2025 AMC 12A Problems/Problem 12
Problem
Solution 1
We will need to determine the sum of the reciprocals of the roots. To find the sum of the reciprocals of the roots 
of the quadratic 
, we use Vieta's formulas. Recall that 
and 
. Therefore, ![\[\frac{1}{p} + \frac{1}{q} = \frac{p+q}{pq} = \frac{-b/a}{c/a} = \frac{-b}{c},\]](http://latex-new.aopstest.com/e/f/6/ef6e050c17f57e240479ac3a1290e33129dafa38.png)
which doesn't depend on 
.
The sum of the reciprocals of the roots of the quadratic 
is 
The same is true for every quadratic in the form 
. The sum of all the reciprocals of the roots of is 
Because we have 
quadratics, there are 
total roots. Our answer is ![\[\frac{1}{\frac{1}{4050}\cdot \frac{-4\cdot 2025}{3}} = \boxed{-\frac{3}{2}}.\]](http://latex-new.aopstest.com/b/5/d/b5d88b2eeaea06eb0abee83bf056178d6076d0a3.png)
~lprado
