knightnemo/LMCompressionDataset · Datasets at Hugging Face

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If strictly more than half of the total population $n$ are unhappy, Robin Hood will appear by popular demand.

Determine the minimum value of $x$ for Robin Hood to appear, or output $-1$ if it is impossible.

$^{\text{∗}}$The average wealth is defined as the total wealth divided by the total population $n$, that is, $\frac{\sum a_i}{n}$, the result is a real number.

The first line of input contains one integer $t$ ($1 \le t \le 10^4$) — the number of test cases.

The first line of each test case contains an integer $n$ ($1 \le n \le 2\cdot10^5$) — the total population.

The second line of each test case contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le 10^6$) — the wealth of each person.

It is guaranteed that the sum of $n$ across all test cases does not exceed $2 \cdot 10^5$.

For each test case, output one integer — the minimum number of gold that the richest person must find for Robin Hood to appear. If it is impossible, output $-1$ instead.

In the first test case, it is impossible for a single person to be unhappy.

In the second test case, there is always $1$ happy person (the richest).

In the third test case, no additional gold are required, so the answer is $0$.

In the fourth test case, after adding $15$ gold, the average wealth becomes $\frac{25}{4}$, and half of this average is $\frac{25}{8}$, result
Impress thy brother, yet fret not thy mother.

Robin's brother and mother are visiting, and Robin gets to choose the start day for each visitor.

All days are numbered from $1$ to $n$. Visitors stay for $d$ continuous days, all of those $d$ days must be between day $1$ and $n$ inclusive.

Robin has a total of $k$ risky 'jobs' planned. The $i$-th job takes place between days $l_i$ and $r_i$ inclusive, for $1 \le i \le k$. If a job takes place on any of the $d$ days, the visit overlaps with this job (the length of overlap is unimportant).

Robin wants his brother's visit to overlap with the maximum number of distinct jobs, and his mother's the minimum.

Find suitable start days for the visits of Robin's brother and mother. If there are multiple suitable days, choose the earliest one.

The first line of the input contains a single integer $t$ ($1\leq t \leq 10^4$) — the number of test cases.

The first line of each test case consists of three integers $n$, $d$, $k$ ($1 \le n \le 10^5, 1 \le d, k \le n$) — the number of total days, duration of the visits, and the number of jobs.

Then follow $k$ lines of each test case, each with two integers $l_i$ and $r_i$ ($1 \le l_i \le r_i \le n$) — the start and end day of each job.

It is guaranteed that the sum of $n$ over all test cases does not exceed $2 \cdot 10^5$.

For each test case, output two integers, the best starting days of Robin's brother and mother respectively. Both visits must fit between day $1$ and $n$ inclusive.

In the first test case, the only job fills all $2$ days, both should visit on day $1$.

In the second test case, day $2$ overlaps with $2$ jobs and day $1$ overlaps with only $1$.

In the third test case, Robert visits for days $[1,2]$, Mrs. Hood visits for days $[4,5]$.
"Why, master," quoth Little John, taking the bags and weighing them in his hand, "here is the chink of gold."

The folk hero Robin Hood has been troubling Sheriff of Nottingham greatly. Sheriff knows that Robin Hood is about to attack his camps and he wants to be prepared.

Sheriff of Nottingham built the camps with strategy in mind and thus there are exactly $n$ camps numbered from $1$ to $n$ and $n-1$ trails, each connecting two camps. Any camp can be reached from any other camp. Each camp $i$ has initially $a_i$ gold.

As it is now, all camps would be destroyed by Robin. Sheriff can strengthen a camp by subtracting exactly $c$ gold from each of its neighboring camps and use it to build better defenses for that camp. Strengthening a camp doesn't change its gold, only its neighbors' gold. A camp can have negative gold.

After Robin Hood's attack, all camps that have been strengthened survive the attack, all others are destroyed.

What's the maximum gold Sheriff can keep in his surviving camps after Robin Hood's attack if he strengthens his camps optimally?

Camp $a$ is neighboring camp $b$ if and only if there exists a trail connecting $a$ and $b$. Only strengthened camps count towards the answer, as others are destroyed.

The first line contains a single integer $t$ ($1 \le t \le 10^4$) — the number of test cases.

Each test case begins with two integers $n$, $c$ ($1 \le n \le 2\cdot10^5, 1 \le c \le 10^9$) — the number of camps and the gold taken from each neighboring camp for strengthening.

The second line of each test case contains $n$ integers $a_1,a_2,\dots,a_n$ ($-10^9 \le a_i \le 10^9$) — the initial gold of each camp.

Then follow $n-1$ lines, each with integers $u$, $v$ ($1 \le u, v \le n$, $u \ne v$) — meaning that there is a trail between $u$ and $v$.

The sum of $n$ over all test cases doesn't exceed $2\cdot10^5$.

It is guaranteed that any camp is reachable from any other camp.

Output a single integer, the maximum gold Sheriff of Nottingham can keep in his surviving camps after Robin Hood
[DJ Genki vs Gram - Einherjar Joker](https://soundcloud.com/leon- hwang-368077289/einherjar-joker-dj-genki-vs-gram)

⠀

You have some cards. An integer between $1$ and $n$ is written on each card: specifically, for each $i$ from $1$ to $n$, you have $a_i$ cards which have the number $i$ written on them.

There is also a shop which contains unlimited cards of each type. You have $k$ coins, so you can buy at most $k$ new cards in total, and the cards you buy can contain any integer between $\mathbf{1}$ and $\mathbf{n}$, inclusive.

After buying the new cards, you must partition all your cards into decks, according to the following rules:

* all the decks must have the same size; * there are no pairs of cards with the same value in the same deck.

Find the maximum possible size of a deck after buying cards and partitioning them optimally.

Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \le t \le 10^4$). The description of the test cases follows.

The first line of each test case contains two integers $n$, $k$ ($1 \leq n \leq 2 \cdot 10^5$, $0 \leq k \leq 10^{16}$) — the number of distinct types of cards and the number of coins.

The second line of each test case contains $n$ integers $a_1, a_2, \ldots, a_n$ ($0 \leq a_i \leq 10^{10}$, $\sum a_i \geq 1$) — the number of cards of type $i$ you have at the beginning, for each $1 \leq i \leq n$.

It is guaranteed that the sum of $n$ over all test cases does not exceed $2 \cdot 10^5$.

If strictly more than half of the total population $n$ are unhappy, Robin Hood will appear by popular demand.

Determine the minimum value of $x$ for Robin Hood to appear, or output $-1$ if it is impossible.

$^{\text{∗}}$The average wealth is defined as the total wealth divided by the total population $n$, that is, $\frac{\sum a_i}{n}$, the result is a real number.

The first line of input contains one integer $t$ ($1 \le t \le 10^4$) — the number of test cases.

The first line of each test case contains an integer $n$ ($1 \le n \le 2\cdot10^5$) — the total population.

The second line of each test case contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le 10^6$) — the wealth of each person.

It is guaranteed that the sum of $n$ across all test cases does not exceed $2 \cdot 10^5$.

For each test case, output one integer — the minimum number of gold that the richest person must find for Robin Hood to appear. If it is impossible, output $-1$ instead.

In the first test case, it is impossible for a single person to be unhappy.

In the second test case, there is always $1$ happy person (the richest).

In the third test case, no additional gold are required, so the answer is $0$.

In the fourth test case, after adding $15$ gold, the average wealth becomes $\frac{25}{4}$, and half of this average is $\frac{25}{8}$, result

Impress thy brother, yet fret not thy mother.

Robin's brother and mother are visiting, and Robin gets to choose the start day for each visitor.

All days are numbered from $1$ to $n$. Visitors stay for $d$ continuous days, all of those $d$ days must be between day $1$ and $n$ inclusive.

Robin has a total of $k$ risky 'jobs' planned. The $i$-th job takes place between days $l_i$ and $r_i$ inclusive, for $1 \le i \le k$. If a job takes place on any of the $d$ days, the visit overlaps with this job (the length of overlap is unimportant).

Robin wants his brother's visit to overlap with the maximum number of distinct jobs, and his mother's the minimum.

Find suitable start days for the visits of Robin's brother and mother. If there are multiple suitable days, choose the earliest one.

The first line of the input contains a single integer $t$ ($1\leq t \leq 10^4$) — the number of test cases.

The first line of each test case consists of three integers $n$, $d$, $k$ ($1 \le n \le 10^5, 1 \le d, k \le n$) — the number of total days, duration of the visits, and the number of jobs.

Then follow $k$ lines of each test case, each with two integers $l_i$ and $r_i$ ($1 \le l_i \le r_i \le n$) — the start and end day of each job.

It is guaranteed that the sum of $n$ over all test cases does not exceed $2 \cdot 10^5$.

For each test case, output two integers, the best starting days of Robin's brother and mother respectively. Both visits must fit between day $1$ and $n$ inclusive.

In the first test case, the only job fills all $2$ days, both should visit on day $1$.

In the second test case, day $2$ overlaps with $2$ jobs and day $1$ overlaps with only $1$.

In the third test case, Robert visits for days $[1,2]$, Mrs. Hood visits for days $[4,5]$.

"Why, master," quoth Little John, taking the bags and weighing them in his hand, "here is the chink of gold."

The folk hero Robin Hood has been troubling Sheriff of Nottingham greatly. Sheriff knows that Robin Hood is about to attack his camps and he wants to be prepared.

Sheriff of Nottingham built the camps with strategy in mind and thus there are exactly $n$ camps numbered from $1$ to $n$ and $n-1$ trails, each connecting two camps. Any camp can be reached from any other camp. Each camp $i$ has initially $a_i$ gold.

As it is now, all camps would be destroyed by Robin. Sheriff can strengthen a camp by subtracting exactly $c$ gold from each of its neighboring camps and use it to build better defenses for that camp. Strengthening a camp doesn't change its gold, only its neighbors' gold. A camp can have negative gold.

After Robin Hood's attack, all camps that have been strengthened survive the attack, all others are destroyed.

What's the maximum gold Sheriff can keep in his surviving camps after Robin Hood's attack if he strengthens his camps optimally?

Camp $a$ is neighboring camp $b$ if and only if there exists a trail connecting $a$ and $b$. Only strengthened camps count towards the answer, as others are destroyed.

The first line contains a single integer $t$ ($1 \le t \le 10^4$) — the number of test cases.

Each test case begins with two integers $n$, $c$ ($1 \le n \le 2\cdot10^5, 1 \le c \le 10^9$) — the number of camps and the gold taken from each neighboring camp for strengthening.

The second line of each test case contains $n$ integers $a_1,a_2,\dots,a_n$ ($-10^9 \le a_i \le 10^9$) — the initial gold of each camp.

Then follow $n-1$ lines, each with integers $u$, $v$ ($1 \le u, v \le n$, $u \ne v$) — meaning that there is a trail between $u$ and $v$.

The sum of $n$ over all test cases doesn't exceed $2\cdot10^5$.

It is guaranteed that any camp is reachable from any other camp.

Output a single integer, the maximum gold Sheriff of Nottingham can keep in his surviving camps after Robin Hood

[DJ Genki vs Gram - Einherjar Joker](https://soundcloud.com/leon- hwang-368077289/einherjar-joker-dj-genki-vs-gram)

⠀

You have some cards. An integer between $1$ and $n$ is written on each card: specifically, for each $i$ from $1$ to $n$, you have $a_i$ cards which have the number $i$ written on them.

There is also a shop which contains unlimited cards of each type. You have $k$ coins, so you can buy at most $k$ new cards in total, and the cards you buy can contain any integer between $\mathbf{1}$ and $\mathbf{n}$, inclusive.

After buying the new cards, you must partition all your cards into decks, according to the following rules:

* all the decks must have the same size; * there are no pairs of cards with the same value in the same deck.

Find the maximum possible size of a deck after buying cards and partitioning them optimally.

Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \le t \le 10^4$). The description of the test cases follows.

The first line of each test case contains two integers $n$, $k$ ($1 \leq n \leq 2 \cdot 10^5$, $0 \leq k \leq 10^{16}$) — the number of distinct types of cards and the number of coins.

The second line of each test case contains $n$ integers $a_1, a_2, \ldots, a_n$ ($0 \leq a_i \leq 10^{10}$, $\sum a_i \geq 1$) — the number of cards of type $i$ you have at the beginning, for each $1 \leq i \leq n$.

It is guaranteed that the sum of $n$ over all test cases does not exceed $2 \cdot 10^5$.