Question Number 215005 by Abdullahrussell last updated on 25/Dec/24
$$\:{a},{b},{c},{d}\in{R}\:{such}\:{that}, \\ $$$$\:\left({a}+{b}\right)\left({c}+{d}\right)=\mathrm{2} \\ $$$$\:\left({a}+{c}\right)\left({b}+{d}\right)=\mathrm{3} \\ $$$$\:\left({a}+{d}\right)\left({b}+{c}\right)=\mathrm{4}\: \\ $$$$\:{find}:\:\:\left({a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} +{d}^{\mathrm{2}} \right)_{{minimum}.} \\ $$
Answered by A5T last updated on 25/Dec/24
$${ac}+{ad}+{bc}+{bd}=\mathrm{2} \\ $$$${ab}+{ad}+{bc}+{cd}=\mathrm{3} \\ $$$${ab}+{ac}+{bd}+{cd}=\mathrm{4} \\ $$$$\Rightarrow\Sigma{ab}={ab}+{ac}+{ad}+{bc}+{bd}+{cd}=\frac{\mathrm{2}+\mathrm{3}+\mathrm{4}}{\mathrm{2}}=\mathrm{4}.\mathrm{5} \\ $$$${a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} +{d}^{\mathrm{2}} \geqslant\frac{\left({a}+{b}+{c}+{d}\right)^{\mathrm{2}} }{\mathrm{4}} \\ $$$$\Rightarrow\mathrm{4}\left({a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} +{d}^{\mathrm{2}} \right)\geqslant{a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} +{d}^{\mathrm{2}} +\mathrm{2}\Sigma{ab} \\ $$$$\Rightarrow\mathrm{3}\left({a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} +{d}^{\mathrm{2}} \right)\geqslant\mathrm{9} \\ $$$$\Rightarrow{a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} +{d}^{\mathrm{2}} \geqslant\mathrm{3} \\ $$
Commented by Abdullahrussell last updated on 25/Dec/24
$$\:{Sir},\:{equality}\:{holds}\:{with}\:\left({a},{b},{c},{d}\right)=? \\ $$
Commented by A5T last updated on 25/Dec/24
$${This}\:{is}\:{a}\:{lower}\:{bound},\:{equality}\:{may}\:{not}\: \\ $$$${necessarily}\:{hold}. \\ $$
Commented by Abdullahrussell last updated on 25/Dec/24
$$\:{Sir},\:{if}\:{a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} +{d}^{\mathrm{2}} =\mathrm{3} \\ $$$$\:\:{then}\:\left({a},{b},{c},{d}\right)=? \\ $$
Commented by A5T last updated on 25/Dec/24
$${The}\:{actual}\:{minimum}\:{value}\:{is}\:\approx\mathrm{7}. \\ $$