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Non-negative real numbers $a, b$ and $c$ satisfy $$a^2+b^2+c^2+70=35ab+7ac+5bc.$$

The least possible value of $a^2+2b^2+3c^2$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. FInd $m+n$

 

So I don't really know how to start on this problem, all I noticed that you can use the factorization $(a+b+c)^2=a^2+b^2+c^2+2(ab+ac+bc),$ other than this, I don't really know how to do this problem. Any solutions and hints would be appreciated!

 
 Jul 27, 2021
 #1
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The minimum is 11200/7, so the answer is 11207.

 
 Jul 27, 2021

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