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

Revision as of 19:20, 23 June 2022

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 $\frac1A + \frac1B + \frac1C = p^2 + q^2 + r^2 - pq - qr - pr$ By Vieta's Formulas, we know that $p^2 + q^2 + r^2 = (p+q+r)^2 - 2(pq + qr + pr) = 324$ and $pq + qr + pr = 80$ . Thus the answer is $324 -80 = \boxed{\textbf{(B) } 244}$ .

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.

-Edit from countmath1

Solution 2 (Pure Elementary Algebra)

Solution 1 uses a trick from Calculus that seemingly contradicts the restriction $s\not\in\{p,q,r\}$ . I am going to provide 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

@@ Line 11: / Line 11: @@
 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>.
-''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. Extensive time is also spent covering this process (partial fraction decomposition) in the Art of Problem Solving's Intermediate Algebra course.
+''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.
 -Edit from countmath1

2019 AMC 10A (Problems • Answer Key • Resources)
Preceded by Problem 23	Followed by Problem 25
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