Let $n\geqslant 1$ be an integer, and let $x_0,x_1,\ldots,x_{n+1}$ be $n+2$ non-negative real numbers that satisfy $x_ix_{i+1}-x_{i-1}^2\geqslant 1$ for all $i=1,2,\ldots,n.$ Show that \[x_0+x_1+\cdots+x_n+x_{n+1}>\bigg(\frac{2n}{3}\bigg)^{3/2}.\]Pakawut Jiradilok and Wijit Yangjit, Thailand

A hunter and an invisible rabbit play a game on an infinite square grid. First the hunter fixes a colouring of the cells with finitely many colours. The rabbit then secretly chooses a cell to start in. Every minute, the rabbit reports the colour of its current cell to the hunter, and then secretly moves to an adjacent cell that it has not visited before (two cells are adjacent if they share an edge). The hunter wins if after some finite time either: the rabbit cannot move; or the hunter can determine the cell in which the rabbit started. Decide whether there exists a winning strategy for the hunter. Proposed by Aron Thomas

Let $I$, $O$, $H$, and $\Omega$ be the incenter, circumcenter, orthocenter, and the circumcircle of the triangle $ABC$, respectively. Assume that line $AI$ intersects with $\Omega$ again at point $M\neq A$, line $IH$ and $BC$ meets at point $D$, and line $MD$ intersects with $\Omega$ again at point $E\neq M$. Prove that line $OI$ is tangent to the circumcircle of triangle $IHE$. Proposed by Li4 and Leo Chang.

A positive integer is said to be palindromic if it remains the same when its digits are reversed. For example, $1221$ or $74847$ are both palindromic numbers. Let $k$ be a positive integer that can be expressed as an $n$-digit number $\overline{a_{n-1}a_{n-2} \cdots a_0}$. Prove that if $k$ is a palindromic number, then $k^2$ is also a palindromic number if and only if $a_0^2 + a^2_1 + \cdots + a^2_{n-1} < 10$. Proposed by Ho-Chien Chen

Determine all functions $f: \mathbb{R}^+ \to \mathbb{R}^+$ satisfying \[f\bigl(x + y^2 f(y)\bigr) = f\bigl(1 + yf(x)\bigr)f(x)\]for any positive reals $x$, $y$, where $\mathbb{R}^+$ is the collection of all positive real numbers. Proposed by Ming Hsiao.

There are $2022$ distinct integer points on the plane. Let $I$ be the number of pairs among these points with exactly $1$ unit apart. Find the maximum possible value of $I$. (Note. An integer point is a point with integer coordinates.) Proposed by CSJL.

Let $ABC$ be a triangle with circumcenter $O$ and orthocenter $H$ such that $OH$ is parallel to $BC$. Let $AH$ intersects again with the circumcircle of $ABC$ at $X$, and let $XB, XC$ intersect with $OH$ at $Y, Z$, respectively. If the projections of $Y,Z$ to $AB,AC$ are $P,Q$, respectively, show that $PQ$ bisects $BC$. Proposed by usjl

For any two coprime positive integers $p, q$, define $f(i)$ to be the remainder of $p\cdot i$ divided by $q$ for $i = 1, 2,\ldots,q -1$. The number $i$ is called a large number (resp. small number) when $f(i)$ is the maximum (resp. the minimum) among the numbers $f(1), f(2),\ldots,f(i)$. Note that $1$ is both large and small. Let $a, b$ be two fixed positive integers. Given that there are exactly $a$ large numbers and $b$ small numbers among $1, 2,\ldots , q - 1$, find the least possible number for $q$. Proposed by usjl

Consider a $100\times 100$ square unit lattice $\textbf{L}$ (hence $\textbf{L}$ has $10000$ points). Suppose $\mathcal{F}$ is a set of polygons such that all vertices of polygons in $\mathcal{F}$ lie in $\textbf{L}$ and every point in $\textbf{L}$ is the vertex of exactly one polygon in $\mathcal{F}.$ Find the maximum possible sum of the areas of the polygons in $\mathcal{F}.$ Michael Ren and Ankan Bhattacharya, USA

A $100 \times100$ chessboard has a non-negative real number in each of its cells. A chessboard is balanced if and only if the numbers sum up to one for each column of cells as well as each row of cells. Find the largest positive real number $x$ so that, for any balanced chessboard, we can find $100$ cells of it so that these cells all have number greater or equal to $x$, and no two of these cells are on the same column or row. Proposed by CSJL.

Let $ABC$ be a triangle with circumcircle $\omega$ and let $\Omega_A$ be the $A$-excircle. Let $X$ and $Y$ be the intersection points of $\omega$ and $\Omega_A$. Let $P$ and $Q$ be the projections of $A$ onto the tangent lines to $\Omega_A$ at $X$ and $Y$ respectively. The tangent line at $P$ to the circumcircle of the triangle $APX$ intersects the tangent line at $Q$ to the circumcircle of the triangle $AQY$ at a point $R$. Prove that $\overline{AR} \perp \overline{BC}$.

Let $r>1$ be a rational number. Alice plays a solitaire game on a number line. Initially there is a red bead at $0$ and a blue bead at $1$. In a move, Alice chooses one of the beads and an integer $k \in \mathbb{Z}$. If the chosen bead is at $x$, and the other bead is at $y$, then the bead at $x$ is moved to the point $x'$ satisfying $x'-y=r^k(x-y)$. Find all $r$ for which Alice can move the red bead to $1$ in at most $2021$ moves.

Let $ABCDE$ be a pentagon inscribed in a circle $\Omega$. A line parallel to the segment $BC$ intersects $AB$ and $AC$ at points $S$ and $T$, respectively. Let $X$ be the intersection of the line $BE$ and $DS$, and $Y$ be the intersection of the line $CE$ and $DT$. Prove that, if the line $AD$ is tangent to the circle $\odot(DXY)$, then the line $AE$ is tangent to the circle $\odot(EXY)$. Proposed by ltf0501.

Let $N>s$ be positive integers. Electricity park has a number of buildings; exactly $N$ of them are power plants, and another one of them is the headquarter. Some pairs of buildings have one-way power cables between them, satisfying: (i) The cables connected to a power plant will only send the power out of the plant. (ii) For each non-headquarter building, there is a unique sequence of cables that can transport the power from that building to the headquarter. A building is $s$-electrifed if, after removing any one cable from the park, the building can still receive power from at least $s$ different power plants. Find the maximum possible number of $s$-electrifed buildings. Note: There seems to be confusion about whether a power plant is $1$-electrified. For the sake of simplicity let's say that any power plant is not $s$-electrified for any $s\geq 1$. Proposed by usjl