Find all polynomials $P$ with integer coefficients such that $P(P(x))-x$ is irreducible over $\mathbb{Z}[x]$.

Let $ABCD$ be a quadtrilateral with no parallel sides. The diagonals intersect at $E$, and $P, Q$ are points on sides $AB, CD$ respectively such that $\frac{AP}{PB} = \frac{CQ}{QD}$. $PQ$ meet $AC$ and $BD$ at $R,S$. Prove that $(EAB),(ECD),(ERS)$ all meet a point other than $E$.

Define the sequence $\{a_n\}_{n=1}^\infty$ as \[ a_1 = a_2 = 1,\quad a_{n+2} = 14a_{n+1} - a_n \; (n \geq 1) \]Prove that if $p$ is prime and there exists a positive integer $n$ such that $\frac{a_n}p$ is an integer, then $\frac{p-1}{12}$ is also an integer.

Find all pairs of positive integers $(m,n)$ such that one can partition a $m\times n$ board with $1\times 2$ or $2\times 1$ dominoes and draw one of the diagonals on each of the dominos so that none of the diagonals share endpoints.

Let $ABCD$ be a convex quadrilateral such that $\angle A, \angle B, \angle C$ are acute. $AB$ and $CD$ meet at $E$ and $BC,DA$ meet at $F$. Let $K,L,M,N$ be the midpoints of $AB,BC,CD,DA$ repectively. $KM$ meets $BC,DA$ at $X$ and $Y$, and $LN$ meets $AB,CD$ at $Z$ and $W$. Prove that the line passing $E$ and the midpoint of $ZW$ is parallel to the line passing $F$ and the midpoint of $XY$.

Find all possible values of $C\in \mathbb R$ such that there exists a real sequence $\{a_n\}_{n=1}^\infty$ such that $$a_na_{n+1}^2\ge a_{n+2}^4 +C$$for all $n\ge 1$.

$2024$ people attended a party. Eunson, the host of the party, wanted to make the participant shake hands in pairs. As a professional daydreamer, Eunsun wondered which would be greater: the number of ways each person could shake hands with $4$ others or the number of ways each person could shake hands with $3$ others. Solve Eunsun's peculiar question.

For a positive integer \( n \), let \( \tau(n) \) denote the number of positive divisors of \( n \). Determine whether there exists a positive integer triple \( a, b, c \) such that there are exactly $1012$ positive integers \( K \) not greater than $2024$ that satisfies the following: the equation \[ \tau(x) = \tau(y) = \tau(z) = \tau(ax + by + cz) = K \]holds for some positive integers $x,y,z$.

Find all $f:\mathbb{R}\rightarrow \mathbb{R}$ such that the equation $$f(x^2+yf(x))=(1-x)f(y-x)$$holds for all $x,y\in\mathbb{R}$.

Find all integer sequences $a_1 , a_2 , \ldots , a_{2024}$ such that $1\le a_i \le 2024$ for $1\le i\le 2024$ and $$i+j|ia_i-ja_j$$for each pair $1\le i,j \le 2024$.

Find all pairs of positive integers $n$ such that one can partition a $n\times (n+1)$ board with $1\times 2$ or $2\times 1$ dominoes and draw one of the diagonals on each of the dominos so that none of the diagonals share endpoints.

Call a set \(\{a,b,c,d\}\) epic if for any four different positive integers \(a, b, c, d\), there is a unique way to select three of them to form the sides of a triangle. Find all positive integers \(n\) such that \(\{1, 2, \ldots, 4n\}\) can be partitioned into \(n\) disjoint epic sets.

Does there exist a real sequence $\{a_n\}_{n=1}^\infty$ such that $$a_na_{n+1}\ge a_{n+2}^2 +1$$for all $n\ge 1$?

For a positive integer \( n \), let \( \tau(n) \) denote the number of positive divisors of \( n \). Determine all positive integers \( K \) such that the equation \[ \tau(x) = \tau(y) = \tau(z) = \tau(2x + 3y + 3z) = K \]holds for some positive integers $x,y,z$.