Difference between revisions of "1989 AIME Problems/Problem 13"

Revision as of 21:12, 17 July 2008

Problem

Let $S$ be a subset of $\{1,2,3,\ldots,1989\}$ such that no two members of $S$ differ by $4$ or $7$ . What is the largest number of elements $S$ can have?

Solution

We first show that we can choose at most 5 numbers from $\{1, 2, \ldots , 11\}$ such that no two numbers have a difference of $4$ or $7$ . We take the smallest number to be $1$ , which rules out $5,8$ . Now we can take at most one from each of the pairs: $[2,9]$ , $[3,7]$ , $[4,11]$ , $[6,10]$ . Now, $1989 = 180\cdot 11 + 9$ , but because this isn't an exact multiple of $5$ , we need to consider the last $9$ numbers.

Now let's examine $\{1, 2, \ldots , 20\}$ . If we pick $1, 3, 4, 6, 9$ from the first $11$ numbers, then we're allowed to pick $11 + 1$ , $11 + 3$ , $11 + 4$ , $11 + 6$ , $11 + 9$ . This means we get 10 members from the 20 numbers. Our answer is thus $179\cdot 5 + 10 = \boxed{905}$ .

@@ Line 1: / Line 1: @@
 == Problem ==
-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?
+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 ==

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

Page

Toolbox

Search

Difference between revisions of "1989 AIME Problems/Problem 13"

Revision as of 21:12, 17 July 2008

Problem

Solution

See also