CMR : a,b,c >0

1,\(\dfrac{a^2}{b+c}+\dfrac{b^2}{a+c}+\dfrac{c^2}{a+b}\dfrac{>}{ }\dfrac{a+b+c}{2}\)

2,\(\dfrac{a+b}{a^2+b^2}+\dfrac{b+c}{b^2+c^2}+\dfrac{a+c}{a^2+c^2}\dfrac{< }{ }\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\)

CMR : a,b,c >0

\(\left(a^3+b^3+c^3\right).\left(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\right)\dfrac{>}{ }\left(a+b+c\right)^2\)

Cho tam giác ABC. Gọi \(l_a\) là độ dài đường phân giác kẻ từ A. CMR:

a,\(l_a=\dfrac{2bc.cos\dfrac{1}{2}}{b+c}\)

b,\(cos\dfrac{1}{2}=\sqrt{\dfrac{p\left(p-a\right)}{bc}}\)

Cho tam giác ABC có AM là trung tuyến, AM=AB. Cmr :

a, sinA=2sin(B-A)

b, cosC=3cotB

Cho tam giác ABC có \(a=6,b=4\sqrt{2},c=2\). M trên BC sao cho BM=2.

a,CosB=?

b,AM=?

Cho tam giác ABC có b-c=\(\dfrac{a}{2}\)

a, SinA=2sinB-2sinC

b, \(\dfrac{1}{h_a}=\dfrac{1}{h_b}-\dfrac{1}{h_c}\)

Cho tam giác ABC có a=3, b=6, c=\(\sqrt[]{17}\)

Cmr : \(\sin^2A+sin^2B=3sin^2C\)

Chứng minh :

a, \(\dfrac{a+b+c}{3}\dfrac{>}{ }\sqrt{\dfrac{ab+bc+ca}{3}}\) với a,b,c>0

b,\(\dfrac{a^2+b^2+c^2}{3}\dfrac{>}{ }\left(\dfrac{a+b+c}{3}\right)^2\)

c,\(\dfrac{x^2+2}{\sqrt{x^2+1}}\dfrac{>}{ }2\)

d,\(\dfrac{a^3+b^3}{2}\dfrac{>}{ }\left(\dfrac{a+b}{2}\right)^3\)

Giải các hpt sau :

\(1,\left\{{}\begin{matrix}x\sqrt{y}+y\sqrt{x}=30\\x\sqrt{x}+y\sqrt{y}=35\end{matrix}\right.\)

\(2,\left\{{}\begin{matrix}\left(x+y\right)\left(1+\dfrac{1}{xy}\right)=5\\\left(x^2+y^2\right)\left(1+\dfrac{1}{x^2y^2}\right)=9\end{matrix}\right.\)

\(3,\left\{{}\begin{matrix}x+y+\dfrac{1}{x}+\dfrac{1}{y}=5\\x^2+y^2+\dfrac{1}{x^2}+\dfrac{1}{y^2}=49\end{matrix}\right.\)