Algebra problem

[compact musk]

Can someone help me to solve this with a method for 10th grade?

[smoky geyser]

Suposse:(ab+1/b)^2+(bc+1/c)^2+(ac+1\a)^2

[normal sparrow]

Can you translate that?

[ivory path]

Ohhh it’s French

[frozen spade]

Idk maybe titu on $\frac{(ab+1)^2}{b^2}+\frac{(bc+1)^2}{c^2}+\frac{(ca+1)^2}{a^2}$

[junior rapids]

BlockLayer2000

[frozen spade]

So $\frac{(ab+1)^2}{b^2}+\frac{(bc+1)^2}{c^2}+\frac{(ca+1)^2}{a^2}\geq \frac{(ab+bc+ca+3)^2}{a^2+b^2+c^2}=\frac{16}{a^2+b^2+c^2}$

[junior rapids]

BlockLayer2000

[frozen spade]

Then we'd want to show $a^2+b^2+c^2\leq1$, to finish the proof

[junior rapids]

BlockLayer2000

[frozen spade]

But for a choice of $a=0.1,b=1.1,c=\frac{ 89 }{120}$, it does not work.

[junior rapids]

BlockLayer2000

[frozen spade]

Dang

[compact musk]

yess

[compact musk]

a , b and c are real numbres such as
ab+bc+ac=1. Poove that (ab+1/b)^2+(bc+1/c)^2+(ac+1\a)^2 >=16