ICPC 2008 - B. Always an Integer
State the problem in your own words. Focus on the mathematical or algorithmic core rather than repeating the full statement.
Source-first archive entry
This page is built from the copied files in competitive_programming/icpc/2008/B-always-an-integer. Edit
competitive_programming/icpc/2008/B-always-an-integer/solution.tex to update the written solution and
competitive_programming/icpc/2008/B-always-an-integer/solution.cpp to update the implementation.
The website does not replace those files with hand-maintained HTML. It reads the copied source tree during the build and exposes the exact files below.
Problem Statement
Copied statement text kept beside the solution archive for direct reference.
Problem B
Always an Integer
Input file: always.in
Combinatorics is a branch of mathematics chiefly concerned with counting discrete objects. For instance, how many
ways can you pick two people out of a crowd of n people? Into how many regions can you divide a circular disk by
connecting n points on its boundary with one another? How many cubes are in a pyramid with square layers ranging
from 1 × 1 to n × n cubes?
Figure 1: If we connect six points on the boundary of a circle, at
T T
most 31 regions are created.
Many questions like these have answers that can be reduced to simple polynomials in n. The answer to the first
question above is n(n-1)/2, or (n^2-n)/2. The answer to the second is (n^4-6n^3+23n^2-18n+24)/24. The answer to
the third is n(n+1)(2n+1)/6, or (2n^3+3n^2+n)/6. We write these polynomials in a standard form, as a polynomial
with integer coefficients divided by a positive integer denominator.
These polynomials are answers to questions that can have integer answers only. But since they have fractional
coefficients, they look as if they could produce non-integer results! Of course, evaluating these particular
polynomials on a positive integer always results in an integer. For other polynomials of similar form, this is not
necessarily true. It can be hard to tell the two cases apart. So that, naturally, is your task.
Input
The input consists of multiple test cases, each on a separate line. Each test case is an expression in the form (P)/D,
where P is a polynomial with integer coefficients and D is a positive integer denominator. P is a sum of terms of the
form Cn^E, where the coefficient C and the exponent E satisfy the following conditions:
1. E is an integer satisfying 0 ≤ E ≤ 100. If E is 0, then Cn^E is expressed as C. If E is 1, then Cn^E is
expressed as Cn, unless C is 1 or -1. In those instances, Cn^E is expressed as n or -n.
2. C is an integer. If C is 1 or -1 and E is not 0 or 1, then the Cn^E will appear as n^E or -n^E.
3. Only non-negative C values that are not part of the first term in the polynomial are preceded by +.
4. Exponents in consecutive terms are strictly decreasing.
5. C and D fit in a 32-bit signed integer.
See the sample input for details.
Input is terminated by a line containing a single period.
Output
For each test case, print the case number (starting with 1). Then print Always an integer if the test case
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polynomial evaluates to an integer for every positive integer n. Print Not always an integer otherwise. Print
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the output for separate test cases on separate lines. Your output should follow the same format as the sample output.
Sample Input Output for the Sample Input
(n^2-n)/2 Case 1: Always an integer
(2n^3+3n^2+n)/6 Case 2: Always an integer
(-n^14-11n+1)/3 Case 3: Not always an integer
.
Editorial
Rendered from the copied solution.tex file. The original TeX source remains
available below.
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
Describe the data structures and the state maintained by the algorithm.
Explain the processing order and why it is sufficient.
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.
Code
Exact copied C++ implementation from solution.cpp.
#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;
}
Source Files
Exact copied source-of-truth files. Edit solution.tex for the write-up and solution.cpp for the implementation.
\documentclass[11pt]{article}
\usepackage[margin=1in]{geometry}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{amsmath,amssymb,amsthm}
\usepackage{enumitem}
\title{ICPC World Finals 2008\\B. Always an Integer}
\author{}
\date{}
\begin{document}
\maketitle
\section*{Problem Summary}
State the problem in your own words. Focus on the mathematical or algorithmic core rather than repeating the full statement.
\section*{Key Observations}
\begin{itemize}[leftmargin=*]
\item Write the structural observations that make the problem tractable.
\item State any useful invariant, monotonicity property, graph interpretation, or combinatorial reformulation.
\item If the constraints matter, explain exactly which part of the solution they enable.
\end{itemize}
\section*{Algorithm}
\begin{enumerate}[leftmargin=*]
\item Describe the data structures and the state maintained by the algorithm.
\item Explain the processing order and why it is sufficient.
\item Mention corner cases explicitly if they affect the implementation.
\end{enumerate}
\section*{Correctness Proof}
We prove that the algorithm returns the correct answer.
\paragraph{Lemma 1.}
State the first key claim.
\paragraph{Proof.}
Provide a concise proof.
\paragraph{Lemma 2.}
State the next claim if needed.
\paragraph{Proof.}
Provide a concise proof.
\paragraph{Theorem.}
The algorithm outputs the correct answer for every valid input.
\paragraph{Proof.}
Combine the lemmas and finish the argument.
\section*{Complexity Analysis}
State the running time and memory usage in terms of the input size.
\section*{Implementation Notes}
\begin{itemize}[leftmargin=*]
\item Mention any non-obvious implementation detail that is easy to get wrong.
\item Mention numeric limits, indexing conventions, or tie-breaking rules if relevant.
\end{itemize}
\end{document}
#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;
}