HMMT 二月 2002 · GEN1 赛 · 第 3 题
HMMT February 2002 — GEN1 Round — Problem 3
题目详情
- How many triples ( A, B, C ) of positive integers (positive integers are the numbers 1 , 2 , 3 , 4 , . . . ) are there such that A + B + C = 10, where order does not matter (for in- stance the triples (2 , 3 , 5) and (3 , 2 , 5) are considered to be the same triple) and where two of the integers in a triple could be the same (for instance (3 , 3 , 4) is a valid triple).
解析
(3) there are at least two professors on each committee; there are at least two committees. What is the smallest number of committees a university can have? Solution: Let C be any committee. Then there exists a professor P not on C (or else there would be no other committees). By axiom 2, P serves on a committee D having no common members with C . Each of these committees has at least two members, and for each Q ∈ C, R ∈ D , there exists (by axiom 1) a committee containing Q and R , which (again by axiom 1) has no other common members with C or D . Thus we have at least 1 2 + 2 · 2 = 6 committees. This minimum is attainable - just take four professors and let any two professors form a committee.