(Analysis by Ethan Guo)

Clearly, Bessie may send an invitation to any individual cow. To gain intuition on how to maximize probability, we will investigate on what circumstances we should extend an interval to increase probability. We will then show the strategy works in general.

Firstly, we will create an expression for the probability of exactly one acceptance. This equals $\sum\limits_{l\leq i\leq r}\left(p_i\prod\limits_{l\leq j\leq r,j\neq i}(1-p_j)\right)$. Letting $P=\prod\limits_{l\leq j\leq r}(1-p_j)$, this expression reduces to $P\sum\limits_{l\leq i\leq r}\frac{p_i}{1-p_i}$.

Consider the expression after a probability $p’$ is added to an end of the interval. For this to increase the answer, we have that $P(1-p’)\left(\frac{p’}{1-p’}+\sum\limits_{l\leq i\leq r}\frac{p_i}{1-p_i}\right)>P\sum\limits_{l\leq i\leq r}\frac{p_i}{1-p_i}$. This, surprisingly, reduces to $\sum\limits_{l\leq i\leq r}\frac{p_i}{1-p_i}<1$. Therefore, as long as the sum of $\frac{p_i}{1-p_i}$ within the interval is less than $1$, it is desirable to extend the interval.

Note that the quantity $\frac{p_i}{1-p_i}$ for any probability is positive. Therefore, each time we extend an interval, we increase $\sum\frac{p_i}{1-p_i}$. Up to a given point, this sum will remain less than $1$, and each augmentation before then is optimal to maximize the answer.

Since we can always extend the interval as long as the condition is satisfied, the correct strategy is to consider the maximal interval subject to the constraint for each starting point. This can be done with the two-pointer method in $O(N)$ since the right end of the maximal interval is nondecreasing (advance the right pointer as long as the sum is still less than $1$, then advance the left pointer once to consider the next starting point). My solution code is shown below.

(Note from Brian Dean: this problem is related to the so-called "Odds Algorithm", for those who want to understand more about its underlying mathematical structure)

#include <iostream>
#include <iomanip>

using namespace std;

int main() {
ios_base::sync_with_stdio(false); cin.tie(0);
freopen("cowdate.in", "r", stdin);
freopen("cowdate.out", "w", stdout);
int n; cin >> n;
double p[n];
for (int i = 0; i < n; i++) {
int pt; cin >> pt;
p[i] = pt * 0.000001;
}
// left and right pointers
int l = 0, r = 0;
// product of 1-p_i and sum of p_i/(1-p_i) within current interval
double prod = 1, sum = 0;
double mxp = -1;
while (l < n) {
// advance right pointer as long as condition is satisfied
while (r < n && sum < 1) {
// update product and sum
prod *= (1 - p[r]);
sum += p[r] / (1 - p[r]);
r++;
}
// expression for probability is prod * sum
mxp = max(mxp, prod * sum);
// advance left pointer by removing from prod and sum
prod /= (1 - p[l]);
sum -= p[l] / (1 - p[l]);
l++;
}
cout << (int) (1000000 * mxp) << endl;
return 0;
}