IMO 2003 Problem 4: Euclidean-geometry translation
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For triangle ABC with circumcircle Ω, take B₁,D₁ on Ω equidistant from C and A. Let X=BB₁∩AC, and let D be the other intersection of XD₁ with Ω. Let P,Q,R be the perpendicular projections of D onto BC, CA, AB. Prove PQ=QR.
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: The problem asks for a metric relation among three pedal points arising from a constrained cyclic construction.
What counts as progress: A derivation of the equal pedal lengths for the specialized translated configuration, rather than for an unrestricted point D.
How the problem developed
International Mathematical Olympiad Published the original IMO 2003 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 2003 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 2003 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