Problem:
Find the number of positive integers less than for which there exists a positive real number such that .
Note: is the greatest integer less than or equal to .
Solution:
Let be a positive integer. Because the distance from the integer to the integer is , there are positive integers that satisfy . For each such there is a rational number such that . The condition implies that . All values of corresponding to satisfy the conditions of the problem, and thus there are such values of .
The problems on this page are the property of the MAA's American Mathematics Competitions