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

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== Problem ==
 
== 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 <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 ==
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== Solution 1 ==
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>:
 
 
 
 
<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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<cmath>\mathcal{S} = \{\, 4,\, 5,\, 4+5, \,5+(4+5),\, \ldots\,\} = \{4, 5, 9, 14, 23, 37, 60, 97, 157, 254\}</cmath>
  
 
<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>.
 
<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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==Solution 2==
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I claim that the answer is <math>253</math>. We will show that no number greater than 253 works. Consider the subset<math> {4, 5, 9, 14, 23, 37, 60, 97, 157, 254}</math>. This subset does not have the property. It is easy to see that any number less than or equal to <math>253</math> works, and hence our answer is <math>253</math>
  
 
== 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 13:27, 2 September 2020

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 1

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 $n = \fbox{253}$.

Solution 2

I claim that the answer is $253$. We will show that no number greater than 253 works. Consider the subset${4, 5, 9, 14, 23, 37, 60, 97, 157, 254}$. This subset does not have the property. It is easy to see that any number less than or equal to $253$ works, and hence our answer is $253$

See also

2001 AIME II (ProblemsAnswer KeyResources)
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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