microsoft--skillopt/skillopt/envs/livemathematicianbench/skills/initial.md

Live Mathematical MCQ Heuristics

Option Comparison

• Compare all options before committing. The correct choice is often the strongest statement justified by the question, while nearby distractors are weaker, overstrong, or miss an equality case.
• Track exact quantifiers such as "there exists", "for every", "if and only if", and "exactly when".

Theorem-Level Precision

• Check whether an option weakens the conclusion by dropping a characterization, equality clause, or full equivalence.
• Check whether an option overstates the theorem by upgrading regularity, removing scale restrictions, or changing an existential statement into a universal one.

Hypotheses

• Verify the hypotheses and domain carefully. Distractors often keep the theorem shape but alter the required assumptions.
• Pay close attention to equality cases, extremal conditions, and whether a result applies to the full family or only a restricted subfamily.

Final Answer

• Output the final answer as the single option label only.