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 3)
m (→‎Solution 1)
 
(18 intermediate revisions by 14 users not shown)
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==
−
===Solution 1===
 
 
Multiplying both sides by <math>(s-p)(s-q)(s-r)</math> yields
 
Multiplying both sides by <math>(s-p)(s-q)(s-r)</math> 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 identity, and it is true for infinitely many <math>s</math>, it must be true for all <math>s</math> (since a polynomial with infinitely many roots must in fact be the constant polynomial <math>0</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 identity, and it is true for infinitely many <math>s</math>, it must be true for all <math>s</math> (since a polynomial with infinitely many roots must in fact be the constant polynomial <math>0</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>. Thus the answer is <math>324 -80 = \boxed{\textbf{(B) } 244}</math>.
+
We can express <math>p^2 + q^2 + r^2 = (p+q+r)^2 - 2(pq + qr + pr)</math>, and by Vieta's Formulas, we know that this expression is equal to <math>484</math>. Vieta's also gives <math>pq + qr + pr = 80</math> (which we also used to find <math>p^2+q^2+r^2</math>), so the answer is <math>484 -240 = \boxed{\textbf{(B) } 244}</math>.  
  
−
''Note'': this process of substituting in the 'forbidden' values in the original identity is a standard technique for partial fraction decomposition, as taught in calculus classes.
+
''Note'': this process of substituting in the 'forbidden' values in the original identity is a standard technique for partial fraction decomposition, as taught in calculus classes.
  
−
===Solution 2 (limits)===
+
-very small latex edit from countmath1 :)
−
Multiplying by <math>(s-p)</math> on both sides, we find that
 
−
<cmath>\frac{s-p}{s^3 - 22s^2 + 80s - 67} = A + \frac{B(s-p)}{s-q} + \frac{C(s-p)}{s-r}</cmath>
 
−
As <math>s\to p</math>, notice that the <math>B</math> and <math>C</math> terms on the right will cancel out and we will be left with only <math>A</math>.  Hence, <math>A = \lim_{s \to p} \frac{s-p}{s^3 - 22s^2 + 80s - 67}</math>, which by L'Hôpital's rule becomes <math>\lim_{s \to p}\frac{1}{3s^2 - 44s + 80} = \frac{1}{3p^2 - 44p + 80}</math>.  We can reason similarly to find <math>B</math> and <math>C</math>.  Adding up the reciprocals and using Vieta's Formulas, we have that
 
−
<cmath>\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}</cmath>
 
  
−
===Solution 3===
+
Minor rephrasing for correctness and clarity ~ Technodoggo
  
−
First multiply both sides by <math>s^3 - 22s^2 + 80s - 67</math> so <math>\sum A(s-q)(s-r)=1</math>. Now <math>s</math> can be arbitrarily close to <math>p,q,r</math>, not affecting anything, so we can set <math>s</math> equal to <math>p,q,r</math> to get <cmath>\sum \frac{1}{A} = \sum (p-q)(p-r) = \sum p^2 - \sum qr = (p+q+r)^2 - 3\sum qr = 484-240=\boxed{244}</cmath> by Vieta's.
+
==Solution 2 (Pure Elementary Algebra)==
 +
Solution 1 uses a trick from Calculus that seemingly contradicts the restriction <math>s\not\in\{p,q,r\}</math>. Here is a solution with pure elementary algebra.
 +
<cmath>A(s-q)(s-r) + B(s-p)(s-r) + C(s-p)(s-q)=1</cmath>
 +
<cmath>s^2(A+B+C)-s(Aq+Ar+Bp+Br+Cp+Cq)+(Aqr+Bpr+Cpq-1)=0</cmath>
 +
<cmath>\begin{cases}
 +
A+B+C=0 & (1)\\
 +
Aq+Ar+Bp+Br+Cp+Cq=0 & (2)\\
 +
Aqr+Bpr+Cpq=1 & (3)
 +
\end{cases}</cmath>
 +
From <math>(1)</math> we get <math>A=-(B+C)</math>, <math>B=-(A+C)</math>, <math>C=-(A+B)</math>, substituting them in <math>(2)</math>, we get <math>Ap + Bq + Cr=0</math>  <math>(4)</math>
  
−
~Lcz
+
<math>(4)- (1) \cdot r</math>, <math>A(p-r)+B(q-r)=0</math>  <math>(5)</math>
 +
 
 +
<math>(3) - (1) \cdot pq</math>, <math>Aq(r-p)+Bp(r-q)=1</math>  <math>(6)</math>
 +
 
 +
<math>(6) + (5) \cdot p</math>, <math>A(r-p)(q-p)=1</math>
 +
 
 +
<math>A = \frac{1}{(r-p)(q-p)}</math>, by symmetry, <math>B = \frac{1}{(r-q)(p-q)}</math>, <math>C = \frac{1}{(q-r)(p-r)}</math>
 +
 
 +
The rest is similar to solution 1, we get <math>\boxed{\textbf{(B) } 244}</math>
 +
 
 +
~[https://artofproblemsolving.com/wiki/index.php/User:Isabelchen isabelchen]
 +
 
 +
==Video Solution by Richard Rusczyk==
 +
https://www.youtube.com/watch?v=GI5d2ZN8gXY
 +
 
 +
==Video Solution by TheBeautyofMath==
 +
https://youtu.be/zw5CCPcT5IU
 +
 
 +
~IceMatrix
  
 
==See Also==
 
==See Also==
−
Video Solution:
 
−
https://www.youtube.com/watch?v=GI5d2ZN8gXY&t=53s
 
−
 
  
 +
[[Category:Intermediate Algebra Problems]]
 
{{AMC10 box|year=2019|ab=A|num-b=23|num-a=25}}
 
{{AMC10 box|year=2019|ab=A|num-b=23|num-a=25}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 23:12, 23 August 2024

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 $(s-p)(s-q)(s-r)$ yields \[1 = A(s-q)(s-r) + B(s-p)(s-r) + C(s-p)(s-q)\] As this is a polynomial identity, and it is true for infinitely many $s$, it must be true for all $s$ (since a polynomial with infinitely many roots must in fact be the constant polynomial $0$). 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 \[\frac1A + \frac1B + \frac1C = p^2 + q^2 + r^2 - pq - qr - pr\] We can express $p^2 + q^2 + r^2 = (p+q+r)^2 - 2(pq + qr + pr)$, and by Vieta's Formulas, we know that this expression is equal to $484$. Vieta's also gives $pq + qr + pr = 80$ (which we also used to find $p^2+q^2+r^2$), so the answer is $484 -240 = \boxed{\textbf{(B) } 244}$.

Note: this process of substituting in the 'forbidden' values in the original identity is a standard technique for partial fraction decomposition, as taught in calculus classes.

-very small latex edit from countmath1 :)

Minor rephrasing for correctness and clarity ~ Technodoggo

Solution 2 (Pure Elementary Algebra)

Solution 1 uses a trick from Calculus that seemingly contradicts the restriction $s\not\in\{p,q,r\}$. Here is a solution with pure elementary algebra. \[A(s-q)(s-r) + B(s-p)(s-r) + C(s-p)(s-q)=1\] \[s^2(A+B+C)-s(Aq+Ar+Bp+Br+Cp+Cq)+(Aqr+Bpr+Cpq-1)=0\] \[\begin{cases} A+B+C=0 & (1)\\ Aq+Ar+Bp+Br+Cp+Cq=0 & (2)\\ Aqr+Bpr+Cpq=1 & (3) \end{cases}\] From $(1)$ we get $A=-(B+C)$, $B=-(A+C)$, $C=-(A+B)$, substituting them in $(2)$, we get $Ap + Bq + Cr=0$ $(4)$

$(4)- (1) \cdot r$, $A(p-r)+B(q-r)=0$ $(5)$

$(3) - (1) \cdot pq$, $Aq(r-p)+Bp(r-q)=1$ $(6)$

$(6) + (5) \cdot p$, $A(r-p)(q-p)=1$

$A = \frac{1}{(r-p)(q-p)}$, by symmetry, $B = \frac{1}{(r-q)(p-q)}$, $C = \frac{1}{(q-r)(p-r)}$

The rest is similar to solution 1, we get $\boxed{\textbf{(B) } 244}$

~isabelchen

Video Solution by Richard Rusczyk

https://www.youtube.com/watch?v=GI5d2ZN8gXY

Video Solution by TheBeautyofMath

https://youtu.be/zw5CCPcT5IU

~IceMatrix

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=225627"