2017/18 AHSMC Part 1 Problem 12

Problem 12 of 16

Let $a$ , $b$ be positive integers. The number of quadratic equations $x^2 - ax - b = 0$ having the positive root less than $10$ is

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Solution

The positive root of the equation is $\dfrac{a + \sqrt{a^2 + 4b}}{2}$ . The condition $\dfrac{a + \sqrt{a^2 + 4b}}{2} < 10$ is equivalent to $10a + b < 100$ and $a \le 20$ . There are $9 + 19 + 29 + 39 + 49 + 59 + 69 + 79 + 89 = 4 \times 98 + 49 = 441$ ordered pairs $(a,b)$ such that $10a + b < 100$ . The answer is (C).

University of Alberta

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