This category tests algebraic rigour rather than numeric approximation. The model must solve a system of non-linear equations over the reals, enumerate every solution pair, and justify each step clearly enough that a reader could verify the derivation.
The scoring rubric penalises missing roots, spurious roots, and hand-wavy transitions. A high score requires the model to treat the task as a proof-like explanation, not just a computation.
Common failure modes
- Only finding the positive roots because the model implicitly assumes symmetry.
- Introducing extraneous solutions when squaring both sides without checking them.
- Stopping at a parametrisation instead of listing explicit pairs.
Task: Equation Solver
Solve the system and . Mathematical correctness, completeness of roots, and clarity of derivation determine the score.