Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2019 AMC 10A Problems/Problem 24"

(→‎Solution)
m (Fixed formatting and solution numbering)
Line 5: Line 5:
 
<math>\textbf{(A) }243\qquad\textbf{(B) }244\qquad\textbf{(C) }245\qquad\textbf{(D) }246\qquad\textbf{(E) } 247</math>
 
<math>\textbf{(A) }243\qquad\textbf{(B) }244\qquad\textbf{(C) }245\qquad\textbf{(D) }246\qquad\textbf{(E) } 247</math>
  
−
==Solution==
+
==Solution 1==
 
Multiplying both sides by <math>(s-p)(s-q)(s-r)</math> on both sides yields
 
Multiplying both sides by <math>(s-p)(s-q)(s-r)</math> on both sides yields
 
<cmath>1 = A(s-q)(s-r) + B(s-p)(s-r) + C(s-p)(s-q)</cmath>
 
<cmath>1 = A(s-q)(s-r) + B(s-p)(s-r) + C(s-p)(s-q)</cmath>
 
As this is a polynomial and is true for infinitely many <math>s</math>, it must be true for all <math>s</math>.  This means we can plug in <math>s = p</math> to find that <math>\frac1A = (p-q)(p-r)</math>.  Similarly, we can find <math>\frac1B = (q-p)(q-r)</math> and <math>\frac1C = (r-p)(r-q)</math>.  Summing them up, we get that <cmath>\frac1A + \frac1B + \frac1C = p^2 + q^2 + r^2 - pq - qr - pr</cmath>
 
As this is a polynomial and is true for infinitely many <math>s</math>, it must be true for all <math>s</math>.  This means we can plug in <math>s = p</math> to find that <math>\frac1A = (p-q)(p-r)</math>.  Similarly, we can find <math>\frac1B = (q-p)(q-r)</math> and <math>\frac1C = (r-p)(r-q)</math>.  Summing them up, we get that <cmath>\frac1A + \frac1B + \frac1C = p^2 + q^2 + r^2 - pq - qr - pr</cmath>
−
By Vieta's formulas, we know that <math>p^2 + q^2 + r^2 = (p+q+r)^2 - 2(pq + qr + pr) = 324</math> and <math>pq + qr + pr = 80</math>.  So the answer is <math>324 -80 = \boxed{\textbf{(B) } 244}</math>
+
By Vieta's formulas, we know that <math>p^2 + q^2 + r^2 = (p+q+r)^2 - 2(pq + qr + pr) = 324</math> and <math>pq + qr + pr = 80</math>.  So the answer is <math>324 -80 = \boxed{\textbf{(B) } 244}</math>.
  
−
(adapted from djmathman's solution)
 
 
==Solution 2 (similar)==
 
==Solution 2 (similar)==
 
Multiplying by <math>(s-p)</math> on both sides we find that
 
Multiplying by <math>(s-p)</math> on both sides we find that

Revision as of 21:45, 17 February 2019

Problem

Let $p$, $q$, and $r$ be the distinct roots of the polynomial $x^3 - 22x^2 + 80x - 67$. It is given that there exist real numbers $A$, $B$, and $C$ such that \[\dfrac{1}{s^3 - 22s^2 + 80s - 67} = \dfrac{A}{s-p} + \dfrac{B}{s-q} + \frac{C}{s-r}\]for all $s\not\in\{p,q,r\}$. What is $\tfrac1A+\tfrac1B+\tfrac1C$?

$\textbf{(A) }243\qquad\textbf{(B) }244\qquad\textbf{(C) }245\qquad\textbf{(D) }246\qquad\textbf{(E) } 247$

Solution 1

Multiplying both sides by $(s-p)(s-q)(s-r)$ on both sides yields \[1 = A(s-q)(s-r) + B(s-p)(s-r) + C(s-p)(s-q)\] As this is a polynomial and is true for infinitely many $s$, it must be true for all $s$. This means we can plug in $s = p$ to find that $\frac1A = (p-q)(p-r)$. Similarly, we can find $\frac1B = (q-p)(q-r)$ and $\frac1C = (r-p)(r-q)$. Summing them up, we get that \[\frac1A + \frac1B + \frac1C = p^2 + q^2 + r^2 - pq - qr - pr\] By Vieta's formulas, we know that $p^2 + q^2 + r^2 = (p+q+r)^2 - 2(pq + qr + pr) = 324$ and $pq + qr + pr = 80$. So the answer is $324 -80 = \boxed{\textbf{(B) } 244}$.

Solution 2 (similar)

Multiplying by $(s-p)$ on both sides we find that \[\frac{s-p}{s^3 - 22s^2 + 80s - 67} = A + \frac{B(s-p)}{s-q} + \frac{C(s-p)}{s-r}\] As $s\to p$, notice that the $B$ and $C$ terms on the right will cancel out and we will be left with only $A$. So, $A = \lim_{s \to p} \frac{s-p}{s^3 - 22s^2 + 80s - 67}$, which by L'Hospital's rule is equal to $\lim_{s \to p}\frac{1}{3s^2 - 44s + 80} = \frac{1}{3p^2 - 44p + 80}$. We can do similarly for $B$ and $C$. Adding up the reciprocals and using Vieta's formulas, we have that \[\frac1A + \frac1B + \frac1C = 3(p^2 + q^2 + r^2) - 44(p + q + r) + 240 = 3(22^2 - 2(80)) - 44(22) + 240 = \boxed{\textbf{(B) } 244}\]

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. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2019_AMC_10A_Problems/Problem_24&oldid=103255"