meetforever/GRPO_CL huggingface dataset:
user: Farmer Feb has three fields with potatoes planted in them. He harvested x potatoes from the first field, y potatoes from the second field and is yet to harvest potatoes from the third field. Feb is very superstitious and believes that if the sum of potatoes he harvests from the three fields is a prime number (http://en.wikipedia.org/wiki/Prime_number), he'll make a huge profit. Please help him by calculating for him the minimum number of potatoes that if harvested from the third field will make the sum of potatoes prime. At least one potato should be harvested from the third field.

-----Input-----
The first line of the input contains an integer T denoting the number of test cases. Each of the next T lines contain 2 integers separated by single space: x and y.

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

-----Constraints-----
- 1 ≤ T ≤ 1000
- 1 ≤ x ≤ 1000
- 1 ≤ y ≤ 1000

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

Output:
1
4

-----Explanation-----

In example case 1: the farmer harvested a potato from the first field and 3 potatoes from the second field. The sum is 4. If he is able to harvest a potato from the third field, that will make the sum 5, which is prime. Hence the answer is 1(he needs one more potato to make the sum of harvested potatoes prime.)
user: Saket loves to play with strings. One day , while he was having fun with Cyclic Permutations of available strings to him, he observed that despite being scarce in numbers Vowels were really clingy.Being clingy means for almost every given string, there was a Cyclic Permutation in which atleast two vowels were together.
So he decided to check this property for all the available strings to him. As the number of strings can be very large, help Saket determine whether the given string is clingy or not.

-----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.
First line of every test case contains an integer N$N$ denoting the length of the string.
Second line contains a string S$S$ of length N$N$, consisting only of uppercase english alphabets.

-----Output:-----
For each test case, print a single line containing "Yes" if any of the cyclic permutations of the string is clingy else print "No".

-----Constraints-----
- 1≤T≤1000$1 \leq T \leq 1000$
- 1≤N≤1000$1 \leq N \leq 1000$
- String S$S$ consists of only upper case english alphabets.

-----Subtasks-----
- 20 points : 1≤N≤5$1 \leq N \leq 5$
- 80 points : Original$Original$ Constraints$Constraints$

-----Sample Input:-----
2
5
AUXFC
6
XBCDEF

-----Sample Output:-----
Yes

No

-----EXPLANATION:-----
Example$Example$ case1:$ case 1: $ One of the cyclic permutation is the original string itself, which has "A" and "U" together.
Example$Example$ case2:$ case 2: $     None of the cyclic permutation will have 2 vowels together.
user: Everyone loves short problem statements.
Given a function $ f(x) $ find its minimum value over the range $ 0 < x < π/2$
$
f(x) = ( x^2 + b*x + c ) / sin( x )
$

-----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, two real numbers $b, c$. 

-----Output:-----
For each test case, output the minimum value of $ f(x) $ over the given range. Absolute error of $10^{-6}$ is allowed.

-----Constraints-----
- $1 \leq T \leq 100000$
- $1 \leq b,c \leq 20$

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

-----Sample Output:-----
5.8831725615
user: Tara was completing an Algorithms assignment and got stuck on a question. She thought of who can help her, and got reminded of Kabir who has good problem solving skills. The question is:
Given N$N$ the number of elements in the sequence A1$A_1$,A2$A_2$ … An$A_n$. Find out the prime factor which occurred maximum number of times among the largest prime factor corresponding to each element. if there are more than one such prime factors print the largest one.
You are friends with Kabir, help him to solve the problem for Tara.

-----Input:-----
- The first line of the input contains a single integer T$T$ denoting the number of test cases. The description of T test cases follows. 
- First line of each test case contains N$N$, the number of elements in the sequence.
- Second line contains N space separated elements A1$A_1$,A2$A_2$ … An$A_n$.

-----Output:-----
- For each test case, print a single line, the number which occurs maximum number of times from the largest prime factor corresponding to each element.

-----Constraints-----
- 1≤T≤10$1 \leq T \leq 10$
- 1≤N≤105$1 \leq N \leq 10^5$
- 2≤A[i]≤105$2 \leq A[i] \leq 10^5$

-----Sample Input:-----
1
7

3 2 15 6 8 5 10

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

-----EXPLANATION:-----
The largest prime factors of numbers are:
3 2  5  3  2  5  5 , of which 5 is most frequent.
user: Sereja has a string A consisting of n lower case English letters. 

Sereja calls two strings X and Y each of length n similar if they can be made equal by applying the following operation at most once in each of them.

- Chose any two position i, j in the string (i can be equal to j too). Swap the characters at position i with character at position j.

For example strings "abcd" and "acbd" are similar, strings "ab" and "ab" are similar, but strings "abcde" and "bcdea" are not similar. Note that strings  "abc" and "cab" are also similar, as you can swap 'a' and 'c' in the first string to get "cba" and 'a' and 'b' in the second string to get "cba". 

Now Sereja is interested in finding number of ordered pairs of non similar strings X and Y such that they can be constructed from a given string A by permutation of its characters. As answer could be large, please output your answer modulo  (109 + 7).

Note 

A string s (of size n) is said to be constructed from string t (also of size n) by permutation of its characters if there exists a permutation P (of length n), such that s[i] = t[P[i]] for each i from 1 to n.

-----Input-----
- First line contain integer T - number of test cases. 
- For each of the next T lines: 
	
- Each line contains a string A as defined in the problem.

-----Output-----
For each test case, output answer modulo 1000000007 (109 + 7) in separate line. 

-----Constraints-----
- 1 ≤ T ≤  10 
- 1 ≤ n ≤  10^5 

-----Constraints-----
- Subtask #1: 1 ≤ n ≤  10  (25 points)
- Subtask #2: 1 ≤ n ≤  100  (25 points)
- Subtask #3: 1 ≤ n ≤  1000  (25 points)
- Subtask #4: original constraints (25 points)

-----Example-----
Input:
2
z
abcd

Output:
0
144
user: Chef spent N days working really hard! He planned loads of tasks: as many as Ai tasks to do on the ith day! Chef's work was brutal, so he only managed to finish Bi tasks on the ith day.
The good news is that Chef has a Time Machine! 
The Time Machine has K white buttons and M black buttons. Each button has a positive integer printed on it. Now Chef goes through all N days consequently and presses buttons. Each day Chef can only press one button (either white or black). After using a button once, it becomes inactive.
Pressing a white button with integer x printed on it reduces the number of planned tasks on the day it was pressed by exactly x. Note that this white button can only be pressed if number of planned tasks on the day are greater than or equal to x.
Pressing a black button with integer x printed on it increases the number of completed tasks on the day it was pressed by exactly x. Note that this black button can only be pressed if after pressing it, number of completed tasks don't exceed the number of tasks.
Chef is interested in finding the minimum possible amount of total uncompleted tasks he will still be left with after N days using the Machine in the best way?
Be careful! Time is sensitive! Chef cannot make a day when he completed more tasks then planned, as this may result in a more-work-than-planned paradox, killing all lazy people on the planet!

-----Input-----
- The first line of input contains a single integer T, denoting the number of test cases. Description of T test cases follows.
- The first line of each test case contains three integers — N, K, M — denoting the number of days, white and black buttons appropriately.
- The second line contains N space-separated integers A1, A2, … , AN, denoting the number of planned tasks.
- The third line contains N space-separated integers B1, B2, … , BN, denoting the number of completed tasks.
- The fourth line contains K space-separated integers C1, C2, … , CK, denoting the integers on white buttons.
- The fifth and last line contains M space-separated integers D1, D2, … , DM, denoting the integers on black buttons.

-----Output-----
- In a single line, output an integer — the minimum possible amount of uncompleted tasks.

-----Constraints-----
- 1 ≤ T ≤ 4
- 1 ≤ N, K, M ≤ 10^5
- 1 ≤ Bi ≤ Ai  ≤ 10^5
- 1 ≤ Ci, Di  ≤ 10^5

-----Subtasks-----
- Subtask N ≤ 10, K, M ≤ 5. Points: 30 
- Subtask Original constraints. Points: 70 

-----Example-----
Input:
1
4 2 2 
5 7 6 1
3 3 1 1
6 3
1 4

Output:
3

-----Explanation-----
Example case 1.
In this example Chef goes through the following steps:
Use black button 1 on the first day.
Use black button 4 on the second day.
Use white button 3 on the third day.
The arrays A and B are now effectively changed to:
5 7 3 1
4 7 1 1
So he will have 3 uncompleted tasks.
user: The chef is trying to decode 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 
1 01 
11 001 
1 01 11 
001 101 011 
111 0001 1001 
1 01 11 001 
101 011 111 0001 
1001 0101 1101 0011 
1011 0111 1111 00001 

-----EXPLANATION:-----
No need, else pattern can be decode easily.
user: You are given two binary strings $S$ and $P$, each with length $N$. A binary string contains only characters '0' and '1'. For each valid $i$, let's denote the $i$-th character of $S$ by $S_i$.
You have to convert the string $S$ into $P$ using zero or more operations. In one operation, you should choose two indices $i$ and $j$ ($1 \leq i < j \leq N$) such that $S_i$ is '1' and $S_j$ is '0', and swap $S_i$ with $S_j$.
Determine if it is possible to convert $S$ into $P$ by performing some operations.

-----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 a single integer $N$.
- The second line contains a single string $S$.
- The third line contains a single string $P$.

-----Output-----
For each test case, print a single line containing the string "Yes" if it is possible to convert $S$ into $P$ or "No" otherwise (without quotes).

-----Constraints-----
- $1 \leq T \leq 10^5$
- $1 \leq N \leq 10^5$
- $S$ and $P$ contain only characters '0' and '1'
- the sum of $N$ over all test cases does not exceed $10^5$

-----Subtasks-----
Subtask #1 (20 points):
- $N \leq 14$
- the sum of $N$ over all test cases does not exceed $100$
Subtask #2 (30 points): the sum of $N$ over all test cases does not exceed $1,000$
Subtask #3 (50 points): original constraints

-----Example Input-----
3
2
00
00
3
101
010
4
0110
0011

-----Example Output-----
Yes
No
Yes

-----Explanation-----
Example case 1: The strings are already equal.
Example case 2: It can be showed that it is impossible to convert $S$ into $P$.
Example case 3: You can swap $S_2$ with $S_4$. The strings will then be equal.
user: Chef has just found a recipe book, where every dish consists of exactly four ingredients.
He is going to choose some two dishes and prepare them for dinner.
Of course, he likes diversity and wants to know whether the two dishes are similar.
Two dishes are called similar if at least half of their ingredients are the same.
In other words, at least two of four ingredients of the first dish should also be present in the second dish.
The order of ingredients doesn't matter.
Your task is to examine T pairs of dishes.
For each pair, check if the two dishes are similar and print "similar" or "dissimilar" accordingly.

-----Input-----
The first line of the input contains an integer T denoting the number of test cases. The description of T test cases follows.

The first line of each test case contains four distinct strings, denoting ingredients needed for the first dish.
Each ingredient is represented by a string of length between 2 and 10 inclusive, consisting of lowercase English letters.

The second line of each test case describes the second dish in the same format.

-----Output-----
For each test case, output a single line containing the answer — "similar" if at least half of the ingredients are same, and "dissimilar" otherwise (without the quotes).

-----Constraints-----
- 1 ≤ T ≤ 200
- The length of each string will be between 2 and 10 inclusive.

-----Example-----
Input:
5
eggs sugar flour salt
sugar eggs milk flour
aa ab ac ad
ac ad ae af
cookies sugar grass lemon
lemon meat chili wood
one two three four
one two three four
gibberish jibberish lalalalala popopopopo
jibberisz gibberisz popopopopu lalalalalu

Output:
similar
similar
dissimilar
similar
dissimilar

-----Explanation-----
Test case 1. The first dish consists of ingredients: eggs, sugar, flour, salt, while the second dish consists of: sugar, eggs, milk, flour. Three of four ingredients are present in both dishes (eggs, sugar, flour) so the two dishes are similar.
Test case 2. This example shows that strings in the input don't necessarily represent real food.
The answer is again "similar", because two ingredients are present in both dishes ("ac" and "ad").
Test case 3. The only common ingredient is lemon, so the answer is "dissimilar".
Remember that at least two ingredients should be present in both dishes.
user: Chef likes to travel a lot. Every day Chef tries to visit as much cities as possible. Recently he had a quite a few trips of great Chefland for learning various recipes. Chefland had N cities numbered from 1 to N. People in Chefland are very friendly, friendliness of i-th city is given by Fi.

Before starting of each trip, Chef's initial enjoyment is 1 unit. Whenever he visits a city with friendliness Fi, his enjoyment gets multiplied by Fi units. 

City 1 is the home city of Chef. He starts each trip from his home city. Before starting a trip, he chooses a parameter R which denotes that he will start from city 1, and go to city 1 + R, then to 1 + 2 * R, then to 1 + 3 * R, till 1 + i * R such that i is largest integer satisfying 1 + i * R ≤ N.

Now, Chef wants you to help him recreate his visit of the cities. Specifically, he will ask you Q queries, each of which can be of following two types.

- 1 p f : friendliness of p-th city changes to f, i.e. Fp = f 
- 2 R   : Find out the total enjoyment Chef will have during this trip. As Chef does not like big numbers, he just asks you to output two things, first digit of the enjoyment and value of enjoyment modulo 109 + 7.

-----Input-----
There is a single test case.
First line of input contains a single integer N, denoting number of cities in Chefland.
Second line of the input contains N space separated integer - F1, F2, ..., FN, denoting the friendliness of the cities in order from 1 to N.
Next line contains an integer Q, denoting number of queries.

For each of the next Q queries, each line corresponds to one of the two types of the query. First there will be an integer denoting the type of the query, followed by the actual query. For query of type 1, there will be three space separated integers "1 p f" as defined above. For query of type 2, there will be two space separated integers "2 R ", as defined above.

-----Output-----
For each query of type 2, output two space separated integers, first digit of Chef's enjoyment in this trip followed by the value of enjoyment modulo 109 + 7.

-----Constraints-----
-  1 ≤ N, Q ≤ 105 
-  1 ≤ Fi ≤ 10^9 
-  1 ≤ f ≤ 10^9 
-  1 ≤ p ≤ N 
-  1 ≤ R ≤ N 

-----Subtasks-----
Subtask #1 (5 points) : 

-  1 ≤ N ≤ 10 
-  1 ≤ Q  ≤ 1000 
-  1 ≤ Fi ≤ 10 
-  1 ≤ f ≤ 10 

Subtask #2 (15 points) : 

-  1 ≤ N ≤ 1000 
-  1 ≤ Q  ≤ 100 
-  1 ≤ Fi ≤ 109 
-  1 ≤ f ≤ 1000 

Subtask #3 (80 points) : 

- original constraints

-----Example-----
Input:
5
1 2 3 4 5
3
2 1
1 3 10
2 2

Output:
1 120
5 50

-----Explanation-----
In the first query, Chef is going to visit cities 1, 2, 3, 4, 5 in order. At the end of the trip, his total enjoyment will be 1 * 2 * 3 * 4 * 5 = 120. First digit of enjoyment is 1 and 120 modulo 109 + 7 is 120.
In the third query, Chef is going to visit cities 1, 3, 5 in order. At the end of the trip, his total enjoyment will be 1 * 10 * 5 = 50.
user: Chef has an array of N natural numbers. Cheffina challenges the chef to choose the two numbers from the array and following the condition as the area of the rectangle formed from the two numbers is maximum. Cheffina also asks the chef to choose two numbers different from the previous two to form the rectangle with a minimum area.

-----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, $N$.
- N space-separated natural numbers. 

-----Output:-----
For each test case, output in one line answers maximum and minimum area of a rectangle.

-----Constraints-----
- $1 \leq T \leq 10$
- $4 \leq N \leq 10^5$
- $1 \leq arr[i] \leq 10^6$

-----Sample Input:-----
1
5
4 2 1 5 3

-----Sample Output:-----
20 2
user: The MEX of a set of integers is defined as the smallest non-negative integer that does not belong to this set. For example, $\mathrm{MEX}(\{0,2,3\}) = 1$ and $\mathrm{MEX}(\{1,3\}) = 0$.
Chef has a tree with $N$ nodes (numbered $1$ through $N$). The tree is rooted at node $1$. Chef wants to assign a non-negative integer to each node in such a way that each integer between $0$ and $N-1$ (inclusive) is assigned to exactly one node.
For each node $u$, consider the integers assigned to the nodes in the subtree of $u$ (including $u$); let $a_u$ denote the MEX of these integers. Chef wants $a_1 + a_2 + \ldots + a_N$ to be as large as possible. Find the maximum possible value of this sum.

-----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 a single integer $N$.
- The second line contains $N-1$ space-separated integers $p_1, p_2, \ldots, p_{N-1}$. For each valid $i$, the node $p_i$ is the parent of the node $i+1$.

-----Output-----
For each test case, print a single line containing one integer ― the maximum sum of subtree MEX-s which can be obtained if you assign the weights optimally.

-----Constraints-----
- $1 \le T \le 5$
- $2 \le N \le 10^5$
- $1 \le p_i < i$ for each valid $i$

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

-----Example Input-----
2
3
1 1
5
1 1 2 2

-----Example Output-----
4
9
user: Pushkar is very good in Number Theory. He takes two numbers $A\ and\ B$ and declares them a Pushkar Pair. Pushkar Pair has a property that $A$ has a $Modular\ Inverse$ modulo $B$.  
He asks you to tell him the largest number $L$ that divides both of them.  

-----Input:-----
- First line will contain $T$, number of testcases. Then the testcases follow. 
- Each testcase contains of a single line of input, two integers $A, B$. 

-----Output:-----
For each testcase, output in a single line the integer $L$.

-----Constraints-----
- $1 \leq T \leq 1000$
- $2 \leq A,B \leq 10^4$

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

-----Sample Output:-----
1
user: You are given an array $a$ of length $n$. A subsequence of this array is valid, if it satisfies these two conditions:
- There shouldn't be any two even numbers within a distance of $K$, both which have been chosen in the subsequence. i.e. there shouldn't be two indices $i, j$ such that $a_i$ and $a_j$ are even, $|i - j| \leq K$ and $a_i$ and $a_j$ are in the subsequence. 
- Similarly, there shouldn't be any two odd numbers within a distance of $K$, both which have been chosen in the subsequence
The sum of a subsequence is the sum of all the numbers in it. Your task is find the maximum sum possible in a valid subsequence of the given array. Print this maximum sum.

-----Input-----
- The first line of the input contains an integer $T$ denoting the number of test cases. The description of the test cases follows.
- The first line of each test case contains two space-separated integers $n, k$.
- The second line of each test case contains $n$ space-separated integers denoting the array $a$.

-----Output-----
For each test case, output an integer corresponding to the answer of the problem.

-----Constraints-----
- $1 \le T \le 10^5$
- $1 \le n \leq 10^5$
- $1 \le k \leq  n$
- $1 \le a_i \leq 10^9$
- Sum of $n$ over all the test cases doesn't exceed $10^6$

-----Example Input-----
3
1 1
3
2 1
2 2
5 2
1 2 3 4 6

-----Example Output-----
3
2
11

-----Explanation:-----
Testcase 2: Only one of the two 2s can be chosen. Hence the answer is 2.
Testcase 3: The subsequence containing the second, third and fifth numbers is a valid subsequence, and its sum is 2+3+6 = 11. You can check that this is the maximum possible, and hence is the answer.
user: Sultan, the freestyle wrestler, you all know him. He broke multiple records in the history of all wrestling leagues. Now 20 years have passed, Sultan has grown old. He has two sons, he wants them to be like him. Sultan being orthodox goes to his astrologer, where he is told that his sons shall be invincible like him. 
Sultan starts to train them. After training, his son Multan & Fultan, having strengths are M and F respectively, decide to fight. They can defeat if the strength of challengers Si is a positive integer multiple of their strength else they lose. Multan fights first, then Fultan. A challenger once knocked out cannot challenge them again. Sultan's sons are still not very good wrestlers. Sultan considers them wrestlers if they both combined are able to win at least 70% of the all the fights. Also, he wants to know who is a better wrestler of the two. Your task is to help Sultan in this venture. Print "Yes" (without quotes) if they are able to win, else print "No" (without quotes). If yes, also name whether, "Multan" or "Fultan" is a better wrestler, if both win equally print “Both”.

-----Input-----
- First line contains single integer T denoting test cases.
- Second Line contains single integer N for number of challengers.
- Third Line contains space separated two integer denoting strength M & F
- Next Line contains strength space separated N integer ith of which denoting Si of N challengers respectively.

-----Output-----
- Yes or No corresponding the result.
- Also, if Yes, print, Multan, Fultan, Both accordingly. 

-----Constraints-----
- 1 ≤ T ≤ 100
- 1 ≤ N ≤ 1000
- 1 ≤ M, F ≤ 109
- 0 ≤ Si ≤ 109

-----Example-----
Input:
2
7
2 3
4 5 7 8 9 10 14
6
5 7
1 2 8 9 10 11

Output:
Yes
Multan
No


-----Explanation-----
Example case 1.

Multan (M) defeats total 4 challengers with strengths 4, 8, 10, 14 and Fultan (F) defeats 1 challenger with strength 9. Their total wins are 5 out of 7 and win accuracy of 71.4%. Hence, 'Yes' and since Multan is better wrestler so 'Multan'

Example case 2.
Multan defeats 1 and Fultan defeat 0 challengers. Total wins 1 out of 6 with accuracy 16.67% Hence, No.