2005 PMWC Problems/Problem T3
Problem
Replace the letters , , and in the following expression with the numbers , , and , without repetition: Find the difference between the maximum value and the minimum value of the expression.
Solution
By the greedy algorithm, the maximum value will occur when . To maximize the fraction, we need to minimize the quantity , so we need to minimize ; . To minimize the remnants, we want to maximize , so we want to maximize ; this leaves .
Following the same pattern of alternating maximums and minimums, the minimum value of the expression occurs when .
Cleaning up the complex fraction, we get
Substituting our values gives us that the maximum is , while the minimum is . Subtracting, our answer is .
See also
2005 PMWC (Problems) | ||
Preceded by Problem T2 |
Followed by Problem T4 | |
I: 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 T: 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 |