Question

1) Cho a,b,c>0 và a²+b²+c²=3

CM \(\frac{1}{1+ab}+\frac{1}{1+bc}+\frac{1}{1+ca}\ge\frac{3}{2}\)

2) Giaỉ pt \(\left(\sqrt{x+1}+1\right)\left(5-x\right)=2x\)

Answer 1

https://i.imgur.com/tTHeSNA.jpg

Answer 2

Bài 1:

Áp dụng BĐT Cauchy-Schwarz:

\(\frac{1}{ab+1}+\frac{1}{bc+1}+\frac{1}{ca+1}\geq \frac{9}{ab+1+bc+1+ca+1}=\frac{9}{ab+bc+ac+3}(1)\)

Theo BĐT AM-GM:

\(ab+bc+ac\leq a^2+b^2+c^2\Leftrightarrow ab+bc+ac\leq 3(2)\)

Từ \((1);(2)\Rightarrow \frac{1}{ab+1}+\frac{1}{bc+1}+\frac{1}{ca+1}\geq \frac{9}{ab+bc+ac+3}\geq \frac{9}{3+3}=\frac{3}{2}\)

Ta có đpcm.

Dấu "=" xảy ra khi $a=b=c=1$