HMMT 十一月 2015 · 冲刺赛 · 第 16 题

英文解析

16. [ 10 ] For how many triples ( x, y, z ) of integers between − 10 and 10 inclusive do there exist reals a, b, cthat satisfyab = xac = ybc = z ?
    Proposed by: Yang Liu
    Answer: 4061
    If none are of x, y, z are zero, then there are 4 · 10 = 4000 ways, since xyz must be positive. Indeed,3
    ( abc ) = xyz. So an even number of them are negative, and the ways to choose an even number of 32
    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 leasttwo of x, y, z must be 0. If all 3 are zero, this gives 1 more solution. If exactly 2 are negative, thenthis gives 3 · 20 more solutions. This comes from choosing one of x, y, z to be nonzero, and choosingits value in 20 ways.
    Our final answer is 4000 + 60 + 1 = 4061 .