OSN 2022

Determine all functions $f: \mathbb{R} \to \mathbb{R}$ such that for any $x,y \in \mathbb{R}$ we have $$f(f(f(x)) + f(y)) = f(y) - f(x)$$

Let $P(x)$ be a polynomial with integer coefficient such that $P(1) = 10$ and $P(-1) = 22$ . (a) Give an example of $P(x)$ such that $P(x) = 0$ has an integer root. (b) Suppose that $P(0) = 4$ , prove that $P(x) = 0$ does not have an integer root.

Let $ABCD$ be a rectangle. Points $E$ and $F$ are on diagonal $AC$ such that $F$ lies between $A$ and $E$ ; and $E$ lies between $C$ and $F$ . The circumcircle of triangle $BEF$ intersects $AB$ and $BC$ at $G$ and $H$ respectively, and the circumcircle of triangle $DEF$ intersects $AD$ and $CD$ at $I$ and $J$ respectively. Prove that the lines $GJ, IH$ and $AC$ concur at a point.

Given a regular $26$ -gon. Prove that for any $9$ vertices of that regular $26$ -gon, then there exists three vertices that forms an isosceles triangle.

Let $N\ge2$ be a positive integer. Given a sequence of natural numbers $a_1,a_2,a_3,\dots,a_{N+1}$ such that for every integer $1\le i\le j\le N+1$ , $$a_ia_{i+1}a_{i+2}\dots a_j \not\equiv1\mod{N}$$ Prove that there exist a positive integer $k\le N+1$ such that $\gcd(a_k, N) \neq 1$

In a triangle $ABC$ , $D$ and $E$ lies on $AB$ and $AC$ such that $DE$ is parallel to $BC$ . There exists point $P$ in the interior of $BDEC$ such that $$\angle BPD = \angle CPE = 90^{\circ}$$Prove that the line $AP$ passes through the circumcenter of triangles $EPD$ and $BPC$ .

Let $A$ be the sequence of zeroes and ones (binary sequence). The sequence can be modified by the following operation: we may pick a block or a contiguous subsequence where there are an unequal number of zeroes and ones, and then flip their order within the block (so block $a_1, a_2, \ldots, a_r$ becomes $a_r, a_{r-1}, \ldots, a_1$ ).

As an example, let $A$ be the sequence $1,1,0,0,1$ . We can pick block $1,0,0$ and flip it, so the sequence $1,\boxed{1,0,0},1$ becomes $1,\boxed{0,0,1},1$ . However, we cannot pick block $1,1,0,0$ and flip their order since they contain the same number of $1$ s and $0$ s.

Two sequences $A$ and $B$ are called related if $A$ can be transformed into $B$ using a finite number the operation mentioned above.

Determine the largest natural number $n$ for which there exists $n$ different sequences $A_1, A_2, \ldots, A_n$ where each sequence consists of 2022 digits, and for every index $i \neq j$ , the sequence $A_i$ is not related to $A_j$ .

Determine the smallest positive real $K$ such that the inequality $$K + \frac{a + b + c}{3} \ge (K + 1) \sqrt{\frac{a^2 + b^2 + c^2}{3}}$$holds for any real numbers $0 \le a,b,c \le 1$ .

Proposed by Fajar Yuliawan, Indonesia