Improving LLM performance on olympiad-level mathematics through cognitive decomposition

Large Language Models often struggle with Olympiad-level mathematics due to logical errors in multi-step proofs. This paper evaluates the impact of cognitive decomposition on reasoning by comparing four architectures: two single-agent systems (Chain-of-Thought and Metacognitive) and two multi-agent systems (Sequential Refinement and a parallel Council of Specialists). Using the OlympiadBench dataset, our results demonstrate that both the Council of Specialists (65.45% accuracy) and the Metacognitive Single-Agent (64.12%) significantly outperform the CoT Single-Agent (55.15%). Critically, the performance difference between the multi-agent Council and the Metacognitive Single-Agent is not statistically significant. This establishes that explicit cognitive decomposition, specifically the structured separation of strategy, pattern seeking, and constraint analysis, is the primary driver of reasoning improvements, rather than multi-agent parallelism. Domain-level analysis reveals that the Sequential architecture remains competitive in Combinatorics. A leave-one-out ablation study further reveals that the Council's performance relies on distinct specialist roles: the Pattern Seeker is critical for Number Theory and Geometry, while the Constraint Analyst is critical for Algebra. We recommend the Metacognitive Single-Agent as the optimal architecture, as it delivers the robust reasoning benefits of a multi-agent council within a single prompt at less than half the computational cost.