HMMT 十一月 2015 · 冲刺赛 · 第 16 题

HMMT November 2015 — Guts Round — Problem 16

[ 10 ] For how many triples ( x, y, z ) of integers between − 10 and 10 inclusive do there exist reals a, b, c that satisfy ab = x ac = y bc = z ?

解析

[ 10 ] For how many triples ( x, y, z ) of integers between − 10 and 10 inclusive do there exist reals a, b, c that satisfy ab = x ac = y bc = z ? Proposed by: Yang Liu Answer: 4061 3 If none are of x, y, z are zero, then there are 4 · 10 = 4000 ways, since xyz must be positive. Indeed, 2 ( abc ) = xyz. So an even number of them are negative, and the ways to choose an even number of 3 variables to be negative is 4 ways. If one of x, y, z is 0, then one of a, b, c is zero at least. So at least two of x, y, z must be 0. If all 3 are zero, this gives 1 more solution. If exactly 2 are negative, then this gives 3 · 20 more solutions. This comes from choosing one of x, y, z to be nonzero, and choosing its value in 20 ways. Our final answer is 4000 + 60 + 1 = 4061 .