meetforever/GRPO_CL huggingface dataset:
user: Cersei wants to be the queen of seven kingdoms.

For this to happen, she needs to address the soldiers in her army. There are n$n$ soldiers in her army (numbered 1$1$ through n$n$). Cersei passes on the message to the first soldier (soldier 1).   
This message needs to reach every soldier in the army. For this, the soldiers communicate among themselves by one soldier passing the message to another soldier through some communication links. It is known that the message could reach every soldier using the given links.
Now, each soldier will receive the message from exactly one soldier or Cersei and could pass on the message to atmost two soldiers. That is each soldier (except soldier 1) has only one incoming link and every soldier (including soldier 1) has atmost two outgoing links.  
Now, the High Sparrow feels that Cersei is planning to kill his people first. Hence, for the sake of his people, he decided to appoint some sparrows to overhear every conversation between the soldiers (The conversation between Cersei and the first soldier needn't be overheard due to the fear of Ser Gregor Clegane).   
To overhear a conversation between soldiers A$A$ and B$B$, there needs to be a sparrow either at soldier A$A$ or soldier B$B$ or both.
Also, by his research, the High Sparrow has found that the soldiers are partitioned into some classes (1$1$ to k$k$). That is, every soldier belongs to exactly one class. He then demands the presence of atleast one sparrow with each class he knows (1$1$ to k$k$).
Find the minimum number of sparrows the High Sparrow needs to recruit for the job or tell that he couldn't.

-----Input:-----
- The first line of the input contains the number of test cases t$t$.
- The first line of each test case gives the number of soldiers n$n$ in the army, the number of communication links m$m$ between the soldiers and the number of classes k$k$ in soldiers.
- The next line of the test case consists of n$n$ integers A1,A2....An$A_1,A_2....A_n$ each denoting the class of the ith$i^{th}$ soldier.
- The next m$m$ lines of the test case contain two integers u$u$ and v$v$, which denotes that soldier u$u$ can pass a message to soldier v$v$ (u≠v$u \neq v$).

-----Output:-----
For each test case, print in a single line the minimum number of sparrows required for the above task or print −1$-1$ if no such way is possible.

-----Constraints-----
- 1≤t≤500$1 \leq t \leq 500$
- 1≤n≤2500$1 \leq n \leq 2500$
- m=n−1$m = n - 1$
- 1≤k≤10$1 \leq k \leq 10$
- 1≤ai≤k$1 \leq a_i \leq k$
- The sum of n$n$ over all test cases is ≤2500$\leq 2500$.

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

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

-----EXPLANATION:-----
Selecting soldiers 1,4,5 would satisfy all the conditions.

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

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

-----EXPLANATION:-----
Classes 4 and 5 are not present. So, there is no way possible.
user: In Ciel's restaurant, a waiter is training.
Since the waiter isn't good at arithmetic, sometimes he gives guests wrong change.
Ciel gives him a simple problem.
What is A-B (A minus B) ?

Surprisingly, his answer is wrong.
To be more precise, his answer has exactly one wrong digit.
Can you imagine this?
Can you make the same mistake in this problem?

-----Input-----

An input contains 2 integers A and B.

-----Output-----

Print a wrong answer of A-B.
Your answer must be a positive integer containing the same number of digits as the correct answer, and exactly one digit must differ from the correct answer.
Leading zeros are not allowed.
If there are multiple answers satisfying the above conditions, anyone will do.

-----Constraints-----

1 ≤ B < A ≤ 10000

-----Sample Input-----
5858 1234

-----Sample Output-----
1624

-----Output details-----

The correct answer of 5858-1234 is 4624.
So, for instance, 2624, 4324, 4623, 4604 and 4629 will be accepted, but 0624, 624, 5858, 4624 and 04624 will be rejected.

-----Notes-----

The problem setter is also not good at arithmetic.
user: Sheldon is a little geek living in Texas. While his friends like to play outside, little Sheldon likes to play around with ICs and lasers in his house. He decides to build N clap activated toggle machines each with one power inlet and one outlet. Each machine works when its power source inlet is receiving power. When the machine is in 'ON' state and is receiving power at its inlet, it makes power available at its power outlet to which a bulb or another ToGgLe machine could be attached.

Suppose Sheldon attached 2 such machines to one another with the power inlet of the first machine attached to a power source at his house and the outlet of the second machine to a bulb. Initially both machines are in 'OFF' state and power source to first machine is off too. Now the power source is switched on. The first machine receives power but being in the 'OFF' state it does not transmit any power. Now on clapping the first ToGgLe machine toggles to 'ON' and the second machine receives power. On clapping once more the first toggles to 'OFF' and the second toggles to 'ON'. But since the second ToGgLe machine receives no power the bulb does not light up yet. On clapping once more, the first machine which is still receiving power from the source toggles to 'ON' and the second which was already 'ON' does not toggle since it was not receiving power. So both the machine are in 'ON' state and the bulb lights up and little Sheldon is happy.

But when Sheldon goes out for a while, his evil twin sister attaches N such ToGgLe machines (after making sure they were all in 'OFF' state) and attaches the first to a power source (the power source is initially switched off) and the last ToGgLe machine to a bulb. Sheldon is horrified to find that his careful arrangement has been disturbed.

Coders, help the poor boy by finding out if clapping k times for the N ToGgLe machines (all in 'OFF' state with the first one connected to a switched off power source and last one to a bulb) would light the bulb. Hurry before Sheldon has a nervous breakdown!

-----Input-----
First line has number of test cases, T.

Following T lines have N, k separated by a single space where N is the number of 
ToGgLe machines and k is the number of times Sheldon clapped.

-----Output-----
T lines with cach line of the form: "ON" (just the word on without the double quotes) if the bulb is 'ON' for the 
test case numbered n and "OFF" (just the word off without the double quotes) if the bulb is 'OFF' for the test case numbered n.


-----Example-----
Input:
4
4 0
4 47
1 0
1 1

Output:
OFF
ON
OFF
ON
user: Well known investigative reporter Kim "Sherlock'' Bumjun needs your help! Today, his mission is to sabotage the operations of the evil JSA. If the JSA is allowed to succeed, they will use the combined power of the WQS binary search and the UFDS to take over the world!
But Kim doesn't know where the base is located. He knows that the base is on the highest peak of the Himalayan Mountains. He also knows the heights of each of the $N$ mountains. Can you help Kim find the height of the mountain where the base is located? 

-----Input:-----
- First line will contain $T$, number of testcases. Then the testcases follow.    
- The first line in each testcase contains one integer, $N$.    
- The following $N$ lines of each test case each contain one integer: the height of a new mountain.

-----Output:-----
For each testcase, output one line with one integer: the height of the tallest mountain for that test case.

-----Constraints-----
- $1 \leq T \leq 10$
- $1 \leq N \leq 100000$
- $0 \leq$ height of each mountain $\leq 10^9$

-----Subtasks:-----
- 100 points: No additional constraints.

-----Sample Input:-----
1
5
4
7
6
3
1

-----Sample Output:-----
7
user: Lyra Belacqua is a very gifted girl. She is one of a very small set of people capable of reading an alethiometer, more commonly known as The Golden Compass. It has one specific use: to tell the truth. The name in fact, is derived from "Aletheia" meaning truth, and "-ometer", meaning "measuring device".

The alethiometer had four needles, out of which the user would direct three of them to lie over symbols on the face of the device to ask a question. The fourth needle then swung into action and pointed to various symbols one after another, thus telling the answer.

For this problem, consider the alethiometer consisting of symbols : digits '0'-'9' and letters 'A'-'Z'. Learned scholars were debating the age of the Universe, and they requested Lyra to find out the age from the alethiometer. Having asked the question, the fourth needle started spouting out symbols, which Lyra quickly recorded. In that long string of characters, she knows that some substring corresponds to the age of the Universe. She also knows that the alethiometer could have wrongly pointed out atmost one digit (0-9) as a letter (A-Z). She then wonders what is the maximum possible age of the Universe.

Given the set of symbols the alethiometer pointed out, help her find the maximum age of the Universe, which could correspond to a substring of the original string with atmost one letter changed.

Note: We consider a substring to be a contiguous part of the string S 

Also, the alethiometer wrongly reports only a letter. All the digits remain as they are.

-----Input-----

Each input consists of a single string S which is what Lyra recorded from the fourth needle's pointing.

-----Output-----

Output one number, the maximum possible answer.

-----Constraints-----
-  1  ≤ |S|  ≤ 1,000
-  S will only contain digits 0-9 and uppercase Latin letters. 

-----Example-----
Input1:
06454

Input2:
C0D3C43F

Output1:
6454

Output2:
3943

-----Explanation-----

In the first example, there is no choice as to what the number can be. It has to be 6,454.

In the second example, there are a total of 41 possible strings (one for the original, and 10 for changing each letter). You can verify that the maximum number as a substring is got by making the string "C0D3943F".
user: Chef is given a sequence of prime numbers $A_1, A_2, \ldots, A_N$. This sequence has exactly $2^N$ subsequences. A subsequence of $A$ is good if it does not contain any two identical numbers; in particular, the empty sequence is good.
Chef has to find the number of good subsequences which contain at most $K$ numbers. Since he does not know much about subsequences, help him find the answer. This number could be very large, so compute it modulo $1,000,000,007$.

-----Input-----
- The first line of the input contains two space-separated integers $N$ and $K$.
- The second line contains $N$ space-separated integers $A_1, A_2, \ldots, A_N$.

-----Output-----
Print a single line containing one integer ― the number of good subsequences with size at most $K$, modulo $1,000,000,007$.

-----Constraints-----
- $1 \le K \le N \le 10^5$
- $2 \le A_i \le 8,000$ for each valid $i$

-----Subtasks-----
Subtask #1 (40 points): $A_1, A_2, \ldots, A_N$ are pairwise distinct
Subtask #2 (60 points): original constraints

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

-----Example Output-----
18

-----Explanation-----
There is $1$ good subsequence with length $0$, $5$ good subsequences with length $1$, $8$ good subsequences with length $2$ and $4$ good subsequences with length $3$.
user: In this problem you are given a sequence of $N$ positive integers $S[1],S[2],\dots,S[N]$. In addition you are given an integer $T$, and your aim  is to find the number of quadruples $(i,j,k,l)$, such that $1 \le i < j < k < l \le N$, and $S[i] + S[j] + S[k] + S[l] = T$.  That is, the number of ways of picking four numbers from the sequence summing up to $T$.

-----Constraints:-----
For all test cases,
- $1 \le N \le 5000$
- $1 \le T \le 10^6$
- $1 \le S[i] \le 10^9$, for all $i$.
Subtask $1:10\%$
It is guaranteed that $N \le 50$.
Subtask $2:50\%$
It is guaranteed that $N \le 300$.
Subtask $3:40\%$
No additional guarantees.

-----Input Format:-----
There is only one line of input with $N+2$ space separated integers. The first two integers are $N$ and $T$. The next $N$ integers are $S[1],S[2],\dots,S[N]$.

-----Output Format:-----
A single integer which is the number of valid quadruples.

-----Sample Input 1:-----
6 20 3 1 1 2 5 10

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

-----Explanation 1:-----
The quadruple $(1,4,5,6)$ is valid  because $S[1]+S[4]+S[5]+S[6]=3+2+5+10=20$. You can check that no other quadruple is valid and hence the answer is $1$.

-----Sample Input 2:-----
6 13 1 2 3 4 5 4 

-----Sample Output 2:-----
3

-----Explanation 2:-----
You can verify that the only quadruples that are valid are $(2,3,4,6),(1,3,4,5)$  and $(1,3,5,6)$. Thus, the answer is $3$.

-----Note:-----
As the answer might be large, please use 64 bit integers (long long int in
C/C++ and long in Java) instead of 32 bit int.
user: A balanced parenthesis string is defined as follows:
- The empty string is balanced
- If P is balanced, (P) is also 
- If P and Q are balanced, PQ is also balanced
You are given two even integers n$n$ and k$k$. Find any balanced paranthesis string of length n$n$ that doesn't contain a balanced substring of length k$k$, or claim that no such string exists.

-----Input-----
- First line will contain T$T$, number of testcases. Then the testcases follow. 
- Each testcase contains of a single line containing n$n$ and k$k$.

-----Output-----
For every testcase, print on a new line, any balanced paranthesis string of length n$n$ that doesn't contain a balanced substring of length k$k$. If there doesn't exist any such string, print −1$-1$ instead.

-----Constraints-----
- 1≤T≤50000$1 \leq T \leq 50000$
- 2≤k≤n≤105$2 \leq k \leq n \leq 10^5$
- Sum of n$n$ over all testcases doesn't exceed 105$10^5$.
- n$n$ and k$k$ are both even integers.

-----Example Input-----
2
4 2
8 6

-----Example Output-----
-1
(())(())

-----Explanation-----
In the first testcase, the only balanced strings of length 4$4$ are (()) and ()(), both of which contain () as a substring.
In the second testcase, (())(()) is a balanced string that doesn't contain any balanced substring of length 6$6$.
user: You will be given m strings. For each of those strings, you need to count the total number of appearances of that string as substrings in all possible strings of length n containing only lower case English letters. 

A string may appear in a string multiple times. Also, these appearances may overlap. All these must be counted separately. For example, aa appears thrice in the string aaacaa: aaacaa, aaacaa and aaacaa.

-----Input-----
- The first line contains one integer, T, the number of test cases. The description of each test case follows:
- The first line of each test case will contain two integers n and m.
- The ith of the next m lines will have one string in each line. All the strings will consist only of lower case English letters.

-----Output-----
- For each test case, print "Case x:" (without quotes. x is the test case number, 1-indexed) in the first line.
- Then print m lines. The ith line should contain the number of appearances of the ith string in all possible strings of length n. As the numbers can be very large, print the answers modulo 109+7.

-----Constraints-----
- 1 ≤ T ≤ 100
- 1 ≤ n ≤ 100000
- 1 ≤ m ≤ 1000 
- 1 ≤ Length of every string in input
- 1 ≤ Total length of all strings in one test case ≤ 5 * 105
- 1 ≤ Total length of all strings in one test file ≤ 5 * 106

-----Example-----
Input:
3
2 1
aa
2 1
d
12 3
cdmn
qweewef
qs

Output:
Case 1:
1
Case 2:
52
Case 3:
443568031
71288256
41317270

-----Explanation:-----
Testcase 1: aa is the only string of length 2 which contains aa as a substring. And it occurs only once. Hence the answer is 1.
user: For a string $S$ let the unique set of characters that occur in it one or more times be $C$. Consider a permutation of the elements of $C$ as $(c_1, c_2, c_3 ... )$. Let $f(c)$ be the number of times $c$ occurs in $S$.
If any such permutation of the elements of $C$ satisfies $f(c_i) = f(c_{i-1}) + f(c_{i-2})$ for all $i \ge 3$, the string is said to be a dynamic string.
Mr Bancroft is given the task to check if the string is dynamic, but he is busy playing with sandpaper. Would you help him in such a state?
Note that if the number of distinct characters in the string is less than 3, i.e. if $|C| < 3$, then the string is always dynamic.

-----Input:-----
- First line will contain $T$, number of testcases. Then the testcases follow. 
- Each testcase contains of a single line of input, a string $S$.

-----Output:-----
For each testcase, output in a single line "Dynamic" if the given string is dynamic, otherwise print "Not". (Note that the judge is case sensitive)

-----Constraints-----
- $1 \leq T \leq 10$
- $1 \leq |S| \leq 10^5$
- $S$ contains only lower case alphabets: $a$, $b$, …, $z$

-----Sample Input:-----
3
aaaabccc
aabbcc
ppppmmnnoooopp

-----Sample Output:-----
Dynamic
Not
Dynamic

-----Explanation:-----
- Testase 1: For the given string, $C = \{a, b, c\}$ and $f(a)=4, f(b)=1, f(c)=3$. $f(a) = f(c) + f(b)$ so the permutation $(b, c, a)$ satisfies the requirement.
- Testcase 2: Here too $C = \{a, b, c\}$ but no permutation satisfies the requirement of a dynamic string.
- Testcase 3: Here $C = \{m, n, o, p\}$ and $(m, n, o, p)$ is a permutation that makes it a dynamic string.
user: Chef made two laddus with sweetness X and Y respectively. Cheffina comes and sees the chef created two laddus with different sweetness (might be same). Cheffina has the magical power to make the sweetness of laddus equal. Cheffina requires 1 unit of power to increase the sweetness of laddu by its original value i.e. 1 unit to convert Z to 2Z and 2 unit to convert Z to 3Z and so on… How many units of power does cheffina want to make the sweetness equal?

-----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 integers $X, Y$. 

-----Output:-----
For each test case, output in a single line answer as power required.

-----Constraints-----
- $1 \leq T \leq 10^5$
- $1 \leq X,Y \leq 10^5$

-----Sample Input:-----
2
2 2
4 6

-----Sample Output:-----
0
3

-----EXPLANATION:-----
For 1) Sweetness are same so no need to use power.
For 2) 
1st laddu
2 Unit power = 4 -> 12
2nd Laddu
1 Unit power = 6 -> 12
After using total 3 unit power sweetness of both laddus are same.
user: A key feature of the Siruseri railway network is that it has exactly one route between any pair of stations.
The government has chosen three contractors to run the canteens at the stations on the railway network. To ensure that there are no disputes between the contractors it has been decided that if two stations, say $A$ and $B$, are assigned to a particular contractor then all the stations that lie on the route from $A$ to $B$ will also be awarded to the same contractor.
The government would like the assignment of stations to the contractors to be as equitable as possible. The government has data on the number of passengers who pass through each station each year. They would like to assign stations so that the maximum number of passengers passing through any contractor's collection of stations is minimized.
For instance, suppose the railway network is as follows, where the volume of passenger traffic is indicated by the side of each station.

One possible assignment would to award stations $1$ and $3$ to one contractor (there by giving him a traffic of $35$ passengers), station $2$ to the second contractor (traffic of $20$) and stations $4, 5$ and $6$ to the third contractor (traffic of $100$). In this assignment, the maximum traffic for any one contractor is 100. On the other hand if we assigned stations $1, 2$ and $3$ to one contractor, station $4$ and $6$ to the second contractor and station $5$ to the third contractor the maximum traffic for any one contractor is $70$. You can check that you cannot do better. (The assignment $1$, $2$ and $3$ to one contractor, $4$ to the second contractor, and $5$ and $6$ to the third contractor has a lower value for the maximum traffic ($55$) but it is not a valid assignment as the route from $5$ to $6$ passes through $4$.)

-----Input:-----
The first line of the input contains one integer $N$ indicating the number of railways stations in the network. The stations are numbered $1,2,..., N$. This is followed by $N$ lines of input, lines $2,3,...,N+1$, indicating the volume of traffic at each station. The volume of traffic at station $i$, $1 \leq i \leq N$, is given by a single integer in line $i+1$. The next $N-1$ lines of input, lines $N+2, N+3, ..., 2 \cdot N$, describe the railway network. Each of these lines contains two integers, denoting a pair of stations that are neighbours.

-----Output:-----
The output should be a single integer, corresponding to the minimum possible value of the maximum traffic of any contractor among all valid assignment of the stations to the three contractors.

-----Constraints:-----
- $1 \leq N \leq 3000$.

-----Sample Input-----
6
10
20
25
40
30
30
4 5
1 3
3 4
2 3
6 4

-----Sample Output-----
70
user: Chef is the leader of Chef's Earth Defense Organization, and his mission is to counter aliens which are threatening the earth. According to information gathered by the organization, there are $N$ alien spaceships (numbered $1$ through $N$) planning to invade the earth. The $i$-th spaceship will appear on the radar at time $C_i$. Each spaceship needs $D$ time to reach earth once it appears on the radar. 
Chef's organization owns a huge laser cannon which can destroy one spaceship in one shot. However, once the cannon is used once it needs some amount of cool-down time in order to be used again (first shot doesn't require cool-down time before it is used). This cool-down time, however, is configurable. It has to be set before the first shot, and cannot be changed after that. If Chef sets a lower cool-down time, that would increase the energy consumed by the cannon, and vice-versa - the higher the cool-down time, the lower the energy consumed.
This might be a long battle, and Chef wants to use as little energy as possible. So Chef wants to maximize the cool-down time while still being able to destroy all spaceships before any of them arrive on earth. In particular, the $i$-th spaceship should be shot between the times $C_i$ and $C_i + D$ (both end points inclusive).

-----Input:-----
- The first line of 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 integers $N$ and $D$. 
- The second line contains $N$ space-separated integers $C_1, C_2, \ldots, C_N$.

-----Output:-----
For each test case, print a single line containing one real number― the maximum cool-down time possible. Your answer will be considered correct if the absolute or relative error of the answer does not exceed $10^{-6}$.

-----Constraints-----
- $1 \leq T \leq 1000$
- $2 \leq N \leq 10^5$
- $1 \leq C_i \leq 10^9$ for each valid $i$
- $1 \leq D \leq 10^9$
- The sum of $N$ over all test cases does not exceed $10^6$

-----Subtasks-----
Subtask #1 (50 points): 
- $N \le 1,000$
- the sum of $N$ over all test cases does not exceed $10,000$
Subtask #2 (50 points): Original constraints

-----Sample Input:-----
2
3 2
3 2 3
2 1
5 6

-----Sample Output:-----
1.5000000000
2.0000000000