IMO 2025 Problem 2: tangency in intersecting circles
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Circles Ω and Γ, centered at M and N, have different radii with Ω smaller, and meet at A,B. Their center line meets Ω at C and Γ at D in the order C,M,N,D. Let P be the circumcenter of ACD; AP meets Ω and Γ again at E,F. If H is the orthocenter of PMN, prove the line through H parallel to AP is tangent to the circumcircle of BEF.
Why it matters and what counts as progress
Why it matters: It links the centers of two intersecting circles to an unexpected tangent of a third circle through a circumcenter and orthocenter construction.
What counts as progress: A proof of the stated tangency for every admissible configuration, or a formal derivation with the circle intersections and point order retained.
How the problem developed
- 15 Jul 2025 · Contest problem [1]
IMO jury The problem was set on this day of the official IMO 2025 contest.
Google DeepMind and Gemini Deep Think DeepMind reported an officially graded five-of-six result (35/42); its published solution collection includes a proof for this problem.
AI contributions 2
Aristotle formalizes five IMO 2025 solutions
Harmonic reports that Aristotle generated machine-verified solutions for 2025 IMO Problems 1–5 from human-formalized problem statements, with the Lean proofs published.
Gemini Deep Think solves five IMO 2025 problems
An advanced Gemini Deep Think model produced natural-language solutions to five of the six 2025 International Mathematical Olympiad problems, earning 35 of 42 points under the official coordinators’ grading.
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