[dektep]

ข้อความเดิมเขียนโดยคุณ tatari/nightmare

a nice inequality!
Problem Let$a,b,c>0$ such that $abc=1$,prove that
$$\sum_{cyc}\frac{a^3(b+c)}{\sqrt[5]{b}+2\sqrt[5]{c}}\geq 2$$
Problem Let$a,b,c$ be distinct non-negative real number.Prove that
$$(\frac{a+b}{a-b})^6+(\frac{b+c}{b-c})^6+(\frac{c+a}{c-a})^6\geq 2$$
(propose by poon thongsai on date 17/2/51)
Problem Let $a,b,c\geq 0$,then show that
$$\sum_{cyc}\sqrt{(a^2+bc)(b^2+ca)}\geq\sqrt{3(a^2b^2+b^2c^2+c^2a^2)}+\sqrt{abc}(\sqrt{a}+\sqrt{b}+\sqrt{c})$$
(propose by wichit yangchit on date 25/1/51)

hint

จาก $$Cauchy;LHS \geq \sum_{cyc}\left(\sqrt {a^3b} + \sqrt {a^3c}\right)$$.
ให้ $$a=x^2,b=y^2,c=z^2$$
แล้วพิสูจน์ว่า $$\sum_{cyc}(x^3y + x^3z - x^2yz)\geq\sqrt {3(x^4y^4 + x^4z^4 + y^4z^4)}$$.