UDET: A Dataset for Detecting Contamination on Competition Math via Underdetermined Problems

Mathematical problem-solving is treated as a proxy for the analytical skills of large language models, yet current benchmarks cannot determine whether a model is genuinely reasoning from first principles, leaning on common mathematical priors, or simply returning a memorized answer. We study what models do when a problem is underdetermined. We represent each problem as a directed acyclic dependency graph (DAG) and remove load-bearing nodes (a given value) or edges (a constraint), producing an underdetermined variant whose answer can no longer be derived from the surviving text. We evaluate 11 models spanning open and closed systems and observe that, under a forced-answer protocol, models recover the original answer in only 181 of 2,728 responses. We then hand-label the mechanism behind every one of the 181 successes, using a four-way taxonomy applied through blind dual annotation and adjudication, and find that 117 represent memorization and only 49 are reconstruction, the rest whose mechanism could not be confidently ascertained were labelled as undetermined. Within memorization, we define two sub classes namely Exact Instance Recall and Prior Math Recall the first category contains traces, which we believe exhibit a pure form of memorization, where the model must have memorized the exact instance of the problem, the second category containts traces where the behaviour still shows memorization, but the model references older problems or math priors in the trace, making it harder to attribute memorization to a single source. Beyond classifying the successes, we analyze model behavior under ambiguity, documenting a tendency, under forced-answer conditions, to fabricate supporting derivations that justify a committed answer. We will release the DAG pipeline, the dataset of underdetermined problems, and all reasoning traces with human labels upon acceptance.