IMO 2025 Problems

A line in the plane is called sunny if it is not parallel to any of the -axis, the -axis, and the line .

Let be a given integer. Determine all nonnegative integers such that there exist distinct lines in the plane satisfying both of the following:

• for all positive integers and with , the point is on at least one of the lines; and
• exactly of the lines are sunny.

Let and be circles with centres and , respectively, such that the radius of is less than the radius of . Suppose circles and intersect at two distinct points and . Line intersects at and at , such that points and lie on the line in that order. Let be the circumcentre of triangle . Line intersects again at . Line intersects again at . Let be the orthocentre of triangle .

Prove that the line through parallel to is tangent to the circumcircle of triangle . (The orthocentre of a triangle is the point of intersection of its altitudes.)

Let denote the set of positive integers. A function is said to be bonza if for all positive integers and .

Determine the smallest real constant such that for all bonza functions and all positive integers .

A proper divisor of a positive integer is a positive divisor of other than itself.

The infinite sequence consists of positive integers, each of which has at least three proper divisors. For each , the integer is the sum of the three largest proper divisors of .

Determine all possible values of .

Alice and Bazza are playing the inekoalaty game, a two-player game whose rules depend on a positive real number which is known to both players. On the turn of the game (starting with ) the following happens:

• If is odd, Alice chooses a nonnegative real number such that
• If is even, Bazza chooses a nonnegative real number such that

If a player cannot choose a suitable number , the game ends and the other player wins. If the game goes on forever, neither player wins. All chosen numbers are known to both players.

Determine all values of for which Alice has a winning strategy and all those for which Bazza has a winning strategy.

Consider a grid of unit squares. Matilda wishes to place on the grid some rectangular tiles, possibly of different sizes, such that each side of every tile lies on a grid line and every unit square is covered by at most one tile.

Determine the minimum number of tiles Matilda needs to place so that each row and each column of the grid has exactly one unit square that is not covered by any tile.