THE 68th ROMANIAN MATHEMATICAL OLYMPIAD

Question

Let f and g be continuous real-valued functions on the closed unit interval [0, 1] such that f(x)g(x) 4x^2 for all x in [0, 1]. Show that (at least) one of the integrals _0^1 f(x) , dx, _0^1 g(x) , dx has an absolute value greater than or equal to 1.

Accepted Answer

The functions f and g vanish at no point in the half-open interval (0, 1], so |_0^1 f(x) , dx| = _0^1 |f(x)| , dx and |_0^1 g(x) , dx| = _0^1 |g(x)| , dx. Consequently, aligned 1 &= _0^1 2x , dx _0^1 f(x)g(x) , dx = _0^1 |f(x)||g(x)| , dx & _0^1 |f(x)| + |g(x)|2 , dx = 12 ( _0^1 |f(x)| , dx + _0^1 |g(x)| , dx ), aligned and the conclusion follows.