It's a little bit crazy for me the problem I propose to you today :
Let $a,b,c\geq 0$ such that $a+b+c=1$ then we have : $$3\cos(\frac{1}{3})^{\cos(\frac{1}{3})^{\cos(\frac{1}{3})}}\geq\cos(a)^{\cos(b)^{\cos(c)}}+\cos(c)^{\cos(a)^{\cos(b)}}+\cos(b)^{\cos(c)^{\cos(a)}}\geq 2+\cos(1)$$
It's nice because we have an upper bound and a lower bound (try with $\sin(x)$ it doesn't works) Furthermore I don't see an inequality of this kind on the website .
I have tested numerically until it becomes obvious for me .
To prove this I would like to use Am-Gm for the RHS but it partial and it becomes very complicated . I'm new with tetration and I haven't the technics to go further . As it's cyclic we can't use Jensen's inequality so it's make the problem harder .
Maybe an angel can prove this...
Many thanks .