EP Ethan Pham Project Euler, competitive programming, and notes on the things I keep studying.
ICPC 2008
ICPC 2008

J. The Sky is the Limit

Input file: skyline.in The city of Banff hired an advertising agency to promote the city’s attractions to potential visitors. One of the planned slogans stated that the mountain ranges around the city form the most beautiful skyline in Canada. But the Insti...

Updated May 21, 2026
Track ICPC
Year 2008
Statement Text + PDF
TeXC++Statement textStatement PDF
Problem statement

Problem Statement

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

Statement text Statement PDF
Input file skyline.in

The city of Banff hired an advertising agency to promote the city’s attractions to potential visitors. One of the planned slogans stated that the mountain ranges around the city form the most beautiful skyline in Canada. But the Institute for Consumer Protection in Canada (ICPC) decided that “the most beautiful skyline” was a subjective and unverifiable claim, and could therefore be considered misleading.

The advertising agency then came up with the slogan “Banff – the longest skyline in Canada.” Although not as catchy, it is hopefully verifiable, and therefore admissible under Canada’s tricky advertising laws.

This is where you come in. What the advertising agency needs is a program that determines the length of a skyline. Consider each mountain as a two-dimensional triangle having two upper sides the same length. A skyline is the outline of one or more mountains. The skyline’s length is the total length of the outline. The left illustration below shows three mountains. The right illustration shows (with bold lines) the skyline and (with dashed lines) the portion of the mountains’ upper edges that are not part of the skyline. Note that parts of the horizon line that lie between mountains are not considered part of the skyline.

Input

Each input file contains one or more test cases, which are descriptions of mountain ranges. Each description starts with a line containing a positive integer N, which specifies the number of mountains in the range. Each of the next N lines describes a mountain with three integers X, H, and B, which specify the horizontal position of the mountain’s peak relative to some fixed point, the height of the peak, and the width of the base of the mountain, respectively. The base of each mountain coincides with a horizontal line. The values satisfy the conditions N ≤ 100, H > 0, and B > 0.

The last test case is followed by a line containing a zero.

Output

For each test case, print the case number (beginning with 1) and the length of the skyline. Print the length rounded to the nearest integer, with 0.5 rounded up. Print a blank line after the output of each test case. Use the format shown in the sample output below.

Sample Tests

Sample
Sample Input 
1
100 50 100
3
20 30 35
37 24    29
60 20 13
0

This page is intentionally blank.
Sample Output 
Case 1: 141

Case 2: 138
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;
}
Resources

Source Files and Assets

Raw files are still available here when you want the original TeX, C++, or statement assets.

Show raw files
TeX write-upcompetitive_programming/icpc/2008/J-the-sky-is-the-limit/solution.texC++ implementationcompetitive_programming/icpc/2008/J-the-sky-is-the-limit/solution.cppStatement textcompetitive_programming/icpc/2008/J-the-sky-is-the-limit/statement.txtStatement PDFcompetitive_programming/icpc/2008/J-the-sky-is-the-limit/statement.pdfMetadatacompetitive_programming/icpc/2008/J-the-sky-is-the-limit/meta.jsonYear packetcompetitive_programming/icpc/2008/contest_problems.pdf