CH-SMO 2023

Let $ABC$ be an acute triangle with incenter $I$ . On its circumcircle, let $M_A$ , $M_B$ and $M_C$ be the midpoints of minor arcs $BC, CA$ and $AB$ , respectively. Prove that the reflection $M_A$ over the line $IM_B$ lies on the circumcircle of the triangle $IM_BM_C$ .

The wizard Albus and Brian are playing a game on a square of side length $2n+1$ meters surrounded by lava. In the centre of the square there sits a toad. In a turn, a wizard chooses a direction parallel to a side of the square and enchants the toad. This will cause the toad to jump $d$ meters in the chosen direction, where $d$ is initially equal to $1$ and increases by $1$ after each jump. The wizard who sends the toad into the lava loses. Albus begins and they take turns. Depending on $n$ , determine which wizard has a winning strategy.

Let $x,y$ and $a_0, a_1, a_2, \cdots$ be integers satisfying $a_0 = a_1 = 0$ , and $$a_{n+2} = xa_{n+1}+ya_n+1$$ for all integers $n \geq 0$ . Let $p$ be any prime number. Show that $\gcd(a_p,a_{p+1})$ is either equal to $1$ or greater than $\sqrt{p}$ .

Determine the smallest possible value of the expression $$\frac{ab+1}{a+b}+\frac{bc+1}{b+c}+\frac{ca+1}{c+a}$$ where $a,b,c \in \mathbb{R}$ satisfy $a+b+c = -1$ and $abc \leqslant -3$

Let $D$ be the set of real numbers excluding $-1$ . Find all functions $f: D \to D$ such that for all $x,y \in D$ satisfying $x \neq 0$ and $y \neq -x$ , the equality $$(f(f(x))+y)f \left(\frac{y}{x} \right)+f(f(y))=x$$ holds.