International Mathematical Olympiad · 2024

Problems

Problem 1Find all real numbers $\alpha$ so that, for every positive integer $n$ , the integer $\lfloor\alpha\rfloor+\lfloor2\alpha\rfloor+\lfloor3\alpha\rfloor+\cdots+\lfloor n\alpha\rfloor$ is divisible by $n$ .Solutions: 1
Problem 2For which pairs of positive integers $(a,b)$ is the sequence $\gcd(a^n+b,b^n+a)$ , $n=1,2,\ldots$ , eventually constant?Solutions: 1
Problem 3Let $a_1,a_2,a_3,\ldots$ be an infinite sequence of positive integers, and let $N$ be a positive integer. Suppose that, for each $n>N$ , the number $a_n$ is equal to the number of times $a_{n-1}$ appears in the list $(a_1,a_2,\ldots,a_{n-1})$ . Prove that at least one of the sequences $a_1,a_3,a_5,\ldots$ and $a_2,a_4,a_6,\ldots$ is eventually periodic.Solutions: 1
Problem 4Let triangle $ABC$ with incenter $I$ satisfy $AB<AC<BC$ . Let $X$ be a point on line $BC$ , different from $C$ , such that the line through $X$ parallel to $AC$ is tangent to the incircle. Similarly, let $Y$ be a point on line $BC$ , different from $B$ , such that the line through $Y$ parallel to $AB$ is tangent to the incircle. Line $AI$ intersects the circumcircle of triangle $ABC$ again at $P$ . Let $K$ and $L$ be the midpoints of $AC$ and $AB$ , respectively. Prove that $\angle KIL+\angle YPX=180^\circ$ .Solutions: 1
Problem 5Turbo the snail is in the top row of a grid with 2024 rows and 2023 columns and wants to get to the bottom row. However, there are 2022 hidden monsters, one in every row except the first and last, with no two monsters in the same column. Turbo makes a series of attempts to go from the first row to the last row. On each attempt, he chooses to start on any cell in the first row, then repeatedly moves to an orthogonal neighbor. (He is allowed to return to a previously visited cell.) If Turbo reaches a cell with a monster, his attempt ends and he is transported back to the first row to start a new attempt. The monsters do not move between attempts, and Turbo remembers whether or not each cell he has visited contains a monster. If he reaches any cell in the last row, his attempt ends and Turbo wins. Find the smallest integer $n$ such that Turbo has a strategy which guarantees being able to reach the bottom row in at most $n$ attempts, regardless of how the monsters are placed.Solutions: 1
Problem 6A function $f:\mathbb Q\to\mathbb Q$ is called aquaesulian if the following property holds: for every $x,y\in\mathbb Q$ , $f(x+f(y))=f(x)+y$ or $f(f(x)+y)=x+f(y)$ . Show that there exists an integer $c$ such that for any aquaesulian function $f$ there are at most $c$ different rational numbers of the form $f(r)+f(-r)$ for some rational number $r$ , and find the smallest possible value of $c$ .Solutions: 1