Nhóm II: BĐT 3 biếnBài 1: \[\begin{array}{l}
\left\{ \begin{array}{l}
a;b;c > 0 \\
ab + bc + ca = 3 \\
\end{array} \right. \\
Prove:\sum {\frac{{{a^2}}}{{2{a^2} + bc}}} \ge abc \\
\end{array}\]
Bài 3:\[\begin{array}{l}
x;y;z > 0 \\
Prove:\sum {\frac{{2\sqrt x }}{{{x^3} + {y^2}}}} \le \sum {\frac{1}{{{x^2}}}} \\
\end{array}\]
Bài 4:\[\begin{array}{l}
\left\{ \begin{array}{l}
x;y;z > 0 \\
xyz = 1 \\
n \in {\mathbb{N}^*} \\
\end{array} \right. \\
Prove:\sum {{{\left( {\frac{{1 + x}}{2}} \right)}^n}} \ge 3 \\
\end{array}\]
Bài 5:\[\begin{array}{l}
\left\{ \begin{array}{l}
a;b;c \ge 0 \\
a + b + c = 3 \\
\end{array} \right. \\
Prove:\prod {\left( {{a^2} - ab + {b^2}} \right)} \le 12 \\
\end{array}\]
Bài 6:\[\begin{array}{l}
\left\{ \begin{array}{l}
a;b;c \ge 0 \\
{a^2} + {b^2} + {c^2} = 3 \\
\end{array} \right. \\
\max \left( {P = a{b^2} + b{c^2} + c{a^2} - abc} \right) = ? \\
\end{array}\]
Bài 7:\[\begin{array}{l}
a;b;c > 0 \\
Prove:\sum {\frac{{ab}}{{a + 3b + 2c}}} \le \frac{{a + b + c}}{6} \\
\end{array}\]
Bài 9:\[\begin{array}{l}
x,y,z:|x| \le 1;|y| \le 1;|z| \le 1 \\
Prove:\sum {\sqrt {1 - {x^2}} } \le \sqrt {9 - {{\left( {x + y + z} \right)}^2}} \\
\end{array}\]
Bài 10:\[\begin{array}{l}
a;b;c > 0 \\
\min \left( {L = \sum {\frac{{\sqrt {{a^2} + 1} .\sqrt {{b^2} + 1} }}{{\sqrt {{c^2} + 1} }}} } \right) = ? \\
\end{array}\]
Bài 11:\[\begin{array}{l}
a;b;c > 0 \\
Prove:\sum {\sqrt {{a^2} + {{\left( {1 - b} \right)}^2}} } \ge \frac{{3\sqrt 2 }}{2} \\
\end{array}\]
Bài 12:\[\begin{array}{l}
x;y > 0 \\
\min \left( {f\left( {x;y} \right) = \frac{{{{\left( {x + y} \right)}^3}}}{{x{y^2}}}} \right) = ? \\
\end{array}\]
Bài 13:\[\begin{array}{l}
\left\{ \begin{array}{l}
x;y;z > 0 \\
x + y + z = \frac{3}{2} \\
\end{array} \right. \\
Proove:\sum {\frac{{\sqrt {{x^2} + xy + {y^2}} }}{{4yz + 1}}} \ge \frac{{3\sqrt 3 }}{4} \\
\end{array}\]
Bài 14:\[\begin{array}{l}
a;b;c > 0 \\
Proove:a + b + c \ge \frac{{a - b}}{{b + 2}} + \frac{{b - c}}{{c + 2}} + \frac{{c - a}}{{a + 2}} \\
\end{array}\]
Bài 15:\[\begin{array}{l}
\left\{ \begin{array}{l}
x;y;z > 0 \\
x + y + z = 3 \\
\end{array} \right. \\
Prove:\sum {{x^2}} + \sum {xy} \ge 6 \\
\end{array}\]
Bài 17:\[\begin{array}{l}
\left\{ \begin{array}{l}
a;b;c > 0 \\
a + b + c = 3 \\
\end{array} \right. \\
Prove:\sum {\frac{{a\left( {a - 2b + c} \right)}}{{ab + 1}}} \ge 0 \\
\end{array}\]
Bài 41:\[\begin{array}{l}
\left\{ \begin{array}{l}
a;b;c > 0 \\
a + b + c = 3 \\
\end{array} \right. \\
\min \left( {\sum {\frac{{{a^5}}}{{{b^3} + {c^2}}}} + \sum {{a^4}} } \right) = ? \\
\end{array}\]
Bài 43:\[\begin{array}{l}
\left\{ \begin{array}{l}
a;b;c > 0 \\
abc = 1 \\
\end{array} \right. \\
Prove:\sum {\frac{1}{{{a^4}(b + c)}}} \ge \frac{3}{2} \\
\end{array}\]
Bài 44:\[\begin{array}{l}
a;b;c > 0 \\
Proove:\sum {\frac{{{a^2}}}{b}} + \sum {\frac{{{b^2}}}{a}} \ge \sum {\sqrt {2({a^2} + {b^2})} } \\
\end{array}\]
Bài 45:\[\begin{array}{l}
\left\{ \begin{array}{l}
a;b;c \in \left[ { - 1;1} \right] \\
a + b + c = 0 \\
\end{array} \right. \\
Proove:{a^4} + {b^3} + {c^2} \le 2 \\
\end{array}\]
Bài 47:\[\begin{array}{l}
x;y;z > 0 \\
Prove:\frac{3}{{\sum x y}} + \frac{2}{{\sum {{x^2}} }} > 14 \\
\end{array}\]
Bài 49: \[\begin{array}{l}
x;y;z > 0:\sum {\frac{1}{x}} = 4 \\
Prove:\sum {\frac{1}{{2x + y + z}}} \le 1 \\
\end{array}\]
Bài 52:\[\begin{array}{l}
x;y;z > 0 \\
Proove:16xyz(x + y + z) \le 3\sqrt[3]{{\prod {{{(x + y)}^4}} }} \\
\end{array}\]
Bài 54: $$a,b,c>0; x+y+z=1$$
$$Prove: \sum \frac{x^2+1}{y^2+1}\leq \frac{7}{2}$$
Bài 56: $$a\geq 8;b\geq 9;c\geq 10; a^2+b^2+c^2=266$$
$$Prove: \sum a\geq 28$$
Bài 58:$$a,b,c>0$$
$$Min(Q=\frac{a+b}{a+b+c}+\frac{b+c}{c+b+4a}+\frac{c+a}{c+a+16b})?$$
Bài 59: $$\sum a=1$$
$$Min(A=\sqrt{8\prod (a^2+b^2)}+\frac{9}{\sum ab})$$
Bài 61: $$min(T=\sum \frac{a^2}{a^2+(b+c)^2})$$
Bài 62: $$a,b,c>0$$
$$Prove: \sum \sqrt{\frac{a^3}{a^3+(b+c)^3}}\geq 1$$
Bài 63:$$min(\sum \frac{x^2}{(x+y)(y+z)}$$
Bài 65: $$a,b,c>0$$
$$Prove: \sum \sqrt[3]{1+a^3}\geq \sqrt[3]{27+(a+b+c)^3}$$
Bài 66: $$a,c,b>0 ;a^2+b^2+c^2=3$$
$$Prove: \sum \frac{1}{a^2}\geq \sum a^2$$
Bài 69: Cho x,y,z thỏa mãn $x^3+y^3+z^3=1$
$$Prove: \sum \frac{x^2}{\sqrt{1-x^2}}\geq 2$$
Bài 72: Cho các số thực x,y,z thỏa $x+y+z+xy+xz+yz=6$
$$Prove: \sum x^2\geq 3$$
Bài 73: a,b,c 3 cạnh $\Delta$
$$Prove: \sum \frac{1}{a+b-c}\geq \sum \frac{1}{a}$$
Bài 75: $$a,b,c>$$
$$Prove: \sum \frac{a^2}{(b-c)^2}\geq 2$$
Bài 81: $$\sum ab=3$$
$$Proove:\sum \frac{1}{a^2+2}\leq 1$$
Bài 88: $$a,b,c>0$$
$$Prove: \frac{2(\sum a^3)}{\prod a}+\frac{9(\sum a)^2}{\sum a^2}\geq 33$$
Bài 91: $$x,y,z \neq 1,0$$
$$Prove: \sum (\frac{x}{x-1})^2\geq 1$$
Bài 93: $$a,b,c>0$$
$$Prove: \sum \sqrt{1+a^2}\geq \sqrt{3^2+(\sum a)^2}$$
Bài 94: $$a,b,c>0$$
$$Prove:\sum \frac{ab}{2c+a+b}\leq \frac{1}{4}(\sum a)$$
Bài 95: $$abc<1$$
$$Prove: \sum \frac{1}{a+1+ab}<1$$
Bài 97: $$Prove: \sum a^2\geq \sum ab+\frac{(a-b)^2}{26}+\frac{(b-c)^2}{6}+\frac{(c-a)^2}{209}$$
Bài 98: $$a>0;b<0$$
$$Proove: \frac{1}{a}\geq \frac{2}{b}+\frac{8}{2a-b}$$
Bài 99: $$\frac{a}{1+a}+\frac{2b}{1+b}=1$$
$$Prove: ab^2\leq \frac{1}{8}$$
Bài 100: $$a>b>c>0$$
$$Prove: \sum a^3b^2>\sum a^2b^3$$
Bài 102: $$a,b,c>0$$
$$Proove:\sum \frac{a^3}{a+b+2c}\geq \frac{1}{4}(\sum a^2)$$
Bài 104: $$a\geq b\geq c> 0$$
$$Prove: \sum \frac{1}{(a+b)^2}\geq \frac{2}{(a+c)(b+c)}+\frac{1}{4ab}$$
Bài 105: $$a+b+c=1$$
$$Prove:\sum \frac{1}{3b+a}\geq \sum \frac{1}{a+2b+c}$$
Bài viết đã được chỉnh sửa nội dung bởi Ispectorgadget: 19-01-2012 - 21:28