Very interesting inequalities

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Very interesting inequalities

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sqing

#1 h Mar 27, 2025, 4:31 AM

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Let $a,b,c\geq 1$ and $abc-\frac{1}{3}( ab+bc+ca)\leq 4.$ Prove that
$2\geq a+b+c-abc\geq-2$ $14\geq ab+bc+ca- abc\geq2$ $2 \geq a+b+c+ abc-(ab+bc+ca)\geq1$ $25 \geq ab+bc+ca+abc-(a+b+c)\geq1$ $10\geq ab+bc+ca+abc - 2(a+b+c)\geq -2$ $3\geq ab+bc+ca+abc- 3(a+b+c)\geq -5$

This post has been edited 2 times. Last edited by sqing, Mar 27, 2025, 5:31 AM

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41325 posts

sqing

#2 h Mar 27, 2025, 4:45 AM

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Let $a,b,c\geq 1$ and $a+b+c+2\leq abc.$ Prove that
$abc-\frac{1}{3}( a+b+c+ab+bc+ca) \geq \frac{5}{3}$ Let $a,b,c\geq 1$ and $a+b+c+2\geq abc.$ Prove that
$abc-\frac{1}{3}( a+b+c+ab+bc+ca) \geq -1$ $abc-\frac{1}{4}( a+b+c+ab+bc+ca) \geq -\frac{1}{2}$

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