Nonosaki Akiho

Cho a,b,c>0 tm a+b+c=6. Cmr

$\frac{a}{\sqrt{b^{3}+1}}+\frac{b}{\sqrt{c^{3}+1}}+\frac{c}{\sqrt{a^{3}+1}}\leq 2$

giai he pt:

$\left\{\begin{matrix}y^{6}+y^{3}+2x^{2}=\sqrt{xy-x^{2}y^{2}} & \\ 8xy^{3}+2y^{3}+1\geq4x^{2}+2\sqrt{1+(2x+y)^{2}} & \end{matrix}\right.$

$\left\{\begin{matrix} x=\sqrt{y+45}-\sqrt{y+5}& \\ y=\sqrt{x+45}-\sqrt{x+5}& \end{matrix}\right.$

ta co

vt =$\sqrt{(x+3)^{2}+(2y)^{2}}+\sqrt{(1-x)^{2}+(3-2y)^{2}}$

$\geq \sqrt{4^{2}+3^{2}}$

Cho a,b,c k am, a+b+c>0. CMR:

$\frac{a}{4(a+b)+c}+\frac{b}{4(b+c)+a}+\frac{c}{4(c+a)+b}\leq \frac{1}{3}$