AMC10 2019 A

AMC10 2019 A · Q24

AMC10 2019 A · Q24. It mainly tests Rational expressions, Vieta / quadratic relationships (basic).

Let $p, q, r$ be the distinct roots of the polynomial $x^3 - 22x^2 + 80x - 67$. There exist real numbers $A, B, C$ such that $$\frac{1}{s^3 - 22s^2 + 80s - 67} = \frac{A}{s-p} + \frac{B}{s-q} + \frac{C}{s-r}$$ for all real numbers $s$ with $s \notin \{p, q, r\}$. What is $\frac{1}{A} + \frac{1}{B} + \frac{1}{C}$?

设 $p, q, r$ 是多项式 $x^3 - 22x^2 + 80x - 67$ 的不同根。存在实数 $A, B, C$ 使得 $$\frac{1}{s^3 - 22s^2 + 80s - 67} = \frac{A}{s-p} + \frac{B}{s-q} + \frac{C}{s-r}$$ 对所有 $s \notin \{p, q, r\}$ 的实数 $s$ 成立。求 $\frac{1}{A} + \frac{1}{B} + \frac{1}{C}$。

(A) 243 243

(B) 244 244

(D) 246 246

(E) 247 247

Answer

Correct choice: (B)

正确答案：（B）

Solution

Answer (B): Because $x^3-22x^2+80x-67=(x-p)(x-q)(x-r),$ multiplying the given equation by the common denominator yields $1=A(s-q)(s-r)+B(s-p)(s-r)+C(s-p)(s-q).$ This is now a polynomial identity that holds for infinitely many values of $s$, so it must hold for all $s$. This means the condition that $s\notin\{p,q,r\}$ can be removed. Setting $s=p$ yields $1=A(p-q)(p-r)$, so $\frac1A=(p-q)(p-r)$. Similarly, $\frac1B=(q-p)(q-r)$ and $\frac1C=(r-p)(r-q)$. Hence $\frac1A+\frac1B+\frac1C=(p-q)(p-r)+(q-p)(q-r)+(r-p)(r-q)$ $=(p+q+r)^2-3(pq+qr+rp).$ By Viète’s Formulas $p+q+r$ is the negative of the coefficient of $x^2$ in the polynomial and $pq+qr+rp$ is the coefficient of the $x$ term, so the requested value is $22^2-3\cdot80=244$. (The numerical values $(p,q,r,A,B,C)$ are approximately $(1.23,3.08,17.7,0.0329,-0.0371,0.00416)$.)

答案（B）：因为 $x^3-22x^2+80x-67=(x-p)(x-q)(x-r),$ 将所给方程乘以公分母得到 $1=A(s-q)(s-r)+B(s-p)(s-r)+C(s-p)(s-q).$ 这现在是一个对无穷多个 $s$ 值成立的多项式恒等式，因此它对所有 $s$ 都成立。这意味着可以去掉条件 $s\notin\{p,q,r\}$。令 $s=p$ 得 $1=A(p-q)(p-r)$，所以 $\frac1A=(p-q)(p-r)$。类似地，$\frac1B=(q-p)(q-r)$ 且 $\frac1C=(r-p)(r-q)$。因此 $\frac1A+\frac1B+\frac1C=(p-q)(p-r)+(q-p)(q-r)+(r-p)(r-q)$ $=(p+q+r)^2-3(pq+qr+rp).$ 由韦达定理，$p+q+r$ 等于多项式中 $x^2$ 项系数的相反数，而 $pq+qr+rp$ 等于 $x$ 项的系数，所以所求值为 $22^2-3\cdot80=244$。（数值 $(p,q,r,A,B,C)$ 近似为 $(1.23,3.08,17.7,0.0329,-0.0371,0.00416)$。）

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