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Ta có: \(\frac{1+3a}{1+b^2}=\left(1+3a\right).\frac{1}{1+b^2}=\left(1+3a\right)\left(1-\frac{b^2}{1+b^2}\right)\)

\(\ge\left(1+3a\right)\left(1-\frac{b^2}{2b}\right)=\left(1+3a\right)\left(1-\frac{b}{2}\right)\)

\(=3a+1-\frac{b}{2}-\frac{3ab}{2}\)(1)

Tương tự ta có: \(\frac{1+3b}{1+c^2}=3b+1-\frac{c}{2}-\frac{3bc}{2}\)(2); \(\frac{1+3c}{1+a^2}=3c+1-\frac{a}{2}-\frac{3ca}{2}\)(3)

Cộng theo vế của 3 BĐT (1), (2), (3), ta được: \(\frac{1+3a}{1+b^2}+\frac{1+3b}{1+c^2}+\frac{1+3c}{1+a^2}\)\(\ge3\left(a+b+c\right)-\frac{a+b+c}{2}-\frac{3\left(ab+bc+ca\right)}{2}+3\)

\(=\frac{5\left(a+b+c\right)}{2}-\frac{3\left(ab+bc+ca\right)}{2}+3\)

\(\ge\frac{5.\sqrt{3\left(ab+bc+ca\right)}}{2}-\frac{3.3}{2}+3=\frac{15}{2}-\frac{9}{2}+3=6\)

Đẳng thức xảy ra khi a = b = c = 1

\(VT=\frac{3a}{1+b^2}+\frac{3b}{1+c^2}+\frac{3c}{1+a^2}+\frac{1}{1+b^2}+\frac{1}{1+c^2}+\frac{1}{1+a^2}\)

Ta tách VT=A+B và xét

\(A=\frac{3a}{1+b^2}+\frac{3b}{1+c^2}+\frac{3c}{1+a^2}=\text{∑}\left(3a-\frac{3ab^2}{1+b^2}\right)\ge\text{∑}\left(3a-\frac{3ab}{2}\right)\)

\(B=\frac{1}{1+b^2}+\frac{1}{1+c^2}+\frac{1}{1+a^2}=\text{∑}\left(1-\frac{b^2}{1+b^2}\right)\ge\text{∑}\left(1-\frac{b}{2}\right)\)

\(\Rightarrow VT=A+B=3+\frac{5}{2}\left(a+b+c\right)-\frac{3}{2}\text{∑}ab=\frac{5}{2}\left(a+b+c\right)-\frac{3}{2}\ge\frac{15}{2}-\frac{3}{2}=6\)

(Do \(a+b+c\ge\sqrt{3\left(ab+bc+ca\right)}=3\))

Dấu = khi a=b=c=1

2 + 2 =22

5555555555555 = 5 x 5 x ........

\(P=\sum\frac{1}{\sqrt{a^2+b^2-ab+b^2+b^2+1}}\le\sum\frac{1}{\sqrt{ab+b^2+2b}}=\sum\frac{2}{\sqrt{4b\left(a+b+2\right)}}\)

\(\Rightarrow P\le\sum\left(\frac{1}{4b}+\frac{1}{a+b+1+1}\right)\le\sum\left(\frac{1}{4b}+\frac{1}{16}\left(\frac{1}{a}+\frac{1}{b}+1+1\right)\right)\)

\(\Rightarrow P\le\frac{3}{8}\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)+\frac{3}{8}\le\frac{3}{2}\)

Dấu "=" xảy ra khi \(a=b=c=1\)

2.

\(1\ge\frac{1}{1+a}+\frac{1}{1+b}+\frac{1}{1+c}\ge\frac{9}{3+a+b+c}\)

\(\Rightarrow a+b+c+3\ge6\Rightarrow a+b+c\ge6\)

\(P=\sum\frac{a^3}{a^2+ab+b^2}=\sum\left(a-\frac{ab\left(a+b\right)}{a^2+ab+b^2}\right)\ge\sum\left(a-\frac{ab\left(a+b\right)}{3ab}\right)\)

\(\Rightarrow P\ge\sum\left(\frac{2a}{3}-\frac{b}{3}\right)=\frac{1}{3}\left(a+b+c\right)\ge\frac{6}{3}=2\)

Dấu "=" xảy ra khi \(a=b=c=2\)

Ta có : \(ab\le\frac{a^2+b^2}{2}\)

\(\Rightarrow a^2-ab+3b^2+1\ge\frac{a^2}{2}+\frac{5}{2}b^2+1\)

Lại có : \(\left(\frac{a^2}{2}+\frac{5}{2}b^2+1\right)\left(\frac{1}{2}+\frac{5}{2}b^2+1\right)\ge\left(\frac{a}{2}+\frac{5}{2}b+1\right)^2\)

\(\Rightarrow\sqrt{a^2-ab+3b^2+1}\ge\frac{a}{4}+\frac{5b}{4}+\frac{1}{2}\)

\(\Rightarrow\frac{1}{\sqrt{a^2-ab+3b^2+1}}\le\frac{4}{a+b+b+b+b+b+1+1}\le\frac{4}{64}\left(\frac{1}{a}+\frac{5}{b}+2\right)\)

Khi đó :

\(P\le\frac{1}{16}\left(6\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)+6\right)\le\frac{3}{2}\)

Dấu " = " xay ra khi a=b=c=1

Vậy \(P_{Max}=\frac{3}{2}\) khi a=b=c=1

khó phết

\(VT=\frac{3a}{1+b^2}+\frac{3b}{1+c^2}+\frac{3c}{1+a^2}+\frac{1}{1+b^2}+\frac{1}{1+c^2}+\frac{1}{1+a^2}\)

Ta tách VT = A + b và xét :

\(A=\frac{3a}{1+b^2}+\frac{3b}{1+c^2}+\frac{3c}{1+a^2}=\Sigma\left(3a-\frac{3ab^2}{1+b^2}\right)\ge\Sigma\left(3a-\frac{3ab}{2}\right)\)\(B=\frac{1}{1+b^2}+\frac{1}{1+c^2}+\frac{1}{1+a^2}=\Sigma\left(1-\frac{b^2}{1+b^2}\right)\ge\Sigma\left(1-\frac{b}{2}\right)\)

\(\Rightarrow VT=A+B=3+\frac{5}{2}\left(a+b+c\right)-\frac{3}{2}\Sigma ab=\frac{5}{2}\left(a+b+c\right)-\frac{3}{2}\ge\frac{15}{2}-\frac{3}{2}=6\)( Do \(a+b+c\ge\sqrt{3\left(ab+bc+ca\right)=3}\))

Dấu = khi a = b = c = 1 .

Yuzuri Yukari:copy câu trả lời của tôi 

Sang học 24 tìm ai tên Perfect Blue nhé t làm bên đó rồi đưa link thì lỗi ==" , tìm tên đăng nhập  springtime ấy

Chào bác Thắng

\(VP=\frac{6}{\sqrt{\left(3a+bc\right)\left(3b+ca\right)\left(3c+ab\right)}}\)

\(=\frac{6}{\sqrt{\left[\left(a+b+c\right)a+bc\right]\left[\left(a+b+c\right)b+ca\right]\left[\left(a+b+c\right)c+ab\right]}}\)

\(=\frac{6}{\sqrt{\left(a+b\right)^2\left(b+c\right)^2\left(c+1\right)^2}}=\frac{6}{\left(a+b\right)\left(b+c\right)\left(a+c\right)}\)

\(VT=\frac{1}{3a+bc}+\frac{1}{3b+ca}+\frac{1}{3c+ab}\)

\(=\frac{1}{\left(a+b+c\right)a+bc}+\frac{1}{\left(a+b+c\right)b+ac}+\frac{1}{\left(a+b+c\right)c+ab}\)

\(=\frac{\left(b+c\right)+\left(a+c\right)+\left(a+b\right)}{\left(a+b\right)\left(b+c\right)\left(a+c\right)}=\frac{6}{\left(a+b\right)\left(b+c\right)\left(a+c\right)}\)

Vậy VT = VP, đẳng thức được chứng minh