Problem Overview

In Codeforces Problem 214A - System of Equations, you are given two integers n and m. The task is to find the number of integer pairs (a, b) that satisfy the equation:

a² + b = n

a + b² = m

You are given n and m, and you need to compute how many valid pairs (a, b) satisfy both equations.

C++ Solution

Here’s a simple and efficient C++ solution to solve the problem:

#include<bits/stdc++.h>
using namespace std;

int main() {
    int n, m, a, b, cnt = 0;
    cin >> n >> m;
    
    for (b = 0; b <= m * m; b++) {
        a = m - b * b;
        if (a * a + b == n && a >= 0) {
            cnt++;
        }
    }
    
    cout << cnt << endl;
    return 0;
}

Detailed Explanation of the Approach

1. Problem Breakdown

You need to find the number of valid integer pairs (a, b) that satisfy two equations:

a² + b = n

a + b² = m

Both equations have to hold true simultaneously for the pair (a, b).

2. Strategy to Solve the Problem

To find all possible pairs:

• We iterate through possible values of b from 0 to m * m. For each b, we compute a using the second equation a = m - b^2.
• For each pair (a, b), we check whether the first equation holds: a^2 + b == n. If it does, increment the count.

3. Optimizations

• We only need to check b from 0 to m * m, because for larger values of b, b^2 would exceed m, making it impossible for the second equation to hold.
• We check for valid pairs and increment the count whenever both conditions are satisfied.

Step-by-Step Dry Run Example

Let’s walk through an example to understand how the solution works:

Input:

n = 5
m = 13

Execution:

For b = 0:

a = m - b² = 13 - 0² = 13

Check: a² + b = 13² + 0 = 169, which is not equal to n = 5. No match.

For b = 1:

a = m - b² = 13 - 1² = 12

Check: a² + b = 12² + 1 = 145, which is not equal to n = 5. No match.

Continue this for each value of b until you find the valid pairs.

Output:

The output is the number of valid pairs.

Time and Space Complexity Analysis

• Time Complexity:
  The time complexity is O(m2)O(m^2)O(m2), where m is the input value. We iterate over all possible values of b and for each, perform constant-time calculations.

• Space Complexity:
  The space complexity is O(1)O(1)O(1), as we only need a few integer variables to store values and do not use any additional memory that scales with the input.

Common Mistakes to Watch Out For

• Incorrect Range for b:
  Ensure that you are iterating through valid values of b from 0 to m * m. If you go beyond this range, you may not get the correct results.

• Misinterpretation of the Equations:
  Double-check the equations before implementation. In this problem, both equations must hold true simultaneously for a valid pair (a, b).

• Overcomplicating the Solution:
  The problem can be solved with a simple loop and checks. Don’t over-engineer the solution by trying to optimize beyond what is necessary.

Related Problems to Practice

After solving this problem, you can practice similar problems that involve system of equations or integer pairs:

• Codeforces 175A - String Task

• Codeforces 116A - Petya and Strings

• Codeforces 313A - Ilya and Bank Account

These problems help improve your understanding of mathematical operations and integer pair problems.

Conclusion

In conclusion, the System of Equations problem is a great exercise in iterating through potential values and checking mathematical conditions. The solution uses basic arithmetic operations and iteration to efficiently compute the number of valid pairs. This problem is a great practice for anyone looking to improve their problem-solving skills in competitive programming.

As you solve more Codeforces problems, you’ll become more adept at handling various types of mathematical operations and optimizations.