ICPC 2010 - K. Paperweight
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/2010/K-paperweight. Edit
competitive_programming/icpc/2010/K-paperweight/solution.tex to update the written solution and
competitive_programming/icpc/2010/K-paperweight/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 K
Paperweight
Problem ID: weight
Your company makes artistic paperweights. Each paperweight is the union of two tetrahedra that share one face.
They are clear solid glass with embedded colored flecks. One of the little flecks in each paperweight is actually
a tiny RFID chip. This chip must be placed close to a partnered computer in order for that computer to operate.
Typically this is accomplished by placing the paperweight on top of the computer. The chip has a limited range,
however, so its distance from the flat computer top is significant. Therefore, it is important to calculate the
minimum and maximum possible distances from the chip to the computer top when the paperweight is in a
sufficiently stable position. A position is considered sufficiently stable if the paperweight would not move if the
center of mass were somehow shifted up to 0.2 units in any direction. You may assume the paperweight has
uniform density, and that the chip is small enough to be considered a point.
z E
D A
F
D
B
A C
B
F
C
B y
x
C
D F E
E
A
Figure 10 Figure 11
As an example, consider a paperweight with the vertices of the common face at A = (0, 0, 0), B = (9, 0, 0),
C = (0, 8, 0), the fourth vertex of one tetrahedron at D = (0, 0, 9), the fourth vertex of the other tetrahedron at
E = (1, 1, -8), and the chip at F = (1, 2, -1). (See Figure 10, where the chip is shown as a red dot.) Placing the
paperweight with face BCD on the computer results in the maximum distance of approximately 3.7 (upper part
of Figure 11). Placing the paperweight with face ACD on the computer results in the minimum distance of 1.0
(lower part of Figure 11). Placing the paperweight with face ACE on the computer results in a distance of only
0.9 but is not sufficiently stable.
Input
The input contains one or more test cases. Each test case is a single line describing the six points A, B, C, D, E,
and F, in that order. Each point is in turn described by three integers x, y and z giving its coordinates. Both of the
two tetrahedra have positive volume, D and E lie on opposite sides of the plane defined by the points A, B, C,
and point F lies strictly inside the paperweight. Each coordinate is bounded by 1000 in absolute value. It is
always possible to put the paperweight in at least one sufficiently stable position.
The input is terminated by a line containing only the integer 0.
Output
For each test case, display the case number followed by the minimum and maximum distances from the chip to
the base plane when the paperweight is sufficiently stable. These numbers should be rounded and displayed to
five digits after the decimal point. Follow the format in the sample output.
Sample Input Output for the Sample Input
0 0 0 9 0 0 0 8 0 0 0 9 1 1 -8 1 2 -1 Case 1: 1.00000 3.73526
0 0 0 7 0 0 0 7 0 0 0 7 -1 -2 -3 2 2 2 Case 2: 0.57735 2.66967
1 2 3 6 2 3 -2 6 3 -1 0 7 4 1 -2 -1 5 3 Case 3: 0.28214 5.00871
0
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 2010\\K. Paperweight}
\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;
}