An odd prime $p$ is called a prime of the year $2022$ if there is a positive integer $n$ such that $p^{2022}$ divides $n^{2022}+2022$. Show that there are infinitely many primes of the year $2022$.

Let $a,b,c>0$ satisfy $a\geq b\geq c$. Prove that $$\frac{4}{a^2(b+c)}+\frac{4}{b^2(c+a)}+\frac{4}{c^2(a+b)} \leq \left(\sum_{cyc} \frac{a^2+1} {b^2} \right)\left(\sum_{cyc} \frac{b^3}{a^2(a^3+2b^3)}\right).$$

An acute scalene triangle $ABC$ with circumcircle $\Omega$ is given. The altitude from $B$ intersects side $AC$ at $B_1$ and circle $\Omega$ at $B_2$. The circle with diameter $B_1B_2$ intersects circle $\Omega$ again at $B_3$. Similarly, the altitude from $C$ intersects side $AB$ at $C_1$ and circle $\Omega$ at $C_2$. The circle with diameter $C_1C_2$ intersects circle $\Omega$ again at $C_3$. Let $X$ be the intersection of lines $B_1B_3$ and $C_1C_3$, and let $Y$ be the intersection of lines $B_3C$ and $C_3B$. Prove that line $XY$ bisects side $BC$.

Let $ABCD$ be a parallelogram with $AC=BC.$ A point $P$ is chosen on the extension of ray $AB$ past $B.$ The circumcircle of $ACD$ meets the segment $PD$ again at $Q.$ The circumcircle of triangle $APQ$ meets the segment $PC$ at $R.$ Prove that lines $CD,AQ,BR$ are concurrent.

Let $S$ be an infinite set of positive integers, such that there exist four pairwise distinct $a,b,c,d \in S$ with $\gcd(a,b) \neq \gcd(c,d)$. Prove that there exist three pairwise distinct $x,y,z \in S$ such that $\gcd(x,y)=\gcd(y,z) \neq \gcd(z,x)$.

Let $S$ be the set of the positive integers greater than $1$, and let $n$ be from $S$. Does there exist a function $f$ from $S$ to itself such that for all pairwise distinct positive integers $a_1, a_2,...,a_n$ from $S$, we have $f(a_1)f(a_2)...f(a_n)=f(a_1^na_2^n...a_n^n)$?

Find all polynomials $f, g, h$ with real coefficients, such that $f(x)^2+(x+1)g(x)^2=(x^3+x)h(x)^2$

Find all positive integers $n\geq1$ such that there exists a pair $(a,b)$ of positive integers, such that $a^2+b+3$ is not divisible by the cube of any prime, and $$n=\frac{ab+3b+8}{a^2+b+3}.$$

An odd positive integer $n$ is called pretty if there exists at least one permutation $a_1, a_2,..., a_n$, of $1,2,...,n$, such that all $n$ sums $a_1-a_2+a_3-...+a_n$, $a_2-a_3+a_4-...+a_1$,..., $a_n-a_1+a_2-...+a_{n-1}$ are positive. Find all pretty integers.

Let $n\geq 3$ be an integer. Each vertex of a regular $n$-gon is labelled with a real number not exceeding $1$. For real numbers $a,b,c$ on any three consecutive vertices which are arranged clockwise in such an order, we have $c=|a-b|$. Determine the maximum value of the sum of all numbers in terms of $n$.

An acute triangle $ABC$ has $AB$ as one of its longest sides. The incircle of $ABC$ has center $I$ and radius $r$. Line $CI$ meets the circumcircle of $ABC$ at $D$. Let $E$ be a point on the minor arc $BC$ of the circumcircle of $ABC$ with $\angle ABE > \angle BAD$ and $E\notin \{B,C\}$. Line $AB$ meets $DE$ at $F$ and line $AD$ meets $BE$ at $G$. Let $P$ be a point inside triangle $AGE$ with $\angle APE=\angle AFE$ and $P\neq F$. Let $X$ be a point on side $AE$ with $XP\parallel EG$ and let $S$ be a point on side $EG$ with $PS\parallel AE$. Suppose $XS$ and $GP$ meet on the circumcircle of $AGE$. Determine the possible positions of $E$ as well as the minimum value of $\frac{BE}{r}$.

Let $n > 1$ be a given integer. Prove that infinitely many terms of the sequence $(a_k )_{k\ge 1}$, defined by \[a_k=\left\lfloor\frac{n^k}{k}\right\rfloor,\] are odd. (For a real number $x$, $\lfloor x\rfloor$ denotes the largest integer not exceeding $x$.) Proposed by Hong Kong