Difference between revisions of "1990 AIME Problems/Problem 11"
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Revision as of 21:07, 24 November 2007
Problem
Someone observed that 
. Find the largest positive integer 
for which 
can be expressed as the product of 
consecutive positive integers.
Solution
The product of 
consecutive integers can be written as 
for some integer 
. Thus, 
, from which it becomes evident that 
. Since 
, we can rewrite this as 
. For 
, we get 
so 
. For greater values of , we need to find the product of 
consecutive integers that equals 
. 
can be approximated as 
, which decreases as increases. Thus, is the greatest possible value to satisfy the given conditions.
See also
| 1990 AIME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 10 |
Followed by Problem 12 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
