131 replies to this topic

[DangHongPhuc]

38. ĐK: $a,b,c> 0$; $abc=1$

$\frac{1}{\sqrt{4a^{2}+a+4}+\frac{3}{2}}+\frac{1}{\sqrt{4b^{2}+b+4}+\frac{3}{2}}+\frac{1}{\sqrt{4c^{2}+c+4}+\frac{3}{2}}\leq \frac{2}{3}$

[DangHongPhuc]

39. ĐK:$a_{1},a_{2},...a_{n}$ là những số nguyên không cùng bằng không; $a_{1}+a_{2}+...+a_{n}=0$ và $k\in {1,2,...n}$

$\left | a_{1}+2a_{2}+...+2^{k-1}a_{k} \right |> \frac{2^{k}}{3}$

[DangHongPhuc]

40. ĐK: $a,b,c > 0$;  $a+b+c=3$

$\frac{a^{2}}{a+2b^{3}}+\frac{b^{2}}{b+2c^{3}}+\frac{c^{2}}{c+2a^{3}}\geq 1$

Edited by DangHongPhuc, 02-08-2016 - 17:12.

[loolo]

40. ĐK: $a,b,c > 0$;  $a+b+c=3$

$\frac{a^{2}}{a+2b^{3}}+\frac{b^{2}}{b+2a^{3}}+\frac{c^{2}}{c+2a^{3}}\geq 1$

hình như chỗ này lộn rồi bạn ơi

[DangHongPhuc]

hình như chỗ này lộn rồi bạn ơi

cảm ơn bạn nhé

[DangHongPhuc]

Mà ai thấy đề bài hay thì like hộ mình cái, ít like quá.

[Shin Janny]

9. ĐK $a,b,c> 0$, $a+b+c=1$

$a\sqrt{b}+b\sqrt{c}+c\sqrt{a}\leq \frac{1}{\sqrt{3}}$

$(a\sqrt{b}+b\sqrt{c}+c\sqrt{a})^{2}=(\sqrt{a}.\sqrt{ab}+\sqrt{b}.\sqrt{bc}+\sqrt{c}.\sqrt{ca})^{2}\leq (a+b+c)(ab+bc+ca)=(ab+bc+ca)\leq \frac{(a+b+c)^{2}}{3}=\frac{1}{3}$

$\Rightarrow$ đpcm

Edited by Shin Janny, 03-08-2016 - 00:29.

[DangHongPhuc]

mình sẽ đăng 54 bài của phần cơ bản lên trước, còn đến cuối tuần bài nào các bạn không giải được mình sẽ up đáp án sau

[DangHongPhuc]

41. ĐK: $a,b,c> 0$

$\frac{a^{2}+b^{2}+c^{2}}{ab+bc+ca}\geq \frac{ab}{b^{2}+bc+c^{2}}+\frac{bc}{c^{2}+ca+a^{2}}+\frac{ca}{a^{2}+ab+b^{2}}$

[DangHongPhuc]

42. ĐK: $a,b,c> 0$; $a^{2}+b^{2}+c^{2}=3$

$\frac{a^{3}}{2b^{2}+c^{2}}+\frac{b^{3}}{2c^{2}+a^{2}}+\frac{c^{3}}{2a^{2}+b^{2}}\geq 1$

[DangHongPhuc]

43. ĐK: $a,b,c> 0$; &a+b+c=1$

$\frac{1-2ab}{c}+\frac{1-2bc}{a}+\frac{1-2ca}{b}\geq 7$

[DangHongPhuc]

44. ĐK:$a_{1}< a_{2}< ...< a_{n}$; $n\geq 2$$a_{1}a_{2}^{4}+a_{2}a_{3}^{4}+...+a_{n}a_{1}^{4}\geq a_{2}a_{1}^{4}+a_{3}a_{2}^{4}+...+a_{1}a_{n}^{4}$

Edited by DangHongPhuc, 03-08-2016 - 16:47.

[DangHongPhuc]

45. ĐK: $x^{2}+y^{2}+z^{2}=2$

$x+y+z\leq xyz+2$

[DangHongPhuc]

46. ĐK: $a,b,c> 0$; $a+b+c=1$

$\frac{a}{\sqrt{b+c}}+\frac{b}{\sqrt{c+a}}+\frac{c}{\sqrt{a+b}}\geq \sqrt{\frac{3}{2}}$

[DangHongPhuc]

47. ĐK: $a,b,c> 0$

$\frac{1}{b(a+b)}+\frac{1}{c(b+c)}+\frac{1}{a(c+a)}\geq \frac{27}{2(a+b+c)^{2}}$

[DangHongPhuc]

48. ĐK: $a_{1},a_{2},...,a_{n}\in \left [ 0,1 \right ]$; $S=a_{1}^{3}+a_{2}^{3}+...+_{n}^{3}$

$\frac{a_{1}}{2n+1+S-a_{1}^{3}}+\frac{a_{2}}{2n+1+S-a_{2}^{3}}+...+\frac{a_{n}}{2n+1+S-a_{n}^{3}}\leq \frac{1}{3}$

[DangHongPhuc]

49. ĐK: $a,b,c> 0$; $(a+b)(b+c)(c+a)=1$

$ab+bc+ca\leq \frac{3}{4}$

[Shin Janny]

15. ĐK $a,b,c> 0$

$\frac{b+c}{a^{2}}+\frac{c+a}{b^{2}}+\frac{a+b}{c^{2}}\geq 2\left ( \frac{1}{a}+\frac{1}{b}+\frac{1}{c} \right )$

$\frac{b+c}{a^{2}}+\frac{c+a}{b^{2}}+\frac{a+b}{c^{2}}+\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$

$=a(\frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}})+b(\frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}})+c(\frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}})$

$=(a+b+c)(\frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}}) $

$\geq (a+b+c)\frac{(\frac{1}{a}+\frac{1}{b}+\frac{1}{c})^{2}}{3}=\frac{(a+b+c)(\frac{1}{a}+\frac{1}{b}+\frac{1}{c})(\frac{1}{a}+\frac{1}{b}+\frac{1}{c})}{3}$

$\geq \frac{3\sqrt[3]{abc}.\frac{3}{\sqrt[3]{abc}}.(\frac{1}{a}.\frac{1}{b}.\frac{1}{c})}{3}=3(\frac{1}{a}+\frac{1}{b}+\frac{1}{c})\Rightarrow$ đpcm

[Shin Janny]

16. ĐK: $a,b,c> 0$

$\frac{a+b+c}{\sqrt[3]{abc}}+\frac{8abc}{(a+b)(b+c)(c+a)}\geq 4$

$\frac{(a+b)+(b+c)+(c+a)}{2\sqrt[3]{abc}}+\frac{8abc}{(a+b)(b+c)(c+a)}$

$\geq \frac{3\sqrt[3]{(a+b)(b+c)(c+a)}}{2\sqrt[3]{abc}}+\frac{8abc}{(a+b)(b+c)(c+a)}$

$= \frac{\sqrt[3]{(a+b)(b+c)(c+a)}}{2\sqrt[3]{abc}}+\frac{\sqrt[3]{(a+b)(b+c)(c+a)}}{2\sqrt[3]{abc}}+\frac{\sqrt[3]{(a+b)(b+c)(c+a)}}{2\sqrt[3]{abc}}+\frac{8abc}{(a+b)(b+c)(c+a)}$

$\geq 4\sqrt[4]{\frac{\sqrt[3]{(a+b)(b+c)(c+a)}}{2\sqrt[3]{abc}}.\frac{\sqrt[3]{(a+b)(b+c)(c+a)}}{2\sqrt[3]{abc}}.\frac{\sqrt[3]{(a+b)(b+c)(c+a)}}{2\sqrt[3]{abc}}.\frac{8abc}{(a+b)(b+c)(c+a)}}$

$= 4$

[loolo]

19. ĐK: $a,b,c> 0$ $abc=1$

$\frac{a}{a^{2}+2}+\frac{b}{b^{2}+2}+\frac{c}{c^{2}+2}\leq 1$

http://diendantoanho...-1/#entry647858