Test collections are information-retrieval tools that allow researchers to quickly and easily evaluate ranking algorithms. While test collections have become an integral part of IR research, the process of data creation involves significant manual-annotation effort, which often makes it very expensive and time-consuming. Consequently, test collections can become too small when the budget is limited, which may lead to unstable evaluations. As a cheaper alternative, recent studies have proposed using large language models (LLMs) to completely replace human assessors. However, while LLMs correlate to some extent with human judgments, their predictions are not perfect and often show bias. Thus, a complete replacement with LLMs is considered too risky and not fully reliable. In this paper, we propose LLM-Assisted Relevance Assessments (LARA), an effective method to balance manual annotations with LLM annotations, helping build a rich and reliable test collection even under a low budget. We use the LLM's predicted relevance probabilities to select the most profitable documents for manual annotation under a budget constraint. Guided by theoretical reasoning, LARA actively learns to calibrate the LLM's predicted relevance probabilities, directing the human-annotation process. Then, using the calibration model learned from the limited manual annotations, LARA debiases the LLM predictions to annotate the remaining non-assessed data. Experiments on TREC-7 Ad Hoc, TREC-8 Ad Hoc, TREC Robust 2004, and TREC-COVID datasets show that LARA outperforms alternative solutions under almost any budget constraint. While the community debates humans versus LLMs in relevance assessments, we contend that, given the same amount of human effort, it is reasonable to leverage LLMs.