# 2019 AMC 10A Problems/Problem 24

## 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.

## 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 $\boxed{\textbf{(B) } 244}$

~IceMatrix