CMO 2022

Assume that real numbers $a$ and $b$ satisfy

$$a b+\sqrt{a b+1}+\sqrt{a^{2}+b} \cdot \sqrt{b^{2}+a}=0$$

Find, with proof, the value of

$$a \sqrt{b^{2}+a}+b \sqrt{a^{2}+b}$$

Let $d(k)$ denote the number of positive integer divisors of $k$. For example, $d(6)=4$ since 6 has 4 positive divisors, namely, $1,2,3$, and 6 . Prove that for all positive integers $n$,

$$d(1)+d(3)+d(5)+\cdots+d(2 n-1) \leq d(2)+d(4)+d(6)+\cdots+d(2 n)$$

Let $n \geq 2$ be an integer. Initially, the number 1 is written $n$ times on a board. Every minute, Vishal picks two numbers written on the board, say $a$ and $b$, erases them, and writes either $a+b$ or $\min \left\{a^{2}, b^{2}\right\}$. After $n-1$ minutes there is one number left on the board. Let the largest possible value for this final number be $f(n)$. Prove that

$$2^{n / 3}<f(n) \leq 3^{n / 3}$$

Let $n$ be a positive integer. A set of $n$ distinct lines divides the plane into various (possibly unbounded) regions. The set of lines is called "nice" if no three lines intersect at a single point. A "colouring" is an assignment of two colours to each region such that the first colour is from the set $\left\{A_{1}, A_{2}\right\}$, and the second colour is from the set $\left\{B_{1}, B_{2}, B_{3}\right\}$. Given a nice set of lines, we call it "colourable" if there exists a colouring such that

1). no colour is assigned to two regions that share an edge; 2). for each $i \in\{1,2\}$ and $j \in\{1,2,3\}$ there is at least one region that is assigned with both $A_{i}$ and $B_{j}$.

Determine all $n$ such that every nice configuration of $n$ lines is colourable.

Let $A B C D E$ be a convex pentagon such that the five vertices lie on a circle and the five sides are tangent to another circle inside the pentagon. There are $\binom{5}{3}=10$ triangles which can be formed by choosing 3 of the 5 vertices. For each of these 10 triangles, mark its incenter. Prove that these 10 incenters lie on two concentric circles.