HMMT 二月 2006 · COMB 赛 · 第 1 题
HMMT February 2006 — COMB Round — Problem 1
题目详情
- Vernonia High School has 85 seniors, each of whom plays on at least one of the school’s three varsity sports teams: football, baseball, and lacrosse. It so happens that 74 are on the football team; 26 are on the baseball team; 17 are on both the football and lacrosse teams; 18 are on both the baseball and football teams; and 13 are on both the baseball and lacrosse teams. Compute the number of seniors playing all three sports, given that twice this number are members of the lacrosse team.
解析
- Vernonia High School has 85 seniors, each of whom plays on at least one of the school’s three varsity sports teams: football, baseball, and lacrosse. It so happens that 74 are on the football team; 26 are on the baseball team; 17 are on both the football and lacrosse teams; 18 are on both the baseball and football teams; and 13 are on both the baseball and lacrosse teams. Compute the number of seniors playing all three sports, given that twice this number are members of the lacrosse team. Answer: 11 Solution: Suppose that n seniors play all three sports and that 2 n are on the lacrosse team. Then, by the principle of inclusion-exclusion, 85 = (74 + 26 + 2 n ) − (17 + 18 + 13) + ( n ) = 100 + 2 n − 48 + n = 52 + 3 n . It is easily seen that n = 11 .