National and Regional Contests Romania Contests District Olympiad 2013 District Olympiad 3 Integral Problem Statement Problem 3.
Let f : [ 0 , π 2 ] → [ 0 , ∞ ) f:\left[ 0,\frac{\pi }{2} \right]\to \left[ 0,\infty \right) f : [ 0 , 2 π ] → [ 0 , ∞ ) ∫ 0 π 2 ( f ( x ) − f ( π 4 ) ) ( sin x − cos x ) d x ≥ 0. \int_{0}^{\frac{\pi }{2}}{\left( f\left( x \right)-f\left( \frac{\pi }{4} \right) \right)}\left( \sin x-\cos x \right)dx\ge 0. ∫ 0 2 π ( f ( x ) − f ( 4 π ) ) ( sin x − cos x ) d x ≥ 0. a ∈ [ π 4 , π 2 ] a\in \left[ \frac{\pi }{4},\frac{\pi }{2} \right] a ∈ [ 4 π , 2 π ] ∫ 0 a f ( x ) sin x d x = ∫ 0 a f ( x ) cos x d x . \int_{0}^{a}{f\left( x \right)\sin x\ dx=}\int_{0}^{a}{f\left( x \right)\cos x\ dx}. ∫ 0 a f ( x ) sin x d x = ∫ 0 a f ( x ) cos x d x .