<s>[INST] Prove the inequality $\sqrt{a^2+b^2}+\sqrt{c^2+b^2}+\sqrt{a^2+c^2}<1+\frac{\sqrt2}{2}$ Let $a,b,c -$ triangle side and $a+b+c=1$. Prove the inequality
$$\sqrt{a^2+b^2}+\sqrt{c^2+b^2}+\sqrt{a^2+c^2}<1+\frac{\sqrt2}{2}$$
My work so far:
1) $a^2+b^2=c^2-2ab\cos \gamma \ge c^2-2ab$
2) $$\sqrt{a^2+b^2}+\sqrt{c^2+b^2}+\sqrt{a^2+c^2}\le3\sqrt{\frac{2(a^2+b^2+c^2)}3}$$
 [/INST] Just a sketch of a possible proof (a variational, mixing variables approach). Assume that $a\leq b\leq c$ and show that if you replace $a$ with $a-\varepsilon$ and $b$ with $b+\varepsilon$, the convexity of $f(x)=\sqrt{1+x^2}$ grants that the LHS increases (after such substitution, the order of $b$ and $c$ may change. That is not an issue, it actually helps us to be sure we always deal with side lengths of a triangle). It follows that the supremum of the LHS is achieved when $a=0$. After that, the problem is trivial.
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