HMMT 二月 2007 · TEAM2 赛 · 第 4 题
HMMT February 2007 — TEAM2 Round — Problem 4
题目详情
- [ 20 ] Thomas and Michael are just two people in a large pool of well qualified candidates for appointmentto a problem writing committee for a prestigious college math contest. It is 40 times more likely thatboth will serve if the size of the committee is increased from its traditional 3 members to a whoppingn members. Determine n. (Each person in the pool is equally likely to be chosen.)
2 2 2
英文原题
[ 25 ] Show that AB + CD = AD + BC . Use the above to conclude that for some positive number α,
AB = α ·
( AI
CI + BI
DI
)
BC = α ·
( BI
DI + CI
AI
)
CD = α ·
( CI
AI + DI
BI
)
DA = α ·
( DI
BI + AI
CI
)
.
解析
英文解析
- [ 20 ] Thomas and Michael are just two people in a large pool of well qualified candidates for appointmentto a problem writing committee for a prestigious college math contest. It is 40 times more likely thatboth will serve if the size of the committee is increased from its traditional 3 members to a whoppingn members. Determine n. (Each person in the pool is equally likely to be chosen.)
Answer: 16 . Suppose there are k candidates. Then the probability that both serve on a 3 membered
( ) ( ) ( )
k k − 2 kcommittee is ( k − 2) / , and the odds that both serve on an n membered committee are / .
3 n − 2 n
The ratio of the latter to the former is
( )( )
k k − 2
k !( k − 2)!1!( k − 3)! n !( k − n )! n · ( n − 1)
3 n − 2
( ) = = .
k !( k − 2)!( n − 2)!( k − n )!3!( k − 3)! 3!k
( k − 2)
Solving n · ( n − 1) = 240 produces n = 16 , − 15 , and we discard the latter.n
2 2 2