ViLQD03

1.Cho a,b,c >0. a+b+c=3.CMR:

a + ab + 2abc $\leq \frac{9}{2}$

2. Cho x,y,z>0. thỏa mãn: x(x + y + z) = 3yz.CMR:

$(x+y)^{3} + (x+z)^{3} + 3(x+y)(y+z)(z+x)\leq 5(y+z)^{3}$

Cho dãy số ( $x_{n}$) xác định như sau:

$\left\{\begin{matrix}x_{1}= \sqrt{30} & & & & \\ x_{n+1}= \sqrt{30x^{2}_ {n} +3x_{n} +2011} & & & \end{matrix}\right.$    $\forall n\notin N*$

Tìm lim$\frac{x_{n+1}}{x_{n}}$

Cho x,y,z>0. CMR:

P= $\frac{2xy}{(z+x)(z+y)} + \frac{2yz}{(x+y)(x+z)} + \frac{3zx}{(y+z)(y+x)} \geq \frac{5}{3}$

Cho x,y,z là các số thực dương, x + y + z = 3 , a $\geq$ 1. CMR

$\frac{1}{a^{x}}+\frac{1}{a^{y}}+\frac{1}{a^{z}} \geq \frac{x}{a^{x}} +\frac{y}{a^{y}}+\frac{z}{a^{z}}$

1)  $sin^{8}x +cos^{8}x=2(sin^{10}x + cos^{10}x)$ + $\frac{5}{4}cos2x$

2)  $cos^{7}x+sin^{4}x=1$

3)  $2sin3x - sin4x - cot^{2}x=3$