meetforever/GRPO_CL huggingface dataset:
user: Tonight, Chef would like to hold a party for his $N$ friends.
All friends are invited and they arrive at the party one by one in an arbitrary order. However, they have certain conditions — for each valid $i$, when the $i$-th friend arrives at the party and sees that at that point, strictly less than $A_i$ other people (excluding Chef) have joined the party, this friend leaves the party; otherwise, this friend joins the party.
Help Chef estimate how successful the party can be — find the maximum number of his friends who could join the party (for an optimal choice of the order of arrivals).

-----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 $N$ space-separated integers $A_1, A_2, \ldots, A_N$.

-----Output-----
For each test case, print a single line containing one integer — the maximum number of Chef's friends who could join the party.

-----Constraints-----
- $1 \le T \le 1,000$
- $1 \le N \le 10^5$
- the sum of $N$ over all test cases does not exceed $10^6$

-----Example Input-----
3
2
0 0
6
3 1 0 0 5 5
3
1 2 3

-----Example Output-----
2
4
0

-----Explanation-----
Example case 1: Chef has two friends. Both of them do not require anyone else to be at the party before they join, so they will both definitely join the party.
Example case 2: At the beginning, friends $3$ and $4$ can arrive and join the party, since they do not require anyone else to be at the party before they join. After that, friend $2$ can arrive; this friend would see that there are two people at the party and therefore also join. Then, friend $1$ will also join, so in the end, there would be $4$ people attending the party.
Example case 3: No one will attend the party because each of Chef's friends will find zero people at the party and leave, regardless of the order in which they arrive.
user: You are given a positive integer $N$. Consider the sequence $S = (1, 2, \ldots, N)$. You should choose two elements of this sequence and swap them.
A swap is nice if there is an integer $M$ ($1 \le M < N$) such that the sum of the first $M$ elements of the resulting sequence is equal to the sum of its last $N-M$ elements. Find the number of nice swaps.

-----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 a single integer $N$.

-----Output-----
For each test case, print a single line containing one integer ― the number of nice swaps.

-----Constraints-----
- $1 \le T \le 10^6$
- $1 \le N \le 10^9$

-----Subtasks-----
Subtask #1 (10 points):
- $T \le 10$
- $N \le 10^3$
Subtask #2 (30 points):
- $T \le 10$
- $N \le 10^6$
Subtask #3 (60 points): original constraints

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

-----Example Output-----
0
0
2
2
3
user: You are playing a Billiards-like game on an $N \times N$ table, which has its four corners at the points $\{(0, 0), (0, N), (N, 0),$ and $(N, N)\}$. You start from a coordinate $(x,y)$,  $(0 < x < N, 0 <  y < N)$ and shoot the ball at an angle $45^{\circ}$ with the horizontal. On hitting the sides, the ball continues to move with the same velocity and ensuring that the angle of incidence is equal to the angle of reflection with the normal, i.e, it is reflected with zero frictional loss. On hitting either of the four corners, the ball stops there and doesn’t move any further. 
Find the coordinates of the point of collision, when the ball hits the sides for the $K^{th}$ time. If the ball stops before hitting the sides $K$ times, find the coordinates of the corner point where the ball stopped instead.

-----Input:-----
- The first line of the input contains an integer $T$, the number of testcases.
- Each testcase contains a single line of input, which has four space separated integers - $N$, $K$, $x$, $y$, denoting the size of the board, the number of collisions to report the answer for, and the starting coordinates.

-----Output:-----
For each testcase, print the coordinates of the ball when it hits the sides for the $K^{th}$ time, or the coordinates of the corner point if it stopped earlier.

-----Constraints-----
- $1 \leq T \leq 10^5$
- $2 \leq N \leq 10^9$
- $1 \leq K \leq 10^9$

-----Subtasks-----
- $30$ points :  Sum of $K$ over all test cases $\le 10^7$
- $70$ points : Original constraints.

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

-----Sample Output:-----
5 5
3 5

-----Explanation:-----
- Sample Case $1$ : 
We are given a $5$ by $5$ board. We shoot the ball from coordinates $(4,4)$, and we need to find its coordinates after it has collided with sides $5$ times. However, after shooting, the ball goes directly to the corner $(5,5)$, and stops there. So we report the coordinates $(5,5)$.

- Sample Case $2$ : 
We are given a $5$ by $5$ board. We shoot the ball from the coordinates $(3,1)$, and we need to find its coordinates after it has collided with the sides twice. After shooting, it first hits the right side at $(5,3)$, and then the top side at $(3,5)$. So, we report $(3,5)$.
user: Chef has a natural number N. Cheffina challenges chef to check whether the given number is divisible by the sum of its digits or not. If the given number is divisible then print "Yes" else "No".

-----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.

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

-----Sample Input:-----
2
16
27

-----Sample Output:-----
No
Yes
user: Gorodetskiy is a university student. He is really good at math and really likes solving engaging math problems. In the last exam, his professor gave him really hard math problems to solve, but Gorodetskiy could not solve them and failed the exam, so the professor told him: "These problems are a piece of cake, you should know them from school!" Here is one of the problems from the exam - you can decide if it really was an easy problem or not.
You are given a positive integer $M$. Let's call a positive integer $A$ an interesting number if there is at least one integer $B$ ($A \le B$) such that $A \cdot B$ is divisible by $M$ and $\frac{A \cdot B}{M} = A+B$.
Find all interesting numbers. Let's denote the number of such integers by $K$; it is guaranteed that $K \le 3*10^5$.

-----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 a single integer $M$.

-----Output-----
For each test case:
- First, print a line containing a single integer $K$.
- Then, print $K$ lines. For each valid $i$, the $i$-th of these lines should contain a single integer ― the $i$-th interesting number in increasing order.

-----Constraints-----
- $1 \le T \le 10$
- $1 \le M \le 10^{14}$

-----Subtasks-----
Subtask #1 (20 points): $1 \le M \le 100$
Subtask #2 (20 points): $1 \le M \le 10^7$
Subtask #3 (60 points): original constraints

-----Example Input-----
2
3
6

-----Example Output-----
2
4
6
5
7
8
9
10
12

-----Explanation-----
Example case 1: There are two interesting numbers, $4$ and $6$. For $A = 4$, we can choose $B = 12$ and for $A = 6$, we can choose $B = 6$.
user: Problem description.

This problem is simple and will introduce you to the Dynamic Programming.

You will be given an array and a key value.

You will have to find out the occurrences of the key value depending upon the query using Brute Force and Top Down Dynamic Programming.

-----Brute-Force: -----
You will check the query, and calculate the occurrences.


-----DP: -----

You will check the query; check whether the memoized solution is already available.

If the memoized solution is available, no need to calculate the number of occurrences again.

If the memoized solution is not available, then calculate the number of  occurrences and memoize it for future use.

-----Pseudo Code for DP:-----
countOccurences(key,from):

if (from = size of array) then 
	return 0
endif
if dp[from] is availabe then 
	return dp[from]
endif
if( array[from] == key) then
	dp[from] = 1+countOccurences(key,from+1)
else
	dp[from] = countOccurences(key,from+1)
endif	
return dp[from]


-----Input:-----

The first line of input is the number of test cases (t).

The first line of each test case is the number of array elements (n).

The next will contain n space separated integers.

The next line will have the key element (k).

The next will have number of queries (q).

The next q lines will contain an integer A such that 0<=A < n. 

You have to find out the number of occurrences from a to end of the array using both brute force and DP. 

Everything will fit into the range of int.

-----Output:-----

For each test case, the output will have q lines with 3 space separated integers.
The first will be the number of occurrences, other will be the loop count/function calls,
using brute force and the last will be the number of loop count/function calls using DP.

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

-----Sample output:-----
3 8 9
2 6 1
1 4 1
1 2 1
3 8 1

-----Explanation:-----

For the first query, we have to find the number of occurrences of 3 from index 2.

Using the brute force, the loop will check each element from index 2 to 9. Thus the loop count is 8.

Using DP, the method countOccurences(key,from) will be called like this : 

countOccurences(3,2)->countOccurences(3,3)->countOccurences(3,4)->countOccurences(3,5)
->countOccurences(3,6)->countOccurences(3,7)->countOccurences(3,8)
->countOccurences(3,9)->countOccurences(3,10).


When the countOccurences(3,10) is called, it will return 0. Total 9 function calls.

For the second query, the brute force will do the same as above.

But the DP will check whether solution for countOccurences(3,4) is available or not.

As it was calculated while solving the first query, it won’t solve it again and will directly give the answer. 

Total 1 function calls.
user: Taru likes reading. Every month he gets a copy of the magazine "BIT". The magazine contains information about the latest advancements in technology.  Taru 

reads the book at night and writes the page number to which he has read on a piece of paper so that he can continue from there the next day. But sometimes 

the page number is not printed or is so dull that it is unreadable.  To make matters worse Taru's brother who is really naughty tears of some of the pages of 

the Magazine and throws them in the dustbin. He remembers the number of leaves he had torn but he does not remember which page numbers got removed. When Taru 

finds this out he is furious and wants to beat him up. His brother apologizes, and says he won't ever do this again. But Taru did not want to be easy on him 

and he says "I will leave you only if you help me find the answer to this. I will tell you how many pages (Printed sides) were there in the Magazine plus the 

pages on which the page numbers were not printed. You already know the number of leaves you tore (T). Can you tell me the expected sum of the page numbers 

left in the Magazine?" Taru's brother replied "huh!! This is a coding problem". Please help Taru's brother.

Note: The magazine is like a standard book with all odd page numbers in front and the successive even page number on its back. If the book contains 6 pages, 

Page number 1 and Page number 2 are front and back respectively. Tearing a leaf removes both the front and back page numbers.


-----Input-----
The first line contains the number of test cases t. 3t lines follow. The first line of each test case contains the number of pages (printed sides) in the 

book. The second line's first integer is F, F integers follow which tell us the numbers of the page numbers not printed. The third line contains a single integer telling us the number of leaves Taru's brother tore.

-----Output-----
Output one real number correct up to 4 decimal digits which is equal to the expected sum of the page numbers left in the book.

-----Constraints-----

Number of printed Sides<=2000. All other values abide by the number of printed sides.

-----Example-----
Input:
2
10
2 1 2
2
10
1 8
0

Output:
31.2000
47.0000
user: Bhallaladeva was an evil king who ruled the kingdom of Maahishmati. He wanted to erect a 100ft golden statue of himself and he looted gold from several places for this. He even looted his own people, by using the following unfair strategy:
There are N houses in Maahishmati, and the ith house has Ai gold plates. Each gold plate costs exactly 1 Nimbda, which is the unit of currency in the kingdom of Maahishmati. Bhallaladeva would choose an integer K, and loots all the houses in several steps. In each step:

- He would choose a house i which hasn't been looted yet, pay the owner exactly Ai Nimbdas, and take away all the gold plates in that house (Hence, he also ends up looting this house).
- He would now choose atmost K houses which haven't been looted yet and take away all the gold plates from these houses without paying a single Nimbda (Yes, he takes all of them for free).

He repeats the above steps until all the N houses have been looted. Your task is to devise a strategy for Bhallaladeva to loot the houses in some order, so that the number of nimbdas he has to pay is minimium. You'll also be given multiple values of K (Q of them to be precise), and you need to find the minimum number of nimbdas for each of these values.

-----Input-----
The first line of input consists of a single integer N denoting the number of houses in Maahishmati. The second line of input consists of N space separated integers denoting A1, A2, ..., AN, where Ai denotes the number of gold plates in the ith house. The third line of input consists of a single integer Q denoting the number of values of K to follow. Each of the following Q lines consist of a single integer, where the value on the ith line denotes the value of K for the ith query.

-----Output-----
Output exactly Q integers on separate lines, where the output on the ith line denotes the answer for the ith value of K.

-----Constraints-----
- 1 ≤ N ≤ 105
- 1 ≤ Q ≤ 105
- 0 ≤ K ≤ N-1
- 1 ≤ Ai ≤ 104

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

Output:
10
3

-----Explanation-----
For the first query, K = 0. Hence, Bhallaladeva cannot take gold plates from any of the houses for free. It will cost him 3 + 2 + 1 + 4 = 10 nimbdas.
For the second query, K = 2. In the first step Bhallaladeva can pay 2 nimbdas for gold plates in house number 2, and take the gold in houses 1 and 4 for free (Note that house 1 has 3 gold plates and house 4 has 4 gold plates). Now, he has looted houses 1, 2 & 4. Now in the second step, he loots house 3, by paying 1 nimbda. Hence, the total cost = 1 + 2 = 3. Note that there might be multiple ways to achieve the minimum cost, and we have explained only one of them.
user: Chef wants to select a subset $S$ of the set $\{1, 2, \ldots, N\}$ such that there are no two integers $x, y \in S$ which satisfy $\frac{x}{y} = M$.
Help Chef find the maximum size of a subset $S$ he can choose and the number of ways in which he can choose a subset $S$ with this maximum size. Since the number of ways to choose $S$ can be very large, calculate it 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 two space-separated integers $N$ and $M$.

-----Output-----
For each test case, print a single line containing two space-separated integers ― the maximum size of a subset Chef can choose and the number of ways to choose a subset with this maximum size modulo $998,244,353$. Note that only the number of ways to choose a subset should be printed modulo $998,244,353$.

-----Constraints-----
- $1 \le T \le 10,000$
- $2 \le M \le N \le 10^{18}$

-----Example Input-----
3
5 2
10 2
100 3

-----Example Output-----
4 1
6 12
76 49152

-----Explanation-----
Example case 1: The only subset $S$ with the maximum size which Chef can choose is $\{1, 3, 4, 5\}$.
user: Chef is a private detective. He was asked to investigate a case of murder in the city of Frangton.
Chef arrived in Frangton to find out that the mafia was involved in the case. Chef spent some time watching for people that belong to the clan and was able to build a map of relationships between them. He knows that a mafia's organizational structure consists of a single Don, heading a hierarchical criminal organization. Each member reports exactly to one other member of the clan. It's obvious that there are no cycles in the reporting system of the mafia.
There are N people in the clan, for simplicity indexed from 1 to N, and Chef knows who each of them report to. Member i reports to member Ri.
Now, Chef needs to identfy all potential killers to continue his investigation. Having considerable knowledge about the mafia's activities, Chef knows that the killer must be a minor criminal, that is, one of the members who nobody reports to. Please find the list of potential killers for Chef. Since Don reports to nobody, his Ri will be equal to 0.

-----Input-----
The first line of input contains one integer N.
Next line has N space-separated integers, the ith integer denotes Ri — the person whom the ith member reports to.

-----Output-----
Output a list of space-separated integers in ascending order — the indices of potential killers.

-----Constraints-----
- 1 ≤ N ≤ 105
- 1 ≤ Ri ≤ N except for Don, whose Ri equals to 0.
- It is guaranteed that there are no cycles in the reporting structure.

-----Subtasks-----
- Subtask #1 [50 points]: N ≤ 10000
- Subtask #2 [50 points]: No additional constraints

-----Example-----
Input:
6
0 1 1 2 2 3

Output:
4 5 6

-----Explanation-----
The reporting structure:
user: You are given a sequence $A_1, A_2, \ldots, A_N$. For each $k$ ($1 \le k \le N$), let's define a function $f(k)$ in the following way:
- Consider a sequence $B_1, B_2, \ldots, B_N$, which is created by setting $A_k = 0$. Formally, $B_k = 0$ and $B_i = A_i$ for each valid $i \neq k$.
- $f(k)$ is the number of ways to split the sequence $B$ into two non-empty contiguous subsequences with equal sums.
Find the sum $S = f(1) + f(2) + \ldots + f(N)$.

-----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 $N$ space-separated integers $A_1, A_2, \ldots, A_N$.

-----Output-----
For each test case, print a single line containing one integer — the sum $S$.

-----Constraints-----
- $1 \le T \le 10$
- $2 \le N \le 2 \cdot 10^5$
- $1 \le |A_i| \le 10^9$ for each valid $i$

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

-----Example Output-----
6
2
user: Tomya is a girl. She loves Chef Ciel very much.

Tomya like a positive integer p, and now she wants to get a receipt of Ciel's restaurant whose total price is exactly p.
The current menus of Ciel's restaurant are shown the following table.
Name of Menupriceeel flavored water1deep-fried eel bones2clear soup made with eel livers4grilled eel livers served with grated radish8savory egg custard with eel16eel fried rice (S)32eel fried rice (L)64grilled eel wrapped in cooked egg128eel curry rice256grilled eel over rice512deluxe grilled eel over rice1024eel full-course2048

Note that the i-th menu has the price 2i-1 (1 ≤ i ≤ 12).

Since Tomya is a pretty girl, she cannot eat a lot.
So please find the minimum number of menus whose total price is exactly p.
Note that if she orders the same menu twice, then it is considered as two menus are ordered. (See Explanations for details)

-----Input-----

The first line contains an integer T, the number of test cases.
Then T test cases follow.
Each test case contains an integer p.

-----Output-----

For each test case, print the minimum number of menus whose total price is exactly p.

-----Constraints-----

1 ≤ T ≤ 5

1 ≤ p ≤ 100000 (105)

There exists combinations of menus whose total price is exactly p.

-----Sample Input-----
4
10
256
255
4096

-----Sample Output-----
2
1
8
2

-----Explanations-----

In the first sample, examples of the menus whose total price is 10 are the following:

1+1+1+1+1+1+1+1+1+1 = 10 (10 menus)

1+1+1+1+1+1+1+1+2 = 10 (9 menus)

2+2+2+2+2 = 10 (5 menus)

2+4+4 = 10 (3 menus)

2+8 = 10 (2 menus)

Here the minimum number of menus is 2.

In the last sample, the optimal way is 2048+2048=4096 (2 menus).
Note that there is no menu whose price is 4096.
user: Sandy is a professor at a very reputed institute. The institute mandates that all the lectures be communicated in English. As Sandy is not very good at English(or anything actually) the presentations he displays in class have a lot of spelling mistakes in them.
As you have a dictionary on you containing $N$ words in it, the responsibility of correctly deducing the misspelt word falls upon your shoulders. Sandy's presentation contains in them, $Q$ words which you $think$ are wrongly spelt. A word is misspelt if a $single$ letter in it is missing or is different from the corresponding correctly spelled word in your dictionary.
For each of the misspelt word in the presentation find out the corresponding correct word from your dictionary.
Note : 
- For each misspelt word in Sandy's presentation, there exists one and only one correctly spelt word in your dictionary that corresponds to it. 
- Out of the $Q$ misspelt words given to you, there might be some which are correctly spelt i.e., that word completely matches a word in your dictionary. (Give Sandy some credit, he's a teacher after all). For such words print the word corresponding to it in your dictionary.
- The maximum length of each string can be $L$.

-----Input:-----
- First line contains a single integer $T$ denoting the number of testcases. Then the testcases follow. 
- The first line of each test case contains two space-separated integers $N, Q$ corresponding to the number of words in your dictionary and the number of misspelt word in Sandy's dictionary respectively.
- $N$ lines follow each containing a single string $S$ denoting a word in your dictionary.
- $Q$ lines follow each containing a single string $M$ denoting a misspelt word in Sandy's presentation.

-----Output:-----
In each testcase, for each of the $Q$ misspelt words, print a single word $S$ corresponding to the correctly spelt word in your dictionary.

-----Constraints-----
- $1 \leq T \leq 100$
- $1 \leq N \leq 200$
- $1 \leq Q \leq 200$
- $1 \leq L \leq 100$
- None of the words contain any spaces in them.
- Each letter in every word is in lower case.

-----Subtasks-----
- 
Subtask 1 : 10 points
- $1 \leq N \leq 50$
- $1 \leq Q \leq 10$
- 
Subtask 2 : 90 points
- Original Constraints

-----Sample Input:-----
1
5 2
szhbdvrngk
qzhxibnuec
jfsalpwfkospl
levjehdkjy
wdfhzgatuh
szhbdvcngk
qzhxbnuec

-----Sample Output:-----
szhbdvrngk
qzhxibnuec

-----EXPLANATION:-----
- In the first misspelt word $szhbdvcngk$, a single letter $c$ is different from the original word $szhbdvrngk$.
- The second misspelt word $qzhxbnuec$ is missing the letter $i$ that  was in the original word $qzhxibnuec$.