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ab+bc+ca=1
Posted: Sun Feb 15, 2009 4:17 pm
by Claudiu Mindrila
Daca \( a,b,c \in (0, \infty) \) astfel incat \( ab+bc+ca=1 \), atunci \( abc(a+\sqrt{a^2+1})(b+\sqrt{b^2+1})(c+\sqrt{c^2+1}) \le 1 \).
Posted: Thu Feb 19, 2009 1:05 am
by Marius Mainea
\( LHS=\prod {\frac{a}{\sqrt{a^2+1}-a}}=\prod{\frac{a}{\sqrt{(a+b)(a+c)}-a}}\le\prod{\frac{a}{a+\sqrt{bc}-a}}=\prod{\frac{a}{\sqrt{bc}}=1=RHS \)