AoPS Community 2025 International Mathematical Olympiad
IMO 2025
www.artofproblemsolving.com/community/c4391531
by Tintarn, sarjinius, KevinYang2.71, vsamc
Day 1 July 15, 2025
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 of the following:
- for all positive integers a and b with a + b ≤ n + 1, the point (a, b) lies on at least one of
the lines; and
- exactly k of the n lines are sunny.
Proposed by Linus Tang, USA
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 M N intersects
Ω at C and Γ at D, so that C, M, N, D lie on M N 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 P M N .
Prove that the line through H parallel to AP is tangent to the circumcircle of triangle BEF .
Proposed by Tran Quang Hung, Vietnam
3 Let N denote the set of positive integers. A function f : N → N is said to be bonza if
f (a) divides 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.
Proposed by Lorenzo Sarria, Colombia
Day 2 July 16, 2025
4 A proper divisor of a positive integer N is a positive divisor of N other than N itself.
The infinite sequence a1 , a2 , · · · consists of positive integers, each of which has at least three
proper divisors. For each n ≥ 1, the integer an+1 is the sum of the three largest proper divisors
of an .
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AoPS Community 2025 International Mathematical Olympiad
Determine all possible values of a1 .
5 Alice and Bazza are playing the inekoalaty game, a two-player game whose rules depend on a
positive real number λ which is known to both players. On the nth turn of the game (starting
with n = 1) the following happens:
- If n is odd, Alice chooses a nonnegative real number xn such that
x1 + x2 + · · · + xn ≤ λn.
-If n is even, Bazza chooses a nonnegative real number xn such that
x21 + x22 + · · · + x2n ≤ n.
If a player cannot choose a suitable xn , 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 λ for which Alice has a winning strategy and all those for which Bazza
has a winning strategy.
Proposed by Massimiliano Foschi and Leonardo Franchi, Italy
6 Answer is 2112
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