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Difference between revisions of "2001 AIME II Problems/Problem 5"

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== Problem ==
 
== Problem ==
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A set of positive numbers has the triangle property if it has three distinct elements that are the lengths of the sides of a triangle whose area is positive. Consider sets <math>\{4, 5, 6, \ldots, n\}</math> of consecutive positive integers, all of whose ten-element subsets have the triangle property. What is the largest possible value of <math>n</math>?
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A [[set]] of positive numbers has the ''triangle property'' if it has three distinct elements that are the lengths of the sides of a [[triangle]] whose area is positive. Consider sets <math>\{4, 5, 6, \ldots, n\}</math> of consecutive positive integers, all of whose ten-element subsets have the triangle property. What is the largest possible value of <math>n</math>?
  
 
== Solution ==
 
== Solution ==
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Out of all ten-element subsets with distinct elements that do not possess the triangle property, we want to find the one with the smallest maximum element. Call this subset <math>\mathcal{S}</math>. Without loss of generality, consider any <math>a, b, c \,\in \mathcal{S}</math> with <math>a < b < c</math>. <math>\,\mathcal{S}</math> does not possess the triangle property, so <math>c \geq a + b</math>. We use this property to build up <math>\mathcal{S}</math> from the smallest possible <math>a</math> and <math>b</math>:
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Out of all ten-element subsets with distinct elements that do not possess the triangle property, we want to find the one with the smallest maximum element. Call this subset <math>\mathcal{S}</math>. Without loss of generality, consider any <math>a, b, c \,\in \mathcal{S}</math> with <math>a < b < c</math>. <math>\,\mathcal{S}</math> does not possess the [[triangle inequality|triangle property]], so <math>c \geq a + b</math>. We use this property to build up <math>\mathcal{S}</math> from the smallest possible <math>a</math> and <math>b</math>:
  
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<cmath>\mathcal{S} = \{\, 4,\, 5,\, 4+5, \,5+(4+5),\, \ldots\,\} = \{4, 5, 9, 14, 23, 37, 60, 97, 157, 254\}</cmath>
  
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<math>\mathcal{S} = \{\, 4,\, 5,\, 4+5, \,5+(4+5),\, \ldots\,\} = \{4, 5, 9, 14, 23, 37, 60, 97, 157, 254\}</math>
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<math>\mathcal{S}</math> is the "smallest" ten-element subset without the triangle property, and since the set <math>\{4, 5, 6, \ldots, 253\}</math> is the largest set of consecutive integers that does not contain this subset, it is also the largest set of consecutive integers in which all ten-element subsets possess the triangle property. Thus, our answer is <math>n = \fbox{253}</math>.
  
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===Note===
  
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<math>\mathcal{S}</math> is the "smallest" ten-element subset without the triangle property, and since the set <math>\{4, 5, 6, \ldots, 253\}</math> is the largest set of consecutive integers that does not contain this subset, it is also the largest set of consecutive integers in which all ten-element subsets possess the triangle property.
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If we wanted to find this for a much larger number (say 2001), we could have noted that this is a "quasi-Fibonacci" sequence with initial terms <math>4,5</math> and built up an explicit function to find the <math>nth</math> term. (The latter part is generally pretty annoying).
  
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~Dhillonr25
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Thus, <math>\fbox{n=253}</math>
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~Minor edit by Yiyj1
  
 
== See also ==
 
== See also ==
 
{{AIME box|year=2001|n=II|num-b=4|num-a=6}}
 
{{AIME box|year=2001|n=II|num-b=4|num-a=6}}
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[[Category:Intermediate Combinatorics Problems]]
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{{MAA Notice}}

Latest revision as of 21:32, 5 January 2024

Problem

A set of positive numbers has the triangle property if it has three distinct elements that are the lengths of the sides of a triangle whose area is positive. Consider sets $\{4, 5, 6, \ldots, n\}$ of consecutive positive integers, all of whose ten-element subsets have the triangle property. What is the largest possible value of $n$?

Solution

Out of all ten-element subsets with distinct elements that do not possess the triangle property, we want to find the one with the smallest maximum element. Call this subset $\mathcal{S}$. Without loss of generality, consider any $a, b, c \,\in \mathcal{S}$ with $a < b < c$. $\,\mathcal{S}$ does not possess the triangle property, so $c \geq a + b$. We use this property to build up $\mathcal{S}$ from the smallest possible $a$ and $b$:

\[\mathcal{S} = \{\, 4,\, 5,\, 4+5, \,5+(4+5),\, \ldots\,\} = \{4, 5, 9, 14, 23, 37, 60, 97, 157, 254\}\]

$\mathcal{S}$ is the "smallest" ten-element subset without the triangle property, and since the set $\{4, 5, 6, \ldots, 253\}$ is the largest set of consecutive integers that does not contain this subset, it is also the largest set of consecutive integers in which all ten-element subsets possess the triangle property. Thus, our answer is $n = \fbox{253}$.

Note

If we wanted to find this for a much larger number (say 2001), we could have noted that this is a "quasi-Fibonacci" sequence with initial terms $4,5$ and built up an explicit function to find the $nth$ term. (The latter part is generally pretty annoying).

~Dhillonr25 ~Minor edit by Yiyj1

See also

2001 AIME II (Problems • Answer Key • Resources)
Preceded by
Problem 4 		Followed by
Problem 6
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

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