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Difference between revisions of "2009 IMO Problems/Problem 1"

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Let <math>n</math> be a positive integer and let <math>a_1,\ldots,a_k (k\ge2)</math> be distinct integers in the set <math>\{1,\ldots,n\}</math> such that <math>n</math> divides <math>a_i(a_{i+1}-1)</math> for <math>i=1,\ldots,k-1</math>. Prove that <math>n</math> doesn't divide <math>a_k(a_1-1)</math>.
 
Let <math>n</math> be a positive integer and let <math>a_1,\ldots,a_k (k\ge2)</math> be distinct integers in the set <math>\{1,\ldots,n\}</math> such that <math>n</math> divides <math>a_i(a_{i+1}-1)</math> for <math>i=1,\ldots,k-1</math>. Prove that <math>n</math> doesn't divide <math>a_k(a_1-1)</math>.
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''Author: Ross Atkins, Australia''
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== Solution ==
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Let <math>n=pq</math> such that <math>p\mid a_1</math> and <math>q\mid a_2-1</math>. Suppose <math>n</math> divides <math>a_k(a_1-1)</math>.
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Note <math>q\mid a_2-1</math> implies <math>(q,a_2)=1</math> and hence <math>q\mid a_3-1</math>. Similarly one has <math>q\mid a_i-1</math> for all <math>i</math>'s, in particular, <math>p\mid a_1</math> and <math>q\mid a_1-1</math> force <math>(p,q)=1</math>. Now <math>(p,a_1-1)=1</math> gives <math>p\mid a_k</math>, similarly one has <math>p\mid a_i</math> for all <math>i</math>'s, that is <math>a_i</math>'s satisfy <math>p\mid a_i</math> and <math>q\mid a_i-1</math>, but there should be at most one such integer satisfies them within the range of <math>1,2,\ldots,n</math> for <math>n=pq</math> and <math>(p,q)=1</math>. A contradiction!!!
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''Solution by ychjae''

Revision as of 06:14, 23 July 2009

Problem

Let $n$ be a positive integer and let $a_1,\ldots,a_k (k\ge2)$ be distinct integers in the set $\{1,\ldots,n\}$ such that $n$ divides $a_i(a_{i+1}-1)$ for $i=1,\ldots,k-1$. Prove that $n$ doesn't divide $a_k(a_1-1)$.

Author: Ross Atkins, Australia

Solution

Let $n=pq$ such that $p\mid a_1$ and $q\mid a_2-1$. Suppose $n$ divides $a_k(a_1-1)$. Note $q\mid a_2-1$ implies $(q,a_2)=1$ and hence $q\mid a_3-1$. Similarly one has $q\mid a_i-1$ for all $i$'s, in particular, $p\mid a_1$ and $q\mid a_1-1$ force $(p,q)=1$. Now $(p,a_1-1)=1$ gives $p\mid a_k$, similarly one has $p\mid a_i$ for all $i$'s, that is $a_i$'s satisfy $p\mid a_i$ and $q\mid a_i-1$, but there should be at most one such integer satisfies them within the range of $1,2,\ldots,n$ for $n=pq$ and $(p,q)=1$. A contradiction!!!

Solution by ychjae

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