cityhuntervp

Chứng minh các phương trình sau không có nghiệm nguyên?

a) $3x^2-4y^2=13$

b) $19x^2+28y^2=2001$

c) $x^2=2y^2-8y^2+3$

d) $x^5-5x^3+4x=24(5y+1)$

e) $3x^5-x^3+6x^2-18x=2001$

Tìm $x\in \mathbb{Z}$: $\sqrt{x+2\sqrt{x+...+2\sqrt{x+2\sqrt{3x}}}}=x$

1) $4\sqrt{x^2+x+1}=1+5x+4x^2-2x^3-x^4$

2) $\sqrt{x(x-1)}+\sqrt{x(x+2)}=2\sqrt{x^2}$

3) $\sqrt{x+3}+2x\sqrt{x+1}=2x+\sqrt{x^2+4x+3}$

4) $729x^4+8\sqrt{1-x^2}=36$

5) $\sqrt{x^2-8x+15}+\sqrt{x^2+2x-15}=\sqrt{x^2-9x+18}$

6) $4x^2+\sqrt{2x+3}=8x+1$

7) $\sqrt{2x^2+8x+6}+\sqrt{x^2-1}=2x+2$

8) $2x^2-6x+10-5(x-2)\sqrt{x+1}=0$

9) $\sqrt{x^2+10x+21}=3\sqrt{x+3}+2\sqrt{x+7}-6$

10) $\sqrt{x^2-x-2}-2\sqrt{x-2}+2=\sqrt{x+1}$

11) $x-2\sqrt{x-1}-(x-1)\sqrt{x}+\sqrt{x^2-x}=0$

12) $\sqrt[3]{x+1}+\sqrt[3]{x+2}=1+\sqrt[3]{x^2+3x+2}$

13) $\sqrt{x}+\sqrt{x+1}-\sqrt{x^2+x}=1$

14) $\sqrt{x+1}+2(x+1)=x-1+\sqrt{1-x}+3\sqrt{1-x^2}$

15) $x+2\sqrt{7-x}=2\sqrt{x-1}+\sqrt{-x^2+8x-7}+1$

16) $\sqrt[3]{x+1}+\sqrt[3]{x^2}=\sqrt[3]{x}+\sqrt[3]{x^2+x}$

17) $(x+3)\sqrt{10-x^2}=x^2-x-12$

18) $\dfrac{x^2}{\sqrt{3x-2}}-\sqrt{3x-2}=1-x$

19) $3x^2+3x+2=(x+6)\sqrt{3x^2-2x-3}$

20) $x^2+x+2=(3x-2)\sqrt{x+1}$

Giải phương trình: $2cos(3x-\frac{\pi}{4})+\frac{\sqrt{2}}{2}-\frac{\sqrt{6}}{2}=0$

Chứng minh rằng $1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{n}}<2\sqrt{n}$ đúng với mọi $n \in N^*$