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

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Go back to the problem. Suppose <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 fact stated above, one can conclude that <math>2P > n</math>, and that <math>4Q = 2(2Q) \ge 2P > n</math>. Let's construct <math>a_n</math>:
 
Go back to the problem. Suppose <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 fact stated above, one can conclude that <math>2P > n</math>, and that <math>4Q = 2(2Q) \ge 2P > n</math>. Let's construct <math>a_n</math>:
  
−
Let <math>a_1=1</math>. There will be three cases: <math>Q>\frac{n}{2}</math>, <math> \frac{n}{2} \ge Q > \frac{n}{3}</math>, and <math>\frac{n}{3} \ge Q > \frac{n}{4}</math>.
+
Let <math>a_1=1</math>. There will be three cases: (i) <math>Q>\frac{n}{2}</math>, (ii) <math>\frac{n}{2} \ge Q > \frac{n}{3}</math>, and (iii) <math>\frac{n}{3} \ge Q > \frac{n}{4}</math>.
  
 
Case (i): <math>2Q>n</math>. Let <math>a_x = 1</math> for all prime numbers <math>x<Q</math>, and <math>a_{xy}=a_xa_y</math>, then (*) becomes:
 
Case (i): <math>2Q>n</math>. Let <math>a_x = 1</math> for all prime numbers <math>x<Q</math>, and <math>a_{xy}=a_xa_y</math>, then (*) becomes:

Revision as of 15:30, 3 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 (*)

\[a_1+2a_2+\cdots+na_n=0\]

for the other equations will be automatically true.

To proceed with the construction, I need the following fact: for any positive integer $m>2$, there exists a prime $p$ such that $\frac{m}{2} <p \le m$.

To prove this, I am going to use Bertrand's Theorem ([1]) 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, for any positive integer $m>2$, if $m=2n$ with $n>1$, then there exists a prime $p$ such that $\frac{m}{2} < p < m$, and if $m=2n-1$ with $n>1$, then there exists a prime $p$ such that $\frac{m+1}{2} <p\le m$, both of which guarantees that for any integer $m>2$, there exists a prime $p$ such that $\frac{m}{2} <p \le m$.



Go back to the problem. Suppose $n\ge 3$. Let the largest two primes not larger than $n$ are $P$ and $Q$, and that $n\ge P > Q$. By the fact stated above, one can conclude that $2P > n$, and that $4Q = 2(2Q) \ge 2P > n$. Let's construct $a_n$:

Let $a_1=1$. There will be three cases: (i) $Q>\frac{n}{2}$, (ii) $\frac{n}{2} \ge Q > \frac{n}{3}$, and (iii) $\frac{n}{3} \ge Q > \frac{n}{4}$.

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

\[Pa_P + Qa_Q = C_1\]

Case (ii): $2Q\le n$ but $3Q > n$. In this case, 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

\[Pa_P - Qa_Q = C_2\]

Case (iii): $3Q\le n$. In this case, 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 \[Pa_P + Qa_Q = C_3\]

In each case, by Bezout's Theorem, there exists non zero integers $a_P$ and $a_Q$ which satisfy the equation. For all other primes $p > P$, just let $a_p=1$ (or any number).

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

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

Since Bertrand's Theorem is not elementary, we still need to wait for a better proof.

--Lightest 21:24, 2 May 2012 (EDT)

This problem needs a solution. If you have a solution for it, please help us out by adding it.

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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