CF 105847A - A + B Problem

Rating: -
Tags: -
Solve time: 26s
Verified: yes

Solution

Problem Understanding

The task is the classic arithmetic problem where the input contains two integer values, representing two numbers that must be combined by addition. The output is the single integer value obtained by adding these two numbers together.

The constraints are intentionally small for this problem, so the main challenge is not performance but correctly handling input and producing the exact required output. Even if the numbers were very large, the intended solution would still be constant time because addition only requires one arithmetic operation. In languages with fixed-size integer types, the only thing to consider would be whether the values fit inside the chosen type. Python integers automatically expand, so overflow is not a concern.

The main mistakes in this problem come from treating the input format incorrectly. For example, if the two numbers appear on one line and the program reads only one value, the result will be wrong.

For input:

5 7

the correct output is:

A careless implementation that only processes the first number would output 5, because it never includes the second value.

Another edge case is when negative values are involved. For input:

-3 8

the correct output is:

A solution that assumes both values are positive and tries to handle the sign manually may introduce unnecessary errors.

Approaches

The brute-force approach would be to search through possible ways to combine the two values or simulate the addition process step by step. While such methods can eventually produce the right answer, they add complexity without using the fact that the operation is directly available in the programming language. A digit-by-digit simulation could require work proportional to the number of digits, and repeated trial approaches would be even slower.

The key observation is that the problem already defines a built-in arithmetic operation. The result depends only on the two input values, and there is no hidden structure, ordering, or search space to explore. Reading the numbers and applying the addition operator immediately gives the answer.

The brute-force method fails because it spends effort recreating a calculation that the processor can perform directly. The observation that the entire problem reduces to one arithmetic expression gives us a constant-time solution.

Approach	Time Complexity	Space Complexity	Verdict
Brute Force	O(d) or worse, where d is the number of digits or attempted operations	O(1)	Unnecessary
Optimal	O(1)	O(1)	Accepted

Algorithm Walkthrough

Read the two integer values from the input. The problem only requires these two values because no other information affects the final result.
Add the two values using the language's arithmetic operator. This directly matches the mathematical definition of the required output.
Print the resulting sum. The answer is complete after this single calculation because there are no additional conditions or transformations.

Why it works:

The algorithm preserves the only necessary property of the problem: the output must equal the mathematical sum of the two input numbers. The addition operation performed by the program has exactly the same meaning as the operation described by the task, so the produced value cannot differ from the required answer.

Python Solution

import sys
input = sys.stdin.readline

a, b = map(int, input().split())
print(a + b)

The program first reads the entire input line and splits it into two pieces. Converting both pieces with int turns the text representation into numeric values that Python can add.

The variables a and b store the two input numbers. The expression a + b performs the required calculation directly, and print writes the answer.

There are no boundary conditions involving indexes or loops, so there are no off-by-one issues. Python's integer type also avoids overflow problems that could appear in languages using fixed-width integer types.

Worked Examples

For the first example:

Input:

2 9

The execution trace is:

Step	a	b	result
Read input	2	9	not computed
Add values	2	9	11
Print answer	2	9	11

This demonstrates the normal case where both values are positive and the answer is a direct sum.

For the second example:

Input:

-10 4

The execution trace is:

Step	a	b	result
Read input	-10	4	not computed
Add values	-10	4	-6
Print answer	-10	4	-6

This confirms that the same operation handles negative numbers without any special cases.

Complexity Analysis

Measure	Complexity	Explanation
Time	O(1)	The algorithm performs one addition operation after reading the input.
Space	O(1)	Only two integer variables and the final result are stored.

The solution easily fits within any reasonable time and memory limits because it performs a fixed amount of work regardless of the input values.

Test Cases

import sys
import io

def solve(data):
    input = io.StringIO(data).readline
    a, b = map(int, input().split())
    return str(a + b)

def run(inp: str) -> str:
    return solve(inp)

assert run("2 9\n") == "11", "sample 1"
assert run("-10 4\n") == "-6", "sample 2"

assert run("0 0\n") == "0", "minimum boundary values"
assert run("1000000000 1000000000\n") == "2000000000", "large values"
assert run("-999999999 999999999\n") == "0", "opposite signs"
assert run("7 7\n") == "14", "equal values"

Test input	Expected output	What it validates
`0 0`	`0`	Checks the smallest natural arithmetic case.
`1000000000 1000000000`	`2000000000`	Confirms large values are handled correctly.
`-999999999 999999999`	`0`	Checks cancellation between negative and positive numbers.
`7 7`	`14`	Checks equal input values.

Edge Cases

For the case where the second number is accidentally ignored, consider:

5 7

The algorithm stores a = 5 and b = 7, then computes a + b, producing 12. A correct implementation never treats the input as a single value.

For the negative number case:

-3 8

the algorithm reads the signed integers exactly as provided. The addition operation computes -3 + 8, which gives 5. No manual sign handling is needed, avoiding the common mistake of assuming all values are positive.

For the large value case:

1000000000 1000000000

the algorithm still performs one addition and prints 2000000000. The number of digits does not change the structure of the solution, so the running time remains constant.