This phase II/III trials studies whether maintenance immunotherapy (nivolumab) following definitive treatment with radiation and chemotherapy (cisplatin) result in significant improvement in overall survival (time being alive) and progression-free survival (time being alive without cancer) for patients with intermediate risk human papillomavirus (HPV) positive oropharynx cancer (throat cancer) that has spread to nearby tissue or lymph nodes. Drugs used in chemotherapy such as cisplatin work in different ways to stop the growth of tumor cells, either by killing the cells, by stopping them from dividing, or by stopping them from spreading. Radiation therapy uses high energy rays to kill tumor cells and shrink tumors. Immunotherapy with monoclonal antibodies, such as nivolumab, may help the body's immune system attack the cancer, and may interfere with the ability of tumor cells to grow and spread. It is not yet known whether chemotherapy and radiation therapy followed by maintenance nivolumab therapy works better than chemotherapy and radiation therapy alone in treating patients with HPV positive oropharyngeal cancer.