CH-SMO 2021

Let $(m,n)$ be pair of positive integers. Julia has carefully planted $m$ rows of $n$ dandelions in an $m \times n$ array in her back garden. Now, Jana un Viviane decides to play a game with a lawnmower they just found. Taking alternating turns and starting with Jana, they can now mow down all the dandelions in a straight horizontal or vertical line (and they must mow down at least one dandelion ). The winner is the player who mows down the final dandelion. Determine all pairs of $(m,n)$ for which Jana has a winning strategy.

Let $\triangle ABC$ be an acute triangle with $AB =AC$ and let $D$ be a point on the side $BC$ . The circle with centre $D$ passing through $C$ intersects $\odot(ABD)$ at points $P$ and $Q$ , where $Q$ is the point closer to $B$ . The line $BQ$ intersects $AD$ and $AC$ at points $X$ and $Y$ respectively. Prove that quadrilateral $PDXY$ is cyclic.

Find all finite sets $S$ of positive integers with at least $2$ elements, such that if $m>n$ are two elements of $S$ , then $$\frac{n^2}{m-n}$$ is also an element of $S$ .

For which integers $n \ge 2$ can we arrange numbers $1,2, \ldots, n$ in a row, such that for all integers $1 \le k \le n$ the sum of the first $k$ numbers in the row is divisible by $k$ ?

Let $\mathbb{N}$ be the set of positive integers. Let $f: \mathbb{N} \rightarrow \mathbb{N}$ be a function such that for every positive integer $n \in \mathbb{N}$ $$f(n) -n<2021 \text{and} f^{f(n)}(n) =n$$ Prove that $f(n)=n$ for infinitely many $n \in \mathbb{N}$

Let $m \ge n$ be positive integers. Frieder is given $mn$ posters of Linus with different integer dimensions of $k \times l$ with $1 \ge k \ge m$ and $1 \ge l \ge n$ . He must put them all up one by one on his bedroom wall without rotating them. Every time he puts up a poster, he can either put it on an empty spot on the wall or on a spot where it entirely covers a single visible poster and does not overlap any other visible poster. Determine the minimal area of the wall that will be covered by posters.

Let $\triangle ABC$ be a triangle with $AB =AC$ and $\angle BAC = 20^{\circ}$ . Let $D$ be point on the side $AB$ such that $\angle BCD = 70^{\circ}$ . Let $E$ be point on the side $AC$ such that $\angle CBE = 60^{\circ}$ . Determine the value of angle $\angle CDE$ .