Given a circle $\Gamma$ with center $I$, and a triangle $\triangle ABC$ with all its sides tangent to $\Gamma$. A line $\ell$ is drawn such that it bisects both the area and the perimeter of $\triangle ABC$. Prove that line $\ell$ passes through $I$.

It is known that there are $n$ integers $a_1, a_2, \cdots, a_n$ such that $$a_1 + a_2 + \cdots + a_n = 0 \qquad \text{and} \qquad a_1 \times a_2 \times \cdots \times a_n = n.$$Determine all possible values of $n$.

Prove that there is a multiple of $2^{2022}$ that has $2022$ digits, and can be written using digits $1$ and $2$ only.

Given a pentagon $ABCDE$ with all its interior angles less than $180^\circ$. Prove that if $\angle ABC = \angle ADE$ and $\angle ADB = \angle AEC$, then $\angle BAC = \angle DAE$.

Let $a, b, r,$ and $s$ be positive integers ($a \ge 2$), where $a$ and $b$ have no common prime factor. Prove that if $a^r + b^r$ is divisible by $a^s + b^s$, then $r$ is divisible by $s$.

A football league has $n$ teams. Each team plays one game with every other team. Each win is awarded $2$ points, each tie $1$ point, and each loss $0$ points. After the league is over, the following statement is true: for every subset $S$ of teams in the league, there is a team (which may or may not be in $S$) such that the total points the team obtained by playing all the teams in $S$ is odd. Prove that $n$ is even.

Given a circle and a quadrilateral $ABCD$ whose vertices all lie on the circle. Let $R$ be the midpoint of arc $AB$. The line $RC$ meets line $AB$ at point $S$, and the line $RD$ meets line $AB$ at point $T$. Prove that $CDTS$ is a cyclic quadrilateral.

Without using a calculator, determine which number is greater: $17^{24}$ or $31^{19}$

Given an integer $n$. We rearrange the digits of $n$ to get another number $m$. Prove that it is impossible to get $m+n = 999999999$.

Given a polygon $ABCDEFGHIJ$. How many diagonals does the polygon have?

The following list shows every number for which more than half of its digits are digits $2$, in increasing order: $$2, 22, 122, 202, 212, 220, 221, 222, 223, 224, \dots$$If the $n$th term in the list is $2022$, what is $n$?

Prove that $$1\cdot 4 + 2\cdot 5 + 3\cdot 6 + \cdots + n(n+3) = \frac{n(n+1)(n+5)}{3}$$for all positive integer $n$.