Livemathbench

LiveMathematicianBench

Each question presents a mathematical theorem statement and asks you to choose the correct answer from options A through E. Reason carefully about the mathematical content, then respond with only the single letter (A, B, C, D, or E) of the correct option -- no explanation, no restating the option text.

General strategies

1. Identify the strongest statement.

   • The correct answer is often the strongest (most precise, most general, most informative) claim among the options that is still provably true.
   • Compare options carefully: look for differences in quantifiers (e.g., “for every” vs. “there exists”), bounds (tight vs. loose), additional conditions, and conclusions.
   • If an option says “One of the remaining options is correct, but a stronger result can be proven,” consider whether it is indeed the strongest among the given choices.

2. Watch for subtle logical distinctions.

   • Pay close attention to logical connectors (“and”, “or”, “if … then”), quantifier order (“for every … there exists” vs. “there exists … for every”), and scope of assumptions.
   • Distinguish between “converges” and “converges uniformly,” “exists” and “exists uniquely,” “O(·)” and “Θ(·),” “almost all” and “all.”

3. Check for consistency with known theory.

   • Use familiarity with standard results (e.g., compactness theorems, classification theorems, asymptotic bounds) to eliminate options that are too weak or obviously false.
   • When options include explicit constants or rates, consider whether they match known extremal examples or sharp bounds.

4. Handle “equivalent” questions.

   • If the question asks for an equivalent formulation, verify that the chosen option is logically equivalent to the given conditions, not just a consequence.
   • Look for bidirectional implications; avoid options that are only necessary or only sufficient.

5. Avoid over-interpreting.

   • Do not read extra assumptions into the problem statement. Use only what is given.
   • Do not rely on task-specific facts from the training trajectories; focus on the mathematical reasoning in the current question.

6. When in doubt, choose the most informative true statement.

   • If multiple options seem plausible, pick the one that gives the most precise description (e.g., exact constant, exact asymptotic, full classification).
   • If an option claims a stronger result than others and appears mathematically sound, it is likely correct.