Show that a^4+b^4+c^4≥ 2b^2+b^2c^2+c^2a^2≥(a+b+c)abc

If a,b,c are positive real numbers such that \dfrac{\sqrt{2}b-2c }{a}=1, show that b^2\geq 4ac

Given a,b,c are positive real numbers such that

a+b+c = \sqrt{10+\sqrt{19} }

and

\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c} = \sqrt{10-\sqrt{19} }

If x = a^2+b^2+c^2, find the value of x+\dfrac{9}{x}

Show that

(x-y)^3+(y-z)^3+(z-x)^3=3(x-y)(y-x)(z-x)

Let (x,y,z) with x\geq y\geq z\geq 0 be integers such that

\dfrac{x^3+y^3+z^3}{3}=xyz+21

Find x.

If x,y,z >0, prove the equality

\sqrt{x+y}+\sqrt{y+z}+\sqrt{z+x}> 2\sqrt{\dfrac{(x+y)(y+z)(z+x)}{xy+yz+zx}}

If n\geq 1 Proof of the square root inequality

2\sqrt{n+1}-2\sqrt{n}<\dfrac{1}{\sqrt{n}}<2\sqrt{n}-2\sqrt{n-1}

Given the equation x+y+z=\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z} =1 . Prove that at least one of x, y, z is equal to 1

If the real number x , y satisfy the condition

\big( x+\sqrt{1+x^2}\big) \big( y+\sqrt{1+y^2} \big) =1 ,

verify the value of the expression (x+y)^2 is equal to 0.

Given a>0,b>0,c>0 , prove a^2+b^2+c^2-ab-bc-ac≥0

If a+b+c=0 , \dfrac{b-c}{a}+\dfrac{c-a}{b}+\dfrac{a-b}{c}=0 , prove \dfrac{bc+b-c}{b^2c^2}+\dfrac{ca+c-a}{c^2a^2}+\dfrac{ab+a-b}{a^2b^2}=0

Prove a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ac)

If abc=1 , prove \dfrac{a}{1+a+ab}+\dfrac{b}{1+b+bc}+\dfrac{c}{1+c+ca} =1