meetforever/GRPO_CL huggingface dataset:
user: You are given a sequence of integers $A_1, A_2, \ldots, A_N$. This sequence is circular ― for each valid $i$, the element $A_{i+1}$ follows after $A_i$, and the element $A_1$ follows after $A_N$.
You may insert any positive integers at any positions you choose in this sequence; let's denote the resulting sequence by $B$. This sequence is also circular. For each pair of its elements $B_s$ and $B_f$, let's denote the (non-circular) sequence created by starting at $B_s$ and moving from each element to the one that follows after it, until we reach $B_f$, by $B(s, f)$. This sequence includes the elements $B_s$ and $B_f$.
For each $K$ from $2$ to $N$ inclusive, find the smallest possible number of elements that need to be inserted into $A$ to form a sequence $B$ for which there is no subsequence $B(p, q)$ such that:
- The size of $B(p, q)$ is at least $K$.
- There is no pair of consecutive elements in $B(p, q)$ such that their GCD is equal to $1$.

-----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 $N-1$ space-separated integers. For each $i$ ($1 \le i \le N-1$), the $i$-th of these integers should be the smallest number of inserted elements in a valid sequence $B$ for $K = i+1$.

-----Constraints-----
- $1 \le T \le 2,000$
- $2 \le N \le 10^5$
- $1 \le A_i \le 10^9$ for each valid $i$
- the sum of $N$ over all test cases does not exceed $2 \cdot 10^6$

-----Example Input-----
1
5
3 6 4 5 9

-----Example Output-----
3 1 1 0
user: There are $N$ robots who work for $Y$ days and on each day they
produce some toys .on some days a few robots are given rest.
So depending on the availability of robots owner has 
made a time table which decides which robots will work on 
the particular day.
Only contiguous robots must be selected as they can form 
a link of communication among themselves.
Initially, all robots have the capacity of one toy.
On each day capacity for the chosen robot is updated i.e
capacity = capacity $+$$ ($minimum capacity of given range % $1000000007)$ .
After calculating the minimum capacity of a given range, compute
it as modulo 1000000007 ($10^9 + 7$).
After $Y$ days find the minimum capacity of the $N$ robots and compute it as modulo 1000000007 ($10^9 + 7$).   

-----Input:-----
- First-line will contain $T$, the number of test cases. Then the test cases follow. 
- Next Line contains a single integer N.
- Next Line contains a single integer Y.
- Next $Y$ lines contains l and r range of chosen robots .

-----Output:-----
For each testcase, output in a single line answer , the minimum capacity of the $N$ robots after $Y$ days and compute it as modulo 1000000007 ($10^9 + 7$) .

-----Constraints-----
- $1 \leq T \leq 100$
- $100 \leq N \leq 10^4$
- $200 \leq Y \leq 1000$
- $0<=l , r<=N-1$  , $l<=r$

-----Sample Input:-----
1
5
4
0 3
1 2
4 4
0 4

-----Sample Output:-----
4

-----EXPLANATION:-----
Initial capacity of the $5$ robots
1 1 1 1 1 
Minimum in range [0,3] = 1
Update the capacity in the range [0,3]
Now capacity becomes, 
Day 1 - 2 2 2 2 1
Similarly capacities changes for each day 
Day 2 - 2 4 4 2 1
Day 3 - 2 4 4 2 2
Day 4 - 4 6 6 4 4
so after 4 days minimum capacity is $4$.
user: Toby has found a game to entertain himself.The game is like this:

You are in a coordinate system initially at (0,0) and you are given a sequence of steps which lead to your destination.The steps are given in the form of directions: ’U’ ,’D’ , ’L’ and ‘R’ for up, down, left and right respectively.If you are at position (x,y) then:
- U:move to (x,y+1)
- D:move to (x,y-1)
- L:move to (x-1,y)
- R:move to (x+1,y)
The sequence is provided as a string ‘s’ of characters where $s_i$ $(1 \leq i \leq N)$ is one of the direction character as mentioned above.An example of a sequence of steps is:   UULRUDR

The destination according to this string is (1,2).

You want to remove maximum number of characters from the string such that the resulting string leads to the same destination as before.
For example in the example above we can remove characters at positions 1,3,4,6 and the resulting path will be UUR which will lead to the same destination i.e (1,2).so we reduced the number of steps by 4,and this is our score.
You need to get maximum score.

-----Input:-----
- First line will contain $T$, number of testcases. Then the testcases follow. 
- First line of each testcase contains a single integer $N$,size of string. 
- Second line of testcase contains a string $s$ of size $N$.

-----Output:-----
For each testcase, output a single line containing the maximum score possible.

-----Constraints-----
- $1 \leq T \leq 1000$
- $1 \leq N \leq 10^5$

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

7

ULUDLLU

4

RUUR

4

LRLR   

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

0

4    

-----EXPLANATION:-----
- 
test case 1:
The final destination after moving according to the sequence is (-3,2).

One way is to remove characters at positions 3,4 and the resulting string will be ULLLU and destination still remains (-3,2).
- 
test case 2: No character can be removed in this case.
user: Chefland has all the cities on a straight line. There are $N$ cities in Chefland numbered $1$ to $N$. City $i$ is located at coordinate $x_i$ on the x-axis. Guru wants to travel from city $A$ to city $B$. He starts at time t=0. He has following choices to travel.
- He can walk $1$ metre in $P$ secs.
- There is a train that travels from city $C$ to city $D$ which travels $1$ metre in $Q$ secs which starts at time t=$Y$ secs. Guru can take the train only at city $C$ and leave the train only at city $D$.
Can you help Guru find the minimum time he will need to travel from city $A$ to $B$. Note that you cannot board the train after time t =$Y$.

-----Input:-----
- First line will contain $T$, number of testcases. Then the testcases follow. 
- First line of each testcase contains eight space separated integers $N, A, B, C, D, P, Q, Y $. 
- Second line of each testcase contains $N$ space-separated integers with the $i$-th integer representing $x_i$.

-----Output:-----
For each testcase, output in a single line containing the minimum travel time.

-----Constraints-----
- $1 \leq T \leq 300$
- $2 \leq N \leq 300$
- $-1000 \leq x_i \leq 1000$
- $0 \leq Y \leq 100000$
- $1 \leq A,B,C,D \leq n $
- $A \neq B$
- $C \neq D$
- $1 \leq P, Q \leq 100$
- $x_i < x_j$  if $i < j$

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

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

-----EXPLANATION:-----
Guru can walk directly in 6 secs.
If Guru takes train, then he will need  atleast 11 secs.
user: Every college has a stud−max$stud-max$ buoy. JGEC$JGEC$ has its own Atul$Atul$ who loves to impress everyone with his smile. A presentation is going on at the auditorium where there are N$N$ rows of M$M$ chairs with people sitting on it. Everyone votes for Atul's presentation skills, but Atul is interested in knowing the maximum$maximum$ amount of vote that he receives either taking K$K$ people vertically$vertically$ or horizontally$horizontally$. Atul$Atul$, however, wants to finish up the presentation party soon, hence asks for your help so that he can wrap up things faster.  

-----Input:-----
- First line will contain T$T$, number of test cases. Then the test cases follow. 
- Each testcase contains of a single line of input, three integers N$N$, M$M$ and K$K$.  
- N lines follow, where every line contains M numbers$numbers$ denoting the size of the sugar cube

-----Output:-----
For each test case, output in a single line the maximum votes Atul can get. 

-----Constraints-----
- 1≤T≤$1 \leq T \leq $5
- 1≤N,M≤500$1 \leq N,M \leq 500$
- K≤min(N,M)$K \leq min(N,M)$ 
- 1≤numbers≤109$1 \leq numbers \leq 10^9$

-----Sample Input:-----
1
4 4 3 
1 4 5 7 
2 3 8 6
1 4 8 9 
5 1 5 6 

-----Sample Output:-----
22

-----EXPLANATION:-----
If Atul starts counting votes from (1,4), then he can take the 3 consecutive elements vertically downwards and those are 7,6 and 9 which is the maximum sum possible.
user: Chef loves saving money and he trusts none other bank than State Bank of Chefland. Unsurprisingly, the employees like giving a hard time to their customers. But instead of asking them to stand them in long queues, they have weird way of accepting money.
Chef did his homework and found that the bank only accepts the money in coins such that the sum of the denomination with any previously deposited coin OR itself can't be obtained by summing any two coins OR double of any coin deposited before. Considering it all, he decided to start with $1$ Chefland rupee and he would keep choosing smallest possible denominations upto $N$ coins. Since chef is busy with his cooking, can you find out the $N$ denomination of coins chef would have to take to the bank? Also find the total sum of money of those $N$ coins.

-----Input:-----
- First line has a single integer $T$ i.e. number of testcases.
- $T$ lines followed, would have a single integer $N$ i.e. the number of coins the chef is taking.

-----Output:-----
- Output for $i$-th testcase ($1 ≤ i ≤ T$) would have 2 lines.
- First line would contain $N$ integers i.e. the denomination of coins chef would deposit to the bank.
- Second line would contain a single integer i.e. the sum of all the coins chef would deposit.

-----Constraints:-----
- $1 ≤ T ≤ 700$
- $1 ≤ N ≤ 700$

-----Subtasks:-----
- $20$ points: $1 ≤ T, N ≤ 80$
- $70$ points: $1 ≤ T, N ≤ 500$
- $10$ points: $1 ≤ T, N ≤ 700$ 

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

-----Sample Output:-----
1
1
1 2
3
1 2 4
7
1 2 4 8
15

-----Explanation:-----
For testcase 1: First coin is stated to be 1, hence for $N$ = 1, 1 is the answer.
For testcase 2: Since chef chooses the lowest possible denomination for each $i$-th coin upto $N$ coins, second coin would be 2. Only sum possible with N = 1 would be 1+1 = 2. For N = 2, $\{1+2,  2+2\}$ $\neq$ $2$.
For testcase 3: With first two coins being 1 and 2, next coin couldn't be 3 because 3+1 = 2+2, but $\{4+1,  4+2,  4+4\}$ $\neq$ $\{1+1,  1+2,  2+2\}$
user: Chef recently took a course in linear algebra and learned about linear combinations of vectors. Therefore, in order to test his intelligence, Raj gave him a "fuzzy" problem to solve.
A sequence of integers $B_1, B_2, \ldots, B_M$ generates an integer $K$ if it is possible to find a sequence of integers $C_1, C_2, \ldots, C_M$ such that $C_1 \cdot B_1 + C_2 \cdot B_2 + \ldots + C_M \cdot B_M = K$.
In this problem, Chef has a sequence $A_1, A_2, \ldots, A_N$ and he should answer $Q$ queries. In each query, he is given an integer $K$; the answer to this query is the number of pairs $(l, r)$ such that $1 \le l \le r \le N$ and the subsequence $(A_l, A_{l+1}, \ldots, A_r)$ generates $K$.
Chef has no idea how to solve this problem ― can you help him find the answers to all queries?

-----Input-----
- The first line of the input contains a single integer $N$.
- The second line contains $N$ space-separated integers $A_1, A_2, \ldots, A_N$.
- The third line contains a single integer $Q$.
- The following $Q$ lines describe queries. Each of these lines contains a single integer $K$.

-----Output-----
For each query, print a single line containing one integer ― the number of contiguous subsequences that generate $K$.

-----Constraints-----
- $1 \le N, Q \le 10^5$
- $1 \le A_i \le 10^9$ for each valid $i$
- $1 \le K \le 10^6$

-----Subtasks-----
Subtask #1 (10 points): $1 \le N \le 1,000$
Subtask #2 (90 points): original constraints

-----Example Input-----
2
2 4
3
1
2
8

-----Example Output-----
0
2
3

-----Explanation-----
The sequence $(2, 4)$ has three contiguous subsequences: $(2)$, $(4)$ and $(2, 4)$. 
- In the first query, $1$ cannot be generated by any subsequence.
- In the second query, $2$ is generated by subsequences $(2)$ and $(2, 4)$. For example, for the subsequence $(2, 4)$, we can choose $C_1 = 1$ and $C_2 = 0$.
- In the third query, $8$ is generated by all three subsequences.
user: In a country called Chef Land, there was a lot of monetary fraud, so Chefu, the head of the country, decided to choose new denominations of the local currency ― all even-valued coins up to an integer $N$ should exist. After a few days, a citizen complained that there was no way to create an odd value, so Chefu decided that he should also introduce coins with value $1$. Formally, you are given an integer $N$; for $v = 1$ and each even positive integer $v \le N$, coins with value $v$ exist.
You are also given an integer $S$. To handle transactions quickly, find the minimum number of coins needed to pay a price $S$.

-----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 $S$ and $N$.

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

-----Constraints-----
- $1 \le T \le 10,000$
- $1 \le S \le 10^9$
- $2 \le N \le 10^9$
- $N$ is even

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

-----Example Input-----
4
2 2
1 14
30 10
31 4

-----Example Output-----
1
1
3
9

-----Explanation-----
Example case 1: One coin with value $2$ is sufficient.
Example case 2: We need to use one coin with value $1$.
Example case 3: We need $3$ coins, each with value $10$.
Example case 4: We can use seven coins with value $4$, one coin with value $2$ and one coin with value $1$.
user: Two sisters, A and B, play the piano every day. During the day, they can play in any order. That is, A might play first and then B, or it could be B first and then A. But each one of them plays the piano exactly once per day. They maintain a common log, in which they write their name whenever they play.
You are given the entries of the log, but you're not sure if it has been tampered with or not. Your task is to figure out whether these entries could be valid or not.

-----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 a string $s$ denoting the entries of the log.

-----Output-----
- For each test case, output yes or no according to the answer to the problem.

-----Constraints-----
- $1 \le T \le 500$
- $2 \le |s| \le 100$
- $|s|$ is even
- Each character of $s$ is either 'A' or 'B'

-----Example Input-----
4
AB
ABBA
ABAABB
AA

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

-----Explanation-----
Testcase 1: There is only one day, and both A and B have played exactly once. So this is a valid log. Hence 'yes'.
Testcase 2: On the first day, A has played before B, and on the second day, B has played first. Hence, this is also a valid log.
Testcase 3: On the first day, A played before B, but on the second day, A seems to have played twice. This cannot happen, and hence this is 'no'.
user: Mark loves eating chocolates and also likes to be fit. Given the calorie count for every chocolate he eats, find what he has to do to burn the calories. 
The name of the chocolates along with its calorie count are given as follows:
Calories per one whole bar:
Dairy milk (D)
238
Twix (T)
244
Milky Bar (M)
138
Bounty (B)
279
Crunchie (C)
186
The exercises preferred by him and the calories burnt are as follows:
Calories burnt per km:
Running
50
Cycling
5
Walking
0.5
Find the number of kilometers he has to run, cycle or walk to burn all of the calories. 
Priority given to the exercises is as follows: 
Running > Cycling > Walking

-----Input:-----
- It is a one line string consisting of the names(initial letter) of all the chocolates he has eaten.

-----Output:-----
Print three lines. First line representing how much he ran(in km), second line representing how much he cycled(in km), third line representing how much he walked(in km).

-----Constraints-----
- 1 <= length of input string <= 10. 

-----Sample Input:-----
DDTM

-----Sample Output:-----
17
1
6

-----EXPLANATION:-----
Calorie intake = 238 + 238 + 244 + 138 = 858
( 17km x 50 ) + ( 1km x 5 ) + ( 6km x 0.5 ) = 858.
user: Given a set of  N  natural numbers 1,2,3........N and Q query.For each query you have to calculate the total number of subset in which  Ith.
number of set come at Kth postion.Elements of every subset should be in sorted order.
The answer could  be very large so you have to print answer modulo 1e9+7.


-----Input:-----
- The first line of input cotains a single integer T denoting the number of test cases.
- For every test case it contains two number N and Q.
- Next Q line contains two number I and K.

-----Output:-----
For each test case print required answer.

-----Constraints and Subtasks:-----
- 1<=T<=5
- 1<=N, K<=4000
- 1<=Q<=1000000
Subtask 3: 5 points
- 1<=T<=5
- 1<=N, K<=16
- 1<=Q<=1000
Subtask 1: 25 points
- T=1
- 1<=N, K<=4000
- 1<=Q<=100000
Subtask 2: 70 points
- Original Constraints.

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

1

3 3

1 2

2 1

3 2

Output:

0

2

2

-----Explanation:-----
For N=3

total subsets are:

{1}

{2}

{3}

{1,2}

{1,3}

{2,3}

{1,2,3}

Now we can see that for I=1 and K=2 there is no subset in which 1 come at 2nd position so the answer is Zero for that query.

For 2nd query I=2 and K=1 there are two subset i.e {2,3} and {2} in which 2 come at 1st position.

Same for 3rd querry there is two subset i.e{1,3} and {2,3}.
user: You are given two strings $S$ and $R$. Each of these strings has length $N$. We want to make $S$ equal to $R$ by performing the following operation some number of times (possibly zero):
- Choose two integers $a$ and $b$ such that $1 \le a \le b \le N$.
- For each $i$ such that $a \le i \le b$, replace the $i$-th character of $S$ by the $i$-th character of $R$.
Suppose that we make $S$ equal to $R$ by performing this operation $k$ times, in such a way that the total number of replaced characters (i.e. the sum of all $k$ values of $b-a+1$) is $l$. Then, the cost of this process is defined as $k \cdot l$.
Find the minimum cost with which we can make $S$ equal to $R$.

-----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 string $S$.
- The second line contains a single string $R$.

-----Output-----
For each test case, print a single line containing one integer ― the minimum cost.

-----Constraints-----
- $1 \le T \le 4,000$
- $1 \le N \le 10^6$
- $|S| = |R| = N$
- $S$ and $R$ contain only lowercase English letters
- the sum of $N$ over all test cases does not exceed $2 \cdot 10^6$

-----Example Input-----
1
adefb
bdefa

-----Example Output-----
4

-----Explanation-----
Example case 1: $S$ can be made equal to $R$ in two moves. First, we replace $S_1$ by $R_1$ and then replace $S_5$ by $R_5$. We have $k = l = 2$, so the cost is $2 \cdot 2 = 4$. If we wanted to perform only one operation, the cost would be $5$.
user: Do you know Professor Saeed? He is the algorithms professor at Damascus University. Yesterday, he gave his students hard homework (he is known for being so evil) - for a given binary string $S$, they should compute the sum of $F(S, L, R)$ over all pairs of integers $(L, R)$ ($1 \le L \le R \le |S|$), where the function $F(S, L, R)$ is defined as follows:
- Create a string $U$: first, set $U = S$, and for each $i$ ($L \le i \le R$), flip the $i$-th character of $U$ (change '1' to '0' or '0' to '1').
- Then, $F(S, L, R)$ is the number of valid pairs $(i, i + 1)$ such that $U_i = U_{i+1}$.
As usual, Professor Saeed will give more points to efficient solutions. Please help the students solve this homework.

-----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 string $S$.

-----Output-----
For each test case, print a single line containing one integer $\sum_{1 \le L \le R \le |S|} F(S, L, R)$.

-----Constraints-----
- $1 \le T \le 100$
- $1 \le |S| \le 3 \cdot 10^6$
- the sum of $|S|$ over all test cases does not exceed $6 \cdot 10^6$

-----Subtasks-----
Subtask #1 (50 points):
- $1 \le |S| \le 300$
- the sum of $|S|$ over all test cases does not exceed $600$
Subtask #2 (50 points): original constraints

-----Example Input-----
1
001

-----Example Output-----
6

-----Explanation-----
Example case 1:
- $L = 1, R = 1$: $U$ is "101", $F = 0$
- $L = 2, R = 2$: $U$ is "011", $F = 1$
- $L = 3, R = 3$: $U$ is "000", $F = 2$
- $L = 1, R = 2$: $U$ is "111", $F = 2$
- $L = 2, R = 3$: $U$ is "010", $F = 0$
- $L = 1, R = 3$: $U$ is "110", $F = 1$
user: As you might remember, the collector of Siruseri had ordered
a complete revision of the Voters List. He knew that constructing
the list of voters is a difficult task, prone to errors. Some
voters may have been away on vacation, others may have moved
during the enrollment and so on. 
To be as accurate as possible, he entrusted the task to three different 
officials. Each of them was to independently record the list of voters and 
send it to the collector. In Siruseri, every one has a ID number and
the list would only list the ID numbers of the voters and not their names.
The officials were expected to arrange the ID numbers in ascending order
in their lists. 
On receiving the lists, the Collector realised that there were
discrepancies - the three lists were not identical.  He decided
to go with the majority. That is, he decided to construct the
final list including only those ID numbers that appeared in at
least 2 out of the 3 lists.  For example if the three lists
were
23  30  42  57  90
21  23  35  57  90  92
21  23  30  57  90 

then the final list compiled by the collector would be:
21  23  30  57  90

The ID numbers 35, 42 and 92 which appeared in only one list
each do not figure in the final list.
Your task is to help the collector by writing a program that
produces the final list from the three given lists.
Input format
The first line of the input contains 3 integers
N1, N2 and
N3.  N1 is the number of
voters in the first list, N2 is the number of
voters in the second list and N3 is the number of
voters in the third list.  The next N1 lines
(lines 2,...,N1+1) contain one positive integer
each and describe the first list in ascending order.  The following

N2 lines (lines
N1+2,...,N1+N2+1)
describe the second list in ascending order and the final
N3 lines (lines

N1+N2+2,...,N1+N2+N3+1)
describe the third list in ascending order.
Output format
The first line of the output should contain a single integer
M indicating the number voters in the final list. The next
M lines (lines 2,...,M+1) should contain one
positive integer each, describing the list of voters in the final
list, in ascending order.
Test data
You may assume that 1 ≤
N1,N2,N3
≤ 50000.  You may also assume that in 50% of the inputs 1 ≤ N1,N2,N3 ≤ 2000.

Example
Sample input:
5 6 5
23
30
42
57
90
21 
23 
35 
57 
90 
92 
21 
23 
30 
57 
90 

Sample output:
5
21 
23 
30 
57 
90