A. Balanced Diet
Every day, Danny buys one sweet from the candy store and eats it. The store has m types of sweets, numbered from 1 to m. Danny knows that a balanced diet is important and is applying this concept to his sweet purchasing. To each sweet type i, he has assigne...
Problem Statement
Formatted from the contest statement text, with sample tests broken out into copyable blocks.
Every day, Danny buys one sweet from the candy store and eats it. The store has m types of sweets,
numbered from 1 to m. Danny knows that a balanced diet is important and is applying this concept to
his sweet purchasing. To each sweet type i, he has assigned a target fraction, which is a real number fi
(0 < fi ≤ 1). He wants the fraction of sweets of type i among all sweets he has eaten to be roughly
fi . To be more precise, let si denote the number of sweets of type i that Danny has eaten, and
equal toP
let n = m i=1 si . We say the set of sweets is balanced if for every i we have
nfi − 1 < si < nfi + 1.
Danny has been buying and eating sweets for a while and during this entire time the set of sweets
has been balanced. He is now wondering how many more sweets he can buy while still fulfilling this
condition. Given the target fractions fi and the sequence of sweets he has eaten so far, determine how
many more sweets he can buy and eat so that at any time the set of sweets is balanced.Input
The input consists of three lines. The first line contains two integers m (1 ≤ m ≤ 105 ), which is the
number of types of sweets, and k (0 ≤ k ≤ 105 ), which is the number of sweets Danny has already
eaten.
The second line contains m positive integers a1 , . . . , am . These numbers are proportional to f1 , . . . , fm ,
that is, fi = Pm . It is guaranteed that the sum of all aj is no larger than 105 .
j=1 jThe third line contains k integers b1 , . . . , bk (1 ≤ bi ≤ m), where each bi denotes the type of sweet Danny bought and ate on the ith day. It is guaranteed that every prefix of this sequence (including the whole sequence) is balanced.
Output
Display the maximum number of additional sweets that Danny can buy and eat while keeping his diet continuously balanced. If there is no upper limit on the number of sweets, display the word forever.
Sample Tests
6 5
2 1 6 3 5 3
1 2 5 3 51 6 4
2 1 6 3 5 3
1 2 5 3foreverEditorial
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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