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Question-203219




Question Number 203219 by hardmath last updated on 12/Jan/24
Answered by witcher3 last updated on 15/Jan/24
evident  (a+b+c+d)^2 =a^2 +b^2 +c^2 +d^2 +2(ab+bc+cd+da)  since a,b,c,d≥0  (a+b+c+d)^2 ≥a^2 +b^2 +c^2 +d^2
$$\mathrm{evident} \\ $$$$\left(\mathrm{a}+\mathrm{b}+\mathrm{c}+\mathrm{d}\right)^{\mathrm{2}} =\mathrm{a}^{\mathrm{2}} +\mathrm{b}^{\mathrm{2}} +\mathrm{c}^{\mathrm{2}} +\mathrm{d}^{\mathrm{2}} +\mathrm{2}\left(\mathrm{ab}+\mathrm{bc}+\mathrm{cd}+\mathrm{da}\right) \\ $$$$\mathrm{since}\:\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d}\geqslant\mathrm{0} \\ $$$$\left(\mathrm{a}+\mathrm{b}+\mathrm{c}+\mathrm{d}\right)^{\mathrm{2}} \geqslant\mathrm{a}^{\mathrm{2}} +\mathrm{b}^{\mathrm{2}} +\mathrm{c}^{\mathrm{2}} +\mathrm{d}^{\mathrm{2}} \\ $$

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