Category Archives: Bất đẳng thức thi HSG Quốc gia

Bất đẳng thức thi HSG Quốc gia

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


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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


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Bất đẳng thức trong đề thi HSG Quốc gia (VMO) 2014

Đề bài
Tìm giá trị lớn nhất của biểu thức $$P=\frac{{{x}^{3}}{{y}^{4}}{{z}^{3}}}{({{x}^{4}}+{{y}^{4}}){{(xy+{{z}^{2}})}^{3}}}+\frac{{{y}^{3}}{{z}^{4}}{{x}^{3}}}{({{y}^{4}}+{{z}^{4}}){{(yz+{{x}^{2}})}^{3}}}+\frac{{{z}^{3}}{{x}^{4}}{{y}^{3}}}{({{z}^{4}}+{{x}^{4}}){{(zx+{{y}^{2}})}^{3}}}$$ với $x,y,z$ là các số thực dương. Continue reading


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