OFM 2021

Let $R$ and $S$ be the numbers defined by $$R = \dfrac{1}{2} \times \dfrac{3}{4} \times \dfrac{5}{6} \times \cdots \times \dfrac{223}{224} \text{ and } S = \dfrac{2}{3} \times \dfrac{4}{5} \times \dfrac{6}{7} \times \cdots \times \dfrac{224}{225}.$$Prove that $R < \dfrac{1}{15} < S$ .

Evariste has drawn twelve triangles as follows, so that two consecutive triangles share exactly one edge. Sophie colors every triangle side in red, green or blue. Among the $3^{24}$ possible colorings, how many have the property that every triangle has one edge of each color?

Every point in the plane was colored in red or blue. Prove that one the two following statements is true: $\bullet$ there exist two red points at distance $1$ from each other; $\bullet$ there exist four blue points $B_1$ , $B_2$ , $B_3$ , $B_4$ such that the points $B_i$ and $B_j$ are at distance $|i - j|$ from each other, for all integers $i$ and $j$ such as $1 \le i \le 4$ and $1 \le j \le 4$ .

Let $\mathbb{N}_{\geqslant 1}$ be the set of positive integers. Find all functions $f \colon \mathbb{N}_{\geqslant 1} \to \mathbb{N}_{\geqslant 1}$ such that, for all positive integers $m$ and $n$ : $$\mathrm{GCD}\left(f(m),n\right) + \mathrm{LCM}\left(m,f(n)\right) = \mathrm{GCD}\left(m,f(n)\right) + \mathrm{LCM}\left(f(m),n\right).$$

Note: if $a$ and $b$ are positive integers, $\mathrm{GCD}(a,b)$ is the largest positive integer that divides both $a$ and $b$ , and $\mathrm{LCM}(a,b)$ is the smallest positive integer that is a multiple of both $a$ and $b$ .

Let $a_1,a_2,a_3,\ldots$ and $b_1,b_2,b_3,\ldots$ be positive integers such that $a_{n+2} = a_n + a_{n+1}$ and $b_{n+2} = b_n + b_{n+1}$ for all $n \ge 1$ . Assume that $a_n$ divides $b_n$ for infinitely many values of $n$ . Prove that there exists an integer $c$ such that $b_n = c a_n$ for all $n \ge 1$ .

Albert and Beatrice play a game. $2021$ stones lie on a table. Starting with Albert, they alternatively remove stones from the table, while obeying the following rule. At the $n$ -th turn, the active player (Albert if $n$ is odd, Beatrice if $n$ is even) can remove from $1$ to $n$ stones. Thus, Albert first removes $1$ stone; then, Beatrice can remove $1$ or $2$ stones, as she wishes; then, Albert can remove from $1$ to $3$ stones, and so on. The player who removes the last stone on the table loses, and the other one wins. Which player has a strategy to win regardless of the other player's moves?

Let $ABCD$ be a square with incircle $\Gamma$ . Let $M$ be the midpoint of the segment $[CD]$ . Let $P \neq B$ be a point on the segment $[AB]$ . Let $E \neq M$ be the point on $\Gamma$ such that $(DP)$ and $(EM)$ are parallel. The lines $(CP)$ and $(AD)$ meet each other at $F$ . Prove that the line $(EF)$ is tangent to $\Gamma$

Let $\mathbb{N}_{\ge 1}$ be the set of positive integers. Find all functions $f \colon \mathbb{N}_{\ge 1} \to \mathbb{N}_{\ge 1}$ such that, for all positive integers $m$ and $n$ :

(a) $n = \left(f(2n)-f(n)\right)\left(2 f(n) - f(2n)\right)$ , (b) $f(m)f(n) - f(mn) = \left(f(2m)-f(m)\right)\left(2 f(n) - f(2n)\right) + \left(f(2n)-f(n)\right)\left(2 f(m) - f(2m)\right)$ , (c) $m-n$ divides $f(2m)-f(2n)$ if $m$ and $n$ are distinct odd prime numbers.