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):
Answer by ikleyn(52903)  (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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