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sqing

#1 h Apr 2, 2025, 4:16 AM

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Let $a,b,c\geq 0$ and $a+b+c=3$ . Prove that
$\frac{a^2+1}{a^2+b+c-\frac{1}{2}}+\frac{b^2+1}{b^2+c+a-\frac{1}{2}}+\frac{c^2+1}{c^2+a+b-\frac{1}{2}} \leq \frac{12}{5}$ $\frac{a^2+2}{a^2+b+c+\frac{1}{2}}+\frac{b^2+2}{b^2+c+a+\frac{1}{2}}+\frac{c^2+2}{c^2+a+b+\frac{1}{2}} \leq \frac{18}{7}$ Equality holds when $(a,b,c)=(0,\frac{3}{2},\frac{3}{2})$ or $(a,b,c)=(1,1,1) .$

This post has been edited 2 times. Last edited by sqing, Apr 2, 2025, 4:52 AM

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sqing

#2 h Apr 2, 2025, 4:23 AM

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Let $a,b,c \geq 0$ and $a+b+c=3$ . Prove that
$\frac{a^2+1}{a^2+b+c-1}+\frac{b^2+1}{b^2+c+a-1}+\frac{c^2+1}{c^2+a+b-1} \leq 3$ $\frac{a^2+1}{a^2+b+c+1}+\frac{b^2+1}{b^2+c+a+1}+\frac{c^2+1}{c^2+a+b+1} \leq \frac{123}{76}$ $\frac{a^2+2}{a^2+b+c-1}+\frac{b^2+2}{b^2+c+a-1}+\frac{c^2+2}{c^2+a+b-1} \leq \frac{9}{2}$ $\frac{a^2+2}{a^2+b+c+1}+\frac{b^2+2}{b^2+c+a+1}+\frac{c^2+2}{c^2+a+b+1} \leq \frac{87}{38}$

This post has been edited 2 times. Last edited by sqing, Apr 2, 2025, 4:55 AM

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