PK 2018 Q18
18. Lonneke drew a regular polygon with 2018 vertices, which she labelled from 1 to 2018, in a clockwise direction. She then drew a diagonal from the vertex labelled 18 to the vertex labelled 1018. She also drew the diagonal from the vertex labelled 1018 to the vertex labelled 2000. This divided the original polygon into three new polygons. How many vertices did each of the resulting three polygons have?
- A 38, 983, 1001
- B 37, 983, 1001
- C 38, 982, 1001
- D 37, 982, 1000
- E 37, 983, 1002
Pink Kangaroo 2018, Question 18. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: A 38, 983, 1001
In between a vertex labelled \(n\) and a vertex labelled \(m\) (with \(m > n\)), there are \(m - n + 1\) vertices (including the end points). Hence between vertex 18 and vertex 1018, there are 1001 vertices; between vertex 1018 and vertex 2000 there are 983 vertices. Continuing clockwise from vertex 2000 to vertex 18, we count 37 vertices (and the polygon also has vertex 1018, making 38). Therefore the resulting polygons have 1001, 983 and 38 vertices.
Solution © UK Mathematics Trust, from the PK 2018 solutions.