Prove that ((2a+b+c)^2)/(2a^2+(b+c)^2)+((2b+c+a)^2)/(2b^2+(c+a)^2)+((2c+a+b))/(2c^2+(a+b)^2)<=8

Tamara Bryan 2022-07-21 Answered
Prove that ( 2 a + b + c ) 2 2 a 2 + ( b + c ) 2 + ( 2 b + c + a ) 2 2 b 2 + ( c + a ) 2 + ( 2 c + a + b ) 2 2 c 2 + ( a + b ) 2 8
Prove that
( 2 a + b + c ) 2 2 a 2 + ( b + c ) 2 + ( 2 b + c + a ) 2 2 b 2 + ( c + a ) 2 + ( 2 c + a + b ) 2 c 2 + ( a + b ) 2 8
MY ATTEMPT:I want to make a relation between a , b , c. By trial I found that if we put a = b = c = 1 then the above inequality holds(equality also holds). So by trial I assume that a + b + c = 3. After that the three functions become of the form of the function below: f ( x ) = ( x + 3 ) 2 2 x 2 + ( 3 x ) 2 . I calculate the function and found that : f ( x ) ( 4 x + 4 ). Am I do right . Anybody has other ideas.
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Answers (1)

kamphundg4
Answered 2022-07-22 Author has 20 answers
For non-negatives a, b and c let a + b + c = 3. Hence,
8 c y c ( 2 a + b + c ) 2 2 a 2 + ( b + c ) 2 = c y c ( 8 3 ( a + 3 ) 2 2 a 2 + ( 3 a ) 2 ) =
= 1 3 c y c ( a 1 ) ( 7 a 15 ) a 2 2 a + 3 = 1 3 ( c y c ( a 1 ) ( 7 a 15 ) a 2 2 a + 3 + 4 ( a 1 ) ) =
= c y c ( a 1 ) 2 ( 4 a + 3 ) 3 ( a 2 2 a + 3 ) 0
Also there is the following.
It's enough to prove that
( 2 a + b + c ) 2 2 a 2 + ( b + c ) 2 4 ( 4 a + b + c ) 3 ( a + b + c ) ,
which is ( 2 a b c ) 2 ( 5 a + b + c ) 0

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