SOLUTION: (a) If 3(a^2 + b^2 + c^2) = (a+b+c)^2, then the relation between a,b,c is ? (b) If P/a + q/b + r/c =1 and a/p + b/q +c/r =0, where p,q,r and a,b,c are non-zero, then the value o

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Question 927383: (a) If 3(a^2 + b^2 + c^2) = (a+b+c)^2, then the relation between a,b,c is ?
(b) If P/a + q/b + r/c =1 and a/p + b/q +c/r =0, where p,q,r and a,b,c are non-zero, then the value of p^2 / a^2 + q^2 / b^2 + r^2/c^2 is
Found 2 solutions by JBnovelwriter, ikleyn:
Answer by JBnovelwriter(34)   (Show Source): You can put this solution on YOUR website!
2014...purging...
Answer by ikleyn(52909)   (Show Source): You can put this solution on YOUR website!
.


                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. 


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 <= ,  ac <= ,  bc <= .


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".



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


Solved and explained.



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