f(x) is a polynomial satisfying the condition below.

f(x)f(1/x) = f(x) + f(1/x)

If f(10) = 1001, find f(20).

If f(10) = 1001,

then 1001*f(1/10) = 1001 + f(1/10).

Solving, f(1/10) = 1001/1000 = 1 + 1/1000.

f(x) could be as simple as x^3 + 1, and that function does satisfy the the first required property. In fact, any function of the form f(x) = x^y + 1 has the first required property.

so f(20) = 8001. Final answer.

*Edited on ***October 1, 2016, 12:51 pm**