E. Need for Speed
event sponsor ICPC 2017 Problem E Need for Speed Time limit: 1 second Sheila is a student and she drives a typical student car: it is old, slow, rusty, and falling apart. Recently, the needle on the speedometer fell off. She glued it back on, but she might...
Problem Statement
Formatted from the contest statement text, with sample tests broken out into copyable blocks.
Sheila is a student and she drives a typical student car: it is old, slow, rusty, and falling apart. Recently, the needle on the speedometer fell off. She glued it back on, but she might have placed it at the wrong angle. Thus, when the speedometer reads s, her true speed is s + c, where c is an unknown constant (possibly negative). Sheila made a careful record of a recent journey and wants to use this to compute c. The journey consisted of n segments. In the ith segment she traveled a distance of di and the speedometer read si for the entire segment. This whole journey took time t. Help Sheila by computing c. Note that while Sheila’s speedometer might have negative readings, her true speed was greater than zero for each segment of the journey.
Input
The first line of input contains two integers n (1 ≤ n ≤ 1 000), the number of sections in Sheila’s journey, and t (1 ≤ t ≤ 106 ), the total time. This is followed by n lines, each describing one segment of Sheila’s journey. The ith of these lines contains two integers di (1 ≤ di ≤ 1 000) and si (|si | ≤ 1 000), the distance and speedometer reading for the ith segment of the journey. Time is specified in hours, distance in miles, and speed in miles per hour.
Output
Display the constant c in miles per hour. Your answer should have an absolute or relative error of less than 10−6 .
Sample Tests
3 5
4 -1
4 0
10 33.000000000 4 10
5 3
2 2
3 6
3 1-0.508653377Editorial
The solution write-up is rendered from the LaTeX source, with equations kept live through MathJax.
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
C++ solution used for this page.
#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;
}
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