AIME I 2010 #1

[stupidityismygam]

Sorry about the delay, but I figure everyone was busy with TMSCA State, so it doesn't really matter.

Maya lists all the positive divisors of [unparseable or potentially dangerous latex formula]. She then randomly selects two distinct divisors from this list. Let [unparseable or potentially dangerous latex formula] be the probability that exactly one of the selected divisors is a perfect square. The probability [unparseable or potentially dangerous latex formula] can be expressed in the form [unparseable or potentially dangerous latex formula], where [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] are relatively prime positive integers. Find [unparseable or potentially dangerous latex formula].

[klebian]

[unparseable or potentially dangerous latex formula] so [unparseable or potentially dangerous latex formula]

Thus [unparseable or potentially dangerous latex formula]has [unparseable or potentially dangerous latex formula] factors.

A factor is a perfect square iff in its prime factorization, the powers of 2,3,5, and 67 are all even. This leaves 2 options for the number of of 2s, 2 options for the number of 3s, etc. So there are [unparseable or potentially dangerous latex formula] perfect square factors. So [unparseable or potentially dangerous latex formula]so [unparseable or potentially dangerous latex formula]

[stupidityismygam]

Not quite.

The question asks for the probability that of two factors exactly one will be a perfect square. So, the probability is actually [unparseable or potentially dangerous latex formula]