IMO 2008 Problem 1a: Euclidean-geometry translation
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Let H be the orthocenter of triangle ABC and D,E,F the midpoints of BC, CA, AB. The circles centered at D,E,F through H meet those respective sidelines in A₁,A₂; B₁,B₂; C₁,C₂. Prove C₁,C₂,B₁,B₂ are concyclic.
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 isolates one four-point cyclicity consequence of a shared orthocenter and three midpoint circles.
What counts as progress: A proof of the specified two-pair cyclicity under the translated construction; the other four-point target is a separate item.
How the problem developed
International Mathematical Olympiad The original contest statement was IMO 2008 P1. IMO-AG-30 represents that one contest problem with separate formal targets P1a and P1b; this record tracks target P1a, not a separately numbered contest question.
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. The paper includes the benchmark proof in its supplement.
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 2008 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