Problem 5. Mom put a vase with 15 tangerines on the table. One of the guests took two tangerines, and mom replaced them with one apple. Other guests also started taking two fruits each. Each time a guest took two of the same fruits (two tangerines or two apples), mom would put one apple back in the vase; if a guest took two different fruits (one tangerine and one apple), mom would put one tangerine back in the vase. In the end, only one fruit was left in the vase. Which one?
Solution. It can be noticed that the number of tangerines either remains the same or decreases by two. Since the number of tangerines was odd, one tangerine will definitely remain at the end.

Answer. A tangerine remained.