2020 AMC 12B Problems/Problem 25
Contents
Problem
For each real number with , let numbers and be chosen independently at random from the intervals and , respectively, and let be the probability that
What is the maximum value of
Solution
Let's start first by manipulating the given inequality.
Let's consider the boundary cases: .
Solving the first case gives us Solving the second case gives us If we graph these equations in , we get a rhombus shape. Testing points in each section tells us that the inside of the rhombus satisfies the inequality in the problem statement.
From the region graph, notice that in order to maximize , . We can solve the rest with geometric probability.
Instead of maximizing , we minimize . consists of two squares (each broken into two triangles), one of area and another of area . To calculate , we divide this area by , so
By AM-GM, , which we can achieve by setting .
Therefore, the maximum value of is , which is
Video Solution
On The Spot STEM: https://www.youtube.com/watch?v=5goLUdObBrY
See Also
2020 AMC 12B (Problems • Answer Key • Resources) | |
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