Reinforcement learning with verifiable rewards (RLVR) has recently advanced the reasoning capabilities of large language models (LLMs). We investigate RLVR from a sample-centric perspective and introduce **LPPO** (Learning-Progress and Prefix-guided Optimization), a framework of progressive optimization techniques. Our work addresses a critical question: how to best leverage a small set of trusted, high-quality demonstrations, rather than simply scaling up data volume. First, motivated by how hints aid human problem-solving, we propose **prefix-guided sampling**, an online data augmentation method that incorporates partial solution prefixes from expert demonstrations to guide the policy, particularly for challenging instances. Second, inspired by how humans focus on important questions aligned with their current capabilities, we introduce **learning-progress weighting**, a dynamic strategy that adjusts each training sample’s influence based on model progression. We estimate sample-level learning progress via an exponential moving average of per-sample pass rates, promoting samples that foster learning and de-emphasizing stagnant ones. Experiments on mathematical-reasoning benchmarks demonstrate that our methods outperform strong baselines, yielding faster convergence and a higher performance ceiling, with these gains proving robust across diverse model architectures, scales, and reinforcement learning optimizers.