meetforever/GRPO_CL huggingface dataset:
user: A sequence of integers ($a_1, a_2, \ldots, a_k$) is said to be UpDown, if these inequalities hold true:
- $a_1 \le a_2$
- $a_2 \ge a_3$
- $a_3 \le a_4$
and so on.
That is, every even-indexed element should be at least as large as its adjacent elements. And every odd-indexed element should be at most as large as its adjacent elements. Formally, $a_{2i} \ge a_{2i+1}$ and $a_{2i+1} \le a_{2i+2}$, for all valid positions.
A subsegment is a consecutive portion of a sequence. That is, a subsegment of ($b_1, b_2, \ldots, b_k$)  will be of the form ($b_i, b_{i+1}, \ldots, b_j$), for some $i$ and $j$.
You are given a sequence ($s_1, s_2, \ldots, s_n$). You can insert at most one integer anywhere in this sequence. It could be any integer. After inserting an integer (or choosing not to), suppose you have the new sequence ($t_1, t_2, \ldots, t_m$). Note that $m$ will either be $n$+1 or $n$. You want to maximize the length of the longest subsegment of ($t_1, t_2, \ldots, t_m$) which is UpDown, and output the length of that.

-----Input-----
- The first line contains a single integer, $T$, which is the number of testcases. The description of each testcase follows.
- The first line of every testcase contains a single integer, $n$, which is the number of integers in the original sequence.
- The second line contains $n$ integers: $s_1, s_2, \ldots, s_n$, which forms the original sequence.

-----Output-----
For each testcase output a single line containing one integer, which should be the length of the longest UpDown subsegment that you can get after inserting at most one integer.

-----Constraints-----
- $1 \le T \le 2$
- $1 \le n \le 10^6$
- $1 \le s_i \le 10^9$

-----Subtasks-----
Subtask #1 (20 points): $1 \le n \le 100$
Subtask #2 (10 points): $1 \le n \le 10000$
Subtask #3 (70 points): Original constraints

-----Sample Input-----
2
7
100 1 10 3 20 25 24
5
3 3 2 4 1

-----Sample Output-----
7
6

-----Explanation-----
Testcase 1: The original sequence is (100, 1, 10, 3, 20, 25, 24). Suppose you insert the element 5 between the elements 20 and 25, you will get the new sequence (100, 1, 10, 3, 20, 5, 25, 24). The longest UpDown subsegment of this sequence is (1, 10, 3, 20, 5, 25, 24), whose length is 7. You can check that you cannot do any better, and hence the answer is 7.
Testcase 2: The original sequence is (3, 3, 2, 4, 1). Suppose you insert the element 4 at the end, you will get the new sequence (3, 3, 2, 4, 1, 4). This entire sequence is UpDown, and so the longest UpDown subsegment of this sequence is (3, 3, 2, 4, 1, 4), whose length is 6. You can check that you cannot do any better, and hence the answer is 6.
user: Binod is a youtuber and he is busy in the fame of social media so he asked you to help him solve a problem.
You have been given an array of $positive$ $integers$ $a_{1},a_{2},a_{3},...,a_{i},...,a_{n}$ of size n.You have to find the smallest length of the subarray such that the length of the subarray must be $strictly$ greater than k and it's sum also must be $strictly$ greater than s. 

-----Input Format :------
- The first line of input contains three space-separated integers n, k and s
- The second line contains n space-separated integers,describing the array a 

-----Output Format:-----
- Print a single integer :- The smallest length of subarray if exists, Otherwise print "-1" (without quotes) 

-----Constraints:------
- $1 \leq n, k \leq 10^{6}$
- $1 \leq  a_{1},a_{2},a_{3},...,a_{i},...,a_{n}\leq 10^{9}$ 
- $1 \leq s \leq 10^{15}$ Subtask #1 (30 points):
- $1 \leq n, k \leq 10^{3}$ Subtask #2 (70 points):
$Original$ $Constraints$ 

-----Sample Test Cases:------

-----Example 1:-----
5 1 5

1 2 3 4 5 

-----Output :-----
2 

-----Explanation-----
$\textbf{There are two possibles answers} :$ 
- Index starts at 3 and ends at 4 have a sum of 7 which is strictly greater than 5 and has a length of subarray greater than 1.  
- Index starts at 4 and ends at 5 have a sum of 9 which is strictly greater than 5 and has a length of subarray greater than 1.
Any of the possible scenarios gives the same answer.

-----Example 2:-----
3 2 1

9 9 1 

-----Output :-----
3

-----Explanation :-----
- Each value in array index satisfies the condition sum greater than 1 but to satisfy the condition of length greater than 2 choose the subarray of length 3
user: The chef was busy in solving algebra, he found some interesting results, that there are many numbers which can be formed by the sum of the factorial of the digits, he wrote all those interesting numbers in the diary(in increasing order) and went to sleep. Cheffina came and stole his diary, in morning chef found that his diary is missing. Now the chef wants your help to find those numbers, Chef asks you whether N is that interesting number or not. If N is an interesting number then print 1. Else print 0. 

-----Input:-----
- First-line will contain $T$, the number of test cases. Then the test cases follow. 
- Each test case contains a single line of input, $N$. 

-----Output:-----
For each test case, output in a single line answer 1 or 0.

-----Constraints-----
- $1 \leq T \leq 10^6$
- $0 \leq N \leq 10^9$

-----Sample Input:-----
2
2
10

-----Sample Output:-----
1
0

-----EXPLANATION:-----
For 1) Factorial of 2 is 2, hence it is an interesting number.
For 2) conversion for 10 is 1! + 0! = 2, which is not equal to 10, hence not an interesting number.
user: Raju has created a program to find the square root of a number. But his program can store only integers. Being a newbie, he didn't know about rounding the numbers. Hence his program returns the absolute value of the result if possible. For example, sqrt(3) = 1.73205080757……. His program will return 1
Given a number $N$, and it's integral square root $S$, His instructor will consider the answer correct if Difference between $N$ and the square of $S$ is within less than or equal to $X$% of $N$.

-----Input:-----
- First line contains $T$ no. of test cases and $X$ separated by space
- For every test case, a line contains an integer $N$

-----Output:-----
For every test case, print yes  if his programs return square root and (N-(S^2)) <= 0.01XN . For everything else, print no on a new line

-----Constraints-----
10 points:
- $1 \leq T \leq 10$
- $0\leq N \leq 10$
20 points:
- $1 \leq T \leq 30000$
- $-10^9 \leq N \leq 10^9$
70 points:
- $1 \leq T \leq 10^6$
- $-10^9 \leq N \leq 10^9$

-----Sample Input:-----
2 20
5
3

-----Sample Output:-----
yes
no

-----EXPLANATION:-----
In #1, sqrt(5) = 2.2360679775. Taking integral value, S = 2.

S2 = 4. Difference=1 which is within 20% of 5
In #1, sqrt(3) = 1.73205080757. Taking integral value, S = 1.

S2 = 1. Difference=2  which is not within 20% of 3
user: Kabir wants to impress Tara by showing her his problem solving skills. He has decided to give the correct answer to the next question which will be asked by his Algorithms teacher.
The question asked is:
Find the sum of alternate consecutive d$d$ odd numbers from the range L$L$ to R$R$ inclusive.
if d$d$ is 3 and L$L$ is 10 and R$R$ is 34, then the odd numbers between 10 and 34 are 11,13,15,17,19,21,23,25,27,29,31,33$11,13,15,17,19,21,23,25,27,29,31,33$, and the d$d$ alternate odd numbers are 11,13,15,23,25,27$11,13,15,23,25,27$.
You are a friend of Kabir, help him solve the question.
Note:$Note:$ Number of odd number between L$L$ and R$R$ (both inclusive) is a multiple of d$d$.

-----Input:-----
- First line will contain T$T$, number of test cases.  
- First line of each test case contains one integer d$d$ . 
- Second line of each test case contains two space separated integer L$L$ and R$R$.

-----Output:-----
For each test case, print the sum modulo 1000000007.

-----Constraints:-----
- 1≤T≤106$1 \leq T \leq 10^6$
- 1≤d≤103$1 \leq d \leq 10^3$
- 1≤L<R≤106$1 \leq L < R \leq 10^6$

-----Sample Input:-----
1
3 

10 33

-----Sample Output:-----
114

-----EXPLANATION:-----
Sum of alternate odd numbers i.e, 11,13,15,23,25,27 is 114
user: You are given a string $S$ and an integer $L$. A operation is described as :- "You are allowed to pick any substring from first $L$ charcaters of $S$, and place it at the end of the string $S$. 
A string $A$ is a substring of an string $B$ if $A$ can be obtained from $B$ by deletion of several (possibly, zero or all) characters from the beginning and several (possibly, zero or all) elements from the end.
Find the lexographically smallest string after performing this opertaion any number of times (possibly zero).
For example $S$ = "codechef" and $L=4$. Then, we can take substring "ode" from S[0-3] and place it at end of the string $S$ = "cchefode".

-----Input:-----
- First line will contain $T$, number of testcases. 
- Then each of the N lines contain an integer $L$ and a string $S$. 

-----Output:-----
For each testcase, output in a single line answer lexographically smallest string.

-----Constraints-----
- $1 \leq T \leq 10^4$
- $2 \leq |S| \leq 10^3$
- $1 \leq L \leq N $

-----Sample Input:-----
2
1 rga
2 cab

-----Sample Output:-----
arg
abc

-----EXPLANATION:-----
In the first testcase:
substring 'r' is picked and placed at the end of the string. rga -> gar
Then performing same operation gives :- gar -> arg
user: This year $p$ footballers and $q$ cricketers have been invited to participate in IPL (Indian Programming League) as guests. You have to accommodate them in $r$ rooms such that-
- No room may remain empty.
- A room may contain either only footballers or only cricketers, not both.
- No cricketers are allowed to stay alone in a room.
Find the number of ways to place the players. Note though, that all the rooms are identical. But each of the cricketers and footballers are unique. 
Since the number of ways can be very large, print the answer modulo $998,244,353$.

-----Input-----
- The first line of the input contains a single integer $T$ denoting the number of test cases. The description of $T$ test cases follows.
- The first and only line of each test case contains three space-separated integers $p$, $q$ and $r$ denoting the number of footballers, cricketers and rooms.

-----Output-----
For each test case, output the number of ways to place the players modulo $998,244,353$.

-----Constraints-----
- $1 \le T \le 100$
- $1 \le p, q, r \le 100$

-----Example Input-----
4
2 1 4
2 4 4
2 5 4
2 8 4

-----Example Output-----
0
3
10
609

-----Explanation-----
Example case 2: Three possible ways are:
- {Footballer 1}, {Footballer 2}, {Cricketer 1, Cricketer 2}, {Cricketer 3, Cricketer 4}
- {Footballer 1}, {Footballer 2}, {Cricketer 1, Cricketer 3}, {Cricketer 2, Cricketer 4}
- {Footballer 1}, {Footballer 2}, {Cricketer 1, Cricketer 4}, {Cricketer 2, Cricketer 3}    
Please note that the rooms are identical.
user: The chef is trying to solve some pattern problems, Chef wants your help to code it. Chef has one number K to form a new pattern. Help the chef to code this pattern problem.

-----Input:-----
- First-line will contain $T$, the number of test cases. Then the test cases follow. 
- Each test case contains a single line of input, one integer $K$. 

-----Output:-----
For each test case, output as the pattern.

-----Constraints-----
- $1 \leq T \leq 100$
- $1 \leq K \leq 100$

-----Sample Input:-----
4
1
2
3
4

-----Sample Output:-----
1
12
34
123
456
789
1234
5678
9101112
13141516

-----EXPLANATION:-----
No need, else pattern can be decode easily.
user: Diana is planning to make a very long journey. Her journey consists of $N$ bus routes, numbered from $1 to N$ in the order she must take them. The buses themselves are very fast but do not run often. The $i-th$ bus route only runs every $Xi$ days.

More specifically, she can only take the $i-th$ bus on day $Xi, 2Xi, 3Xi$, and so on. Since the buses are very fast, she can take multiple buses on the same day.

Diana must finish her journey by day D, but she would like to start the journey as late as possible. What is the latest day she could take the first bus, and still finish her journey by day $D$?
It is guaranteed that it is possible for Diana to finish her journey by day $D$.  

-----Input:-----
The first line of the input gives the number of test cases, $T$. $T$ test cases follow. Each test case begins with a line containing the two integers N and D. Then, another line follows containing $N$ integers, the $i-th$ one is $Xi$.

-----Output:-----
For each test case, output one line containing an integer $Y$, where $Y$ is the latest day she could take the first bus, and still finish her journey by day $D$.

-----Constraints-----
- $1 \leq T \leq 100$
- $1 \leq Xi \leq D.$
- $1 \leq N \leq 1000.$
It is guaranteed that it is possible for Diana to finish her journey by day $D$.

-----Sample Input:-----
3

3 10

3 7 2

4 100

11 10 5 50

1 1

1  

-----Sample Output:-----
6  

99

1  

-----EXPLANATION:-----
In Sample $Case 1$, there are $N = 3$ bus routes and Bucket must arrive by day $D = 10$. She could:
- Take the 1st bus on day 6 $(X1 = 3)$,
- Take the 2nd bus on day 7 $(X2 = 7)$ and
- Take the 3rd bus on day 8 $(X3 = 2)$.  
In Sample $Case 2$, there are $N = 4$ bus routes and Bucket must arrive by day $D = 100$. She could:
- Take the 1st bus on day 99 $(X1 = 11)$,
- Take the 2nd bus on day 100$ (X2 = 10)$,
- Take the 3rd bus on day 100 $(X3 = 5)$ and
- Take the 4th bus on day 100 $(X4 = 50)$,  
In Sample Case 3, there is $N = 1$ bus route, and Bucket must arrive by day $D = 1$. She could:
- Take the 1st bus on day 1 $(X1 = 1)$.
user: You are given an array A with size N (indexed from 0) and an integer K. Let's define another array B with size N · K as the array that's formed by concatenating K copies of array A.
For example, if A = {1, 2} and K = 3, then B = {1, 2, 1, 2, 1, 2}.
You have to find the maximum subarray sum of the array B. Fomally, you should compute the maximum value of Bi + Bi+1 + Bi+2 + ... + Bj, where 0 ≤ i ≤ j < N · K.

-----Input-----

- The first line of the input contains a single integer T denoting the number of test cases. The description of T test cases follows.
- The first line of each test case contains two space-separated integers N and K.
- The second line contains N space-separated integers A0, A1, ..., AN-1.

-----Output-----
For each test case, print a single line containing the maximum subarray sum of B.

-----Constraints-----
- 1 ≤ T ≤ 10
- 1 ≤ N ≤ 105
- 1 ≤ K ≤ 105
- -106 ≤ Ai ≤ 106 for each valid i

-----Subtasks-----
Subtask #1 (18 points): N · K ≤ 105
Subtask #2 (82 points): original constraints

-----Example-----
Input:

2
2 3
1 2
3 2
1 -2 1

Output:

9
2

-----Explanation-----
Example case 1: B = {1, 2, 1, 2, 1, 2} and the subarray with maximum sum is the whole {1, 2, 1, 2, 1, 2}. Hence, the answer is 9.
Example case 2: B = {1, -2, 1, 1, -2, 1} and the subarray with maximum sum is {1, 1}. Hence, the answer is 2.
user: Nadaca is a country with N$N$ cities. These cities are numbered 1$1$ through N$N$ and connected by M$M$ bidirectional roads. Each city can be reached from every other city using these roads.
Initially, Ryan is in city 1$1$. At each of the following K$K$ seconds, he may move from his current city to an adjacent city (a city connected by a road to his current city) or stay at his current city. Ryan also has Q$Q$ conditions (a1,b1),(a2,b2),…,(aQ,bQ)$(a_1, b_1), (a_2, b_2), \ldots, (a_Q, b_Q)$ meaning that during this K$K$-second trip, for each valid i$i$, he wants to be in city ai$a_i$ after exactly bi$b_i$ seconds.
Since you are very good with directions, Ryan asked you to tell him how many different trips he could make while satisfying all conditions. Compute this number modulo 109+7$10^9 + 7$. A trip is a sequence of Ryan's current cities after 1,2,…,K$1, 2, \ldots, K$ seconds.

-----Input-----
- The first line of the input contains a single integer T$T$ denoting the number of test cases. The description of T$T$ test cases follows.
- The first line of each test case contains three space-separated integers N$N$, M$M$ and K$K$.
- Each of the next M$M$ lines contains two space-separated integers u$u$ and v$v$ denoting a road between cities u$u$ and v$v$.
- The next line contains a single integer Q$Q$.
- Q$Q$ lines follow. For each i$i$ (1≤i≤Q$1 \le i \le Q$), the i$i$-th of these lines contains two space-separated integers ai$a_i$ and bi$b_i$.

-----Output-----
For each test case, print a single line containing one integer — the number of trips Ryan can make, modulo 109+7$10^9+7$.

-----Constraints-----
- 1≤T≤50$1 \le T \le 50$
- 1≤N,M,K,Q≤9,000$1 \le N, M, K, Q \le 9,000$
- 1≤ui,vi≤N$1 \le u_i, v_i \le N$ for each valid i$i$
- ui≠vi$u_i \neq v_i$ for each valid i$i$
- there is at most one road between each pair of cities
- each city is reachable from every other city
- 1≤ai≤N$1 \le a_i \le N$ for each valid i$i$
- 0≤bi≤K$0 \le b_i \le K$ for each valid i$i$
- the sum of N$N$ over all test cases does not exceed 9,000$9,000$
- the sum of K$K$ over all test cases does not exceed 9,000$9,000$
- the sum of M$M$ over all test cases does not exceed 9,000$9,000$
- the sum of Q$Q$ over all test cases does not exceed 9,000$9,000$

-----Subtasks-----
Subtask #1 (20 points):
- the sum of N$N$ over all test cases does not exceed 400$400$
- the sum of K$K$ over all test cases does not exceed 400$400$
- the sum of M$M$ over all test cases does not exceed 400$400$
- the sum of Q$Q$ over all test cases does not exceed 400$400$
Subtask #2 (80 points): original constraints

-----Example Input-----
3
4 3 3
1 2
1 3
1 4
0
4 3 3
1 2
1 3
1 4
1
2 2
4 3 3
1 2
1 3
1 4
1
2 1

-----Example Output-----
28
4
6
user: Sereja has an array A of N positive integers : A[1], A[2], A[3], ... , A[N]. 

In a single operation on the array, he performs the following two steps :  

- Pick two indices i, j s.t. A[i] > A[j]
- A[i] -= A[j]

Sereja can apply these operations any number of times (possibly zero), such that the sum of resulting elements of the array is as small as possible.

Help Sereja find this minimum sum.

-----Input-----

First line of input contains an integer T - the number of test cases. T test cases follow.

First line of each test case contains the integer N. The next line contains N integers — A[1], A[2], A[3], ... , A[N].

-----Output-----
For each test case, output a single line with the answer.

-----Constraints-----
- 1 ≤ T ≤ 10
- 1 ≤ N ≤ 105
- 1 ≤ A[i] ≤ 109

-----Example-----
Input:
2
1
1
3
2 4 6

Output:
1
6

-----Explanation-----
Example case 2. In this case, one possible way in which Sereja can perform the operations could be as follows. 

-  Pick i = 2, j = 1. A[2] -= A[1]. Now the resulting array would be [2, 2, 6].
-  Pick i = 3, j = 2. A[3] -= A[2]. Now the resulting array would be [2, 2, 4].
-  Pick i = 3, j = 2. A[3] -= A[2]. Now the resulting array would be [2, 2, 2]. 

As the resulting array is [2 2 2], so the sum is 6.
user: Chef is going to start playing Fantasy Football League (FFL) this season. In FFL, each team consists of exactly $15$ players: $2$ goalkeepers, $5$ defenders, $5$ midfielders and $3$ forwards. Chef has already bought $13$ players; he is only missing one defender and one forward.
There are $N$ available players (numbered $1$ through $N$). For each valid $i$, the $i$-th player is either a defender or a forward and has a price $P_i$. The sum of prices of all players in a team must not exceed $100$ dollars and the players Chef bought already cost him $S$ dollars.
Can you help Chef determine if he can complete the team by buying one defender and one forward in such a way that he does not exceed the total price limit?

-----Input-----
- The first line of the input contains a single integer $T$ denoting the number of test cases. The description of $T$ test cases follows.
- The first line of each test case contains two space-separated integers $N$ and $S$.
- The second line contains $N$ space-separated integers $P_1, P_2, \ldots, P_N$.
- The last line contains $N$ space-separated integers. For each valid $i$, the $i$-th of these integers is $0$ if the $i$-th player is a defender or $1$ if the $i$-th player is a forward.

-----Output-----
For each test case, print a single line containing the string "yes" if it is possible to build a complete team or "no" otherwise (without quotes).

-----Constraints-----
- $1 \le T \le 100$
- $1 \le N \le 100$
- $13 \le S \le 100$
- $1 \le P_i \le 100$ for each valid $i$

-----Subtasks-----
Subtask #1 (100 points): original constraints

-----Example Input-----
2
4 90
3 8 6 5
0 1 1 0
4 90
5 7 6 5
0 1 1 0

-----Example Output-----
yes
no

-----Explanation-----
Example case 1: If Chef buys the $1$-st and $3$-rd player, the total price of his team is $90 + 9 = 99$, which is perfectly fine. There is no other valid way to pick two players.
Example case 2: Chef cannot buy two players in such a way that all conditions are satisfied.
user: In my town ,there live a coder named Chef . He is a cool programmer . One day , he participate in a programming contest ,the contest give him only one problem . If he can’t solve the problem ,the problem setter will kill him . But the round allow you to help Chef. Can you save the life of Chef from problem setter ? :p
You are given two point of a straightline in X and Y axis and they are A(x1 , y1) and B(x2 ,y2) . Problem setter will give you another point C(x3 , y3) . If C exist in AB straightline ,then print “YES” . Otherwise ,print “NO” in first line and print the minimum distance from C to AB straightline in second line .
Please , save the life of Chef .
Note : It is not possible that A and B point is similar .

-----Input:-----
The first line of the input contains a single integer t (1≤t≤100) — the number of test cases .
Each test case starts with four integers( x1, y1 , x2 , y2 ) in first line .
Next line contains a single number q ,the number of queries . Each query contains two integers ( x3 ,y3 )

-----Output:-----
Print , q number of “YES” or “NO” (as it mentioned above) in each test case .For every test case , print “Test case : i ” ( 1<= i <=T )

-----Constraints-----
-1000 <=  x1 , y1 , x2 , y2 , x3 , y3 <= 1000

-----Sample Input:-----
2
3  5  6  5
2
4  5
6  8
3  4  7  10
1
7  4

-----Sample Output:-----
Test case : 1
YES
NO
3.000000
Test case : 2
NO
3.328201
user: Its Christmas time and Santa has started his ride to deliver gifts to children waiting for him in a 1-dimentional city. All houses in this city are on a number line numbered as 1, 2, 3… and so on. Santa wants to deliver to houses from n to m, but he found that all the kids living at positions that are divisible by a, a+d, a+2d, a+3d or a+4d are naughty and he does not want to deliver them any gifts. Santa wants to know how many gifts he has to carry before leaving to the city given that there is only one kid in a house. Help him out!
Formally, Given $m, n, a, d \in \mathbb{N}$ where $n < m$, find the number of $x \in \{n, n+1, ..., m-1, m\}$ such that $x$ is not divisible by $a$, $a+d$, $a+2d$, $a+3d$ or $a+4d$

-----Input-----
The first line is the number $t$, corresponding to number of test cases\
This is followed by $t$ lines of the format: $n$ $m$ $a$ $d$

-----Output-----
For each test case, print a single number that is the number of gifts Santa should pack.

-----Constraints-----
- $1 < m, n, a \leq 2^{32}$
- $1 < d \leq 2^{10}$

-----Sample Input:-----
1
2 20 2 1

-----Sample Output:-----
5

-----Explanation:-----
In the range {2, 3, 4, …, 19, 20}, only {7, 11, 13, 17, 19} are not divisible by 2, 3, 4, 5, or 6