Beyond Pass@ 1: Self-Play with Variational Problem Synthesis Sustains RLVR

˚Equal contribution, work done during internships at Microsoft. : Corresponding authors Reinforcement Learning with Verifiable Rewards (RLVR) has recently emerged as a key paradigm for post-training Large Language Models (LLMs), particularly for complex reasoning tasks. However, standard RLVR training has been shown to improve Pass@1 performance at the expense of policy entropy, leading to reduced generation diversity and limiting the Pass@k performance, which typically represents the upper bound of LLM reasoning capability. In this paper, we systematically analyze the policy's generation diversity from the perspective of training data and find that augmenting and updating training problems helps mitigate entropy collapse during training. Based on these observations, we propose an online Self-play with Variational problem Synthesis (SvS) strategy for RLVR training, which uses the policy's correct solutions to synthesize variational problems while ensuring their reference answers remain identical to the originals. This self-improving strategy effectively preserves policy entropy during training and substantially improves Pass@k compared with standard RLVR, sustaining long-term improvements and achieving absolute gains of 18.3% and 22.8% in Pass@32 performance on the competition-level AIME 24 and AIME 25 benchmarks, as well as on code generation tasks. Experiments on 12 reasoning benchmarks across varying model sizes from 3B to 32B consistently demonstrate the generalizability and robustness of SvS.