MK-MO 2022

Let $(x_n)_{n=1}^\infty$ be a sequence defined recursively with: $x_1=2$ and $x_{n+1}=\frac{x_n(x_n+n)}{n+1}$ for all $n \ge 1$ . Prove that $$n(n+1) >\frac{(x_1+x_2+ \ldots +x_n)^2}{x_{n+1}}.$$ Proposed by Nikola Velov

Let $ABCD$ be cyclic quadrilateral and $E$ the midpoint of $AC$ . The circumcircle of $\triangle CDE$ intersect the side $BC$ at $F$ , which is different from $C$ . If $B'$ is the reflection of $B$ across $F$ , prove that $EF$ is tangent to the circumcircle of $\triangle B'DF$ .

Proposed by Nikola Velov

The sequence $(a_n)_{n \ge 1}^\infty$ is given by: $a_1=2$ and $a_{n+1}=a_n^2+a_n$ for all $n \ge 1$ . For an integer $m \ge 2$ , $L(m)$ denotes the greatest prime divisor of $m$ . Prove that there exists some $k$ , for which $L(a_k) > 1000^{1000}$ .

Proposed by Nikola Velov

Sofia and Viktor are playing the following game on a $2022 \times 2022$ board:

• Firstly, Sofia covers the table completely by dominoes, no two are overlapping and all are inside the table;
• Then Viktor without seeing the table, chooses a positive integer $n$ ;
• After that Viktor looks at the table covered with dominoes, chooses and fixes $n$ of them;
• Finally, Sofia removes the remaining dominoes that aren't fixed and tries to recover the table with dominoes differently from before. If she achieves that, she wins, otherwise Viktor wins. What is the minimum number $n$ for which Viktor can always win, no matter the starting covering of dominoes.

Proposed by Viktor Simjanoski

An acute $\triangle ABC$ with circumcircle $\Gamma$ is given. $I$ and $I_a$ are the incenter and $A-$ excenter of $\triangle ABC$ respectively. The line $AI$ intersects $\Gamma$ again at $D$ and $A'$ is the antipode of $A$ with respect to $\Gamma$ . $X$ and $Y$ are point on $\Gamma$ such that $\angle IXD = \angle I_aYD = 90^\circ$ . The tangents to $\Gamma$ at $X$ and $Y$ intersect in point $Z$ . Prove that $A', D$ and $Z$ are collinear.

Proposed by Nikola Velov