2023/24 AHSMC Part 1 Problem 13

Problem 13 of 16

In a regular polygon $A_1A_2\ldots A_{21}$ the number of triangles $A_i A_j A_k$ having an angle greater than $90^\circ$ is

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Solution

The number of triangles $A_1 A_j A_k$ with $\angle A_1 = \frac{180^\circ}{21}(k-j) > 90^\circ$ is the number of pairs $(j,k)$ with $k-j > 21/2$ and $2 \le j < k \le 21$ . This yields $9+8+\cdots+1 = 45$ triangles with an obtuse angle at $A_1$ . The same count holds at each of the $21$ vertices, so the total is $45 \cdot 21 = 945$ . The answer is (B).

University of Alberta

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