Let $\alpha$ be real and define $\varphi(x)=x^2e^{\alpha x}$ for $x\in\mathbb R$. Find all functions $f:\mathbb R\to\mathbb R$ satisfying, for every $x,y\in\mathbb R$, $$f(\varphi(x)+f(y))=y+\varphi(f(x)).$$
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A convex polyhedron has 2022 faces. Three of its faces already carry the numbers 26, 4 and 2022, one number on each face. Fill every remaining face with a real number so that each newly entered number equals the arithmetic mean of the numbers on the faces sharing an edge with its face. Prove that exactly one such filling exists.
Let $ABCD$ be a parallelogram whose diagonals $AC,BD$ meet at $I$. Let $G$ lie inside triangle $IAB$ and satisfy $$\angle IAG=\angle IBG=45^\circ-\frac{\angle AIB}{4}.$$ Let $E,F$ be the perpendicular projections of $C$ onto $AG$ and of $D$ onto $BG$, respectively. The median from $E$ in triangle $BEF$ and the median from $F$ in triangle $AEF$ meet at $H$.
(a) Prove that $AF,BE,IH$ are concurrent; denote their common point by $L$.
(b) Let $K=CE\cap DF$. Let $J$ be the circumcenter of $LAB$, and let $M,N$ be the circumcenters of $EIJ,FIJ$, respectively. Prove that $EM,FN$, and the line joining the circumcenters of $GAB$ and $KCD$ are concurrent.
Let $ABC$ be an acute non-isosceles triangle inscribed in circle $(O)$. The line through $O$ and the midpoint $I$ of $BC$ meets lines $AB,AC$ at $E,F$, respectively. Let $D,G$ be the reflections of $A$ in $O$ and in the circumcenter of $AEF$, respectively. Let $K$ be the reflection of $O$ in the circumcenter of $OBC$.
(a) Prove that $D,G,K$ are collinear.
(b) Choose $M,N$ on lines $KB,KC$, respectively, such that $IM\perp AC$ and $IN\perp AB$. The perpendicular bisector of $IK$ meets $MN$ at $H$. Let $P,Q$ be the intersections of $IH$ with $AB,AC$, respectively. Prove that the circumcircle of $APQ$ meets $(O)$ again at a point on $AI$.
A rational number $x$ is called beautiful if it has a finite expansion in some base $b$, where $b$ is an integer in $[2,2022]$. Prove that there are only finitely many positive integers $n\ge4$ such that, for every integer $m$ in the open interval $(2n/3,n)$, at least one of $$\frac m{n-m},\qquad\frac{n-m}{m}$$ is beautiful.
Let $A=\{1,2,\ldots,4044\}$. Color 2022 elements of $A$ white and the other 2022 elements black. For $i\in A$, define its weight as the total number of white elements smaller than $i$ and black elements greater than $i$. For each natural number $m$, determine all positive integers $k$ for which a coloring exists with exactly $k$ elements of weight $m$.
Let $a\ge0$ be real and define
$$u_1=6,\qquad u_{n+1}=\frac{2n+a}{n}+\sqrt{\frac{n+a}{n}u_n+4}\quad(n\ge1).$$
(a) For $a=0$, prove that $(u_n)$ has a finite limit and determine it.
(b) For every $a\ge0$, prove that $(u_n)$ has a finite limit.
Let $ABC$ be acute. Points $E,F$ vary on the rays opposite to $BA,CA$, respectively, with $BF=CE$, $E\ne B$, $F\ne C$. Let $M,N$ be the midpoints of $BE,CF$, and $D=BF\cap CE$.
(a) Let $I,J$ be the circumcenters of $DBE,DCF$, respectively. Prove $MN\parallel IJ$.
(b) Let $K$ be the midpoint of $MN$, and let $H$ be the orthocenter of $AEF$. Prove that $HK$ passes through a fixed point.
For positive integers $n\lt m$, let $s(n,m)$ be the number of positive integers in $[n,m]$ coprime to $m$. Find all integers $m\ge2$ satisfying both:
(i) $\dfrac{s(n,m)}{m-n}\ge\dfrac{s(1,m)}m$ for every $n=1,2,\ldots,m-1$;
(ii) $m^2$ divides $2022^m+1$.
Let $P,Q$ be nonconstant polynomials with nonnegative integer coefficients. Every coefficient of $P$ is at most 2021, while at least one coefficient of $Q$ exceeds 2021. Suppose $P(2022)=Q(2022)$ and $P,Q$ have a common nonzero rational root $p/q$, where $p,q\in\mathbb Z$ are coprime. Prove that
$$|p|+n|q|\le Q(n)-P(n)\qquad(n=1,2,\ldots,2021).$$
Four fair, identical dice are rolled. Let $x_i\in\{1,2,\ldots,6\}$ be the result of die $i$, for $i=1,2,3,4$.
(a) How many ordered outcomes $(x_1,x_2,x_3,x_4)$ are possible?
(b) Find the probability that one result equals the sum of the other three.
(c) Find the probability that the four results can be divided into two groups of equal sum.
Let $B,C$ be fixed points on a circle $(O)$, with $BC$ not passing through $O$. Let $A$ vary on the major arc $BC$, with $AB\ne AC$. The incircle $(I)$ of $ABC$ touches $BC$ at $D$. Let $I_a$ be the $A$-excenter, let $L=I_aD\cap OI$, and choose $E$ on $(I)$ such that $DE\parallel AI$.
(a) Let $F=LE\cap AI$. Prove that $AF=AI$.
(b) On the circumcircle $(J)$ of $I_aBC$, choose $M$ such that $I_aM\parallel AD$, and let $MD$ meet $(J)$ again at $N$. Prove that the midpoint $T$ of $MN$ lies on a fixed circle.