<p>Let <img alt="$P(x) = x^2 - 3x - 7$" height="19" src="https://latex.artofproblemsolving.com/3/3/4/3344584cff763edf07dd431a60fd0d44a386a384.png" width="153" />, and let <img alt="$Q(x)$" height="18" src="https://latex.artofproblemsolving.com/d/9/d/d9dfe7c7a0b887097fa2227a230fc915ba2e5477.png" width="37" /> and <img alt="$R(x)$" height="20" src="https://latex.artofproblemsolving.com/a/3/e/a3eeefa367883375994fd2e1f575d088bce9d8b1.png" width="40" /> be two quadratic polynomials also with the coefficient of <img alt="$x^2$" height="15" src="https://latex.artofproblemsolving.com/7/6/b/76b2878564ab19b80abb21ba964abe90fe120846.png" width="16" /> equal to <img alt="$1$" height="14" src="https://latex.artofproblemsolving.com/d/c/e/dce34f4dfb2406144304ad0d6106c5382ddd1446.png" width="11" />. David computes each of the three sums <img alt="$P + Q$" height="16" src="https://latex.artofproblemsolving.com/d/e/2/de2bc4d2fa383396c8f16726de1edc73fc2eec3a.png" width="50" />, <img alt="$P + R$" height="14" src="https://latex.artofproblemsolving.com/c/4/d/c4dac70f8cc88cf25fadf97649cae05e11084d66.png" width="50" />, and <img alt="$Q + R$" height="16" src="https://latex.artofproblemsolving.com/8/1/8/8182014118c638fd3197c77fc5a9fd926f17ece3.png" width="50" /> and is surprised to find that each pair of these sums has a common root, and these three common roots are distinct. If <img alt="$Q(0) = 2$" height="18" src="https://latex.artofproblemsolving.com/e/0/8/e083287a774dd17d99d62a61f54c8820b954becd.png" width="70" />, then <img alt="$R(0) = \frac{m}{n}$" height="33" src="https://latex.artofproblemsolving.com/6/f/5/6f5fb24086d4fcf71217890f0fc5842f885a7de1.png" width="79" />, where <img alt="$m$" height="8" src="https://latex.artofproblemsolving.com/f/5/0/f5047d1e0cbb50ec208923a22cd517c55100fa7b.png" width="15" /> and <img alt="$n$" height="8" src="https://latex.artofproblemsolving.com/1/7/4/174fadd07fd54c9afe288e96558c92e0c1da733a.png" width="10" /> are relatively prime positive integers. Find <img alt="$m + n$" height="12" src="https://latex.artofproblemsolving.com/5/3/b/53bf1afcb274a3bb84338109fffec8a1f1c02008.png" width="48" />.</p>

Question No: 375

Let $P(x) = x^2 - 3x - 7$ , and let $Q(x)$ and $R(x)$ be two quadratic polynomials also with the coefficient of $x^2$ equal to $1$ . David computes each of the three sums $P + Q$ , $P + R$ , and $Q + R$ and is surprised to find that each pair of these sums has a common root, and these three common roots are distinct. If $Q(0) = 2$ , then $R(0) = \frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$ .

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Tags: Algebra

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