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

Revision as of 21:57, 1 August 2020

Let $a_1,\ldots,a_8$ be reals, not all equal to zero. Let $c_n = \sum^8_{k=1} a^n_k$ for $n=1,2,3,\ldots$ . Given that among the numbers of the sequence $(c_n)$ , there are infinitely many equal to zero, determine all the values of $n$ for which $c_n = 0.$

Solution

It can be found here [1]

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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.
+Let <math>a_1,\ldots,a_8</math> be reals, not all equal to zero. Let <cmath>c_n = \sum^8_{k=1} a^n_k</cmath> for <math>n=1,2,3,\ldots</math>. Given that among the numbers of the sequence <math>(c_n)</math>, there are infinitely many equal to zero, determine all the values of <math>n</math> for which <math>c_n = 0.</math>
-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>.
+==Solution==
+It can be found here [https://artofproblemsolving.com/community/c6h21159p137339]

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

Revision as of 21:57, 1 August 2020

Solution