Three athletes and the mystery scoring system

Question

Adam, Bob and Charlie are the only competitors in a series of track and field events. In every event, first place scores x points, second scores y and third scores z, where x, y and z are whole numbers with x greater than y greater than z greater than 0, the same in every event. There are no ties. At the end, Adam has 22 points, Bob has 9 and Charlie has 9. Bob won the javelin. Who came second in the 100 metres? Explain how you can be sure, and find the number of events and the scoring system along the way.

Accepted Answer

Charlie came second in the 100 metres. There were 5 events, scored 5, 2, 1. How many events? Each event awards x + y + z points, and the total awarded is 22 + 9 + 9 = 40. So (number of events) × (x + y + z) = 40, with x + y + z at least 1 + 2 + 3 = 6. The factor pairs of 40 with the second factor at least 6 are: 1 event × 40, 2 × 20, 4 × 10, 5 × 8. - 1 event: Adam, Bob and Charlie would have scored x, y, z, all different, but Bob and Charlie both have 9. Impossible. - 2 events: Adam's 22 requires x at least 11; with x + y + z = 20 that means Adam won both events (22 = 11 + 11), but Bob won th…

Three athletes and the mystery scoring system

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