Suppose $F_1(x)$ and $F_2(x)$ are two cumulative distribution functions of a positive random variable. That is, ($i = 1,2$), $F_i(0) = 0$, $F_i(x) \leq 1$, and is non-decreasing.
Prove (or disprove) that at least one of these two inequalities is true: $$\int_0^\infty [1-F_1(x)F_2(x)]dx \leq \frac{1}{2}\int_0^\infty [1-F_1^2(x)]dx + \int_0^\infty [1-F_2(x)]dx$$ or $$\int_0^\infty [1-F_1(x)F_2(x)]dx \leq \frac{1}{2}\int_0^\infty [1-F_2^2(x)]dx + \int_0^\infty [1-F_1(x)]dx$$ It's okay to assume that the support is bounded, i.e., change the limit of integration from $\infty$ to $t$.