Prove that 1 1 + a 2 + 1 1 + b 2 + 1 1 + c 2 ≤ 3 2 For positive real numbers with a + b + c = a b c prove that 1 1 + a 2 + 1 1 + b 2 + 1 1 + c 2 ≤ 3 2 I made the substitution a = tan ⁡ ( α ) , b = tan ⁡ ( β ) , c = tan ⁡ ( γ ) with the constraint α + β + γ = πMy inequality reduces to proving, cos 2 ⁡ ( α ) + cos 2 ⁡ ( β ) + cos 2 ⁡ ( γ ) ≤ 3 2 But I am stuck on it. Any help with either inequality would be appreciated.

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Prove that       1          1      +              a        2              +      1          1      +              b        2              +      1          1      +              c        2              ≤      3    2  For positive real numbers with   a  +  b  +  c  =  a  b  c prove that      1          1      +              a        2              +      1          1      +              b        2              +      1          1      +              c        2              ≤      3    2  I made the substitution   a  =  tan  ⁡  (  α  )  ,  b  =  tan  ⁡  (  β  )  ,  c  =  tan  ⁡  (  γ  ) with the constraint   α  +  β  +  γ  =  πMy inequality reduces to proving,       cos    2    ⁡  (  α  )  +      cos    2    ⁡  (  β  )  +      cos    2    ⁡  (  γ  )  ≤      3    2  But I am stuck on it. Any help with either inequality would be appreciated.

Kamora Greer · Accepted Answer

Nope, reduces to   ∑  cos not squared. Then just a standard Jensen&#039;s argument. ;)

Mohammad Cannon · Accepted Answer

By AM-GM      ∑          c      y      c            1          1      +              a        2              =      ∑          c      y      c            1                            a          b          c                          a          +          b          +          c                    +              a        2              =      ∑          c      y      c                          a        +        b        +        c                    a        (        a        +        b        )        (        a        +        c        )              =  =      ∑          c      y      c                          b        c                    (        a        +        b        )        (        a        +        c        )              ≤      1    2        ∑          c      y      c            (          b              a        +        b              +          c              a        +        c              )    =      1    2        ∑          c      y      c            (          b              a        +        b              +          a              a        +        b              )    =      3    2    .Done!

Prove that 1 <msqrt> 1 + a 2 </msup> </msqr

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