ab+bc+ca=1

[Claudiu Mindrila]

Daca \( a,b,c \in (0, \infty) \) astfel incat \( ab+bc+ca=1 \), atunci \( abc(a+\sqrt{a^2+1})(b+\sqrt{b^2+1})(c+\sqrt{c^2+1}) \le 1 \).

[Marius Mainea]

\( LHS=\prod {\frac{a}{\sqrt{a^2+1}-a}}=\prod{\frac{a}{\sqrt{(a+b)(a+c)}-a}}\le\prod{\frac{a}{a+\sqrt{bc}-a}}=\prod{\frac{a}{\sqrt{bc}}=1=RHS \)