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Difference between revisions of "1967 IMO Problems/Problem 5"

(Created page with "Take |a1| >= |a2| >= ... >= |a8|. Suppose that |a1|, ... , |ar| are all equal and greater than |ar+1|. Then for sufficiently large n, we can ensure that |as|n < 1/8 |a1|n for ...")
 
(Latexed the document.)
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Take |a1| >= |a2| >= ... >= |a8|. Suppose that |a1|, ... , |ar| are all equal and greater than |ar+1|. Then for sufficiently large n, we can ensure that |as|n < 1/8 |a1|n for s > r, and hence the sum of |as|n for all s > r is less than |a1|n. Hence r must be even with half of a1, ... , ar positive and half negative.
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Take <math>|a_1| >= |a_2| >= ... >= |a_8|</math>. Suppose that <math>|a_1|, ... , |a_r|</math> are all equal and greater than <math>|a_{r+1}|</math>. Then for sufficiently large <math>n</math>, we can ensure that <math>|a_s|n < \frac{1}{8} |a_1|n</math> for <math>s > r</math>, and hence the sum of <math>|a_s|n</math> for all <math>s > r</math> is less than <math>|a_1|n</math>. Hence <math>r</math> must be even with half of <math>a_1, ... , a_r</math> positive and half negative.
  
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If that does not exhaust the ai, then in a similar way there must be an even number of ai with the next largest value of |ai|, with half positive and half negative, and so on. Thus we find that cn = 0 for all odd n.
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If that does not exhaust the <math>a_i</math>, then in a similar way there must be an even number of <math>a_i</math> with the next largest value of <math>|a_i|</math>, with half positive and half negative, and so on. Thus we find that <math>cn = 0</math> for all odd <math>n</math>.

Revision as of 10:18, 30 June 2020

Take $|a_1| >= |a_2| >= ... >= |a_8|$. Suppose that $|a_1|, ... , |a_r|$ are all equal and greater than $|a_{r+1}|$. Then for sufficiently large $n$, we can ensure that $|a_s|n < \frac{1}{8} |a_1|n$ for $s > r$, and hence the sum of $|a_s|n$ for all $s > r$ is less than $|a_1|n$. Hence $r$ must be even with half of $a_1, ... , a_r$ positive and half negative.

If that does not exhaust the $a_i$, then in a similar way there must be an even number of $a_i$ with the next largest value of $|a_i|$, with half positive and half negative, and so on. Thus we find that $cn = 0$ for all odd $n$.

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