Difference between revisions of "2002 AIME I Problems/Problem 14"
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== Problem == | == Problem == | ||
| + | A set <math>\mathcal{S}</math> of distinct positive integers has the following property: for every integer <math>x</math> in <math>\mathcal{S},</math> the arithmetic mean of the set of values obtained by deleting <math>x</math> from <math>\mathcal{S}</math> is an integer. Given that 1 belongs to <math>\mathcal{S}</math> and that 2002 is the largest element of <math>\mathcal{S},</math> what is the greatest number of elements that <math>\mathcal{S}</math> can have? | ||
== Solution == | == Solution == | ||
| − | {{ | + | Let the sum of the integers in <math>\mathcal{S}</math> be <math>N</math>, and let the size of <math>|\mathcal{S}|</math> be <math>n+1</math>. After any element <math>x</math> is removed, we are given that <math>n|N-x</math>, so <math>x\equiv N\pmod{n}</math>. Since <math>1\in\mathcal{S}</math>, <math>N\equiv1\pmod{n}</math>, and all elements are congruent to 1 mod <math>n</math>. Since they are positive integers, the largest element is at least <math>n^2+1</math>, the <math>(n+1)</math>th positive integer congruent to 1 mod <math>n</math>. |
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| + | We are also given that this largest member is 2002, so <math>2002\equiv1\pmod{n}</math>, and <math>n|2001=3\cdot23\cdot29</math>. Also, we have <math>n^2+1\le2002</math>, so <math>n \leq 44</math>. The largest factor of 2001 less than 45 is 29, so <math>n=29</math> and <math>n+1</math> <math>\Rightarrow{\fbox{30}}</math> is the largest possible. This can be achieved with <math>\mathcal{S}=\{1,30,59,88,\ldots,813,2002\}</math>, for instance. | ||
== See also == | == See also == | ||
| − | + | {{AIME box|year=2002|n=I|num-b=13|num-a=15}} | |
| − | |||
| − | |||
| − | + | [[Category:Intermediate Number Theory Problems]] | |
| + | {{MAA Notice}} | ||
Latest revision as of 18:18, 21 June 2021
Problem
A set 
of distinct positive integers has the following property: for every integer 
in 
the arithmetic mean of the set of values obtained by deleting from is an integer. Given that 1 belongs to and that 2002 is the largest element of what is the greatest number of elements that can have?
Solution
Let the sum of the integers in be 
, and let the size of 
be 
. After any element is removed, we are given that 
, so 
. Since 
, 
, and all elements are congruent to 1 mod 
. Since they are positive integers, the largest element is at least 
, the 
th positive integer congruent to 1 mod .
We are also given that this largest member is 2002, so 
, and 
. Also, we have 
, so 
. The largest factor of 2001 less than 45 is 29, so 
and 
is the largest possible. This can be achieved with 
, for instance.
See also
| 2002 AIME I (Problems • Answer Key • Resources) | ||
| Preceded by Problem 13 |
Followed by Problem 15 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
