Drugs used in chemotherapy work in different ways to stop tumor cells from dividing so they stop growing or die. Combining more than one drug may kill more tumor cells. Chemoprotective drugs such as amifostine may protect normal cells from the side effects of chemotherapy. It is not yet known which chemotherapy regimen is most effective for children and young adults with liver cancer. This randomized phase III trial is studying giving combination chemotherapy together with amifostine to see how well it works compared to combination chemotherapy alone in treating patients with liver cancer.