Let $n$ be a positive integer. Chef Kao has $n$ different flavors of ice cream. He wants to serve one small cup and one large cup for each flavor. He arranges the $2n$ ice cream cups into two rows of $n$ cups on a tray. He wants the tray to be colorful, so he arranges the ice cream cups with the following conditions: each row contains all ice cream flavors, and each column has different sizes of ice cream cup. Determine the number of ways that Chef Kao can arrange cups of ice cream with the above conditions.

Let $P(x)$ be a polynomial with real coefficients. Prove that not all roots of $x^3P(x)+1$ are real.

Let $a$ and $n$ be positive integers such that the greatest common divisor of $a$ and $n!$ is $1$. Prove that $n!$ divides $a^{n!}-1$.

Let $ABC$ be a triangle, and let $D$ and $D_1$ be points on segment $BC$ such that $BD = CD_1$. Construct point $E$ such that $EC\perp BC$ and $ED\perp AC$. Similarly, construct point $F$ such that $FB\perp BC$ and $FD\perp AB$. Prove that $EF\perp AD_1$.

For each positive integer $k$, let $d(k)$ be the number of positive divisors of $k$ and $\sigma(k)$ be the sum of positive divisors of $k$. Let $\mathbb N$ be the set of all positive integers. Find all functions $f: \mathbb{N} \to \mathbb N$ such that \begin{align*} f(d(n+1)) &= d(f(n)+1)\quad \text{and} \\ f(\sigma(n+1)) &= \sigma(f(n)+1) \end{align*}for all positive integers $n$.

Let $ABC$ be a triangle. Construct point $X$ such that $BX=BA$ and $X$ and $C$ lies on the same side of line $AB$. Construct point $Y$ such that $CY=CA$ and $Y$ and $B$ lies on different sides of line $AC$. Suppose that triangle $BAX$ and triangle $CAY$ are similar, prove that the circumcenter of triangle $AXY$ lies on the circumcircle of triangle $ABC$.

Let $a_0,a_1,\dots$ be a sequence of positive reals such that $$ a_{n+2} \leq \frac{2023a_n}{a_na_{n+1}+2023}$$for all integers $n\geq 0$. Prove that either $a_{2023}<1$ or $a_{2024}<1$.

Let $ABC$ be an acute triangle with $AB<AC$. Let $M$ be the midpoint of $BC$ and $E$ be the foot of altitude from $B$ to $AC$. The point $C'$ is the reflection of $C$ across $AM$. The point $D$ not equal to $C$ is placed on line $BC$ such that $AD=AC$. Prove that $B$ is the incenter of triangle $DEC'$.

Find all sequences of positive integers $a_1,a_2,\dots$ such that $$(n^2+1)a_n = n(a_{n^2}+1)$$for all positive integers $n$.

Let $n$ be an even positive integer. Alice and Bob play the following game. Before the start of the game, Alice chooses a set $S$ containing $m$ integers and announces it to Bob. The players then alternate turns, with Bob going first, choosing $i\in\{1,2,\dots, n\}$ that has not been chosen and setting the value of $v_i$ to either $0$ or $1$. At the end of the game, when all of $v_1,v_2,\dots,v_n$ have been set, the expression $$E=v_1\cdot 2^0 + v_2 \cdot 2^1 + \dots + v_n \cdot 2^{n-1}$$is calculated. Determine the minimum $m$ such that Alice can always ensure that $E\in S$ regardless of how Bob plays.