HMMT 二月 2004 · COMB 赛 · 第 3 题

英文解析

3. A class of 10 students took a math test. Each problem was solved by exactly 7 of thestudents. If the first nine students each solved 4 problems, how many problems didthe tenth student solve?
   Solution: 6
   Suppose the last student solved n problems, and the total number of problems onthe test was p . Then the total number of correct solutions written was 7 p (seven perproblem), and also equal to 36 + n (the sum of the students’ scores), so p = (36 + n ) / 7.
   The smallest n ≥ 0 for which this is an integer is n = 6. But we also must have n ≤ p ,
   so 7 n ≤ 36 + n , and solving gives n ≤ 6. Thus n = 6 is the answer.