cho a b c la số dương, biết a+b+c<=3. Tìm Pmin

\(P=\frac{a^2}{\left[\sqrt{b^3+8}-\left(c-1\right)^2\right]}+\frac{b^2}{\left[\sqrt{c^3+8}-\left(a-1\right)^2\right]}+\frac{c^2}{\left[\sqrt{a^3+8}-\left(b-1\right)^2\right]}\)

Tìm a, b biết

\(\left(a,b\right)=1\)và\(\frac{a+b}{a^2-ab+b^2}=\frac{8}{73}\)

cho a,b ,c deu duong .cmr

\(\frac{8}{\left(a+b\right)^2+4abc}+\frac{8}{\left(b+c\right)^2+4abc}+\frac{8}{\left(a+c\right)^2+4abc}+a^2+b^2+c^2\ge\frac{8}{a+3}+\frac{8}{b+3}+\frac{8}{c+3}\)

cho số thực a,b,c>0. CMR

\(\frac{8}{\left(a+b\right)^2+4abc}+\frac{8}{\left(b+c\right)^2+4abc}+\frac{8}{\left(c+a\right)^2+4abc}+a^2+b^2+c^2\ge\frac{8}{a+3}+\frac{8}{b+3}+\frac{8}{c+3}\)

Tìm min,max của P=xyz biết A= \(\frac{8-x^2}{16+x^4}+\frac{8-y^2}{16+y^4}+\frac{8-z^2}{16+z^4}\ge0.\)

Cho a;b;c >0 thỏa mã \(a+b+c\le3\)Tìm min P \(=\left(3+\frac{1}{a}+\frac{1}{b}\right)\left(3+\frac{1}{b}+\frac{1}{c}\right)\left(3+\frac{1}{c}+\frac{1}{a}\right)\)

Cho a, b, c > 0. Tìm k lớn nhất để:
a) \(\frac{k}{a^2+b^2}+\frac{1}{a^2}+\frac{1}{b^2}\ge\frac{8+2k}{\left(a+b\right)^2}\)
b) \(\frac{k}{a^3+b^3}+\frac{1}{a^3}+\frac{1}{b^3}\ge\frac{16+4k}{\left(a+b\right)^3}\)

\(\frac{6+a}{9-a^2}+...=\frac{6}{9-a^2}+...+\frac{a^2}{9a-a^3}\ge\frac{54}{27-a^2-b^2-c^2}+\frac{\left(a+b+c\right)^2}{9\left(a+b+c\right)-\left(a^3+b^3+c^3\right)}\)

\(\ge\frac{54}{27-2\left(a+b+c\right)+3}+\frac{9}{27-3\left(a+b+c\right)+6}=\frac{54}{24}+\frac{9}{24}=\frac{21}{8}\)

Cho các số thực a, b, c thỏa mãn a+b+c=3. Tìm GTLN của biểu thức:\(P=3\left(ab+bc+ca\right)+\frac{1}{2}\left(a-b\right)^2+\frac{1}{4}\left(b-c\right)^2+\frac{1}{8}\left(c-a\right)^2\)

Cho a,b,c là các số thỏa \(\left(\frac{-a}{2}+\frac{b}{3}+\frac{c}{6}\right)^3+\left(\frac{a}{3}+\frac{b}{6}-\frac{c}{2}\right)^3+\left(\frac{a}{6}-\frac{b}{2}+\frac{c}{3}\right)^3=\frac{1}{8}\)

Chứng minh rằng: \(\left(a-3b+2c\right)\left(2a+b-3c\right)\left(-3a+2b+c\right)=9\)