1989 AIME Problems/Problem 13
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
can have the numbers
through
, but it can't have numbers
through
, because no two numbers can have a difference of
or
. So,
through
work, but
through
don't work, and so on. Now notice that this list contains only numbers
through
.
is
, so
is
. We now have the sequence
We add 7 to each term to get
We divide by 11 to get
So there are 181 numbers in S. We multiply by 4 to account for
, and
:
.
See also
1989 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 12 |
Followed by Problem 14 | |
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