jmc

Question

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<y is the harmonic mean of x and y equal to 6^20?

Accepted Answer

The harmonic mean of x and y is equal to 11x+1y2 = 2xyx+y, so we have xy=(x+y)(3^202^19), and by SFFT, (x-3^202^19)(y-3^202^19)=3^402^38. Now, 3^402^38 has 4139=1599 factors, one of which is the square root (3^202^19). Since x<y, the answer is half of the remaining number of factors, which is 1599-12= 799.