Find all functions $f:\mathbb{N}\to\mathbb{N}$ s.t. for all $A\subset \mathbb{N}$ with 2024 elements, the set $$S_A:=\{f^{(k)}(x)\mid k=1,...,2024,x\in A\}$$also has 2024 elements. ($f^{(k)}=f\circ f\circ...\circ f$ is the $k$-th iteration of $f$.)

Let $ABC$ be a triangle with $O$ as its circumcenter. A circle $\Gamma$ tangents $OB, OC$ at $B, C$, respectively. Let $D$ be a point on $\Gamma$ other than $B$ with $CB=CD$, $E$ be the second intersection of $DO$ and $\Gamma$, and $F$ be the second intersection of $EA$ and $\Gamma$. Let $X$ be a point on the line $AC$ so that $XB\perp BD$. Show that one half of $\angle ADF$ is equal to one of $\angle BDX$ and $\angle BXD$. Proposed by usjl

Let $\mathbb{R}_+$ be the set of positive real numbers. Find all functions $f\colon \mathbb{R}_+ \to \mathbb{R}_+$ such that \[f(xy + x + y) + f \left( \frac1x \right) f\left( \frac1y \right) = 1\]for every $x$, $y\in \mathbb{R}_+$. Proposed by Li4 and Untro368.

Let $k$ be a positive integer. The little one and the magician on the skywalk play a game. Initially, there are $N = 2^k$ distinct balls line up in a row, with each of the ball covered by a cup. On each turn, the little one chooses two cups, then the magician can either swap the balls in the two cups, or do a fake move so that the balls in the two cups stay the same. The little one cannot distinguish whether the magician fakes a move on not, nor can she observe the balls inside the cups. After $M = k \times 2^{k-1}$ turns, the magician opens all cups so the little one can check the ball in each of the cups. If the little one can identify whether the magician fakes a move or not for each of the $M$ turns, then the little one win. Prove that the little one has a winning strategy. Proposed by usjl

Let $ABC$ be a triangle and $D$ be a point inside triangle $ABC$. $\Gamma$ is the circumcircle of triangle $ABC$, and $DB$, $DC$ meet $\Gamma$ again at $E$, $F$ , respectively. $\Gamma_1$, $\Gamma_2$ are the circumcircles of triangle $ADE$ and $ADF$ respectively. Assume $X$ is on $\Gamma_2$ such that $BX$ is tangent to $\Gamma_2$. Let $BX$ meets $\Gamma$ again at $Z$. Prove that the line $CZ$ is tangent to $\Gamma_1$ . Proposed by HakureiReimu.

For any positive integer $n$, consider its binary representation. Denote by $f(n)$ the number we get after removing all the $0$'s in its binary representation, and $g(n)$ the number of $1$'s in the binary representation. For example, $f(19) = 7$ and $g(19) = 3.$ Find all positive integers $n$ that satisfy $$n = f(n)^{g(n)}.$$ Proposed by usjl

Given triangle $ABC$. Let $BPCQ$ be a parallelogram ($P$ is not on $BC$). Let $U$ be the intersection of $CA$ and $BP$, $V$ be the intersection of $AB$ and $CP$, $X$ be the intersection of $CA$ and the circumcircle of triangle $ABQ$ distinct from $A$, and $Y$ be the intersection of $AB$ and the circumcircle of triangle $ACQ$ distinct from $A$. Prove that $\overline{BU} = \overline{CV}$ if and only if the lines $AQ$, $BX$, and $CY$ are concurrent. Proposed by Li4.

Let $n$ be a positive integer. Prove that the inequality \[n \sum_{i=1}^n \sum_{j = 1}^n \sum_{k=1}^n \frac{3}{a_ja_k + a_ka_i + a_i a_j} \ge \left(\sum_{j=1}^n \sum_{k=1}^n \frac{2}{a_j + a_k}\right)^2 \]holds for any positive real numbers $a_1$, $a_2$, $\dots$, $a_n$. Proposed by Li4 and Ming Hsiao.

Let $\mathbb{N}$ be the set of all positive integers. Find all functions $f\colon \mathbb{N}\to \mathbb{N}$ such that $mf(m)+(f(f(m))+n)^2$ divides $4m^4+n^2f(f(n))^2$ for all positive integers $m$ and $n$.