2022 EGMO Problems

Day 1

Problem 1

Let $ABC$ be an acute-angled triangle in which $BC<AB$ and $BC<CA$ . Let point $P$ lie on segment $AB$ and point $Q$ lie on segment $AC$ such that $P \neq B$ , $Q \neq C$ and $BQ = BC = CP$ . Let $T$ be the circumcenter of triangle $APQ$ , $H$ the orthocenter of triangle $ABC$ , and $S$ the point of intersection of the lines $BQ$ and $CP$ . Prove that $T$ , $H$ , and $S$ are collinear.

Solution

Problem 2

Let $\mathbb{N}=\{1, 2, 3, \dots\}$ be the set of all positive integers. Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that for any positive integers $a$ and $b$ , the following two conditions hold: (i) $f(ab) = f(a)f(b)$ , and

(ii) at least two of the numbers $f(a)$ , $f(b)$ , and $f(a+b)$ are equal.

Solution

Problem 3

An infinite sequence of positive integers $a_1, a_2, \dots$ is called $good$ if

(i) $a_1$ is a perfect square, and

(ii) for any integer $n \ge 2$ , $a_n$ is the smallest positive integer such that $na_1 + (n-1)a_2 + \dots + 2a_{n-1} + a_n$ is a perfect square.

Prove that for any good sequence $a_1, a_2, \dots$ , there exists a positive integer $k$ such that $a_n=a_k$ for all integers $n \ge k$ .

Solution

Day 2

Problem 4

Given a positive integer $n \ge 2$ , determine the largest positive integer $N$ for which there exist $N+1$ real numbers $a_0, a_1, \dots, a_N$ such that $(1)$ $a_0+a_1 = -\frac{1}{n},$ and $(2)$ $(a_k+a_{k-1})(a_k+a_{k+1})=a_{k-1}-a_{k+1}$ for $1 \le k \le N-1$ .

Solution

Problem 5

For all positive integers $n$ , $k$ , let $f(n, 2k)$ be the number of ways an $n \times 2k$ board can be fully covered by $nk$ dominoes of size $2 \times 1$ . (For example, $f(2, 2)=2$ and $f(3, 2)=3$ .) Find all positive integers $n$ such that for every positive integer $k$ , the number $f(n, 2k)$ is odd.

Solution

Problem 6

Let $ABCD$ be a cyclic quadrilateral with circumcenter $O$ . Let the internal angle bisectors at $A$ and $B$ meet at $X$ , the internal angle bisectors at $B$ and $C$ meet at $Y$ , the internal angle bisectors at $C$ and $D$ meet at $Z$ , and the internal angle bisectors at $D$ and $A$ meet at $W$ . Further, let $AC$ and $BD$ meet at $P$ . Suppose that the points $X$ , $Y$ , $Z$ , $W$ , $O$ , and $P$ are distinct.