Although $k$ is rather small, trying all possible positions for the doors is $O(\binom n k)$ and is still way too slow.

Sometimes it's helpful in solving a problem to think about simpler cases of the problem first. When you're given an optimization problem on a circular array, one easy simplification is to think about it in a linear setting instead. So, if Farmer John's barn was linear, then how could we solve it? For starters, we clearly need to put a barn at the left-most position. It's more apparent now that we can do dynamic programming, since if we iterate from left to right, the only state that we need to keep track of is the number of exterior doors used so far, and the position of the last exterior door. This algorithm would have a running time of $O(n^2k)$, which is somewhat fast, but we can make it faster. Let's say we want to find the best way to allocate $k$ doors from spot $1$ to spot $i$. If we find this number and then want to allocate the $k$ doors between spot $1$ and spot $i+1$, then it follows that the last door that we allocate will have an equal or higher index than the last door in the problem that we've just solved. In other words, the index of the last door in the first problem is less than or equal to the index of the last door in the second problem. This can be shown by assuming the opposite; if we could obtain a better solution by moving a door to a spot with a lower index, then we could just do that in our first problem as well.

We can compute the best solution using $k$ doors via dynamic programming, iterating on $k$ and adding one door at a time. We can use a divide-and-conquer algorithm to solve the linear version of this problem. The idea is that we can compute the middle values of the dynamic program first: if we compute the position to put our $k$-th door in $dp[k][n/2]$, then $dp[k][i]$ for $i < n/2$ must be to the left of that and $dp[k][i]$ for $i > n/2$ must be to the right of that. In this way, we can on average halve the search space at each level of the recursion, so we don't have to do $O(n)$ work to compute each value of the dynamic program. At each level of the recursion, we do $O(nk)$ work, meaning the total runtime of this algorithm ends up being $O(nk \log n)$ for the linear version of this problem.

Since we have a lot of running time to spare, we can simply iterate over all possible locations of the first interior door. This turns the problem into effectively a linear one, although a bit of care is needed when processing the last door. We obtain an $O(n^2k \log n)$ algorithm, which is fast enough to receive full credit.

(Note from problem author: a running time of $O(n^2k)$ is also possible by further exploiting the monotonic structure of the underlying DP formulation.)

Here's Mark Gordon's solution:

#include <iostream>
#include <vector>
#include <algorithm>
#include <cstring>

using namespace std;

#define MAXN 1010
#define MAXK 10
#define INF 0x3FFFFFFFFFFFFFFFLL

int rot;
int N, K;
long long A[MAXN];
long long DP[MAXK][MAXN];
long long WSUM[MAXN][MAXN];

int madd(int x, int y) {
  x += y;
  if (x >= N) {
    x -= N;
  }
  return x;
}

long long wsum(int a, int b) {
  return WSUM[madd(a, rot)][madd(b, N - a)];
}

void solve(int k, int ia, int ib, int ja, int jb) {
  if (ia == ib) {
    return;
  }

  int i = (ia + ib) / 2;
  int arg_j = -1;

  DP[k][i] = INF;
  for (int j = max(i + 1, ja); j < jb; j++) {
    long long v = wsum(i, j) + DP[k - 1][j];
    if (v < DP[k][i]) {
      arg_j = j;
      DP[k][i] = v;
    }
  }

  solve(k, ia, i, ja, arg_j + 1);
  solve(k, i + 1, ib, arg_j, jb);
}

int main() {
  cin >> N >> K;
  for (int i = 0; i < N; i++) {
    cin >> A[i];
  }
  reverse(A, A + N);

  for (int i = 0; i < N; i++) {
    long long sm = 0;
    for (int j = 1; j <= N; j++) {
      WSUM[i][j] = WSUM[i][j - 1] + sm;
      sm += A[madd(i, j - 1)];
    }
  }

  long long result = INF;
  memset(DP, 0x3F, sizeof(DP));
  DP[0][N] = 0;
  for (rot = 0; rot < N; rot++) {
    for (int i = 1; i <= K; i++) {
      solve(i, 0, N, 1, N + 1);
    }
    result = min(result, DP[K][0]);
  }
  cout << result << endl;
}