Difference between revisions of "2019 AMC 10A Problems/Problem 24"
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| − | + | ==Problem== | |
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| + | Let <math>p</math>, <math>q</math>, and <math>r</math> be the distinct roots of the polynomial <math>x^3 - 22x^2 + 80x - 67</math>. It is given that there exist real numbers <math>A</math>, <math>B</math>, and <math>C</math> such that <cmath>\dfrac{1}{s^3 - 22s^2 + 80s - 67} = \dfrac{A}{s-p} + \dfrac{B}{s-q} + \frac{C}{s-r}</cmath>for all <math>s\not\in\{p,q,r\}</math>. What is <math>\tfrac1A+\tfrac1B+\tfrac1C</math>? | ||
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| + | <math>\textbf{(A) }243\qquad\textbf{(B) }244\qquad\textbf{(C) }245\qquad\textbf{(D) }246\qquad\textbf{(E) } 247</math> | ||
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| + | ==Solution== | ||
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| + | ==See Also== | ||
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| + | {{AMC10 box|year=2019|ab=A|num-b=23|num-a=25}} | ||
| + | {{MAA Notice}} | ||
Revision as of 16:56, 9 February 2019
Problem
Let 
, 
, and 
be the distinct roots of the polynomial 
. It is given that there exist real numbers 
, 
, and 
such that ![\[\dfrac{1}{s^3 - 22s^2 + 80s - 67} = \dfrac{A}{s-p} + \dfrac{B}{s-q} + \frac{C}{s-r}\]](http://latex.artofproblemsolving.com/e/d/4/ed44e7481e867b959ea84d896b30b222fe5efea3.png)
for all 
. What is 
?
Solution
See Also
| 2019 AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by Problem 23 |
Followed by Problem 25 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AMC 10 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
