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  Consider n numbers x1, x2, . . . , xn laid out on a circle...

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Advance Math

 

Consider n numbers x1, x2, . . . , xn laid out on a circle and some value α. Consider the requirement that every number equals α times the sum of its two neighbors. For example, if α were zero, this would force all the numbers to be zero.

(a) Show that, no matter what α is, the system has a solution.

(b) Show that if α = 1/2 , then the system has a nontrivial solution.

(c) Show that if α = − 1/2 , then there is a nontrivial solution if and only if n is even.

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