IMC 1998 · Problem 6

Let $f : [0,1] \to \mathbb{R}$ be a continuous function with the property that for any $x$ and $y$ in the interval,

$$xf(y) + yf(x) \le 1.$$

a) Show that

$$\int_0^1 f(x)\,dx \le \frac{\pi}{4}.$$

b) Find a function, satisfying the condition, for which there is equality.