if-the-sum-of-three-positive-real-numbers-is-equal-to-their-product-prove-that-at-least-one-of-the-numbers-is-larger-than-1-7-

Question Number 206355 by mr W last updated on 12/Apr/24

i f t h e s u m o f t h r e e p o s i t i v e r e a l

n u m b e r s i s e q u a l t o t h e i r p r o d u c t,

p r o v e t h a t a t l e a s t o n e o f t h e

n u m b e r s i s l a r g e r t h a n 1.7 .

Answered by A5T last updated on 12/Apr/24

a + b + c = a b c ⩾ 3 \sqrt[3]{a b c} \Rightarrow (a b c) ⩾ 3^{\frac{3}{2}}

O n e m u s t b e a t l e a s t \sqrt[3]{3^{\frac{3}{2}}} = \sqrt{3} \approx 1.732

Answered by mr W last updated on 12/Apr/24

s = a + b + c ⩾ 3 \sqrt[3]{a b c} = 3 \sqrt[3]{a + b + c} = 3 \sqrt[3]{s}

s^{\frac{2}{3}} ⩾ 3 \Rightarrow s ⩾ \sqrt{3^{3}} = \sqrt{27}

a t l e a s t o n e n u m b e r m u s t b e

⩾ \frac{s}{3} ⩾ \sqrt{3} > 1.7

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