IMO 2007 Problem 4: Euclidean-geometry translation
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In triangle ABC with circumcenter O, take R on the circumcircle with RA=RB. Let L and K be the midpoints of CA and CB. Define P=OK∩CR and Q=OL∩CR, and let L₁,K₁ be the perpendicular feet of L,K on CR. Prove KK₁/LL₁=RQ/RP.
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: It asks for an exact link between projected midpoint distances and intersections with circumcenter lines.
What counts as progress: A ratio proof with the translated circle and projection hypotheses and all required nonzero denominators.
How the problem developed
International Mathematical Olympiad Published the original IMO 2007 Problem 4 geometry problem at the annual contest. The AG30 file later gives its specialized translated statement.
Trieu H. Trinh and coauthors Included a translated version in the 30-problem AlphaGeometry benchmark; the study reports 25/30 solved overall, which does not identify this task’s individual result.
- 2025 · Problem level benchmark [4]
Yuri Chervonyi and coauthors Figure 8 of the AlphaGeometry 2 study lists the original IMO 2007 Problem 4 as solved by DDAR; for Problem 2, the chart does not separate the translated 2a and 2b tasks.
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 2007 official problem statement booklet
- Google DeepMind, AlphaGeometry 30 translated problem statements
- AlphaGeometry study and benchmark results
- Chervonyi et al., Gold-medalist Performance in Solving Olympiad Geometry with AlphaGeometry2
- Trinh and collaborators: Solving olympiad geometry without human demonstrations
- AlphaGeometry source and proof artifacts
- DeepMind account of computer checking
- Precise translated question and assumptions
Website timeline
Publication dates and editorial changes.
AlphaGeometry solves 25 translated Olympiad geometry problems