## Cross Country Race

#### Problem

Alice, Belinda, and Clara were the three representatives for their school in a team cross country race. Alice finished the race in middle position, Belinda finished after Alice, in 19th position, and Clara finished 28th.

How many schools took part in the race?

#### Solution

As Alice finished in middle position we know that there must be an odd number of runners in the race, but because there are three girls representing each school, the number of runners must be an odd multiple of three. In addition, we know that Clara, who finished 28th, was not last, as there are an odd number of runners. So there must be at least 29 runners.

If there were 33 runners, the middle position would be 17th.
If there were 39 runners, the middle position would be 20th.

However, we are told that Belinda, who finished 19th, finished after Alice.

Hence there must have been 33 runners in the race and we deduce that eleven schools must have taken part.

Problem ID: 182 (01 Nov 2004)     Difficulty: 1 Star

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