\(VT=\frac{25a}{b+c}+25+\frac{16b}{a+c}+16+\frac{c}{a+b}+1-42\)

\(VT=\frac{25\left(a+b+c\right)}{b+c}+\frac{16\left(a+b+c\right)}{a+c}+\frac{a+b+c}{a+b}-42\)

\(VT=\left(a+b+c\right)\left(\frac{25}{b+c}+\frac{16}{a+c}+\frac{1}{a+b}\right)-42\)

\(VT\ge\left(a+b+c\right).\frac{\left(5+4+1\right)^2}{b+c+a+c+a+b}-42=\frac{100\left(a+b+c\right)}{2\left(a+b+c\right)}-42=8\)

Dấu "=" xảy ra khi: \(\frac{b+c}{5}=\frac{a+c}{4}=\frac{a+b}{1}=\frac{2\left(a+b+c\right)}{5+4+1}=\frac{a+b+c}{5}\)

\(\Rightarrow a=0\) trái giả thiết a dương, vậy dấu "=" không xảy ra

\(\Rightarrow\frac{25a}{b+c}+\frac{16b}{a+c}+\frac{c}{a+b}>8\)

Ta có : \(\frac{a}{1+9b^2}=\frac{a+9ab^2-9ab^2}{1+9b^2}=a-\frac{9ab^2}{1+9b^2}\ge a-\frac{9ab^2}{6b}=a-\frac{3ab}{2}\)

Tương tự : \(\frac{b}{1+9c^2}\ge b-\frac{3bc}{2}\); \(\frac{c}{1+9a^2}\ge c-\frac{3ac}{2}\)

\(\Rightarrow Q\ge a+b+c-\frac{3ab+3bc+3ac}{2}\ge a+b+c-\frac{3.\frac{\left(a+b+c\right)^2}{3}}{2}=1-\frac{1}{2}=\frac{1}{2}\)

Dấu "=" xảy ra khi \(a=b=c=\frac{1}{3}\)

Ta có: \(Q=\frac{a}{1+9b^2}+\frac{b}{1+9c^2}+\frac{c}{9a^2}=\frac{a+9ab^2-9ab^2}{1+9b^2}+\frac{b+9bc^2-9bc^2}{1+9b^2}+\frac{c+9ca^2-9ca^2}{1+9c^2}\)

\(=1-\frac{9ab^2}{1+9b^2}+b-\frac{9bc^2}{1+9c^2}+c-\frac{9ca^2}{1+9a^2}=1-\left(\frac{9ab^2}{1+9b^2}+\frac{9bc^2}{1+9c^2}+\frac{9ca^2}{1+9a^2}\right)\)

Áp dụng BĐT AM-GM ta có:

\(\frac{9ab^2}{1+9b^2}\le\frac{9ab^2}{2\sqrt{1\cdot9b^2}}=\frac{9ab^2}{2\cdot3b}=\frac{3ab}{2}\)

Tương tự ta có: \(\hept{\begin{cases}\frac{9bc^2}{1+9c^2}\le\frac{3ab}{2}\\\frac{9ca^2}{1+9a^2}\le\frac{3ab}{2}\end{cases}}\)

\(\Rightarrow\frac{9ab^2}{1+9b^2}+\frac{9bc^2}{1+9c^2}+\frac{9ac^2}{1+9a^2}\le\frac{3\left(ab+bc+ca\right)}{2}\le\frac{\left(a+b+c\right)^2}{2}=\frac{1}{2}\)

Hay \(Q=1-\left(\frac{9ab^2}{1+9b^2}+\frac{9bc^2}{1+9c^2}+\frac{9ca^2}{1+9a^2}\right)\ge1-\frac{1}{2}=\frac{1}{2}\)

Dấu "=" xảy ra <=> \(a=b=c=\frac{1}{3}\)

Vậy \(Min_P=\frac{1}{2}\)đạt được khi \(a=b=c=\frac{1}{3}\)

\(\frac{a}{b}=\frac{c}{d}\Rightarrow\frac{a}{c}=\frac{b}{d}=\frac{9a}{9c}=\frac{16b}{16d}=\frac{9a-16b}{9c-16d}=\frac{9a+16b}{9c+16d}\Rightarrow\frac{9a-16b}{9a+16b}=\frac{9c-16d}{9c+16d}\)

P = 4a + 7b + 10c + \(\frac{4}{a}+\frac{1}{4b}+\frac{1}{9c}\)

P = \(3\left(a+2b+3c\right)+\left(a+\frac{4}{a}\right)+\left(b+\frac{1}{4b}\right)+\left(c+\frac{1}{9c}\right)\)

\(\ge3.4+2\sqrt{a.\frac{4}{a}}+2\sqrt{b.\frac{1}{4b}}+2\sqrt{c.\frac{1}{9c}}=\frac{53}{3}\)

Vây GTNN của P là \(\frac{53}{3}\)khi  \(a=1;b=\frac{1}{2};c=\frac{1}{3}\)

n=2 mới đúng

quên a=2 mới đúng, vì bđt côsi đ/k là a=b

Bít làm thì đã làm rùi

khó quá

\(\frac{a}{9b^2+1}=\frac{a\left(9b^2+1\right)-9ab^2}{9b^2+1}=a-\frac{9ab^2}{9b^2+1}\ge a-\frac{9ab^2}{2\sqrt{9b^2.1}}=\)

\(=a-\frac{9ab^2}{6b}=a-\frac{3ab}{2}\)

Tương tự với các biểu thức còn lại, kết hợp với 

\(ab+bc+ca\le\frac{1}{3}\left(a+b+c\right)^2\)

là được đáp án.

Bài 2 :

Ta có :

\(2a^2+16ab+7b^2=\left(2a+3b\right)^2-2\left(a-b\right)^2\le\left(2a+3b\right)^2\)

\(\Rightarrow P\ge\frac{25a^2}{2a+3b}+\frac{25b^2}{2b+3c}+\frac{c^2\left(a+3\right)}{a}\)

Áp dụng BĐT Cô - si ta có :

\(\frac{25a^2}{2a+3b}+2a+3b\ge10a\)

\(\frac{25b^2}{2b+3c}+2b+3c\ge10b\)

\(\frac{c^2\left(a+3\right)}{a}=\left(c^2+1\right)+\left(\frac{3c^2}{a}+3a\right)-3a-1\ge2c+6c-3a-1=8c-3a-1\)

Khi đó :

\(P\ge\left(10-2a-3b\right)+\left(10b-2b-3c\right)+\left(8c-3a-1\right)\)

\(\Rightarrow P\ge5\left(a+b+c\right)-1=14\)

Vậy \(MinP=14\) khi a=b=c=1