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[SOLVED] Find a+b+c

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anemone

MHB POTW Director
Staff member
Feb 14, 2012
3,802
$a,\,b$ and $c$ are positive integers that satisfy the inequality $ab+3b+2c>a^2+b^2+c^2+3$. Evaluate $a+b+c$.
 

kaliprasad

Well-known member
Mar 31, 2013
1,331
We have


$a^2 + b^2 + c^2 + 3 < ab + 3b +2c$


or $(a^2-ab) + (b^2 - 3b) + (c^2 - 2c) + 3 < 0$


or $(a-\frac{b}{2})^2 - \frac{b^2}{4} + (b^2 - 3b) + (c^2 - 2c) + 3< 0$


Or $(a-\frac{b}{2})^2 + (3 \frac{b^2}{4} - 3b) + (c^2 - 2c) +3 < 0$


Or $(a-\frac{b}{2})^2 + 3(\frac{b^2}{4} - b + 1) + (c^2 - 2c) < 0$


Or $(a-\frac{b}{2})^2 + 3(\frac{b}{2} -1)^2 + (c^2 - 2c+1) < 1$


Or $(a-\frac{b}{2})^2 + 3(\frac{b}{2} -1)^2 + (c-1)^2 < 1$


For these to be true each of the terms has to be zero.


(it can be seen that each of the terms which is squared if not zero has to be minimum $\frac{1}{4}$ giving sum 1 which is not true)


So b = 2 a = 1 and c = 1 is the solution giving a+b+c = 4
 
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