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41457 posts

sqing

#1 h Mar 30, 2025, 9:22 AM

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Let $a,b,c$ be reals such that $a+b+c=0.$ Prove that
$\frac{a+b+1}{c^2+c+2} + \frac{a+c+1}{b^2+b+2} \leq 2$ $\frac{a+b+1}{c^2+c+1} + \frac{a+c+1}{b^2+b+1} \leq 2+\frac{4}{\sqrt{3}}$

This post has been edited 1 time. Last edited by sqing, Mar 30, 2025, 9:27 AM

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sqing

#2 h Mar 30, 2025, 9:45 AM

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Let $a,b,c$ be reals such that $a+b+c=0.$ Prove that
$\frac{a+b+1}{c^2+c+k} + \frac{a+c+1}{b^2+b+k} \leq \frac{2(3+2\sqrt{k+2})}{4k-1}$ $\frac{a+b+2}{c^2+c+k} + \frac{a+c+2}{b^2+b+k} \leq \frac{2(5+2\sqrt{k+6})}{4k-1}$ Where $k\geq 1.$

This post has been edited 1 time. Last edited by sqing, Mar 30, 2025, 10:01 AM

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#5 h Mar 31, 2025, 12:21 PM

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Let $a,b,c$ be reals such that $a+b+c=0.$ Prove that :
$\frac{a+b+m}{c^2+c+n} + \frac{a+c+m}{b^2+b+n} \leq \frac{2(2m+1+2\sqrt{m^2+m+n})}{4n-1}$ Where $m\in R$ and $n> \frac{1}{4}.$

This post has been edited 2 times. Last edited by SunnyEvan, Mar 31, 2025, 12:38 PM

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sqing

#6 h Mar 31, 2025, 12:32 PM

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Very nice.Thanks.
Let $a,b,c$ be reals such that $a+b+c=0.$ Prove that :
$\frac{a+b+m}{c^2+c+n} + \frac{a+c+m}{b^2+b+n} \leq \frac{2(2m+1+2\sqrt{m^2+m+n})}{4n-1}$ Where $m \in R$ and $n> \frac{1}{4}.$

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#7 h Mar 31, 2025, 12:34 PM

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Thanks for your rigorous corrections and guidance ! :-D

:-D

$m\in R$

This post has been edited 1 time. Last edited by SunnyEvan, Mar 31, 2025, 12:36 PM

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sqing

#8 h Mar 31, 2025, 12:46 PM

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You're welcome.

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