2022 Canada National Olympiad

1

Assume that real numbers $a$ and $b$ satisfy $$ab+\sqrt{ab+1}+\sqrt{a^2+b}\sqrt{a+b^2}=0.$$Find, with proof, the value of $$b\sqrt{a^2+b}+a\sqrt{b^2+a}.$$

2

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 $n$, $$d(1) + d(3) + d(5) +...+ d(2n - 1)\le d(2) + d(4) + d(6) + ... + d(2n).$$

3

Vishal starts with $n$ copies of the number $1$ written on the board. Every minute, he takes two numbers $a, b$ and replaces them with either $a+b$ or $\min(a^2, b^2)$. After $n-1$ there is $1$ number on the board. Let the maximal possible value of this number be $f(n)$. Prove $2^{n/3}<f(n)\leq 3^{n/3}$.

4

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 $\{A_1, A_2\}$, and the second colour is from the set $\{B_1, B_2, B_3\}$. Given a nice set of lines, we call it “colourable” if there exists a colouring such that (a) no colour is assigned to two regions that share an edge; (b) 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.

5

Let $ABCDE$ 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 ${5 \choose 3}= 10$ 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.