2006 SMT/General Problems/Problem 16

Problem

Points $A_1, A_2, \cdots$ are placed on a circle with center $O$ such that $\angle OA_nA_{n+1}=35^\circ$ and $A_n\not=A_{n+2}$ for all positive integers $n$ . What is the smallest $n>1$ for which $A_n=A_1$ ?

Solution

The condition we are looking for is $110(n-1)\equiv 0 \mod 360$ , because if any other $A_n = A_1$ , than $110(n-1)$ must be a multiple of 360. This is because the central angle formed between two consecutive points is 110 degrees, and because we have rotated $(n-1)$ times by the time we place $A_n$ .

$110(n-1)\equiv 0\mod 360 \Rightarrow 10(n-1)\equiv 0\mod 360 \Rightarrow x\equiv 37\mod 360$ We want the first $A_n$ that satisfies the conditions from the problem, therefore, our answer is $n=37$ or $\boxed{A_{37}}$

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2006 SMT/General Problems/Problem 16

Problem

Solution