1989 AIME Problems/Problem 13
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Problem
Let 
be a subset of 
such that no two members of differ by 
or 
. What is the largest number of elements can have?
Solution
We first show that we can choose at most 5 numbers from 
such that no two numbers have a difference of or . We take the smallest number to be 
, which rules out 
. Now we can take at most one from each of the pairs: 
, 
, 
, 
. Now, 
, but because this isn't an exact multiple of 
, we need to consider the last 
numbers.
Now let's examine 
. If we pick 
from the first 
numbers, then we're allowed to pick 
, 
, 
, 
, 
. This means we get 10 members from the 20 numbers. Our answer is thus 
.
See also
| 1989 AIME (Problems • Resources) | ||
| Preceded by Problem 12 | Followed by Problem 14 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
