Find maximum value (1 Viewer)

TuePhuong · Jul 10, 2016

Give three positive numbers a, b, c such that $a+b+c=3$ . Find maximum value of following expression
$\frac{a}{a+\sqrt{3a+bc}}+\frac{b}{b+\sqrt{3b+ca}}+\frac{c}{c+\sqrt{3c+ab}}.$

seanieg89 · Jul 11, 2016

I think it is just 1. (Achieved for example, when a=b=c=1.)

$P=\sum_{cyc}\frac{a}{a+\sqrt{3a+bc}}\\ \\ =\sum_{cyc}\frac{a(a-\sqrt{3a+bc})}{a^2-(a+b+c)a-bc}\\ \\ =\sum_{cyc}\frac{a\sqrt{(a+b)(a+c)}-a^2}{ab+ac+bc}\\ \\ \leq \sum_{cyc} \frac{a(2a+b+c)/2-a^2}{ab+ac+bc}\\ \\=1.$

Find maximum value (1 Viewer)

TuePhuong

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seanieg89

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