6? By using AM-GM

the result is ...

$$LHS=\sqrt{\sum\frac{a^2b^2}{c^2}+2\sum a^2}\geq\sqrt{\sum\frac{a^2b^2}{b^2}+2}=\sqrt{3}$$where we used rearrangement inequality

parmenides51 wrote: Determine the smallest possible value of the sum $S (a, b, c) = \frac{ab}{c}+\frac{bc}{a}+\frac{ca}{b}$ where $a, b, c$ are three positive real numbers with $a^2 + b^2 + c^2 = 1$ Stronger \[\frac{ab}{c}+\frac{bc}{a}+\frac{ca}{b} \geqslant \sqrt{4(a^2+b^2+c^2)-ab-bc-ca}.\]

parmenides51 wrote: Determine the smallest possible value of the sum $S (a, b, c) = \frac{ab}{c}+\frac{bc}{a}+\frac{ca}{b}$ where $a, b, c$ are three positive real numbers with $a^2 + b^2 + c^2 = 1$ Resemble to France MO 2005.

Nguyenhuyen_AG wrote: parmenides51 wrote: Determine the smallest possible value of the sum $S (a, b, c) = \frac{ab}{c}+\frac{bc}{a}+\frac{ca}{b}$ where $a, b, c$ are three positive real numbers with $a^2 + b^2 + c^2 = 1$ Stronger \[\frac{ab}{c}+\frac{bc}{a}+\frac{ca}{b} \geqslant \sqrt{4(a^2+b^2+c^2)-ab-bc-ca}.\] Nice, but easy! Let $c=\min\{a,b,c\}$. It's equivalent to: $${c}^{4} \left( a-b \right) ^{2} \left\{ 3\,{c}^{2}+5\, \left( a-c \right) \left( -c+b \right) \right\} +2\, \left( -2\,c+b+a \right) ^{3}{c}^{5}+M(a-c)(b-c)\geqq 0$$But: $M=3\,{c}^{6}+35\,{c}^{4} \left( a-c \right) \left( -c+b \right) \\+20\, \left( a-c \right) \left( -c+b \right) \left( -2\,c+b+a \right) {c} ^{3}\\+ \left( a-c \right) \left( -c+b \right) \left( a-5\,c+4\,b \right) \left( 4\,a-5\,c+b \right) {c}^{2}\\+4\, \left( a-c \right) ^{ 2} \left( -c+b \right) ^{2} \left( -2\,c+b+a \right) c\\+ \left( a-c \right) ^{3} \left( -c+b \right) ^{3} \geqq 0$ Done!

This problem is also there in evan otis application this year!!!

also we can solve it with lagrange multipliers,if a,b,c are not equal the system of equations is impossible.and the minimum is sqrt(3) for a=b=c=1/sqrt(3)

ektorasmiliotis wrote: also we can solve it with lagrange multipliers,if a,b,c are not equal the system of equations is impossible.and the minimum is sqrt(3) for a=b=c=1/sqrt(3) You need to prove the existence of the minimum.