Maximize the value of the function z = a b + c a + b + c ,where a , b , c are natural numbers and are all lesser than 2010 and not necessarily distinct from each other. Please provide a proof, and if possible a general technique. Thank you.

Question

Maximize the value of the function  z  =            a      b      +      c              a      +      b      +      c        ,where   a  ,  b  ,  c are natural numbers and are all lesser than 2010 and not necessarily distinct from each other. Please provide a proof, and if possible a general technique. Thank you.

Kendrick Fritz · Accepted Answer

z  =            a      b      +      c              a      +      b      +      c        =            a      b      −      a      −      b              a      +      b      +      c        +  1Clearly, this will be maximum if c minimum =1So,  z  ≤            a      b      −      a      −      b              a      +      b      +      1        +  1  =            a      b      +      1              a      +      b      +      1                          a        1                    b        1            +      1                      a        1            +              b        1            +      1       will be greater than                     a        2                    b        2            +      1                      a        2            +              b        2            +      1      if   (      a    1        a    2    −  1  )  (      b    1    −      b    2    )  +  (      b    1        b    2    −  1  )  (      a    1    −      a    2    )  +      a    1        b    1    −      a    2        b    2    &amp;gt;  0if   (      a    1        a    2    −  1  )  (      b    1    −      b    2    )  +  (      b    1        b    2    −  1  )  (      a    1    −      a    2    )  +  (      a    1    −      a    2    )      b    1    +      a    2    (      b    1    −      b    2    )  &amp;gt;  0if   (      a    1        a    2    −  1  +      a    2    )  (      b    1    −      b    2    )  +  (      b    1        b    2    −  1  +      b    1    )  (      a    1    −      a    2    )  &amp;gt;  0Clearly,             a      b      +      1              a      +      b      +      1       increases with the increment of   a  ,  b or both.So,             a      b      +      1              a      +      b      +      1       will be maximum if   a  ,  b are maximum.

Maximize the value of the function z = <mfrac> <mrow> a b +

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