IMO 2015 Problem 3: Euclidean-geometry translation
Follow the latest progress by email.
Let H,O be the orthocenter and circumcenter of ABC, F=AH∩BC and M the midpoint of BC. Choose Q on the circumcircle with AQ perpendicular to QH, then K on the circumcircle with HK perpendicular to KQ. Let O₁,O₂ be the centers of circles KQH and FKM. Prove O₁,O₂,K are collinear.
This page concerns the specialized translated task used in the IMO-AG-30 benchmark. The study’s aggregate result does not establish whether this individual task was solved.
Why it matters and what counts as progress
Why it matters: This links two diameter-circle conditions to a relation between auxiliary circumcenters.
What counts as progress: A proof of the center collinearity for the translated choices of Q and K and the stated altitude foot.
How the problem developed
International Mathematical Olympiad The official archive lists this as IMO 2015 P3; it is one of the contest problems later encoded in the specialized IMO-AG-30 geometry language.
Trieu H. Trinh, Yuhuai Wu, Quoc V. Le, He He, and Thang Luong (AlphaGeometry) The Nature paper reports AlphaGeometry solved this translated benchmark statement using language-model auxiliary constructions and symbolic deduction/algebra. For the 2015 geometry problems, the authors submitted the solutions to the USA IMO team coach, who recommended full scores; this is an expert score evaluation, not a proof audit of the original wording.
Yuri Chervonyi, Trieu H. Trinh, Miroslav Olšák, Xiaomeng Yang, Hoang H. Nguyen, Marcelo Menegali, Junehyuk Jung, Junsu Kim, Vikas Verma, Quoc V. Le, and Thang Luong (AlphaGeometry2) The AlphaGeometry2 paper reports solving all 30 statements in the original IMO-AG-30 benchmark, which includes this formalized target. This is a system result on the benchmark encoding, not an independent review of the original contest wording.
AI contribution
AlphaGeometry solves 25 translated Olympiad geometry problems
AlphaGeometry solved 25/30 specialized IMO-AG-30 classical Euclidean geometry translations using learned auxiliary-point constructions plus a symbolic deduction and algebra engine; the paper does not claim all natural-language IMO geometry conditions are represented.
Sources
- IMO 2015 Problems (official archive)
- Trinh et al., Solving olympiad geometry without human demonstrations
- AlphaGeometry IMO-AG-30 formalized problem set
- Chervonyi et al., Gold-medalist Performance in Solving Olympiad Geometry with AlphaGeometry2
- AlphaGeometry2 paper, arXiv record
- Trinh and collaborators: Solving olympiad geometry without human demonstrations
- AlphaGeometry source and proof artifacts
- DeepMind account of computer checking
- Precise translated question and assumptions
- AlphaGeometry study and benchmark results
Website timeline
Publication dates and editorial changes.
AlphaGeometry solves 25 translated Olympiad geometry problems