INEQUALITY'S BLOG

December 16, 2010

Problem 74: van khea

Filed under: Uncategorized — KKKVVV @ 10:35 am

Let a, b, c>0. Prove that:  \displaystyle \frac{ab}{\sqrt{c+a}}+\frac{bc}{\sqrt{a+b}}+\frac{ca}{\sqrt{b+c}}\leq \frac{\sqrt{2}}{\sqrt{3}}(a+b+c)^{\frac{3}{2}}-\frac{3}{\sqrt{2}}\sqrt{abc}

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Problem 73: van khea

Filed under: Uncategorized — KKKVVV @ 10:31 am

Let a, b, c>0. Prove that \displaystyle \biggl(\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+a}\biggl)\biggl(a^2+b^2+c^2-\frac{3abc}{a+b+c}\biggl)\geq ab+bc+ca

November 22, 2010

Problem 69: van khea

Filed under: Uncategorized — KKKVVV @ 2:37 pm

Let a, b, c>0. Prove that:
3(a^4+b^4+c^4)+a^2b^2+b^2c^2+c^2a^2\geq 4(a^3b+bc^3+c^3a)

Problem 68: van khea

Filed under: Uncategorized — KKKVVV @ 1:10 am

Let a, b, c>0; abc=8. Prove that:
\displaystyle \frac{a^3\sqrt{3a+4b+3c}}{4a+3b+4c}+\frac{b^3\sqrt{3a+3b+4c}}{4a+4b+3c}+\frac{c^3\sqrt{4a+3b+3c}}{3a+4b+4c}\geq \frac{24\sqrt{5}}{11}

Problem 67: van khea

Filed under: Uncategorized — KKKVVV @ 1:08 am

Let a, b, c>0 ; abc=8. Prove that
\displaystyle \frac{\sqrt[3]{7a+5b}}{c^4(2b+3c)}+\frac{\sqrt[3]{7b+5c}}{a^4(2c+3a)}+\frac{\sqrt[3]{7c+5a}}{b^4(2a+3b)}\geq \frac{3\sqrt[3]{3}}{40}

Problem 66: van khea

Filed under: Uncategorized — KKKVVV @ 1:07 am

Let a, b, c>0 ; abc=8. Prove that:
\displaystyle \frac{\sqrt[3]{a+2b}}{c^3\sqrt{2a+b}}+\frac{\sqrt[3]{b+2c}}{a^3\sqrt{2b+c}}+\frac{\sqrt[3]{c+2a}}{b^3\sqrt{2c+a}}\geq \frac{3}{8\sqrt[6]{6}}

Problem 65: van khea

Filed under: Uncategorized — KKKVVV @ 1:05 am

Let a, b, c be positive real numbers and satisfying abc=8. Prove that:
\displaystyle \frac{a^{10}\sqrt{a+b}}{b^2+c^2}+\frac{b^{10}\sqrt{b+c}}{c^2+a^2}+\frac{c^{10}\sqrt{c+a}}{a^2+b^2}\geq 3.2^8

Problem 64: van khea

Filed under: Uncategorized — KKKVVV @ 1:03 am

8) Let a, b, c be positive real numbers and satisfying abc=8. Prove that:
\displaystyle \frac{a^3\sqrt{b+c}}{a+b}+\frac{b^3\sqrt{c+a}}{b+c}+\frac{c^3\sqrt{a+b}}{c+a}\geq 12

Problem 63: van khea

Filed under: Uncategorized — KKKVVV @ 1:00 am

7) Let a, b, c>0 ដែល abc=1. Prove that:
\displaystyle P=\frac{a^4\sqrt[3]{5a+7b}}{\sqrt{a+b}}+\frac{b^4\sqrt[3]{5b+7c}}{\sqrt{b+c}}+\frac{c^4\sqrt[3]{5c+7a}}{\sqrt{c+a}}\geq 3\sqrt[3]{3}.\sqrt[6]{2}

Problem 62: van khea

Filed under: Uncategorized — KKKVVV @ 12:48 am

Let a, b, c>0 ; abc=1 . Prove that
\displaystyle \frac{a^3\sqrt{2a+b}}{(a+b)^2}+\frac{b^3\sqrt{2b+c}}{(b+c)^2}+\frac{c^3\sqrt{2c+a}}{(c+a)^2}\geq \frac{3\sqrt{3}}{4}

November 21, 2010

Problem 62: van khea

Filed under: Uncategorized — KKKVVV @ 6:29 am

6) Let a, b, c be positive real numbers satisfying abc=8 ។ Find Min(P):
\displaystyle P=\frac{a^5\sqrt{2a+b}}{a+b}+\frac{b^5\sqrt{2b+c}}{b+c}+\frac{c^5\sqrt{2c+a}}{c+a}

Problem 61: van khea

Filed under: Uncategorized — KKKVVV @ 6:20 am

5) Let a, b, c>0 ; a^3+b^3+c^3=3 ; x, y, z\in (0, 1) ; x^3+y^3+z^3=2 ។ Prove that:
\displaystyle \frac{a^3}{\sqrt{(1+ab^2)(1+xy^2)}}+\frac{b^3}{\sqrt{(1+bc^2)(1+yz^2)}}+\frac{c^3}{\sqrt{(1+ca^2)(1+zx^2)}}\geq \frac{9\sqrt{2}}{10}\sqrt{1+xyz}

Problem 60: van khea

Filed under: Uncategorized — KKKVVV @ 6:16 am

Let a, b, c>0 ; x, y, z\in [0, 1] .Prove that:
\displaystyle a^2\sqrt{2+x+y}+b^2\sqrt{2+y+z}+c^2\sqrt{2+z+x}\geq \frac{\sqrt{2}}{3}(a+b+c)^2\sqrt{1+\sqrt[3]{xyz}}

November 20, 2010

Problem 59: van khea

Filed under: Uncategorized — KKKVVV @ 9:12 am

Let a, b, c>0 ; ab+bc+ca=1. Prove that:
\displaystyle \frac{a}{\sqrt[3]{1+a^2}}+\frac{b}{\sqrt[3]{1+b^2}}+\frac{c}{\sqrt[3]{1+c^2}}\leq \sqrt[3]{\frac{9}{4}(a+b+c)}

August 10, 2010

Problem 048

Filed under: Uncategorized — KKKVVV @ 1:31 pm

(Van Khea): Let a, b, c>0. Prove that:

\displaystyle \biggl(\frac{a^3}{bc}+\frac{b^3}{ca}+\frac{c^3}{ab}\biggl)(a^3+b^3+c^3)\geq (ab+bc+ca)^2

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