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20

Part of 2016 Math Prize for Girls Problems

Problems(1)

Math Prize 2016 Problem 20

Source:

9/12/2016
Let a1a_1, a2a_2, a3a_3, a4a_4, and a5a_5 be random integers chosen independently and uniformly from the set {0,1,2,…,23}\{ 0, 1, 2, \dots, 23 \}. (Note that the integers are not necessarily distinct.) Find the probability that ∑k=15cis⁡(akπ12)=0. \sum_{k=1}^{5} \operatorname{cis} \Bigl( \frac{a_k \pi}{12} \Bigr) = 0. (Here cis⁡θ\operatorname{cis} \theta means cos⁡θ+isin⁡θ\cos \theta + i \sin \theta.)
Math Prize for Girls