[Everyday Mathematics]20150121

设 $f\in C[0,1]$ 适合 $$\bex xf(y)+yf(x)\leq 1,\quad\forall\ x,y\in [0,1]. \eex$$ 试证: $$\bex \int_0^1 f(x)\rd x\leq \frac{\pi}{4}. \eex$$