¶ 2025 AMC 12A Problem 22
Problem:
Three real numbers are chosen independently and uniformly at random between and . What is the probability that the greatest of these three numbers is greater than times each of the other two numbers? (If the chosen numbers are , then .)
Answer Choices:
A.
B.
C.
D.
E.
Solution:
The region bounded by the planes and has base area and height , so it has volume (it is pictured above). The regions where and are the greatest have the same volume. Since these regions are disjoint, the total volume is .
Alternate Solution
As before, assume that is the largest. Then, the volume is given by:
Hence, the total volume is , which yields the same answer as before.
The problems on this page are the property of the MAA's American Mathematics Competitions