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ICPC 2011
ICPC 2011

A. To Add or to Multiply

Problem ID: addmul The Industrial Computer Processor Company offers very fast, special purpose processing units tailored to customer needs. Processors of the a-C-m family (such as the 1-C-2 and the 5-C-3) have an instruction set with only two different oper...

Updated May 21, 2026
Track ICPC
Year 2011
Statement Text + PDF
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Problem statement

Problem Statement

Formatted from the contest statement text, with sample tests broken out into copyable blocks.

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The Industrial Computer Processor Company offers very fast, special purpose processing units tailored to customer needs. Processors of the a-C-m family (such as the 1-C-2 and the 5-C-3) have an instruction set with only two different operations:

A add a
M multiply by m

The processor receives an integer, executes a sequence of A and M operations (the program) that modifies the input, and outputs the result. For example, the 1-C-2 processor executing the program AAAM with the input 2 yields the output 10 (the computation is 2 → 3 → 4 → 5 → 10), while the 5-C-3 processor yields 51 with the same program and input (2 → 7 → 12 → 17 → 51). You are an a-C-m programmer assigned to a top secret project. This means that you have not been told the precise computation your program should perform. But you are given particular values p, q, r, and s and the following conditions:

  1. The input is guaranteed to be a number between p and q.
  2. The output must be some number between r and s.

Given an a-C-m processor and the numbers p, q, r, and s, your job is to construct the shortest a-C-m program which, for every input x such that p ≤ x ≤ q, yields some output y such that r ≤ y ≤ s. If there is more than one program of minimum length, choose the one that come first lexicographically, treating each program as a string of As and Ms.

Input

The input contains several test cases. Each test case is given by a line with the six integers a, m, p, q, r, and s as described above (1 ≤ a, m, p, q, r, s ≤ 109 , p ≤ q and r ≤ s). The last test case is followed by a line with six zeros.

Output

For each test case, display its case number followed by the best program as described above. Display the word “empty” if the best program uses no operations. Display the word “impossible” if there is no program meeting the specifications. Display the program as a sequence of space-separated strings, alternating between strings of the form “nA” and strings of the form “nM”, where n > 0. Strings of the former type indicate n consecutive A operations, and strings of the latter type indicate n consecutive M operations. Follow the format of the sample output.

ICPC 2011 World Finals Problem A: To Add or to Multiply

Sample Tests

Sample
Sample Input 
 1   2   2   3   10 20
 1   3   2   3   22 33
 3   2   2   3   4 5
 5   3   2   3   2 3
 0   0   0   0   0 0

ICPC 2011 World Finals Problem A: To Add or to Multip
Sample Output 
Case   1:   1A 2M
Case   2:   1M 2A 1M
Case   3:   impossible
Case   4:   empty


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Editorial

Editorial

The solution write-up is rendered from the LaTeX source, with equations kept live through MathJax.

Raw TeX

Key Observations

  • Write the structural observations that make the problem tractable.

  • State any useful invariant, monotonicity property, graph interpretation, or combinatorial reformulation.

  • If the constraints matter, explain exactly which part of the solution they enable.

Algorithm

  1. Describe the data structures and the state maintained by the algorithm.

  2. Explain the processing order and why it is sufficient.

  3. Mention corner cases explicitly if they affect the implementation.

Correctness Proof

We prove that the algorithm returns the correct answer.

Lemma 1.

State the first key claim.

Proof.

Provide a concise proof.

Lemma 2.

State the next claim if needed.

Proof.

Provide a concise proof.

Theorem.

The algorithm outputs the correct answer for every valid input.

Proof.

Combine the lemmas and finish the argument.

Complexity Analysis

State the running time and memory usage in terms of the input size.

Implementation Notes

  • Mention any non-obvious implementation detail that is easy to get wrong.

  • Mention numeric limits, indexing conventions, or tie-breaking rules if relevant.

Implementation

Code

C++ solution used for this page.

C++

Clean code view with a raw-file link when you want the original source.

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

namespace {

void solve() {
    // Fill in the full solution logic for the problem here.
}

}  // namespace

int main() {
    ios::sync_with_stdio(false);
    cin.tie(nullptr);

    solve();
    return 0;
}
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