IZhO 2022

Non-zero polynomials $P(x)$ , $Q(x)$ , and $R(x)$ with real coefficients satisfy the identities $$P(x) + Q(x) + R(x) = P(Q(x)) + Q(R(x)) + R(P(x)) = 0.$$ Prove that the degrees of the three polynomials are all even.

A ten-level $2$ -tree is drawn in the plane: a vertex $A_1$ is marked, it is connected by segments with two vertices $B_1$ and $B_2$ , each of $B_1$ and $B_2$ is connected by segments with two of the four vertices $C_1, C_2, C_3, C_4$ (each $C_i$ is connected with one $B_j$ exactly); and so on, up to $512$ vertices $J_1, \ldots, J_{512}$ . Each of the vertices $J_1, \ldots, J_{512}$ is coloured blue or golden. Consider all permutations $f$ of the vertices of this tree, such that (i) if $X$ and $Y$ are connected with a segment, then so are $f(X)$ and $f(Y)$ , and (ii) if $X$ is coloured, then $f(X)$ has the same colour. Find the maximum $M$ such that there are at least $M$ permutations with these properties, regardless of the colouring.

In parallelogram $ABCD$ with acute angle $A$ a point $N$ is chosen on the segment $AD$ , and a point $M$ on the segment $CN$ so that $AB = BM = CM$ . Point $K$ is the reflection of $N$ in line $MD$ . The line $MK$ meets the segment $AD$ at point $L$ . Let $P$ be the common point of the circumcircles of $AMD$ and $CNK$ such that $A$ and $P$ share the same side of the line $MK$ . Prove that $\angle CPM = \angle DPL$ .

In triangle $ABC$ , a point $M$ is the midpoint of $AB$ , and a point $I$ is the incentre. Point $A_1$ is the reflection of $A$ in $BI$ , and $B_1$ is the reflection of $B$ in $AI$ . Let $N$ be the midpoint of $A_1B_1$ . Prove that $IN > IM$ .

A polynomial $f(x)$ with real coefficients of degree greater than $1$ is given. Prove that there are infinitely many positive integers which cannot be represented in the form $$f(n+1)+f(n+2)+\cdots+f(n+k)$$ where $n$ and $k$ are positive integers.

Do there exist two bounded sequences $a_1, a_2,\ldots$ and $b_1, b_2,\ldots$ such that for each positive integers $n$ and $m>n$ at least one of the two inequalities $|a_m-a_n|>1/\sqrt{n},$ and $|b_m-b_n|>1/\sqrt{n}$ holds?