2009 IMO Shortlist A4

BLR Let $a, b, c$ be positive real numbers such that $a b+b c+c a \leq 3 a b c$. Prove that $$\sqrt{\frac{a^{2}+b^{2}}{a+b}}+\sqrt{\frac{b^{2}+c^{2}}{b+c}}+\sqrt{\frac{c^{2}+a^{2}}{c+a}}+3 \leq \sqrt{2}(\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a}) .$$

BLR 设 $a, b, c$ 为正实数，使得 $a b+b c+c a \leq 3 a b c$。证明 $$\sqrt{\frac{a^{2}+b^{2}}{a+b}}+\sqrt{\frac{b^{2}+c^{2}}{b+c}}+\sqrt{\frac{c^{2}+a^{2}}{c+a}}+3 \leq \sqrt{2}(\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a}) 。$$