Cho a,b,c không âm thỏa mãn \(a^2+b^2+c^2=3\). Tìm GTNN, GTLN của \(P=\dfrac{a}{4-ab}+\dfrac{b}{4-bc}+\dfrac{c}{4-ca}\)

Cho x,y>0. CMR: \(\left(x+y\right)^2\left(\dfrac{1}{x^2}+\dfrac{1}{y^2}+\dfrac{1}{x^2+y^2}\right)\ge10\)

Tìm tất cả các số nguyên tố a,b,c,d,e thỏa mãn \(a^4+b^4+c^4+d^4+e^4=abcde\)

Cmr: \(\left(a^3+b^3+c^3-3abc\right)^2\le\left(a^2+b^2+c^2\right)^3\) với mọi số thực a,b,c.

Cmr nếu a+b+c=0 thì:

a) \(10\left(a^7+b^7+c^7\right)=7\left(a^2+b^2+c^2\right)\left(a^5+b^5+c^5\right)\)

b) \(a^5\left(b^2+c^2\right)+b^5\left(c^2+a^2\right)+c^5\left(a^2+b^2\right)=\dfrac{1}{2}\left(a^3+b^3+c^3\right)\left(a^4+b^4+c^4\right)\)

1.Gpt: \(\dfrac{6}{x-3\sqrt{x-2}+7}=\dfrac{1}{\sqrt{x-2}}+\dfrac{\sqrt{3}}{3\sqrt{2\sqrt{x-2}}-3}\)

2.Ghpt: \(\left\{{}\begin{matrix}x^2-y-z=0\\x^3-y^2-z^2+2=0\end{matrix}\right.\)

Cho a,b,c>0 thỏa mãn ab+bc+ca=1. CMR:

\(\left(\dfrac{a}{\sqrt{a^2+1}}+\dfrac{b}{\sqrt{b^2+1}}+\dfrac{c}{\sqrt{c^2+1}}\right)^3\le\dfrac{3}{2}\left(\dfrac{1}{a^2+1}+\dfrac{1}{b^2+1}+\dfrac{1}{c^2+1}\right)\)