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?
The number of gymnasts is some integer $n$, so that the number of gymnast-gymnast handshakes is ${n \choose 2}$ for some $n$. Also, the coach must participate in an integer $k<n$ number of handshakes. So, ${n \choose 2} + k = 281$. If we want to minimize $k$, we need the maximal $n$ such that ${n \choose 2} \le 281$, which implies $\frac{n(n-1)}{2} \le 281$ or $n^2 - n - 562 \le 0 $. So, the maximal $n$ is 24. So, $k = 281 - {24 \choose 2} = 281 - 12 \cdot 23 = 281 - 276 = \boxed{5}$.
5
Level 5
Counting & Probability
Problem: 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?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>Let the number of gymnasts be denoted as n. Each gymnast shakes hands with n-1 other gymnasts, leading to a total of n(n-1)/2 handshakes among the gymnasts. The coach shakes hands with k gymnasts from her team, contributing k handshakes. The total number of handshakes is given as 281, leading to the equation:

n(n-1)/2 + k = 281.

To minimize k, we need to maximize n. The expression n(n-1)/2 must be less than or equal to 281. Solving for n, we rearrange the equation:

n(n-1) ≤ 562.

Testing integer values for n, we find:

For n = 23: 23(22)/2 = 253, thus k = 281 - 253 = 28.
For n = 24: 24(23)/2 = 276, thus k = 281 - 276 = 5.
For n = 25: 25(24)/2 = 300, which exceeds 281.

The maximum feasible n is 24, yielding the minimum k of 5. Therefore, the fewest number of handshakes the coach could have participated in is 5.<|END_OF_LATENT|> The final answer is $5$.