WakeUp wrote: Let $x,y,z$ be real numbers such that $0\le x,y,z\le\frac{\pi}{2}$. Prove the inequality \[\frac{\pi}{2}+2\sin x\cos y+2\sin y\cos z\ge\sin 2x+\sin 2y+\sin 2z.\] You probably mean \[2\sin z\cos x+2\sin x\cos y+2\sin y\cos z\ge\sin 2x+\sin 2y+\sin 2z.\] Equivalently \[\sin(x+y)+\sin(y+z)+\sin(z+x)\ge\sin 2x+\sin 2y+\sin 2z\] which is clear from Karamata, as $(2x,2y,2z)\succ (x+y,x+z,y+z)$ and $\sin$ is concave in the range $[0,{\pi}]$.

spanferkel wrote: You probably mean \[2\sin z\cos x+2\sin x\cos y+2\sin y\cos z\ge\sin 2x+\sin 2y+\sin 2z.\] Actually I stand by the original problem, it does however seem strange.