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As stated in the title, my problem is quite simple.

Minimize $ab+bc+ca$ under these three constraints:

$$ a^2+b^2=1 $$$$ b^2+c^2=2 $$$$ c^2+a^2=2 $$

I can brute force it, with some intelligence of course, and calculate the value of only three possibilities. But is there any way to do it with even more intelligence?

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2 Answers 2

First step is that you can get the possible values of $a,b,c$ according to $$ a^2+b^2=1 $$$$ b^2+c^2=2 $$$$ c^2+a^2=2 $$ Which are $$ a=\pm\sqrt{1/2} $$$$ b=\pm\sqrt{1/2} $$$$ c=\pm\sqrt{3/2} $$ Notice that among the three products, $ab, bc$, and $ca$, at most 2 of them are negative, because at least 2 numbers among $a,b$ and $c$ have the same sign (i.e., these 2 numbers are either both negative, or positive). So if we can maximize the absolute values of these 2 negative products, we get the minimum value of $ab+bc+ca$. Which will be $$ a=\sqrt{1/2} $$$$ b=\sqrt{1/2} $$$$ c=-\sqrt{3/2} $$ and under this condition, $$ ab+bc+ca=1/2-\sqrt{3} $$

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$\newcommand{\angles}[1]{\left\langle #1 \right\rangle}% \newcommand{\braces}[1]{\left\lbrace #1 \right\rbrace}% \newcommand{\bracks}[1]{\left\lbrack #1 \right\rbrack}% \newcommand{\dd}{{\rm d}}% \newcommand{\ds}[1]{\displaystyle{#1}}% \newcommand{\equalby}[1]{{#1 \atop {= \atop \vphantom{\huge A}}}}% \newcommand{\expo}[1]{\,{\rm e}^{#1}\,}% \newcommand{\ic}{{\rm i}}% \newcommand{\imp}{\Longrightarrow}% \newcommand{\pars}[1]{\left( #1 \right)}% \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\pp}{{\cal P}}% \newcommand{\sgn}{\,{\rm sgn}}% \newcommand{\ul}[1]{\underline{#1}}% \newcommand{\verts}[1]{\left\vert #1 \right\vert}% \newcommand{\yy}{\Longleftrightarrow}$ With the constraints: $$ b^{2} - c^{2} = \overbrace{\pars{a^{2} + b^{2}}}^{=\ 1}\ -\ \overbrace{\pars{c^{2} + a^{2}}}^{=\ 2} = -1 $$

$$ b^{2} = {1 \over 2}\,\bracks{\overbrace{\pars{b^{2} + c^{2}}}^{=\ 2}\ +\ \overbrace{\pars{b^{2} - c^{2}}}^{=\ -1}} = {1 \over 2} $$

$$ a^{2} = 1 - b^{2} = {1 \over 2}\,, \qquad\qquad c^{2} = 2 - b^{2} = {3 \over 2} $$

Then, $$ a_{\pm} = \pm\,{\sqrt{2\,} \over 2}\,, \qquad b_{\pm} = \pm\,{\sqrt{2\,} \over 2} \,, \qquad c_{\pm} = \pm\,{\sqrt{6\,} \over 2} $$

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