Tag Archives: AM-GM

Bất đẳng thức trong đề thi HSG Quốc gia (VMO) 2015

Đề bài

Cho $a,b,c$ là các số thực không âm. Chứng minh rằng $$3(a^2+b^2+c^2) \ge (a+b+c)(\sqrt{ab} + \sqrt{bc} + \sqrt{ca}) + (a-b)^2+(b-c)^2+(c-a)^2 \ge (a+b+c)^2.$$ Continue reading


Deprecated: category_link is deprecated since version 2.5.0! Use term_link instead. in /home/nginx/domains/tanghaituan.com/public/wp-includes/functions.php on line 5088

Deprecated: tag_link is deprecated since version 2.5.0! Use term_link instead. in /home/nginx/domains/tanghaituan.com/public/wp-includes/functions.php on line 5088

Deprecated: tag_link is deprecated since version 2.5.0! Use term_link instead. in /home/nginx/domains/tanghaituan.com/public/wp-includes/functions.php on line 5088

Deprecated: tag_link is deprecated since version 2.5.0! Use term_link instead. in /home/nginx/domains/tanghaituan.com/public/wp-includes/functions.php on line 5088

Deprecated: tag_link is deprecated since version 2.5.0! Use term_link instead. in /home/nginx/domains/tanghaituan.com/public/wp-includes/functions.php on line 5088
This entry was posted in Bất đẳng thức thi HSG Quốc gia and tagged , , , on by .

Bất đẳng thức chọn HSG Quốc Gia – Hà Nội – 2014 – 2015

Bài toán: Cho $a,b,c$ là các số thực dương thỏa mãn $$a^2+b^2+c^2 = 2(ab+bc+ca).$$ Tìm giá trị nhỏ nhất của $$P=a+b+c+\dfrac{1}{abc}-\dfrac{9}{a+b+c}.$$

Trích đề thi chọn HSG Quốc Gia môn Toán – TP Hà Nội – 2014-2015 Continue reading


Deprecated: category_link is deprecated since version 2.5.0! Use term_link instead. in /home/nginx/domains/tanghaituan.com/public/wp-includes/functions.php on line 5088

Deprecated: tag_link is deprecated since version 2.5.0! Use term_link instead. in /home/nginx/domains/tanghaituan.com/public/wp-includes/functions.php on line 5088

Deprecated: tag_link is deprecated since version 2.5.0! Use term_link instead. in /home/nginx/domains/tanghaituan.com/public/wp-includes/functions.php on line 5088

Deprecated: tag_link is deprecated since version 2.5.0! Use term_link instead. in /home/nginx/domains/tanghaituan.com/public/wp-includes/functions.php on line 5088
This entry was posted in Bất đẳng thức thi HSG Quốc gia and tagged , , on by .

Đề bài: Cho các số thực không âm $a,b,c$. Chứng minh rằng :
$$\sqrt{\dfrac{a(b+c)}{a^2+bc}}+\sqrt{\dfrac{b(c+a)}{b^2+ca}}+\sqrt{\dfrac{c(a+b)}{c^2+ab}}\ge 2$$ Continue reading

Bất đẳng thức thi thử lần 5 – 2013 – Diễn đàn Toán phổ thông

Đề bài
Cho $a,b,c$ là các số thực dương. Tìm giá trị nhỏ nhất của
\[P=\sqrt{\frac{(a+b+c)(ab+bc+ca)}{abc}}+\frac{4bc}{{{(b+c)}^{2}}}\] Continue reading


Deprecated: category_link is deprecated since version 2.5.0! Use term_link instead. in /home/nginx/domains/tanghaituan.com/public/wp-includes/functions.php on line 5088

Deprecated: tag_link is deprecated since version 2.5.0! Use term_link instead. in /home/nginx/domains/tanghaituan.com/public/wp-includes/functions.php on line 5088
This entry was posted in Bất đẳng thức trong các đề thi thử Đại học and tagged on by .

Đề bài: Chứng minh rằng nếu $a,b,c$ là các số thực dương thì: $$\dfrac{(a+b+c)^2}{a^2+b^2+c^2}+\dfrac{1}{2} \left( \dfrac{a^3+b^3+c^3}{abc}-\dfrac{a^2+b^2+c^2}{ab+bc+ca} \right)\geq4 .$$ Continue reading

Đề bài: Chứng minh rằng nếu $a,b,c$ là các số thực dương thì: $$\dfrac{(a+b+c)^2}{a^2+b^2+c^2}+\dfrac{1}{2} \left( \dfrac{a^3+b^3+c^3}{abc}-\dfrac{a^2+b^2+c^2}{ab+bc+ca} \right)\geq4 .$$

Continue reading

Olympic 30-4 THPT chuyên Lê Quý Đôn, Bình Định

Đề bài:
Cho $x, y, z$ là ba số thực dương. Chứng minh rằng: $$\frac{1}{3}\left(\frac{yz}{x^2}+\frac{zx}{y^2}+ \frac{xy}{z^2}\right)+\left(\frac{xyz(x+y+z)}{x^2y^2+y^2z^2+z^2x^2}\right)^2 \ge 2$$ Continue reading

Deprecated: category_link is deprecated since version 2.5.0! Use term_link instead. in /home/nginx/domains/tanghaituan.com/public/wp-includes/functions.php on line 5088

Deprecated: tag_link is deprecated since version 2.5.0! Use term_link instead. in /home/nginx/domains/tanghaituan.com/public/wp-includes/functions.php on line 5088

Deprecated: tag_link is deprecated since version 2.5.0! Use term_link instead. in /home/nginx/domains/tanghaituan.com/public/wp-includes/functions.php on line 5088
This entry was posted in Bất đẳng thức and tagged , on by .

Arhimede International Mathematics Competition 2008

Problem
Let $x,y$ be reals s.t. $x^2y^2\leq1$ and $n$ a natural number. Prove that:
$$ (x^n+y)^2+y^2\geq\dfrac{1}{n+2}(x^2+y^2)^n$$ Continue reading


Deprecated: category_link is deprecated since version 2.5.0! Use term_link instead. in /home/nginx/domains/tanghaituan.com/public/wp-includes/functions.php on line 5088

Deprecated: tag_link is deprecated since version 2.5.0! Use term_link instead. in /home/nginx/domains/tanghaituan.com/public/wp-includes/functions.php on line 5088

Deprecated: tag_link is deprecated since version 2.5.0! Use term_link instead. in /home/nginx/domains/tanghaituan.com/public/wp-includes/functions.php on line 5088
This entry was posted in Bất đẳng thức trong các kì thi HSG and tagged , on by .