66th International Mathematical Olympiad (IMO) 2025 - Day 1 Questions

Problem 1

A line in the plane is called sunny if it is not parallel to any of the x-axis, the y-axis, or the line $x + y = 0$. Let $n ≥ 3$ be a given integer. Determine all nonnegative integers k such that there exist n distinct lines in the plane satisfying both conditions:

• For all positive integers a and b with $a + b ≤ n + 1$, the point (a,b) lies on at least one of the lines
• Exactly k of the n lines are sunny

Problem 2

Let Ω and Γ be circles with centres M and N, respectively, such that the radius of Ω is less than the radius of Γ.

Suppose Ω and Γ intersect at two distinct points A and B. Line MN intersects Ω at C and Γ at D, so that C, M, N, D lie on MN in that order. Let P be the circumcentre of triangle ACD. Line AP meets Ω again at E ≠ A and meets Γ again at F ≠ A. Let H be the orthocentre of triangle PMN.

Prove that the line through H parallel to AP is tangent to the circumcircle of triangle BEF.

Problem 3

Let ℕ denote the set of positive integers. A function f : ℕ → ℕ is said to be bonza if $f(a) | bᵃ$ - f(b)f(a) for all positive integers a and b.

Determine the smallest real constant c such that $f(n) ≤ cn$ for all bonza functions f and all positive integers n.