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Part of 2001 Singapore Senior Math Olympiad

Problems(1)

sum sin x_i=0, sum n sin x_i=100, system

Source: Singapore Senior Math Olympiad 2001 2nd Round p1 SMO

4/4/2020
Let nn be a positive integer. Suppose that the following simultaneous equations {sin⁡x1+sin⁡x2+...+sin⁡xn=0sin⁡x1+2sin⁡x2+...+nsin⁡xn=100\begin{cases} \sin x_1 + \sin x_2+ ...+ \sin x_n = 0 \\ \sin x_1 + 2\sin x_2+ ...+ n \sin x_n = 100 \end{cases} has a solution, where x1x2,..,xnx_1 x_2,.., x_n are the unknowns. Find the smallest possible positive integer nn. Justify your answer.
system of equationstrigonometryalgebra