We present **AMO-Bench**, an **A**dvanced **M**athematical reasoning benchmark with **O**lympiad level or even higher difficulty, comprising 50 human-crafted problems. Existing benchmarks have widely leveraged high school math competitions for evaluating mathematical reasoning capabilities of large language models (LLMs). However, many existing math competitions are becoming less effective for assessing top-tier LLMs due to performance saturation (e.g., AIME24/25). To address this, AMO-Bench introduces more rigorous challenges by ensuring all 50 problems are (1) cross-validated by experts to meet at least the International Mathematical Olympiad (IMO) difficulty standards, and (2) entirely original problems to prevent potential performance leakages from data memorization. Experimental results across 36 LLMs on AMO-Bench highlights three key findings: (1) high-level mathematical reasoning remains challenging for current LLMs, with even the best-performing model achieving only 63.1% accuracy and most LLMs scoring below 50%; (2) scaling test-time compute remains a highly effective strategy for substantially improving reasoning performances, and (3) open-source models are progressively narrowing the performance gap with proprietary models. Additionally, we conduct further analysis about reasoning efficiency, volatility, and cross-lingual robustness, providing deeper insights behind the reasoning performances.

Retrieval-Augmented Generation (RAG) grounds language models in external evidence, but multi-hop question answering remains difficult because iterative pipelines must control what to retrieve next and when the available evidence is adequate. In practice, systems may answer from incomplete evidence chains, or they may accumulate redundant or distractor-heavy text that interferes with later retrieval and reasoning. We propose S2G-RAG (Structured Sufficiency and Gap-judging RAG), an iterative framework with an explicit controller, S2G-Judge. At each turn, S2G-Judge predicts whether the current evidence memory supports answering and, if not, outputs structured gap items that describe the missing information. We map these gap items into the next retrieval query, producing stable multi-turn retrieval trajectories. To reduce noise accumulation, we maintain a sentence-level Evidence Context by extracting a compact set of relevant sentences from retrieved documents. Experiments on TriviaQA, HotpotQA, and 2WikiMultiHopQA show that S2G-RAG improves multi-hop QA performance and robustness under multi-turn retrieval. Furthermore, S2G-RAG can be integrated into existing RAG pipelines with a lightweight component, without modifying the search engine or retraining the generator.