https://artofproblemsolving.com/wiki/api.php?action=feedcontributions&user=Goblashapa&feedformat=atomAoPS Wiki - User contributions [en]2024-03-29T06:53:30ZUser contributionsMediaWiki 1.31.1https://artofproblemsolving.com/wiki/index.php?title=2002_AIME_I_Problems/Problem_14&diff=794672002 AIME I Problems/Problem 142016-07-19T00:22:27Z<p>Goblashapa: /* Problem */</p>
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<div>== Problem ==<br />
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?<br />
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== Solution ==<br />
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>.<br />
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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<45</math>. The largest factor of 2001 less than 45 is 29, so <math>n=29</math> and <math>n+1=\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.<br />
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== See also ==<br />
{{AIME box|year=2002|n=I|num-b=13|num-a=15}}<br />
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[[Category:Intermediate Number Theory Problems]]<br />
{{MAA Notice}}</div>Goblashapahttps://artofproblemsolving.com/wiki/index.php?title=BMO&diff=72877BMO2015-11-11T03:45:23Z<p>Goblashapa: Grammar error</p>
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<div>The abbreviation '''BMO''' is used to refer to all the [[Balkan Mathematical Olympiad]], the [[Belgian Mathematical Olympiad]], and the [[British Mathematical Olympiad]]. Outside of the United Kingdom, the abbreviation usually refers to the Balkan Olympiad when no further explanation is given.<br />
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