Difference between revisions of "1989 AIME Problems/Problem 13"
(solution by 4everwise/kalva, could use some further explanation of inspiration?) |
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== Problem == | == Problem == | ||
− | Let <math>S | + | Let <math>S</math> be a [[subset]] of <math>\{1,2,3,\ldots,1989\}</math> such that no two members of <math>S</math> differ by <math>4</math> or <math>7</math>. What is the largest number of [[element]]s <math>S</math> can have? |
== Solution == | == Solution == |
Revision as of 21:12, 17 July 2008
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 • Answer Key • Resources) | ||
Preceded by Problem 12 |
Followed by Problem 14 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |