Find all functions $f:\mathbb{N} \rightarrow \mathbb{N}$ such that for all positive integers $n$, there exists an unique positive integer $k$, satisfying $f^k(n)\leq n+k+1$.

Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$, such that $$f(f(x)+y)(x-f(y)) = f(x)^2-f(y^2).$$

Given positive reals $x,y,z$ satisfying $x+y+z=3$, prove that \[\sum_{cyc}\left( x^2+y^2+x^2y^2+\frac{y^2}{x^2}\right)\geq 4\sum_{cyc}\frac{y}{x}.\] Proposed by chengbilly.

Find all functions $f:\mathbb{R^{+}} \rightarrow \mathbb{R^{+}}$, such that $$xf(1+xf(y))=f(f(x)+f(y))$$for all positive reals $x, y$.

We can conduct the following moves to a real number $x$: choose a positive integer $n$, and positives reals $a_1,a_2,\cdots, a_n$ whose reciprocals sum up to $1$. Let $x_0=x$, and $x_k=\sqrt{x_{k-1}a_k}$ for all $1\leq k\leq n$. Finally, let $y=x_n$. We said $M>0$ is "tremendous" if for any $x\in \mathbb{R}^+$, we can always choose $n,a_1,a_2,\cdots, a_n$ to make the resulting $y$ smaller than $M$. Find all tremendous numbers. Proposed by ckliao914.

We define \[f(x,y,z)=|xy|\sqrt{x^2+y^2}+|yz|\sqrt{y^2+z^2}+|zx|\sqrt{z^2+x^2}.\]Find the best constants $c_1,c_2\in\mathbb{R}$ such that \[c_1(x^2+y^2+z^2)^{3/2}\leq f(x,y,z)\leq c_1(x^2+y^2+z^2)^{3/2}\]hold for all reals $x,y,z$ satisfying $x+y+z=0$. Proposed by Untro368.

There are $n$ cards on a table in a line, with a positive real written on eachcard. LTF and Sunny are playing a game where they take turns taking away the first or the last card in line. The player that has the bigger sum of all the numberson his cards wins. If LTF goes first, find all $n$ such that LTF can always prevent Sunny from winning, regardless of the numbers written on the cards.

A square house is partitioned into an $n \times n$ grid, where each cell is a room. All neighboring rooms have a door connecting them, and each door can either be normalor inversive. If USJL walks over an inversive door, he would become inverted-USJL,and vice versa. USJL must choose a room to begin and walk pass each room exactly once. If it is inverted-USJL showing up after finishing, then he would be trapped for all eternity. Prove that USJL could always escape.

Graph $G$ has $n\geq 2$ vertices. Find the largest $m$ such that one of the following is true for always: 1. There exists a cycle with $k\geq m$ vertices. 2. There exists an independent set with $m$ vertices.

A ghost leg is a game with some vertical lines and some horizontal lines. A player starts at the top of the vertical line and go downwards, and always walkthrough a horizontal line if he encounters one. We define a layer is some horizontal line with the same height and has no duplicated endpoints. Find the smallest number of layers needed to grant that you can walk from $(1, 2, \ldots , n)$ on the top to any permutation $(\sigma_1, \sigma_2, \ldots, \sigma_n)$ on the bottom.

In an $2023\times 2023$ grid we fill in numbers $1,2,\cdots,2023^2$ without duplicating. Find the largest integer $M$ such that there exists a way to fill the numbers, satisfying that any two adjacent numbers has a difference at least $M$ (two squares $(x_1,y_1),(x_2,y_2)$ are adjacent if $x_1=x_2$ and $y_1-y_2\equiv \pm1\pmod{2023}$ or $y_1=y_2$ and $x_1-x_2\equiv \pm1\pmod{2023}$). Proposed by chengbilly.

Given integer $n \geq 3$. $1, 2, \ldots, n$ were written on the blackboard. In each move, one could choose two numbers $x, y$, erase them, and write down $x + y, |x-y|$ in the place of $x, y$. Find all integers $X$ such that one could turn all numbers into $X$ within a finite number of moves.

Triangle $ABC$ has circumcenter $O$. $M$ is the midpoint of arc $BC$ not containing $A$. $S$ is a point on $(ABC)$ such that $AS$ and $BC$ intersect on the line passing through $O$ and perpendicular to $AM$. $D$ is a point such that $ABDC$ is a parallelogram. Prove that $D$ lies on the line $SM$.

$P$ is a point inside $\triangle ABC$. $AP, BP, CP$ intersects $BC, CA, AB$ at $D, E, F$, respectively. $AD$ meets $(ABC)$ again at $D_1$. $S$ is a point on $(ABC)$. Lines $AS$, $EF$ intersect at $T$, lines $TP, BC$ intersect at $K$, and $KD_1$ meets $(ABC)$ again at $X$. Prove that $S, D, X$ are colinear.

$ABCD$ is a cyclic quadrilateral with circumcenter $O$. The lines $AC, BD$ intersect at $E$ and $AD, BC$ intersect at $F$. $O_1$ and $O_2$ are the circumcenters of $\triangle ABE$ and $\triangle CDE$, respectively. Assume that $(ABCD)$ and $(OO_1O_2)$ intersect at two points $P, Q$. Prove that $P, Q, F$ are collinear.

Given triangle $ABC$. $D$ is a point on $BC$. $AC$ meets $(ABD)$ again at $E$,and $AB$ meets $(ACD)$ again at $F$. $M$ is the midpoint of $EF$. $BC$ meets $(DEF)$ again at $P$. Prove that $\angle BAP = \angle MAC$.

$ABCDEF$ is a cyclic hexagon with circumcenter $O$, and $AD, BE, CF$ are concurrent at $X$. $P$ is a point on the plane. The circumenter of $PAB$ is $O_{AB}$. Define $O_{BC}, O_{CD}$, $O_{DE}, O_{EF}, O_{FA}$ similarly. Prove that $O_{AB} O_{DE}, O_{BC}O_{EF}, O_{CD}O_{FA}$, $OX$ are concurrent.

Triangle $ABC$ has circumcenter $O$. $D$ is the foot from $A$ to $BC$, and $P$ is apoint on $AD$. The feet from $P$ to $CA, AB$ are $E, F$, respectively, and the foot from $D$ to $EF$ is $T$. $AO$ meets $(ABC)$ again at $A'$. $A'D$ meets $(ABC)$ again at $R$. If $Q$ is a point on $AO$ satisfying $\angle ABP = \angle QBC$, prove that $D, P, T, R$ lie on acircle and $DQ$ is tangent to it.

Find all positive integers $k$ satisfying: there is only a finite number of positive integers $n$, such that the positive integer solution $x$ of $xn+1\mid n^2+kn+1$ is not unique.

Find all pairs of positive integers $(a, b)$ such that $a^b+b^a=a!+b^2+ab+1$.

Find all functions $f:\mathbb{N} \rightarrow \mathbb{N}$, such that $f(a)+f(b)+ab \mid a^2f(a)+b^2f(b)+f(a)f(b)$ for all positive integers $a,b$.

Find all functions $f:\mathbb{N} \rightarrow \mathbb{N}$, such that $af(a)^3+2abf(a)+bf(b)$ is a perfect square for all positive integers $a,b$.

Let $p=4k+1$ be a prime and let $|x| \leq \frac{p-1}{2}$ such that $\binom{2k}{k}\equiv x \pmod p$. Show that $|x| \leq 2\sqrt{p}$.

Let $S(b)$ be the number of nonuples of positive integers $(a_1, a_2, \ldots , a_9)$ satisfying $3b-1=a_1+a_2+\ldots+a_9$ and $b^2+1=a_1^2+\ldots+a_9^2$. Prove that for all $\epsilon>0$, there exists $C_{\epsilon}>0$ such that $S(b)\leq C_{\epsilon}b^{3+\epsilon}$.