $H$ except the nth coin. Jack can repeatedly perform the following operation: he chooses a coin showing $T$, and turns over the coins next to it to the left and to the right (if any). Determine all $n$ such that Jack can make all coins show $T$ after a finite number of operations.

Let $n$ be a positive integer. Find the number of permutations $a_1$, $a_2$, $\dots a_n$ of the sequence $1$, $2$, $\dots$ , $n$ satisfying $$a_1 \le 2a_2\le 3a_3 \le \dots \le na_n$$. Proposed by United Kingdom

Find all positive integers $n$ such that $2021^n$ can be expressed in the form $x^4-4y^4$ for some integers $x,y$.

A triangle $ABC$ with $AB<AC<BC$ is given. The point $P$ is the center of an excircle touching the line segment $AB$ at $D$. The point $Q$ is the center of an excircle touching the line segment $AC$ at $E$. The circumcircle of the triangle $ADE$ intersects $\overline{PE}$ and $\overline{QD}$ again at $G$ and $H$ respectively. The line perpendicular to $\overline{AG}$ at $G$ intersects the side $AB$ at $R$. The line perpendicular to $\overline{AH}$ at $H$ intersects the side $AC$ at $S$. Prove that $\overline{DE}$ and $\overline{RS}$ are parallel.

For each positive integer $n$, let $\rho(n)$ be the number of positive divisors of $n$ with exactly the same set of prime divisors as $n$. Show that, for any positive integer $m$, there exists a positive integer $n$ such that $\rho(202^n+1)\geq m.$

Let $d\geq 1$ and $n\geq 0$ be integers. Find the number of ways to write down a nonnegative integer in each square of a $d\times d$ grid such that the numbers in any set of $d$ squares, no two in the same row or column, sum to $n$.

Let $m, n$ be positive integers. Show that the polynomial $$f(x)=x^m(x^2-100)^n-11$$cannot be expressed as a product of two non-constant polynomials with integral coefficients.

Let $a,b,c$ be distinct positive real numbers such that $\frac{1}{1+a}+\frac{1}{1+b}+\frac{1}{1+c}\leq 1$. Prove that $$2\left(\sqrt{\frac{a+b}{ac}}+\sqrt{\frac{b+c}{ba}}+\sqrt{\frac{c+a}{cb}}\right)<\frac{a^3}{(a-b)(a-c)}+\frac{b^3}{(b-c)(b-a)}+\frac{c^3}{(c-a)(c-b)}.$$

Let $n$ be a positive integer and let $0\leq k\leq n$ be an integer. Show that there exist $n$ points in the plane with no three on a line such that the points can be divided into two groups satisfying the following properties. $\text{(i)}$ The first group has $k$ points and the distance between any two distinct points in this group is irrational. $\text{(ii)}$ The second group has $n-k$ points and the distance between any two distinct points in this group is an integer. $\text{(iii)}$ The distance between a point in the first group and a point in the second group is irrational.

A finite sequence of integers $a_0,,a_1,\dots,a_n$ is called quadratic if for each $i\in\{1,2,\dots n\}$ we have the equality $|a_i-a_{i-1}|=i^2$. $\text{(i)}$ Prove that for any two integers $b$ and $c$, there exist a positive integer $n$ and a quadratic sequence with $a_0=b$ and $a_n = c$. $\text{(ii)}$ Find the smallest positive integer $n$ for which there exists a quadratic sequence with $a_0=0$ and $a_n=2021$.