AMC 10B 2023 (Problem 22)
How many distinct values of satisfy , where denotes the greatest integer less than or equal to ?
Use 29. Integer and Fractional Parts.
Let with , . Then .
Substitute:
Since ,
From the left inequality: Using 31. Signs and Roots of Quadratics: Since k is an integer, can be any integer.
From the right inequality: Use 21. Quadratic Formula: roots . By 31. Signs and Roots: .
Compute for each : ; ; ; . Thus — four distinct values.
(B) 4