A panel of five judges—J, K, L, M, N—are rating six gymnasts—P, Q, R, S, T, U—in a competition. Each gymnast receives exactly one rating from one judge. Each judge rates at least one gymnast, and no judge rates more than two gymnasts. The following conditions apply: 1. Judge J rates exactly one gymnast. 2. Judge K rates more gymnasts than Judge L. 3. Judge M rates either gymnast S or gymnast T, but not both. 4. If Judge N rates gymnast P, then Judge K rates gymnast R. 5. Judge L does not rate gymnast U. If Judge M rates gymnast T, and Judge K rates gymnast U, which of the following must be true?

Correct answer: C. Judge K rates exactly two gymnasts.

Let's break down the given information and conditions: Judges: J, K, L, M, N (5 judges) Gymnasts: P, Q, R, S, T, U (6 gymnasts) Constraints: Each judge rates 1 or 2 gymnasts. Total ratings = 6. Given: M rates T, K rates U. 1. **Judge J rates exactly one gymnast.** (J=1) 2. **Judge K rates more gymnasts than Judge L.** (K > L) 3. **Judge M rates either gymnast S or gymnast T, but not both.** (M rates S or T) * Since M rates T, M cannot rate S. So M rates 1 gymnast (T). 4. **If Judge N rates gymnast P, then Judge K rates gymnast R.** (N:P -> K:R) 5. **Judge L does not rate gymnast U.** (L != U) From given: M rates T (1 gymnast). K rates U (1 gymnast). Total gymnasts rated = 6. J rates 1. M rates 1. So, K, L, N must rate the remaining 4 gymnasts (6 - 1 - 1 = 4). Possible ratings per judge: 1 or 2. We have: K > L. K and L can be 1 or 2. N can be 1 or 2. Possible distributions for (K, L, N) that sum to 4, with K > L: * K=2, L=1, N=1 (K > L is satisfied, sum = 4) * K=2, L=2, N=0 (N must rate at least one, so N=0 is impossible) * K=1, L=1, N=2 (K > L is NOT satisfied) * K=2, L=1, N=1 is the only valid distribution for (K, L, N) that sums to 4 AND satisfies K > L, and each judge rates at least one. Therefore, K must rate 2 gymnasts. Since K already rates U, K must rate one more gymnast. Let's check the options: * A: Judge J rates gymnast P. We don't know who J rates. Could be P, Q, R, S. (Not necessarily true) * B: Judge L rates gymnast S. We know M doesn't rate S. L could rate S. (Not necessarily true, L could rate Q or R). * C: Judge N rates gymnast Q. We don't know who N rates. (Not necessarily true) * D: Judge K rates exactly two gymnasts. This must be true based on the distribution derived above. K=2, L=1, N=1. K already rates U, so K rates one more. (Must be true) Therefore, D is the correct answer.