51st Ukrainian National Mathematical Olympiad, 3rd Round

For $x=0$ we have $f_1(0)=f_2(0)=b$, thus all $b\le 0$ does not satisfy the condition of the problem.

Let $b>0$. We add two values up and get $f_1(x)+f_2(x)=2x^2+2b>0$, thus at least one function is positive.