Let $x,y,z$ be three positive integers with $\gcd(x,y,z)=1$. If \[x\mid yz(x+y+z),\]\[y\mid xz(x+y+z),\]\[z\mid xy(x+y+z),\]and \[x+y+z\mid xyz,\]show that $xyz(x+y+z)$ is a perfect square. Proposed by usjl

There are $2022$ black balls numbered $1$ to $2022$ and $2022$ white balls numbered $1$ to $2022$ as well. There are also $1011$ black boxes and white boxes each. In each box we put two balls that are the same color as the the box. Prove that no matter how the balls are distributed, we can always pick one ball from each box such that the $2022$ balls we chose have all the numbers from $1$ to $2022$.

Find all functions $f,g:\mathbb{R}^2\to\mathbb{R}$ satisfying that \[|f(a,b)-f(c,d)|+|g(a,b)-g(c,d)|=|a-c|+|b-d|\]for all real numbers $a,b,c,d$. Proposed by usjl

Two babies A and B are playing a game with $2022$ bottles of milk. Each bottle has a maximum capacity of $200$ml, and initially each bottle holds $30$ml of milk. Starting from A, they take turns and do one of the following: (1) Pick a bottle with at least $100$ml of milk, and drink half of it. (2) Pick two bottles with less than $100$ml of milk, pour the milk of one bottle into the other one, and toss away the empty bottle. Whoever cannot do any operations loses the game. Who has a winning strategy? Proposed by Chu-Lan Kao and usjl

Let $J$ be the $A$-excenter of an acute triangle $ABC$. Let $X$, $Y$ be two points on the circumcircle of the triangle $ACJ$ such that $\overline{JX} = \overline{JY} < \overline{JC}$. Let $P$ be a point lies on $XY$ such that $PB$ is tangent to the circumcircle of the triangle $ABC$. Let $Q$ be a point lies on the circumcircle of the triangle $BXY$ such that $BQ$ is parallel to $AC$. Prove that $\angle BAP = \angle QAC$. Proposed by Li4.