jmc

Question

After a gymnastics meet, each gymnast shook hands once with every gymnast on every team (except herself). Afterwards, a coach came down and only shook hands with each gymnast from her own team. There were a total of 281 handshakes. What is the fewest number of handshakes the coach could have participated in?

Accepted Answer

The number of gymnasts is some integer n, so that the number of gymnast-gymnast handshakes is n 2 for some n. Also, the coach must participate in an integer k<n number of handshakes. So, n 2 + k = 281. If we want to minimize k, we need the maximal n such that n 2 281, which implies n(n-1)2 281 or n^2 - n - 562 0 . So, the maximal n is 24. So, k = 281 - 24 2 = 281 - 12 23 = 281 - 276 = 5.