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Interesting inequality
sqing   1
N Mar 21, 2025 by sqing
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Let $ a,b,c\geq \frac{1}{3}$ and $ a+b+c=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+8 . $ Prove that
$$ ab+bc +ca\leq 17+2\sqrt{73}$$Let $ a,b,c\geq \frac{1}{2}$ and $ a+b+c=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+8 . $ Prove that
$$ ab+bc +ca\leq \frac{469+115\sqrt{17}}{32}$$Let $ a,b,c\geq \frac{1}{5}$ and $ a+b+c=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+8 . $ Prove that
$$ ab+bc +ca\leq \frac{569+34\sqrt{281}}{25}$$
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sqing
Mar 21, 2025
sqing
Mar 21, 2025
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Interesting inequality
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sqing
#1 h  Mar 21, 2025, 2:37 AM
Y by
Let $ a,b,c\geq \frac{1}{3}$ and $ a+b+c=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+8 . $ Prove that
$$ ab+bc +ca\leq 17+2\sqrt{73}$$Let $ a,b,c\geq \frac{1}{2}$ and $ a+b+c=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+8 . $ Prove that
$$ ab+bc +ca\leq \frac{469+115\sqrt{17}}{32}$$Let $ a,b,c\geq \frac{1}{5}$ and $ a+b+c=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+8 . $ Prove that
$$ ab+bc +ca\leq \frac{569+34\sqrt{281}}{25}$$
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sqing
41299 posts
sqing
#2 h  Mar 21, 2025, 2:58 AM
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Let $ a,b,c\geq \frac{71}{90}$ and $ a+b+c=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+8 . $ Prove that
$$ ab+bc +ca\leq 27 $$
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