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

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== Problem ==
 
== Problem ==
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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 elements <math>S^{}_{}</math> can have?
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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 ==
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S can have the numbers 1 through 4, but it can't have numbers 5 through 11, because no two numbers can have a difference of 4 or 7. So, 12 through 15 work, but 16 through 22 don't work. So on. Now notice that this list contains only numbers 1 through 4 mod 11. 1989 is 9 mod 11, so 1984 is 4 mod 11. We now have the sequence
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We first show that we can choose at most 5 numbers from <math>\{1, 2, \ldots , 11\}</math> such that no two numbers have a difference of <math>4</math> or <math>7</math>. We take the smallest number to be <math>1</math>, which rules out <math>5,8</math>. Now we can take at most one from each of the pairs: <math>[2,9]</math>, <math>[3,7]</math>, <math>[4,11]</math>, <math>[6,10]</math>. Now, <math>1989 = 180\cdot 11 + 9</math>. Because this isn't an exact multiple of <math>11</math>, we need to consider some numbers separately.  
  
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{4,15,26,...,1984}
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Notice that <math>1969 = 180\cdot11 - 11 = 179\cdot11</math> (*). Therefore we can put the last <math>1969</math> numbers into groups of 11. Now let's examine <math>\{1, 2, \ldots , 20\}</math>. If we pick <math>1, 3, 4, 6, 9</math> from the first <math>11</math> numbers, then we're allowed to pick <math>11 + 1</math>, <math>11 + 3</math>, <math>11 + 4</math>, <math>11 + 6</math>, <math>11 + 9</math>. This means we get 10 members from the 20 numbers. Our answer is thus <math>179\cdot 5 + 10 = \boxed{905}</math>.
  
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We add 7 to each term to get
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Remarks
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(*) Suppose that you choose the values 1, 2, 4, 7, and 10. Because <math>1989 = 180 \times 11 + 9</math>, this is not maximized. It is only maximized if we include the last element of the final set of 11, which is 10 (this is <math>\text{mod}(11)</math> btw). To include the final element, we have to rewrite 1989 as <math>179 \times 11 + 20</math>, which includes the final element and increases our set by 1 element.
  
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{11,22,33,...,1991}
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~Brackie 1331
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We divide by 11 to get
 
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{1,2,3,...,181}
 
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So there are 181 numbers 4 mod 11 in S. We multiply by 4 to account for 1, 2, and 3 mod 11:
 
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<math>181*4=\boxed{724}</math>
 
  
 
== See also ==
 
== See also ==
 
{{AIME box|year=1989|num-b=12|num-a=14}}
 
{{AIME box|year=1989|num-b=12|num-a=14}}
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[[Category:Intermediate Combinatorics Problems]]
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[[Category:Intermediate Number Theory Problems]]
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{{MAA Notice}}

Latest revision as of 03:07, 17 December 2023

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, $1989 = 180\cdot 11 + 9$. Because this isn't an exact multiple of $11$, we need to consider some numbers separately.

Notice that $1969 = 180\cdot11 - 11 = 179\cdot11$ (*). Therefore we can put the last $1969$ numbers into groups of 11. 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 $179\cdot 5 + 10 = \boxed{905}$.

Remarks (*) Suppose that you choose the values 1, 2, 4, 7, and 10. Because $1989 = 180 \times 11 + 9$, this is not maximized. It is only maximized if we include the last element of the final set of 11, which is 10 (this is $\text{mod}(11)$ btw). To include the final element, we have to rewrite 1989 as $179 \times 11 + 20$, which includes the final element and increases our set by 1 element.

~Brackie 1331

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

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