(Analysis by Nick Wu)

The most direct solution involves taking each photo and then checking every flower in that photo for an average flower.

There are $\mathcal{O}(N^2)$ photos, and checking each flower in the photo takes $\mathcal{O}(N)$, so this solution runs in $\mathcal{O}(N^3)$ time.

Code is as follows.

import java.io.*;
import java.util.*;
public class Main {
public static void main(String[] args) throws IOException {
PrintWriter pw = new PrintWriter(new BufferedWriter(new OutputStreamWriter(System.out)));
int[] petals = new int[n];
for(int i = 0; i < n; i++) {
petals[i] = Integer.parseInt(st.nextToken());
}
int photos = 0;
for(int i = 0; i < n; i++) {
for(int j = i; j < n; j++) {
int totalPetals = 0;
for(int k = i; k <= j; k++) {
totalPetals += petals[k];
}
boolean present = false;
for(int k = i; k <= j; k++) {
if(petals[k] * (j-i+1) == totalPetals) {
present = true;
}
}
if(present) {
photos++;
}
}
}
pw.println(photos);
pw.close();
}
}


It's possible (but not necessary) to optimize this down to $\mathcal{O}(N^2)$. One observation we can make is that all photos with more than one flower consist of taking a smaller photo and then including the next rightmost flower.

Let us consider processing all photos that use flower $i$ as the leftmost flower, starting by using the photo that contains only flower $i$ and then adding flowers one by one to the right. We can maintain a collection of all the flowers we have seen so far. As we do this, we also need to keep track of the total sum of petals we have seen so far so we know what the average petal count is, and then we can check whether we have seen that petal count before. Because the number of petals is bounded above by $10^3$, we can use a boolean array to track which petal counts we have seen. Alternatively, we can use a set.

import java.io.*;
import java.util.*;
public class Main {
public static void main(String[] args) throws IOException {
PrintWriter pw = new PrintWriter(new BufferedWriter(new OutputStreamWriter(System.out)));
int[] petals = new int[n];
for(int i = 0; i < n; i++) {
petals[i] = Integer.parseInt(st.nextToken());
}
int photos = 0;
for(int i = 0; i < n; i++) {
boolean[] present = new boolean;
int petalsSeen = 0;
for(int j = i; j < n; j++) {
petalsSeen += petals[j];
present[petals[j]] = true;
if(petalsSeen % (j-i+1) == 0 && present[petalsSeen / (j-i+1)]) {
photos++;
}
}
}
pw.println(photos);
pw.close();
}
}