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#1 h Nov 17, 2017, 4:02 PM • 2 Y

Y by Adventure10, Mango247

Let $a,b,c > 0$ such that $ab + bc + ca = 3$ . Prove that ${a^3} + {b^3} + {c^3} \ge {a^3}{b^3} + {b^3}{c^3} + {c^3}{a^3}$

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#2 h Nov 17, 2017, 4:17 PM • 1 Y

Y by Adventure10

False. Let $a = b = \frac{3}{2}$ and $c = \frac{1}{4}$ .

Or better yet, let $a = b = \sqrt{3} - \varepsilon_1$ , $c = \varepsilon_2$ for some small $\varepsilon_i$ .

This post has been edited 1 time. Last edited by scrabbler94, Nov 17, 2017, 4:17 PM

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sqing

#3 h Apr 17, 2025, 1:20 PM

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Let $a,b,c\ge 0$ and $a^2+b^2+c^2\le 3.$ Prove that $a^3+b^3+c^3\ge a^3b^3+b^3c^3+c^3a^3.$ Let $a,b,c\geq0$ and $a+b+c=3$ . Prove that
$9(a^2+b^2+c^2) \geq 8(a^2b^2+c^2a^2+c^2b^2)$

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#4 h Apr 19, 2025, 9:27 AM

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sqing wrote:

Let $a,b,c\ge 0$ and $a^2+b^2+c^2\le 3.$ Prove that $a^3+b^3+c^3\ge a^3b^3+b^3c^3+c^3a^3.$ How to do this, I have tried $P.Q.R$ but can't solve the problem

This post has been edited 1 time. Last edited by MathsII-enjoy, Apr 19, 2025, 9:27 AM

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