UNO High School Problem of the Week Competition
Problem 4
If f(x) satisfies 2f(x) + f(1-x) = x2 for all x, then f(x) = ?
Solution
Since the above equation holds for all x, we can replace x with 1-x.
When we do this we find:
2f(1-x) + f(x) = (1-x)2.
If we subtract our new equation twice from the first equation and then substitute x with 1-x, we find:
3f(x) = 2x2 - (1-x)2
= x2 + 2x -1
Dividing through by three, we find that f(x) = (x2 + 2x -1)/3
