The rapid advancement of large language models has opened new avenues for automating complex problem-solving tasks such as algorithmic coding and competitive programming. This paper introduces a novel evaluation technique, LLM-ProS, to assess the performance of state-of-the-art LLMs on International Collegiate Programming Contest (ICPC) problems. Using a curated dataset of 166 World Finals problems from 2011 to 2024, we benchmark the models' reasoning, accuracy, and efficiency. We evaluate the five models-GPT-4o, Mistral Large, Llama-3.1-405B, and the o1 family, consisting of o1-mini and o1-preview, across critical metrics like correctness, resource utilization, and response calibration. Our results reveal significant differences in the models' abilities to generalize, adapt, and solve novel problems. We also investigated the impact of training methodologies, dataset contamination, and chain-of-thought reasoning on model performance. The findings provide new insights into optimizing LLMs for algorithmic tasks, highlighting both strengths and limitations of current models.