This problem is a problem in the last selection phase of the math olympiads in my country.
If $\alpha, \beta,\gamma$ are angles $\in[0,\frac\pi2]$ such that $\sin^2(\alpha)+\sin^2(\beta)+\sin^2(\gamma)=1$.Minimize $\cos(\alpha)+\cos(\beta)+\cos(\gamma)$
I started by $$\cos^2(\alpha)+\cos^2(\beta)+\cos^2(\gamma)=2$$ Then, how can I minimize it? According to Wolfram Alpha, the anwer is $2$ for $(0,1,1)$ and permutations. Just squaring the desired value does not help, what can I do then?
Another additional thought about the problem is that if we take the solutions by Wolfram as true, this hints us that inequalities like $AM\ge GM$ are very unlikely to help.
We can show that for every $a \in [0,1]$ $$a^2\le a$$ Then, we use that three times and show we can achieve equality. Is this proof right?