Question 927383
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                I will solve part  (a)  of this problem.



If   3(a^2 + b^2 + c^2) = (a+b+c)^2,   then the relation between  a,  b,  c  is ?
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            It is nice  Math problem on equalities and inequalities,  pretty educative.

            Good for a  Math circle at a local high school.

            See the solution below.



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From  3(a^2+b^2+c^2) = (a+b+c)^2,  you can easily deduce, making FOIL, that

    a^2 + b^2 + c^2 = ab + ac + bc.     (1)


Next, take into account these well known remarkable inequalities

   ab <= {{{(a^2+b^2)/2}}},  ac <= {{{(a^2+c^2)/2}}},  bc <= {{{(b^2+c^2)/2}}}.


Each of these inequalities becomes EQUALITY if and only if the participating quantities are equal:

   a = b;  a = c;  b = c.



THEREFORE, (1) implies that  a = b = c.



It is the seeking relation between  "a", "b" and "c".



<U>ANSWER</U>.  The given equality  3(a^2 + b^2 + c^2) = (a+b+c)^2   is possible if and only if   a = b = c.
</pre>


Solved and explained.