2005 PMWC Problems/Problem T3
From AoPSWiki
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 | ||
