Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for all real numbers $x,y$, we have $$f(xf(x)+2y)=f(x)^2+x+2f(y)$$

It is known that a polynomial $P$ with integer coefficients has degree $2022$. What is the maximum $n$ such that there exist integers $a_1, a_2, \cdots a_n$ with $P(a_i)=i$ for all $1\le i\le n$? [Extra: What happens if $P \in \mathbb{Q}[X]$ and $a_i\in \mathbb{Q}$ instead?]

Find all functions $ f : \mathbb{R} \rightarrow \mathbb{R} $ such that for all reals $ x, y $,$$ f(x^2+f(x+y))=y+xf(x+1) $$

Given a polynomial $P\in \mathbb{Z}[X]$ of degree $k$, show that there always exist $2d$ distinct integers $x_1, x_2, \cdots x_{2d}$ such that $$P(x_1)+P(x_2)+\cdots P(x_{d})=P(x_{d+1})+P(x_{d+2})+\cdots + P(x_{2d})$$for some $d\le k+1$. [Extra: Is this still true if $d\le k$? (Of course false for linear polynomials, but what about higher degree?)]

Given $k\ge 2$, for which polynomials $P\in \mathbb{Z}[X]$ does there exist a function $h:\mathbb{N}\rightarrow\mathbb{N}$ with $h^{(k)}(n)=P(n)$?

Given a graph $G$, consider the following two quantities, $\bullet$ Assign to each vertex a number in $\{0,1,2\}$ such that for every edge $e=uv$, the numbers assigned to $u$ and $v$ have sum at least $2$. Let $A(G)$ be the minimum possible sum of the numbers written to each vertex satisfying this condition. $\bullet$ Assign to each edge a number in $\{0,1,2\}$ such that for every vertex $v$, the sum of numbers on all edges containing $v$ is at most $2$. Let $B(G)$ be the maximum possible sum of the numbers written to each edge satisfying this condition. Prove that $A(G)=B(G)$ for every graph $G$. [Note: This question is not original] [Extra: Show that this statement is still true if we replace $2$ to $n$, if and only if $n$ is even (where we replace $\{0,1,2\}$ to $\{0,1,\cdots, n\}$)]

A pentagon $ABCDE$ is such that $ABCD$ is cyclic, $BE\parallel CD$, and $DB=DE$. Let us fix the points $B,C,D,E$ and vary $A$ on the circumcircle of $BCD$. Let $P=AC\cap BE$, and $Q=BC\cap DE$. Prove that the second intersection of circles $(ABE)$ and $(PQE)$ lie on a fixed circle.

Let $ABCD$ be a circumscribed quadrilateral with incircle $\gamma$. Let $AB\cap CD=E, AD\cap BC=F, AC\cap EF=K, BD\cap EF=L$. Let a circle with diameter $KL$ intersect $\gamma$ at one of the points $X$. Prove that $(EXF)$ is tangent to $\gamma$.

Let $\omega$ be the circumcircle of an actue triangle $ABC$ and let $H$ be the feet of aliitude from $A$ to $BC$. Let $M$ and $N$ be the midpoints of the sides $AC$ and $AB$. The lines $BM$ and $CN$ intersect each other at $G$ and intersect $\omega$ at $P$ and $Q$ respectively. The circles $(HMG)$ and $(HNG)$ intersect the segments $HP$ and $HQ$ again at $R$ and $S$ respectively. Prove that $PQ\parallel RS$.

Let $ABC$ be a triangle, and let $BE, CF$ be the altitudes. Let $\ell$ be a line passing through $A$. Suppose $\ell$ intersect $BE$ at $P$, and $\ell$ intersect $CF$ at $Q$. Prove that: i) If $\ell$ is the $A$-median, then circles $(APF)$ and $(AQE)$ are tangent. ii) If $\ell$ is the inner $A$-angle bisector, suppose $(APF)$ intersect $(AQE)$ again at $R$, then $AR$ is perpendicular to $\ell$.

Let $n$, $k$ be fixed integers. On a $n \times n$ board, label each square $0$ or $1$ such that in each $2k \times 2k$ sub-square of the board, the number of $0$'s and $1$'s written are the same. What is the largest possible sum of numbers written on the $n\times n$ board?

Let $a, b, c,$ be nonnegative reals with $ a+b+c=3 $, find the largest positive real $ k $ so that for all $a,b,c,$ we have $$ a^2+b^2+c^2+k(abc-1)\ge 3 $$

Given a four digit string $ k=\overline{abcd} $, $ a, b, c, d\in \{0, 1, \cdots, 9\} $, prove that there exist a $n<20000$ such that $2^n$ contains $k$ as a substring when written in base $10$. [Extra: Can you give a better bound? Mine is $12517$]

Find all positive integer $n$ such that for all positive integers $ x $, $ y $, $ n \mid x^n-y^n \Rightarrow n^2 \mid x^n-y^n $.

Given an acute triangle $ABC$, mark $3$ points $X, Y, Z$ in the interior of the triangle. Let $X_1, X_2, X_3$ be the projections of $X$ to $BC, CA, AB$ respectively, and define the points $Y_i, Z_i$ similarly for $i=1, 2, 3$. a) Suppose that $X_iY_i<X_iZ_i$ for all $i=1,2,3$, prove that $XY<XZ$. b) Prove that this is not neccesarily true, if triangle $ABC$ is allowed to be obtuse. Proposed by Ivan Chan Kai Chin

Let $\mathcal{S}$ be a set of $2023$ points in a plane, and it is known that the distances of any two different points in $S$ are all distinct. Ivan colors the points with $k$ colors such that for every point $P \in \mathcal{S}$, the closest and the furthest point from $P$ in $\mathcal{S}$ also have the same color as $P$. What is the maximum possible value of $k$? Proposed by Ivan Chan Kai Chin

Given a positive integer $n$, suppose that $P(x,y)$ is a real polynomial such that \[P(x,y)=\frac{1}{1+x+y} \hspace{0.5cm} \text{for all $x,y\in\{0,1,2,\dots,n\}$} \]What is the minimum degree of $P$? Proposed by Loke Zhi Kin

Find all functions $f : \mathbb{Z}\rightarrow \mathbb{Z}$ such that for all prime $p$ the following condition holds: $$p \mid ab + bc + ca \iff p \mid f(a)f(b) + f(b)f(c) + f(c)f(a)$$ Proposed by Anzo Teh Zhao Yang

Given a triangle $ABC$ with $AB=AC$ and circumcenter $O$. Let $D$ and $E$ be midpoints of $AC$ and $AB$ respectively, and let $DE$ intersect $AO$ at $F$. Denote $\omega$ to be the circle $(BOE)$. Let $BD$ intersect $\omega$ again at $X$ and let $AX$ intersect $\omega$ again at $Y$. Suppose the line parallel to $AB$ passing through $O$ meets $CY$ at $Z$. Prove that the lines $FX$ and $BZ$ meet at $\omega$. Proposed by Ivan Chan Kai Chin