abcde0101

Bai 1: Cho $x,y\in \mathbb{Q},\neq0$ va $n\in \mathbb{Z}^{+}$
CM: $x\sqrt{n}+y\sqrt{n+1}\in \mathbb{I}$
Bai 2: CM: Khong co so $x,y\in \mathbb{Q}$ nao thoa man dieu kien sau:
$\left ( x+y\sqrt{3} \right )^{n}=\sqrt{1+\sqrt{3}}$ $(n\in \mathbb{N})$

Bai 1:Cho $\left ( x+\sqrt{1+y^{2}} \right ) \left ( y+\sqrt{1+x^{2}} \right )=1$
CM:$\left ( x+\sqrt{1+x^{2}} \right ) \left ( y+\sqrt{1+y^{2}} \right )=1$
Bai 2:Tim cac so $x$ sao cho:
$F=\frac{\sqrt{x}}{x\sqrt{x}-3\sqrt{x}+3}$$\in \mathbb{Z}$
Bai 3:CM: Voi moi $n\in \mathbb{Z}^{+}$ ta co:

a,$\left [ \sqrt{n}+\sqrt{n+1} \right ]=\left [ \sqrt{4n+2} \right ]$

b,$\left [ \sqrt{n}+\sqrt{n+1}\sqrt{n+2} \right ]=\left [ \sqrt{9n+8} \right ]$

Tim 4 c/s tan cung cua:
a, 1354768368999x93664736753x4663648386x75765x878x98
b, 1996²⁰¹²

Bài 1 :
Cho $P\left ( x \right )=ax^{2}+bx+c$ thỏa mãn:
$\left | P\left ( x \right ) \right |\leq 1$ khi $\left | x \right |\leq 1$
CM: Nếu $\left | x \right |\leq 1$ thì $\left | cx^{2}+bx+a \right |\leq 2$
Bài 2 :
Cho $x\in\mathbb{R}$ thỏa mãn: $x^{5}-x^{3}+x=2$
CM: $3< x^{6}< 4$
Bài 3 :
Cho $a,b> 0$ và $a^{3}+b^{3}=a-b$
CM: $a^{2}+b^{2}+\frac{1}{2}.ab< 1$
Bài 4 :
Cho $ad-bc=1$.
CM: $\left ( a^{2}+b^{2}+c^{2}+d^{2}+ac+bd \right )^{2}\geq 3$