Yunanistan 2020 JBMO TST #2 {çözüldü}

[Hüseyin Yiğit EMEKÇİ]

$a,b,c\in \mathbf{R^+}$ olmak üzere $\frac{1}{a}+ \frac{1}{b}+ \frac{1}{c}=3$ ise


$$\frac{a+b}{a^2+ab+b^2}+ \frac{b+c}{b^2+bc+c^2}+ \frac{c+a}{c^2+ca+a^2}\le 2$$

olduğunu gösteriniz

[Hüseyin Yiğit EMEKÇİ]

$$\sum_{cyc}{\frac{a+b}{a^2+ab+b^2}}\leq \sum_{cyc}{\frac{a+b}{3ab}}$$
$$=\sum_{cyc}{\frac{a+b}{\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)ab}}$$
$$=\sum_{cyc}{\frac{c(a+b)}{ab+bc+ca}}=2$$