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ICPC 2015
ICPC 2015

J. Tile Cutting

Youssef is a Moroccan tile installer who specializes in mosaics like the one shown on the right. He has rectangular tiles of many dimensions at his disposal, and the dimensions of all his tiles are integer numbers of centime- ters. When Youssef needs parall...

Updated May 21, 2026
Track ICPC
Year 2015
Statement Text + PDF
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Problem statement

Problem Statement

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

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Time limit 15 seconds

Youssef is a Moroccan tile installer who specializes in mosaics like the one shown on the right. He has rectangular tiles of many dimensions at his disposal, and the dimensions of all his tiles are integer numbers of centime- ters. When Youssef needs parallelogram-shaped tiles, he cuts them from his supply on hand. To make this work easier, he invented a tile cutting machine that superimposes a centimeter grid on the cutting surface to guide the cuts on the tiles. Due to machine limitations, aesthetic sensibilities, and Youssef’s dislike of wasted tiles, the following rules determine the possible cuts.

  1. The rectangular tile to be cut must be positioned in the bottom left corner of the cutting surface and the edges must be aligned with the grid lines.
  2. The cutting blade can cut along any line connecting two different grid points on the tile boundary as long as the points are on adjacent boundary edges.
  3. The four corners of the resulting parallelogram tile must lie on the four sides of the original rectangular tile.
  4. No edge of the parallelogram tile can lie along an edge of the rectangular tile.

Figure J.1 shows the eight different ways in which a parallelogram tile of area 4 square centimeters can be cut out of a rectangular tile, subject to these restrictions.

Figure J.1: The eight different ways for cutting a parallelogram of area 4.

Youssef needs to cut tiles of every area between alo and ahi . Now he wonders, for which area a in this range can he cut the maximum number of different tiles?

Input

The input consists of multiple test cases. The first line of input contains an integer n (1 ≤ n ≤ 500), the number of test cases. The next n lines each contain two integers alo , ahi (1 ≤ alo ≤ ahi ≤ 500 000), the range of areas of the tiles.

Output

For each test case alo , ahi , display the value a between alo and ahi such that the number of possible ways to cut a parallelogram of area a is maximized as well as the number of different ways w in which such a parallelogram can be cut. If there are multiple possible values of a display the smallest one.

Sample Tests

Sample 1
Sample Input 
 2
 4 4
 2 6
Sample Output 
4 8
6 20
Editorial

Editorial

The solution write-up is rendered from the LaTeX source, with equations kept live through MathJax.

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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

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