# Difference between revisions of "1996 AIME Problems/Problem 8"

## Problem

The harmonic mean of two positive integers is the reciprocal of the arithmetic mean of their reciprocals. For how many ordered pairs of positive integers $(x,y)$ with $x is the harmonic mean of $x$ and $y$ equal to $6^{20}$?

## Solution

The harmonic mean of $x$ and $y$ is equal to $2xy/(x+y)$, so we have $xy=(x+y)(3^{20}\cdot2^{19})$, and $(x-3^{20}\cdot2^{19})(y-3^{20}\cdot2^{19})=3^{40}\cdot2^{38}$. $3^{40}\cdot2^{38}$ has $41\cdot39=1599$ factors, one of which is the square root. Since $x, the answer is half of the rest of them, which is $799$.