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1993 IMO Problems/Problem 6

Revision as of 14:36, 6 October 2023 by Tomasdiaz (talk | contribs) (Created page with "==Problem == There are <math>n</math> lamps <math>L_0, \ldots , L_{n-1}</math> in a circle (<math>n > 1</math>), where we denote <math>L_{n+k} = L_k</math>. (A lamp at all tim...")
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Problem

There are $n$ lamps $L_0, \ldots , L_{n-1}$ in a circle ($n > 1$), where we denote $L_{n+k} = L_k$. (A lamp at all times is either on or off.) Perform steps $s_0, s_1, \ldots$ as follows: at step $s_i$, if $L_{i-1}$ is lit, switch $L_i$ from on to off or vice versa, otherwise do nothing. Initially all lamps are on. Show that:


(a) There is a positive integer $M(n)$ such that after $M(n)$ steps all the lamps are on again;

(b) If $n = 2^k$, we can take $M(n) = n^2 - 1$;

(c) If $n = 2^k + 1$, we can take $M(n) = n^2 - n + 1.$

Solution

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