Find all values of $n$ such that there exists a rectangle with integer side lengths, perimeter $n$, and area $2n$.

Six teams participate in a hockey tournament. Each team plays once against every other team. In each game, a team is awarded $3$ points for a win, $1$ point for a draw, and none for a loss. After the tournament the teams are ranked by total points. No two teams have the same total points. Each team (except the bottom team) has $2$ points more than the team ranking one place lower. Prove that the team that finished fourth has won two games and lost three games.

Let $x$ and $y$ be two rational numbers such that $$x^5 + y^5 = 2x^2y^2.$$Prove that $\sqrt{1-xy}$ is also a rational number.

Given an octagon such that all its interior angles are equal, and all its sides have integer lengths. Prove that any pair of opposite sides have equal lengths.

There are $n$ guests at a gathering. Any two guests are either friends or not friends. Every guest is friends with exactly four of the other guests. Whenever a guest is not friends with two other guests, those two other guests cannot be friends with each other either. Determine all possible values of $n$.

Prove that there is a positive integer $m$ such that the number $5^{2021}m$ has no even digits (in its decimal representation).

Given a circle with center $O$. Points $A$ and $B$ lie on the circle such that triangle $OBA$ is equilateral. Let $C$ be a point outside the circle with $\angle ACB = 45^{\circ}$. Line $CA$ intersects the circle at point $D$, and the line $CB$ intersects the circle at point $E$. Find $\angle DBE$.

Can we find positive integers $a$ and $b$ such that both $(a^2 + b)$ and $(b^2 + a)$ are perfect squares?

Given a sequence of positive integers $$a_1, a_2, a_3, a_4, a_5, \dots$$such that $a_2 > a_1$ and $a_{n+2} = 3a_{n+1} - 2a_n$ for all $n \geq 1$. Prove that $a_{2021} > 2^{2019}$.

An equilateral triangle $ABC$ is divided by nine lines parallel to $BC$ into ten bands that are equally wide. We colour the bands alternately red and blue, with the smallest band coloured red. The difference between the total area in red and the total area in blue is $20$ $\text{cm}^2$. What is the area of triangle $ABC$?

The five numbers $a, b, c, d,$ and $e$ satisfy the inequalities $$a+b < c+d < e+a < b+c < d+e.$$Among the five numbers, which one is the smallest, and which one is the largest?

Given a cube. On each edge of the cube, we write a number, either $1$ or $-1$. For each face of the cube, we multiply the four numbers on the edges of this face, and write the product on this face. Finally, we add all the eighteen numbers that we wrote down on the edges and face of the cube. What is the smallest possible sum that we can get?