In cyclic quadrilateral $ABCD$ with circumcenter $O$ and circumradius $R$, define $X=\overline{AB}\cap\overline{CD}$, $Y=\overline{AC}\cap \overline{BD}$, and $Z=\overline{AD}\cap\overline{BC}$. Prove that \[OX^2+OY^2+OZ^2\ge 2R^2+2[ABCD].\]Rohan Bodke

On an infinite square grid, Gru and his $2022$ minions play a game, making moves in a cyclic order with Gru first. On any move, the current player selects $2$ adjacent cells of their choice, and paints their shared border. A border cannot be painted over more than once. Gru wins if after any move there is a $2 \times 1$ or $1 \times 2$ subgrid with its border (comprising of $6$ segments) completely colored, but the $1$ segment inside it uncolored. Can he guarantee a win? Evan Chang (oops)

Find all functions $f\colon\mathbb R^+\to\mathbb R^+$ such that \[(f(x)+f(y)+f(z))(xf(y)+yf(z)+zf(x))>(f(x)+y)(f(y)+z)(f(z)+x)\]for all $x,y,z\in\mathbb R^+$. Alexander Wang (oops)

On a $5\times 5$ grid $\mathcal A$ of integers, each with absolute value $<10^9$, define a flip to be the operation of negating each element in a row / column with negative sum. For example, $(-1,-4,3,-4,1) \to (1,4,-3,4,-1)$. Determine whether there exists an $\mathcal A$ so that it's possible to perform $90$ flips on it. Alex Chen

Complex numbers $a,b,w,x,y,z,p$ satisfy \begin{align*} \frac{(x-w)\lvert a-w \rvert}{(a-w)\lvert x-w \rvert}&=\text{(cyclic variants)};\\ \frac{(z-w)\lvert b-w \rvert}{(b-w)\lvert z-w \rvert}&=\text{(cyclic variants)};\\ p &= \frac{\sum_{\text{cyc}} \frac w{\lvert p-w \rvert}}{\sum_{\text{cyc}}\frac1{\lvert p-w \rvert}}; \end{align*}where cyclic sums, equations, etc. are wrt $w,x,y,z$. Prove that there exists a real $k$ such that \[\sum_{\text{cyc}} \frac{(x-w)(a-w)}{\lvert x-w\rvert (p-w)} =k\sum_{\text{cyc}} \frac{(z-w)(b-w)}{\lvert z-w\rvert(p-w)}.\]Neal Yan