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Difference between revisions of "2012 USAMO Problems/Problem 3"

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holds for every positive integer <math>k</math>.
 
holds for every positive integer <math>k</math>.
  
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==Solution==
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==Partial Solution==
 
For <math>n</math> equal to any odd prime <math>p</math>, the sequence <math>\left\{a_i = \left(\frac{1-n}{2}\right)^{m_p\left(i\right)}\right\}</math>, where <math>p^{m_p\left(i\right)}</math> is the greatest power of <math>p</math> that divides <math>i</math>, gives a valid sequence. Therefore, the set of possible values for <math>n</math> is at least the set of odd primes.
 
For <math>n</math> equal to any odd prime <math>p</math>, the sequence <math>\left\{a_i = \left(\frac{1-n}{2}\right)^{m_p\left(i\right)}\right\}</math>, where <math>p^{m_p\left(i\right)}</math> is the greatest power of <math>p</math> that divides <math>i</math>, gives a valid sequence. Therefore, the set of possible values for <math>n</math> is at least the set of odd primes.
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==Solution that involves a non-elementary result==
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For <math>n=2</math>, <math> |a_1| = 2 |a_2| = \cdots = 2^m |a_{2^m}| </math> implies that for any positive integer <math>m</math>, <math>|a_1| \ge 2^m</math>, which is impossible.
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We proceed to prove that the infinite sequence exists for all <math>n\ge 3</math>.
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First, one notices that if we have <math>a_{xy} = a_x a_y</math> for any integers <math>x</math> and <math>y</math>, then it is suffice to define all <math>a_x</math> for <math>x</math> prime, and one only needs to verify the equation
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<cmath>a_1+2a_2+\cdots+na_n    (*)</cmath>
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for the other equations will be automatically true.
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 +
In the following construction, I am using Bertrand's Theorem without proof. The Theorem states that, for any integer <math>n>1</math>, there exists a prime <math>p</math> such that <math>n<p\le 2n-1</math>.
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In other words, if <math>m=2n</math> with <math>n>1</math>, then there exists a prime <math>p</math> such that <math>m/2 < p <  m</math>, and if <math>m=2n-1</math> with <math>n>1</math>, then there exists a prime <math>p</math> such that <math>(m+1)/2 <p\le m</math>, both of which guarantees that for any integer <math>m>2</math>, there exists a prime <math>p</math> such that <math>m/2 <p \le m</math>
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So, for <math>n\ge 3</math>, let the largest two primes not larger than <math>n</math> are <math>P</math> and <math>Q</math>, and that <math>n\ge P > Q</math>. By the Theorem stated above, one can conclude that <math>2P > n</math>, and that <math>4Q = 2(2Q) \ge 2P > n</math>. Using this fact, I'm going to construct the sequence <math>a_n</math>.
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Let <math>a_1=1</math>. If <math>2Q>n</math>, then let <math>a_x = 1</math> for all prime numbers <math>x<Q</math>, and <math>a_{xy}=a_xa_y</math>, then (*) becomes:
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<cmath> Pa_P + Qa_Q = C_1  (1)</cmath>
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If <math>2Q\le n</math> but <math>3Q > n</math>, then let <math>a_2=-1</math>, and <math>a_x = 1</math> for all prime numbers <math>2<x<Q</math>, and <math>a_{xy}=a_xa_y</math>, then (*) becomes:
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<cmath> Pa_P + Qa_Q - Qa_{2Q} = C_2 </cmath>
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or
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<cmath> Pa_P - Qa_Q = C_2    (2)</cmath>
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If <math>3Q\le n</math>, let <math>a_2=3</math>, <math>a_3=-2</math>, and <math>a_x = 1</math> for all prime numbers <math>3<x<Q</math>, and <math>a_{xy}=a_xa_y</math>, then (*) becomes:
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<cmath> Pa_P + Qa_Q + 3Qa_{2Q} - 2Qa_{3Q} = C_3 </cmath>
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or
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<cmath> Pa_P + Qa_Q = C_3  (3)</cmath>
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In each of (1), (2), (3), there exists non zero integers <math>a_P</math> and <math>a_Q</math> which satisfy the equation. Then for other primes <math>p > P</math>, just let <math>a_p=1</math> (or any number).
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This construction is correct because, for any <math>k> 1</math>,
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<cmath>a_k + 2 a_{2k} + \cdots n a_{nk} =  a_k (1 + 2 a_2 + \cdots n a_n ) = 0 </cmath>
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21:22, 2 May 2012 (EDT)
  
 
{{solution}}
 
{{solution}}

Revision as of 21:23, 2 May 2012

Problem

Determine which integers $n > 1$ have the property that there exists an infinite sequence $a_1$, $a_2$, $a_3$, $\dots$ of nonzero integers such that the equality \[a_k + 2a_{2k} + \dots + na_{nk} = 0\] holds for every positive integer $k$.

Partial Solution

For $n$ equal to any odd prime $p$, the sequence $\left\{a_i = \left(\frac{1-n}{2}\right)^{m_p\left(i\right)}\right\}$, where $p^{m_p\left(i\right)}$ is the greatest power of $p$ that divides $i$, gives a valid sequence. Therefore, the set of possible values for $n$ is at least the set of odd primes.


Solution that involves a non-elementary result

For $n=2$, $|a_1| = 2 |a_2| = \cdots = 2^m |a_{2^m}|$ implies that for any positive integer $m$, $|a_1| \ge 2^m$, which is impossible.

We proceed to prove that the infinite sequence exists for all $n\ge 3$.

First, one notices that if we have $a_{xy} = a_x a_y$ for any integers $x$ and $y$, then it is suffice to define all $a_x$ for $x$ prime, and one only needs to verify the equation

\[a_1+2a_2+\cdots+na_n (*)\]

for the other equations will be automatically true.

In the following construction, I am using Bertrand's Theorem without proof. The Theorem states that, for any integer $n>1$, there exists a prime $p$ such that $n<p\le 2n-1$.

In other words, if $m=2n$ with $n>1$, then there exists a prime $p$ such that $m/2 < p < m$, and if $m=2n-1$ with $n>1$, then there exists a prime $p$ such that $(m+1)/2 <p\le m$, both of which guarantees that for any integer $m>2$, there exists a prime $p$ such that $m/2 <p \le m$

So, for $n\ge 3$, let the largest two primes not larger than $n$ are $P$ and $Q$, and that $n\ge P > Q$. By the Theorem stated above, one can conclude that $2P > n$, and that $4Q = 2(2Q) \ge 2P > n$. Using this fact, I'm going to construct the sequence $a_n$.

Let $a_1=1$. If $2Q>n$, then let $a_x = 1$ for all prime numbers $x<Q$, and $a_{xy}=a_xa_y$, then (*) becomes:

\[Pa_P + Qa_Q = C_1 (1)\]

If $2Q\le n$ but $3Q > n$, then let $a_2=-1$, and $a_x = 1$ for all prime numbers $2<x<Q$, and $a_{xy}=a_xa_y$, then (*) becomes:

\[Pa_P + Qa_Q - Qa_{2Q} = C_2\]

or

\[Pa_P - Qa_Q = C_2 (2)\]

If $3Q\le n$, let $a_2=3$, $a_3=-2$, and $a_x = 1$ for all prime numbers $3<x<Q$, and $a_{xy}=a_xa_y$, then (*) becomes:

\[Pa_P + Qa_Q + 3Qa_{2Q} - 2Qa_{3Q} = C_3\]

or \[Pa_P + Qa_Q = C_3 (3)\]

In each of (1), (2), (3), there exists non zero integers $a_P$ and $a_Q$ which satisfy the equation. Then for other primes $p > P$, just let $a_p=1$ (or any number).

This construction is correct because, for any $k> 1$,

\[a_k + 2 a_{2k} + \cdots n a_{nk} = a_k (1 + 2 a_2 + \cdots n a_n ) = 0\]

21:22, 2 May 2012 (EDT)

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See Also

2012 USAMO (Problems • Resources)
Preceded by
Problem 2 		Followed by
Problem 4
1 • 2 • 3 • 4 • 5 • 6
All USAMO Problems and Solutions
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