dapo-math-base

# DAPO-Math Base Skill Solve one DAPO-Math prompt carefully, efficiently, and in the required answer format. Treat the task as a contest-style math problem with hidden evaluation against a reward-model ground truth. Optimize for a correct final answer, not for lengthy exposition. ## Required Output Contract End the final response with exactly one standalone line: ```text Answer: <final answer> ``` Use the simplest equivalent form for `<final answer>`: - Use an integer for integer answers. - Use a reduced fraction like `17/12` for rational answers. - Use exact symbolic forms when needed, such as `sqrt(3)`, `2*pi`, or a compact expression. - Avoid trailing punctuation on the answer line. - Do not include multiple competing final answers. ## Core Workflow 1. **Classify the problem.** Identify whether the prompt is mainly algebra, number theory, combinatorics, probability, geometry, sequences, inequalities, or optimization. 2. **Extract givens and the target.** Rewrite the key quantities, constraints, and requested output before calculating. 3. **Choose the shortest reliable method.** Prefer invariants, symmetry, counting complements, modular arithmetic, factorization, recurrence simplification, or direct algebra over brute-force reasoning. 4. **Carry exact quantities.** Avoid decimal approximations unless the final answer is explicitly numerical or approximation-based. 5. **Check edge cases.** Verify boundary values, positivity/integrality constraints, parity, divisibility, and whether objects are counted once or multiple times. 6. **Sanity-check the result.** Substitute the answer back into the original conditions when possible. For counting/probability, confirm totals and denominators. For geometry, confirm units and scale. 7. **Finish with the required answer line.** Provide concise reasoning above it, then the final `Answer: ...` line. ## Problem-Type Tactics ### Number Theory - Factor early. For expressions involving divisors, gcd, lcm, valuations, or unit fractions, write prime factorizations and track exponents. - Use p-adic valuations to control divisibility and denominators. - For modular constraints, reduce to the smallest relevant modulus and check invertibility before dividing. - For divisor counts or sums, separate independent prime exponents. - For Diophantine conditions, derive bounds before enumerating cases. ### Algebra and Equations - Normalize equations before expanding: move all terms to one side, factor if possible, and look for substitutions. - Preserve domain restrictions from square roots, logarithms, denominators, and inverse functions. - For symmetric polynomials, switch to sums/products or known identities. - For systems, eliminate variables in the order that keeps expressions smallest. - After solving, reject extraneous roots introduced by squaring or multiplying by variable expressions. ### Combinatorics and Probability - Define the sample space explicitly before counting favorable cases. - Prefer complements when forbidden configurations are simpler. - Use bijections or recursive states when direct enumeration becomes messy. - Check whether order matters, whether repetition is allowed, and whether rotations/reflections are considered identical. - For probability, reduce the final fraction. ### Geometry - Draw a mental diagram and name key points, lengths, and angles. - Look for similarity, cyclic quadrilaterals, power of a point, angle bisectors, area ratios, and coordinate placements. - Choose coordinates when the diagram has perpendiculars, circles, or many length constraints. - Use vectors or complex numbers only when they shorten the solution. - Verify that the final answer has the requested dimension: length, area, angle, ratio, or count. ### Sequences, Recurrences, and Functional Patterns - Compute a few initial terms to identify structure, then prove or justify the pattern. - Solve linear recurrences with characteristic roots when applicable. - For floor/ceiling problems, split into intervals where the expression is constant or monotone. - For iterative definitions, search for invariants, fixed points, or telescoping expressions. ## Arithmetic and Verification Discipline - Keep fractions exact and reduce only when helpful. - Label intermediate variables clearly. - Recalculate the final arithmetic independently once before finishing. - If terminal access is available, use Python only for exact arithmetic, symbolic checks, or bounded enumeration; do not rely on floating-point approximations for proof. - If terminal access is unavailable, keep computations small and structured enough to audit manually. ## DAPO-Specific Notes - The prompt already contains the required final-answer instruction. Follow it exactly even if the system prompt repeats it. - The hidden scorer usually compares the extracted final answer to `reward_model.ground_truth`; equivalent simple forms are safest. - Avoid saying that the ground truth is known. Derive the result from the prompt. - Prefer a complete but compact solution. Long exploratory text increases risk of contradictions and token cost. - If the problem is difficult, still return the best derived answer with the required `Answer:` line rather than ending without a parseable answer. ## Final Response Template Use this structure unless the problem requires an unusual format: ```text [Concise derivation with exact arithmetic and a quick check.] Answer: <final answer> ```