Question

How many distinct values of $x$ satisfy $\lfloor x \rfloor^2 - 3x + 2 = 0,$ where $\lfloor x \rfloor$ denotes the largest integer less than or equal to $x?$

an infinite number

$4$

$2$

$3$

$0$

Answer: B

Concepts:floor and ceiling functions substitution bounding to limit cases

Difficulty rating: 2120

Solution:

Set $n = \lfloor x \rfloor.$ Then $n^2 - 3x + 2 = 0$ gives $x = \frac{n^2 + 2}{3}.$ For this to be consistent we need $n \le \frac{n^2 + 2}{3} \lt n + 1.$ The left side, $n^2 - 3n + 2 \ge 0,$ holds for every integer $n.$ The right side, $n^2 - 3n - 1 \lt 0,$ holds only for $n \in \{0, 1, 2, 3\}.$ Those give $x = \frac{2}{3}, 1, 2, \frac{11}{3},$ so there are $4$ distinct values. Therefore, the answer is B.

2023 AMC 10B Problem 22

Answer: B

Difficulty rating: 2120

Solution:

Problem 22 in Other Years