ICPC 2013 - J. Pollution Solution
State the problem in your own words. Focus on the mathematical or algorithmic core rather than repeating the full statement.
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competitive_programming/icpc/2013/J-pollution-solution/solution.tex to update the written solution and
competitive_programming/icpc/2013/J-pollution-solution/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.
ICPC 2013
2013 World Finals
St. Petersburg
HOSTED BY ITMO
Problem J
Pollution Solution
Time Limit: 1 second
As an employee of Aqueous Contaminate Management, you must monitor the pollution that gets dumped
(sometimes accidentally, sometimes purposefully) into rivers, lakes and oceans. One of your jobs is to
measure the impact of the pollution on various ecosystems in the water such as coral reefs, spawning
grounds, and so on.
Figure J.1: Illustration of Sample Input 1.
The model you use in your analysis is illustrated in Figure J.1. The shoreline (the horizontal line in the
figure) lies on the x-axis with the source of the pollution located at the origin (0,0). The spread of the
pollution into the water is represented by the semicircle, and the polygon represents the ecosystem of
concern. You must determine the area of the ecosystem that is contaminated, represented by the dark
blue region in the figure.
Input
The input consists of a single test case. A test case starts with a line containing two integers n and r,
where 3 ≤ n ≤ 100 is the number of vertices in the polygon and 1 ≤ r ≤ 1 000 is the radius of the
pollution field. This is followed by n lines, each containing two integers xi , yi , giving the coordinates of
the polygon vertices in counter-clockwise order, where −1 500 ≤ xi ≤ 1 500 and 0 ≤ yi ≤ 1 500. The
polygon does not self-intersect or touch itself. No vertex lies on the circle boundary.
Output
Display the area of the polygon that falls within the semicircle centered at the origin with radius r. Give
the result with an absolute error of at most 10−3 .
ICPC 2013
2013 World Finals
St. Petersburg
HOSTED BY ITMO
Sample Input 1 Sample Output 1
6 10 101.576437872
-8 2
8 2
8 14
0 14
0 6
-8 14
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 2013\\J. Pollution Solution Time Limit: 1 second}
\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;
}