HMMT 十一月 2022 · THM 赛 · 第 5 题
HMMT November 2022 — THM Round — Problem 5
题目详情
- Alice is once again very bored in class. On a whim, she chooses three primes p , q , r independently and 2 uniformly at random from the set of primes of at most 30. She then calculates the roots of px + qx + r . What is the probability that at least one of her roots is an integer?
解析
- Alice is once again very bored in class. On a whim, she chooses three primes p , q , r independently and 2 uniformly at random from the set of primes of at most 30. She then calculates the roots of px + qx + r . What is the probability that at least one of her roots is an integer? Proposed by: Eric Shen 3 Answer: 200 Solution: Since all of the coefficients are positive, any root x must be negative. Moreover, by the rational root theorem, in order for x to be an integer we must have either x = − 1 or x = − r . So we 2 must have either pr − qr + r = 0 ⇐⇒ pr = q − 1 or p − q + r = 0. Neither of these cases are possible if all three primes are odd, so we know so we know that one of the primes is even, hence equal to