knightnemo/LMCompressionDataset · Datasets at Hugging Face

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The first line contains an integer $n$ ($1\le n\le 700$).

The second line contains $n$ integers $a_1,\ldots,a_n$ ($0\le a_i<998\,244\,353$).

The third line contains $n$ integers $b_1,\ldots,b_n$ ($0\le b_i<998\,244\,353$).

The fourth line contains $n$ integers $c_0,\ldots,c_{n-1}$ ($0\le c_i<998\,244\,353$).

Print $n$ integers: the $i$-th one is $f(i)$ modulo $998\,244\,353$.

In the second example:

* Consider permutation $[1,2,3]$. Indices $1,2,3$ are prefix maximums. Index $3$ is the only suffix maximum. Indices $2,3$ are ascents. In conclusion, it has $3$ prefix maximums, $1$ suffix maximums, and $2$ ascents. Therefore, its cost is $a_3b_1c_2=12$. * Permutation $[1,3,2]$ has $2$ prefix maximums, $2$ suffix maximums, and $1$ ascent. Its cost is $6$. * Permutation $[2,1,3]$ has $2$ prefix maximums, $1$ suffix maximum, and $1$ ascent. Its cost is $4$. * Permutation $[2,3,1]$ has $2$ prefix maximums, $2$ suffix maximums, and $1$ ascent. Its cost is $6$. * Permutation $[3,1,2]$ has $1$ prefix maximum, $2$ suffix maximums, and $1$ ascent. Its cost is $3$. * Permutation $[3,2,1]$ has $1$ prefix maximum, $3$ suffix maximums, and $0$ ascents. Its cost is $3$.

The sum of all permutations' costs is $34$, so $f(3)=34$.
The two versions are different problems. You may want to read both versions. You can make hacks only if both versions are solved.

You are given two positive integers $n$, $m$.

Calculate the number of ordered pairs $(a, b)$ satisfying the following conditions:

* $1\le a\le n$, $1\le b\le m$; * $b \cdot \gcd(a,b)$ is a multiple of $a+b$.

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 $n$, $m$ ($1\le n,m\le 2 \cdot 10^6$).

It is guaranteed that neither the sum of $n$ nor the sum of $m$ over all test cases exceeds $2 \cdot 10^6$.

For each test case, print a single integer: the number of valid pairs.

In the first test case, no pair satisfies the conditions.

In the fourth test case, $(2,2),(3,6),(4,4),(6,3),(6,6),(8,8)$ satisfy the conditions.
A movie company has released $2$ movies. These $2$ movies were watched by $n$ people. For each person, we know their attitude towards the first movie (liked it, neutral, or disliked it) and towards the second movie.

If a person is asked to leave a review for the movie, then:

* if that person liked the movie, they will leave a positive review, and the movie's rating will increase by $1$; * if that person disliked the movie, they will leave a negative review, and the movie's rating will decrease by $1$; * otherwise, they will leave a neutral review, and the movie's rating will not change.

Every person will review exactly one movie — and for every person, you can choose which movie they will review.

The company's rating is the minimum of the ratings of the two movies. Your task is to calculate the maximum possible rating of the company.

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

The first line of each test case contains a single integer $n$ ($1 \le n \le 2 \cdot 10^5$).

The second line contains $n$ integers $a_1, a_2, \dots, a_n$ ($-1 \le a_i \le 1$), where $a_i$ is equal to $-1$ if the first movie was disliked by the $i$-th viewer; equal to $1$ if the first movie was liked; and $0$ if the attitude is neutral.

The third line contains $n$ integers $b_1, b_2, \dots, b_n$ ($-1 \le b_i \le 1$), where $b_i$ is equal to $-1$ if the second movie was disliked by the $i$-th viewer; equal to $1$ if the second movie was liked; and $0$ if the attitude is neutral.

Additional constraint on the input: the sum of $n$ over all test cases does not exceed $2 \cdot 10^5$.

For each test case, print a single integer — the maximum possible rating of the company, if for each person, choose which movie to leave a review on.

You are given a matrix, consisting of $n$ rows and $m$ columns.

You can perform two types of actions on it:

* paint the entire column in blue; * paint the entire row in red.

Note that you cannot choose which color to paint the row or column.

In one second, you can perform either one action or multiple actions at the same time. If you perform one action, it will be free. If you perform $k > 1$ actions at the same time, it will cost $k^2$ coins. When multiple actions are performed at the same time, for each cell affected by actions of both types, the color can be chosen independently.

You are asked to process $q$ queries. Before each query, all cells become colorless. Initially, there are no restrictions on the color of any cells. In the $i$-th query, a restriction of the following form is added:

* $x_i~y_i~c_i$ — the cell in row $x_i$ in column $y_i$ should be painted in color $c_i$.

Thus, after $i$ queries, there are $i$ restrictions on the required colors of the matrix cells. After each query, output the minimum cost of painting the matrix according to the restrictions.

The first line contains three integers $n, m$ and $q$ ($1 \le n, m, q \le 2 \cdot 10^5$) — the size of the matrix and the number of queries.

In the $i$-th of the next $q$ lines, two integers $x_i, y_i$ and a character $c_i$ ($1 \le x_i \le n$; $1 \le y_i \le m$; $c_i \in$ {'R', 'B'}, where 'R' means red, and 'B' means blue) — description of the $i$-th restriction. The cells in all queries are pairwise distinct.

Print $q$ integers — after each query, output the minimum cost of painting the matrix according to the restrictions.

We define the $\operatorname{MAD}$ (Maximum Appearing Duplicate) in an array as the largest number that appears at least twice in the array. Specifically, if there is no number that appears at least twice, the $\operatorname{MAD}$ value is $0$.

For example, $\operatorname{MAD}([1, 2, 1]) = 1$, $\operatorname{MAD}([2, 2, 3, 3]) = 3$, $\operatorname{MAD}([1, 2, 3, 4]) = 0$.

You are given an array $a$ of size $n$. Initially, a variable $sum$ is set to $0$.

The following process will be executed in a sequential loop until all numbers in $a$ become $0$:

1. Set $sum := sum + \sum_{i=1}^{n} a_i$; 2. Let $b$ be an array of size $n$. Set $b_i :=\ \operatorname{MAD}([a_1, a_2, \ldots, a_i])$ for all $1 \le i \le n$, and then set $a_i := b_i$ for all $1 \le i \le n$.

Find the value of $sum$ after the process.

The first line contains an integer $t$ ($1 \leq t \leq 2 \cdot 10^4$) — the number of test cases.

For each test case:

* The first line contains an integer $n$ ($1 \leq n \leq 2 \cdot 10^5$) — the size of the array $a$; * The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \leq a_i \leq n$) — the elements of the array.

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

For each test case, output the value of $sum$ in a new line.

The first line contains an integer $n$ ($1\le n\le 700$).

The second line contains $n$ integers $a_1,\ldots,a_n$ ($0\le a_i<998\,244\,353$).

The third line contains $n$ integers $b_1,\ldots,b_n$ ($0\le b_i<998\,244\,353$).

The fourth line contains $n$ integers $c_0,\ldots,c_{n-1}$ ($0\le c_i<998\,244\,353$).

Print $n$ integers: the $i$-th one is $f(i)$ modulo $998\,244\,353$.

In the second example:

* Consider permutation $[1,2,3]$. Indices $1,2,3$ are prefix maximums. Index $3$ is the only suffix maximum. Indices $2,3$ are ascents. In conclusion, it has $3$ prefix maximums, $1$ suffix maximums, and $2$ ascents. Therefore, its cost is $a_3b_1c_2=12$. * Permutation $[1,3,2]$ has $2$ prefix maximums, $2$ suffix maximums, and $1$ ascent. Its cost is $6$. * Permutation $[2,1,3]$ has $2$ prefix maximums, $1$ suffix maximum, and $1$ ascent. Its cost is $4$. * Permutation $[2,3,1]$ has $2$ prefix maximums, $2$ suffix maximums, and $1$ ascent. Its cost is $6$. * Permutation $[3,1,2]$ has $1$ prefix maximum, $2$ suffix maximums, and $1$ ascent. Its cost is $3$. * Permutation $[3,2,1]$ has $1$ prefix maximum, $3$ suffix maximums, and $0$ ascents. Its cost is $3$.

The sum of all permutations' costs is $34$, so $f(3)=34$.

The two versions are different problems. You may want to read both versions. You can make hacks only if both versions are solved.

You are given two positive integers $n$, $m$.

Calculate the number of ordered pairs $(a, b)$ satisfying the following conditions:

* $1\le a\le n$, $1\le b\le m$; * $b \cdot \gcd(a,b)$ is a multiple of $a+b$.

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 $n$, $m$ ($1\le n,m\le 2 \cdot 10^6$).

It is guaranteed that neither the sum of $n$ nor the sum of $m$ over all test cases exceeds $2 \cdot 10^6$.

For each test case, print a single integer: the number of valid pairs.

In the first test case, no pair satisfies the conditions.

In the fourth test case, $(2,2),(3,6),(4,4),(6,3),(6,6),(8,8)$ satisfy the conditions.

A movie company has released $2$ movies. These $2$ movies were watched by $n$ people. For each person, we know their attitude towards the first movie (liked it, neutral, or disliked it) and towards the second movie.

If a person is asked to leave a review for the movie, then:

* if that person liked the movie, they will leave a positive review, and the movie's rating will increase by $1$; * if that person disliked the movie, they will leave a negative review, and the movie's rating will decrease by $1$; * otherwise, they will leave a neutral review, and the movie's rating will not change.

Every person will review exactly one movie — and for every person, you can choose which movie they will review.

The company's rating is the minimum of the ratings of the two movies. Your task is to calculate the maximum possible rating of the company.

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

The first line of each test case contains a single integer $n$ ($1 \le n \le 2 \cdot 10^5$).

The second line contains $n$ integers $a_1, a_2, \dots, a_n$ ($-1 \le a_i \le 1$), where $a_i$ is equal to $-1$ if the first movie was disliked by the $i$-th viewer; equal to $1$ if the first movie was liked; and $0$ if the attitude is neutral.

The third line contains $n$ integers $b_1, b_2, \dots, b_n$ ($-1 \le b_i \le 1$), where $b_i$ is equal to $-1$ if the second movie was disliked by the $i$-th viewer; equal to $1$ if the second movie was liked; and $0$ if the attitude is neutral.

Additional constraint on the input: the sum of $n$ over all test cases does not exceed $2 \cdot 10^5$.

For each test case, print a single integer — the maximum possible rating of the company, if for each person, choose which movie to leave a review on.

You are given a matrix, consisting of $n$ rows and $m$ columns.

You can perform two types of actions on it:

* paint the entire column in blue; * paint the entire row in red.

Note that you cannot choose which color to paint the row or column.

In one second, you can perform either one action or multiple actions at the same time. If you perform one action, it will be free. If you perform $k > 1$ actions at the same time, it will cost $k^2$ coins. When multiple actions are performed at the same time, for each cell affected by actions of both types, the color can be chosen independently.

You are asked to process $q$ queries. Before each query, all cells become colorless. Initially, there are no restrictions on the color of any cells. In the $i$-th query, a restriction of the following form is added:

* $x_i~y_i~c_i$ — the cell in row $x_i$ in column $y_i$ should be painted in color $c_i$.

Thus, after $i$ queries, there are $i$ restrictions on the required colors of the matrix cells. After each query, output the minimum cost of painting the matrix according to the restrictions.

The first line contains three integers $n, m$ and $q$ ($1 \le n, m, q \le 2 \cdot 10^5$) — the size of the matrix and the number of queries.

In the $i$-th of the next $q$ lines, two integers $x_i, y_i$ and a character $c_i$ ($1 \le x_i \le n$; $1 \le y_i \le m$; $c_i \in$ {'R', 'B'}, where 'R' means red, and 'B' means blue) — description of the $i$-th restriction. The cells in all queries are pairwise distinct.

Print $q$ integers — after each query, output the minimum cost of painting the matrix according to the restrictions.

We define the $\operatorname{MAD}$ (Maximum Appearing Duplicate) in an array as the largest number that appears at least twice in the array. Specifically, if there is no number that appears at least twice, the $\operatorname{MAD}$ value is $0$.

For example, $\operatorname{MAD}([1, 2, 1]) = 1$, $\operatorname{MAD}([2, 2, 3, 3]) = 3$, $\operatorname{MAD}([1, 2, 3, 4]) = 0$.

You are given an array $a$ of size $n$. Initially, a variable $sum$ is set to $0$.

The following process will be executed in a sequential loop until all numbers in $a$ become $0$:

1. Set $sum := sum + \sum_{i=1}^{n} a_i$; 2. Let $b$ be an array of size $n$. Set $b_i :=\ \operatorname{MAD}([a_1, a_2, \ldots, a_i])$ for all $1 \le i \le n$, and then set $a_i := b_i$ for all $1 \le i \le n$.

Find the value of $sum$ after the process.

The first line contains an integer $t$ ($1 \leq t \leq 2 \cdot 10^4$) — the number of test cases.

For each test case:

* The first line contains an integer $n$ ($1 \leq n \leq 2 \cdot 10^5$) — the size of the array $a$; * The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \leq a_i \leq n$) — the elements of the array.

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

For each test case, output the value of $sum$ in a new line.