2024/25 AHSMC Part 1 Problem 10

Problem 10 of 16

What is the probability that the equation $x^2 + ax + b = 0$ has exactly one root, when $a$ and $b$ are (not necessarily different) numbers chosen randomly from the set $\{1,2,3,\ldots,100\}$ ?

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Solution

The equation $x^2 + ax + b = 0$ has exactly one root if and only if $a^2 = 4b$ . Since $b \le 100$ , this means $(a,b) = (2k, k^2)$ for some $1 \le k \le 10$ . There are $100^2$ possible pairs $(a,b)$ and only $10$ favourable ones, so the probability is $\dfrac{1}{1000}$ . The answer is (A).

University of Alberta

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