IMO 2024 Problems

Determine all real numbers such that, for every positive integer , the integer is a multiple of .

Determine all pairs of positive integers for which there exist positive integers and such that holds for all integers .

Let , , , be an infinite sequence of positive integers, and let be a positive integer. Suppose that, for each , is equal to the number of times appears in the list .

Prove that at least one of the sequences , , , and , , , is eventually periodic.

Let be a triangle with . Let the incenter and the incircle of triangle be and , respectively. Let be a point on line , different from such that the line through and parallel to is tangent to . Similarly, let be a point on line different from such that the line through and parallel to is tangent to . Let intersect the circumcircle of triangle at . Let and be the midpoints of and , respectively.

Prove that .

Turbo the snail plays a game on a board with rows and columns. There are hidden monsters in of the cells. Initially, Turbo does not know where any of the monsters are, but he knows that there is exactly one monster in each row except the first row and the last row, and that each column contains at most one monster.

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 adjacent cell sharing a common side. (He is allowed to return to a previously visited cell.) If he 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, 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 the game is over.

Determine the minimum value of for which Turbo has a strategy that guarantees reaching the last row on the -th attempt or earlier, regardless of the locations of the monsters.

Let be the set of rational numbers. A function is called aquaesulian if the following property holds: for every , Show that there exists an integer such that for any aquaesulian function there are at most different rational numbers of the form for some rational number , and find the smallest possible value of .