PK 2018 Q20
20. Omar marks a sequence of 12 points on a straight line beginning with a point \(O\), followed by a point \(P\) with \(OP = 1\). He chooses the points so that each point is the midpoint of the two immediately following points. For example \(O\) is the midpoint of \(PQ\), where \(Q\) is the third point he marks. What is the distance between the first point \(O\) and the 12th point \(Z\)?
- A 171
- B 341
- C 512
- D 587
- E 683
Pink Kangaroo 2018, Question 20. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: E 683
Let the points \(O\), \(P\) etc, be labelled as \(A_1\), \(A_2\), etc, in the order in which they are drawn. Without loss of generality, we can draw the line horizontally with \(A_2\) on the right of \(A_1\). Since \(A_1\) is the midpoint of \(A_2\) and \(A_3\), we then put \(A_3\) to the left of \(A_1\) at a distance of 1 from \(A_1\). Since \(A_2\) is the midpoint of \(A_3\) and \(A_4\), we put \(A_4\) on the right of \(A_2\) at a distance of 2 from \(A_2\). Proceeding in this fashion, the odd points end up further to the left of \(A_1\), and the even points further to the right, with the distances doubling at each stage. The diagram shows the points and the distances between neighbouring points, though not to scale! The distance \(A_1A_{12}\) is \(1 + 2 + 8 + 32 + 128 + 512 = 683\).
Solution © UK Mathematics Trust, from the PK 2018 solutions.