21. Martin is standing in a queue. The number of people in the queue is a multiple of 3 . He notices that he has as many people in front of him as behind him. He sees two friends, both standing behind him in the queue, one in 19th place and the other in 28 th place. In which position in the queue is Martin?
A 14
B 15
C 16
D 17
E 18
Solution
D

Let the number of people in front of Martin in the queue be $N$. Therefore, he is in place $N+1$ in the queue and since his friend in 19th place is behind him we have $N+1<19$ and hence $N \leq 17$. Since there are also $N$ people behind him in the queue, the total number of people in the queue is $2 N+1$ and since his other friend is in 28 th place, we have $2 N+1 \geq 28$ and hence $N \geq 14$ as $N$ is an integer.

When we combine these two results for $N$, we find that the only possible values of $N$ are $14,15,16$ and 17 with corresponding values for the total number of people in the queue of $29,31,33$ and 35 . Of these, the only total that is a multiple of 3 is 33 and hence $N=16$ and Martin is in 17 th position in the queue.