GK 2015 Q23
23. Five points lie on a straight line. Alex finds the distances between every pair of points. He obtains, in increasing order, 2, 5, 6, 8, 9, \(k\), 15, 17, 20 and 22. What is the value of \(k\)?
- A 14
- B 13
- C 12
- D 11
- E 10
Grey Kangaroo 2015, Question 23. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: A 14
Let the five points be \(P\), \(Q\), \(R\), \(S\) and \(T\) with individual distances between them of \(w\), \(x\), \(y\) and \(z\) as shown.
The maximum distance between any two points is 22 so \(PT = 22\). The next largest distance is 20 and, since no distance is 1, this is either \(PS\) or \(QT\). Assume \(QT = 20\) so that \(PQ = w = 2\). The next largest distance is 17 and, since no distance is 3 this cannot be \(QS\) or \(RT\) and so is \(PS\). Hence \(ST = z = 5\) and \(QS = 15\). The remaining distances are 6, 8, 9 and \(k\) which represent the lengths of \(PR\), \(QR\), \(RS\) and \(RT\) in some order with \(QR + RS = 15\). Since \(k > 9\), the only possible pair of distances adding to 15 is 6 and 9 and so \(QR = x = 6\) and \(RS = y = 9\) (since \(QR = 9\) and \(RS = 6\) would mean two distances are 11) leaving \(PR = 8\) and \(RT = k = 14\). Hence the value of \(k\) is 14 with the distances between the points taking the values shown below.
(Note: The same distances would result but in the reverse order if we assumed \(PS = 20\).)
Solution © UK Mathematics Trust, from the GK 2015 solutions.