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Rimba Erlangga

OSP SMA 2017 - Bagian Uraian No. 4

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Misalkan $a$, $b$, dan $c$ adalah bilangan-bilangan real yang nilai mutlaknya tidak lebih besar dari $1$. Buktikan bahwa \[\sqrt{|a-b|}+\sqrt{|b-c|}+\sqrt{|c-a|} \leq 2+\sqrt{2}\].

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Posted (edited)

WLOG $a\ge b\ge c$. Perhatikan bahwa $a- c \le 2$. Dengan CS atau QM-AM, perhatikan bahwa

 

$$ \sqrt{a-b} + \sqrt{b-c} + \sqrt{a-c} \le \sqrt{2 \{(a-b)+(b-c)\}}+\sqrt{a-c} = (1+\sqrt{2})\sqrt{a-c} \le 2+\sqrt{2}$$

Edited by blajaran
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