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Ellie Benjamin 2022-07-05 Answered
Prove 1 ( x + 2 y ) 2 1 x y + y z + z x
Let x , y , z > 0 Prove that:
1 ( x + 2 y ) 2 + 1 ( y + 2 z ) 2 + 1 ( z + 2 x ) 2 1 x y + y z + z x
I tried to apply Cauchy - Schwarz's inequality but I couldn't prove this inequality!
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Answers (1)

gozaderaradiox5
Answered 2022-07-06 Author has 19 answers
a = x + 2 y , b = y + 2 z , c = z + 2 x x = a 2 b + 4 c 9 , y = b 2 c + 4 a 9 , z = c 2 a + 4 b 9
the inequality become:
1 a 2 + 1 b 2 + 1 c 2 27 5 ( a b + b c + a c ) 2 ( a 2 + b 2 + c 2 )
now use UVW method:
3 u = a + b + c , 3 v 2 = a b + b c + a c , w 3 = a b c u v w
( 3 v 2 ) 2 6 u w 3 w 6 3 3 v 2 2 u 2 w 6 + 2 u ( 3 v 2 2 u 2 ) w 3 3 v 4 ( 3 v 2 u 2 ) 0
let w 3 = x , f ( x ) = x 2 + 2 u ( 3 v 2 2 u 2 ) x 3 v 4 ( 3 v 2 2 u 2 )
2 u ( 3 v 2 2 u 2 ) 0
f m a x ( x ) = f ( w 3 | w = v ) = f ( v 3 ), when
w = v u = v = w f ( v 3 ) = 0 f ( x ) 0
when u = v = w a = b = c x = y = z
QED.

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