1993 IMO Problems/Problem 6

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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1993 IMO (Problems) • Resources
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1993 IMO Problems/Problem 6

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