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Question Number 58462 by naka3546 last updated on 23/Apr/19

a, b, c, d \in R^{+}

a + b + c + d = 1

P r o v e t h a t :

a b c + b c d + c d a + d a b ⩽ \frac{1}{27} + \frac{176}{27} a b c d

Answered by tanmay last updated on 24/Apr/19

\frac{a + b + c + d}{4} ⩾ {(a b c d)}^{\frac{1}{4}}

a b c d ⩽ {(\frac{1}{4})}^{4} \to a b c d ⩽ \frac{1}{256}

c o n s i d e r i n g a b c d = \frac{1}{256}

(a b c + b c d + c d a + d a b) \times \frac{1}{4} ⩾ {(a^{3} b^{3} c^{3} d^{3})}^{\frac{1}{4}}

(a b c + b c d + c d a + d a b) ⩾ 4 {(a b c d)}^{\frac{3}{4}}

(a b c + b c d + c d a + d a b) ⩾ 4 {(\frac{1}{256})}^{\frac{3}{4}}

(a b c + b c d + c d a + d a b) ⩾ \frac{1}{16}

a b c + b c d + c d a + d a b = \frac{1}{16}

R H S

\frac{1}{27} + \frac{176}{27} a b c d

\frac{1}{27} + \frac{176 \times 1}{27 \times 256}

\frac{1}{27} (1 + \frac{11}{16})

= \frac{1}{16}

L H S = R H S

i h a v e s o l v e d c o n s i d e r i n g = s i g n i n ⩾ s i g n

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