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Ill-defined problems (IDPs) are pervasive in science and engineering, where solutions may fail to exist, be non-unique, or lack a meaningful interpretation. Despite recent advances in large language models (LLMs) in mathematical reasoning, their ability to reason about IDPs remains poorly understood. To address this gap, we introduce Ill-Defined Math (IDM), a benchmark of 1,300 mathematically ill-defined problems whose ill-definedness stems from fundamental mathematical flaws. IDM consists of a test set of 300 expert-constructed and verified problems, along with a larger training set generated via a verified multi-agent pipeline. We further propose a fine-grained, three-stage LLM judge that evaluates responses from three complementary perspectives: final answer statements, issue recognition, and issue fixing. Evaluations on recent LLMs expose clear performance gaps between well-defined and ill-defined problems. Moreover, models frequently recognize ill-definedness during reasoning yet still produce definitive final answers, revealing a systematic disconnect between issue recognition and faithful response behavior. Project page: \url{https://idmath.github.io}