ICPC 2011 - G. Magic Sticks
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/2011/G-magic-sticks. Edit
competitive_programming/icpc/2011/G-magic-sticks/solution.tex to update the written solution and
competitive_programming/icpc/2011/G-magic-sticks/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 G
Magic Sticks
Problem ID: magicsticks
Magic was accepted by all ancient peoples as a technique to compel the help of divine powers. In a well-known story,
one group of sorcerers threw their walking sticks on the floor where they magically appeared to turn into live serpents.
In opposition, another person threw his stick on the floor, where it turned into a serpent which then consumed the
sorcerers’ serpents!
The only magic required for this problem is its solution. You are given a magic stick that has several straight segments,
with joints between the segments that allow the stick to be folded. Depending on the segment lengths and how they are
folded, the segments of the stick can be arranged to produce a number of polygons. You are to determine the maximum
area that could be enclosed by the polygons formed by folding the stick, using each segment in at most one polygon.
Segments can touch only at their endpoints. For example, the stick shown below on the left has five segments and four
joints. It can be folded to produce a polygon as shown on the right.
Input
The input contains several test cases. Each test case describes a magic stick. The first line in each test case contains
an integer n (1 ≤ n ≤ 500) which indicates the number of the segments in the magic stick. The next line contains n
integers S1 , S2 , . . . , Sn (1 ≤ Si ≤ 1000) which indicate the lengths of the segments in the order they appear in the
stick.
The last test case is followed by a line containing a single zero.
Output
For each case, display its case number followed by the maximum total enclosed area that can be obtained by folding
the magic stick at the given points. Answers within an absolute or relative error of 10−4 will be accepted.
Follow the format of the sample output.
Sample input Output for the Sample Input
4 Case 1: 4.898979
1 2 3 4 Case 2: 19.311
8
3 4 5 33 3 4 3 5
0
ICPC 2011 World Finals Problem G: Magic Sticks
This page is intentionally left blank.
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 2011\\G. Magic Sticks}
\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;
}