HMMT 二月 2006 · GEN2 赛 · 第 4 题

HMMT February 2006 — GEN2 Round — Problem 4

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. ∣ ∣ ∣ ∣ a + b 2 2 ∣ ∣

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. 1 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 . ∣ ∣ ∣ ∣ a + b 2 2 ∣ ∣