IMC 2004 · Problem 2

Let $f, g : [a,b] \to [0,\infty)$ be continuous and non-decreasing functions such that for each $x \in [a,b]$ we have

$$\int_a^x \sqrt{f(t)}\,dt \le \int_a^x \sqrt{g(t)}\,dt$$

and $\int_a^b \sqrt{f(t)}\,dt = \int_a^b \sqrt{g(t)}\,dt$.

Prove that

$$\int_a^b \sqrt{1 + f(t)}\,dt \ge \int_a^b \sqrt{1 + g(t)}\,dt.$$