2022 AIME II Problems/Problem 5

Problem

Twenty distinct points are marked on a circle and labeled $1$ through $20$ in clockwise order. A line segment is drawn between every pair of points whose labels differ by a prime number. Find the number of triangles formed whose vertices are among the original $20$ points.

Solution 1

Let $a$ , $b$ , and $c$ be the vertex of a triangle that satisfies this problem, where $a < b < c$ . $a - b = p_1$ $b - c = p_2$ $a - b = p_3 = a - b + b - c = p_1 + p_2$

Because $p_3 = p_1 + p_2$ , $p_1$ or $p_2$ must be $2$ . Let $p_1 = 2$ , then $p_2 \in \{ 3, 5, 11, 17 \}$ .

Once $a$ is determined, $b$ and $c$ are determined. There are $18$ values of $a$ where $a+2 \le 20$ . Therefore the answer is $18 \cdot 4 = \boxed{\textbf{72}}$

~isabelchen

2022 AIME II (Problems • Answer Key • Resources)
Preceded by Problem 4	Followed by Problem 6
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
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2022 AIME II Problems/Problem 5

Problem

Solution 1

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