alexjercan/bugnet huggingface dataset:
Score : 700 points

Problem Statement

There are N panels arranged in a row in Takahashi's house, numbered 1 through N. The i-th panel has a number A_i written on it. Takahashi is playing by throwing balls at these panels.

Takahashi threw a ball K times. Let the panel hit by a boll in the i-th throw be panel p_i. He set the score for the i-th throw as i \times A_{p_i}.

He was about to calculate the total score for his throws, when he realized that he forgot the panels hit by balls, p_1,p_2,...,p_K. The only fact he remembers is that for every i (1 ≦ i ≦ K-1), 1 ≦ p_{i+1}-p_i ≦ M holds. Based on this fact, find the maximum possible total score for his throws.

Constraints

1 ≦ M ≦ N ≦ 100,000

1 ≦ K ≦ min(300,N)

1 ≦ A_i ≦ 10^{9}

Partial Scores

In the test set worth 100 points, M = N.

In the test set worth another 200 points, N ≦ 300 and K ≦ 30.

In the test set worth another 300 points, K ≦ 30.

Input

The input is given from Standard Input in the following format:

N M K
A_1 A_2 … A_N

Output

Print the maximum possible total score for Takahashi's throws.

Sample Input 1

5 2 3
10 2 8 10 2

Sample Output 1

56

The total score is maximized when panels 1,3 and 4 are hit, in this order.

Sample Input 2

5 5 2
5 2 10 5 9

Sample Output 2

28

This case satisfies the additional constraint M = N for a partial score.

Sample Input 3

10 3 5
3 7 2 6 9 4 8 5 1 1000000000

Sample Output 3

5000000078
Score : 1400 points

Problem Statement

Takahashi found an integer sequence (A_1,A_2,...,A_N) with N terms. Since it was too heavy to carry, he decided to compress it into a single integer.

The compression takes place in N-1 steps, each of which shorten the length of the sequence by 1. Let S be a string describing the steps, and the sequence on which the i-th step is performed be (a_1,a_2,...,a_K), then the i-th step will be as follows:

When the i-th character in S is M, let b_i = max(a_i,a_{i+1}) (1 ≦ i ≦ K-1), and replace the current sequence by (b_1,b_2,...,b_{K-1}).

When the i-th character in S is m, let b_i = min(a_i,a_{i+1}) (1 ≦ i ≦ K-1), and replace the current sequence by (b_1,b_2,...,b_{K-1}).

Takahashi decided the steps S, but he is too tired to carry out the compression. On behalf of him, find the single integer obtained from the compression.

Constraints

2 ≦ N ≦ 10^5

1 ≦ A_i ≦ N

S consists of N-1 characters.

Each character in S is either M or m.

Partial Scores

In the test set worth 400 points, there exists i (1 ≦ i < N-1) such that the first i characters in S are M, and the remaining characters are m, that is, S is in the form M...Mm...m.

In the test set worth 800 points, the odd-number-th characters in S are M, and the even-number-th characters are m, that is, S is in the form MmMmMm....

Input

The input is given from Standard Input in the following format:

N
A_1 A_2 … A_N
S

Output

Print the single integer obtained from the compression.

Sample Input 1

4
1 2 3 4
MmM

Sample Output 1

3

The initial sequence ( 1 , 2 , 3 , 4 ) is compressed as follows:

After the first step, the sequence is ( 2 , 3 , 4 ).

After the second step, the sequence is ( 2 , 3 ).

After the third step, the sequence is ( 3 ).

Thus, the single integer obtained from the compression is 3.

Sample Input 2

5
3 4 2 2 1
MMmm

Sample Output 2

2

Sample Input 3

10
1 8 7 6 8 5 2 2 6 1
MmmmmMMMm

Sample Output 3

5

Sample Input 4

20
12 7 16 8 7 19 8 19 20 11 7 13 20 3 4 11 19 11 15 5
mMMmmmMMMMMMmMmmmMM

Sample Output 4

11
Score : 700 points

Problem Statement

Aoki is in search of Takahashi, who is missing in a one-dimentional world.
Initially, the coordinate of Aoki is 0, and the coordinate of Takahashi is known to be x, but his coordinate afterwards cannot be known to Aoki.

Time is divided into turns. In each turn, Aoki and Takahashi take the following actions simultaneously:

Let the current coordinate of Aoki be a, then Aoki moves to a coordinate he selects from a-1, a and a+1.

Let the current coordinate of Takahashi be b, then Takahashi moves to the coordinate b-1 with probability of p percent, and moves to the coordinate b+1 with probability of 100-p percent.

When the coordinates of Aoki and Takahashi coincide, Aoki can find Takahashi.
When they pass by each other, Aoki cannot find Takahashi.

Aoki wants to minimize the expected number of turns taken until he finds Takahashi. Find the minimum possible expected number of turns.

Constraints

1 ≦ x ≦ 1,000,000,000

1 ≦ p ≦ 100

x and p are integers.

Partial Scores

In the test set worth 200 points, p=100.

In the test set worth 300 points, x ≦ 10.

Input

The input is given from Standard Input in the following format:

x
p

Output

Print the minimum possible expected number of turns.
The output is considered correct if the absolute or relative error is at most 10^{-6}.

Sample Input 1

3
100

Sample Output 1

2.0000000

Takahashi always moves by -1.
Thus, by moving to the coordinate 1 in the 1-st turn and staying at that position in the 2-nd turn, Aoki can find Takahashi in 2 turns.

Sample Input 2

6
40

Sample Output 2

7.5000000

Sample Input 3

101
80

Sample Output 3

63.7500000
Score : 1200 points

Problem Statement

Takahashi is a magician. He can cast a spell on an integer sequence (a_1,a_2,...,a_M) with M terms, to turn it into another sequence (s_1,s_2,...,s_M), where s_i is the sum of the first i terms in the original sequence.

One day, he received N integer sequences, each with M terms, and named those sequences A_1,A_2,...,A_N. He will try to cast the spell on those sequences so that A_1 < A_2 < ... < A_N will hold, where sequences are compared lexicographically.
Let the action of casting the spell on a selected sequence be one cast of the spell. Find the minimum number of casts of the spell he needs to perform in order to achieve his objective.

Here, for two sequences a = (a_1,a_2,...,a_M), b = (b_1,b_2,...,b_M) with M terms each, a < b holds lexicographically if and only if there exists i (1 ≦ i ≦ M) such that a_j = b_j (1 ≦ j < i) and a_i < b_i.

Constraints

1 ≦ N ≦ 10^3

1 ≦ M ≦ 10^3

Let the j-th term in A_i be A_{(i,j)}, then 1 ≦ A_{(i,j)} ≦ 10^9.

Partial Scores

In the test set worth 200 points, Takahashi can achieve his objective by at most 10^4 casts of the spell, while keeping the values of all terms at most 10^9.

In the test set worth another 800 points, A_{(i,j)} ≦ 80.

Input

The input is given from Standard Input in the following format:

N M
A_{(1,1)} A_{(1,2)} … A_{(1,M)}
A_{(2,1)} A_{(2,2)} … A_{(2,M)}
:
A_{(N,1)} A_{(N,2)} … A_{(N,M)}

Output

Print the minimum number of casts of the spell Takahashi needs to perform. If he cannot achieve his objective, print -1 instead.

Sample Input 1

3 3
2 3 1
2 1 2
2 6 3

Sample Output 1

1

Takahashi can achieve his objective by casting the spell on A_2 once to turn it into (2 , 3 , 5).

Sample Input 2

3 3
3 2 10
10 5 4
9 1 9

Sample Output 2

-1

In this case, Takahashi cannot achieve his objective by casting the spell any number of times.

Sample Input 3

5 5
2 6 5 6 9
2 6 4 9 10
2 6 8 6 7
2 1 7 3 8
2 1 4 8 3

Sample Output 3

11
Score : 700 points

Problem Statement

Takahashi found an undirected connected graph with N vertices and M edges. The vertices are numbered 1 through N. The i-th edge connects vertices a_i and b_i, and has a weight of c_i.

He will play Q rounds of a game using this graph. In the i-th round, two vertices S_i and T_i are specified, and he will choose a subset of the edges such that any vertex can be reached from at least one of the vertices S_i or T_i by traversing chosen edges.

For each round, find the minimum possible total weight of the edges chosen by Takahashi.

Constraints

1 ≦ N ≦ 4,000

1 ≦ M ≦ 400,000

1 ≦ Q ≦ 100,000

1 ≦ a_i,b_i,S_i,T_i ≦ N

1 ≦ c_i ≦ 10^{9}

a_i \neq b_i

S_i \neq T_i

The given graph is connected.

Partial Scores

In the test set worth 200 points, Q = 1.

In the test set worth another 300 points, Q ≦ 3000.

Input

The input is given from Standard Input in the following format:

N M
a_1 b_1 c_1
a_2 b_2 c_2
:
a_M b_M c_M
Q
S_1 T_1
S_2 T_2
:
S_Q T_Q

Output

Print Q lines. The i-th line should contain the minimum possible total weight of the edges chosen by Takahashi.

Sample Input 1

4 3
1 2 3
2 3 4
3 4 5
2
2 3
1 4

Sample Output 1

8
7

We will examine each round:

In the 1-st round, choosing edges 1 and 3 achieves the minimum total weight of 8.

In the 2-nd round, choosing edges 1 and 2 achieves the minimum total weight of 7.

Sample Input 2

4 6
1 3 5
4 1 10
2 4 6
3 2 2
3 4 5
2 1 3
1
2 3

Sample Output 2

8

This input satisfies the additional constraints for both partial scores.
Score : 1000 points

Problem Statement

You are given a string S consisting of digits between 1 and 9, inclusive.
You will insert at most K commas (,) into this string to separate it into multiple numbers.

Your task is to minimize the maximum number among those produced by inserting commas. Find minimum possible such value.

Constraints

0 ≦ K < |S| ≦ 100,000

S consists of digits between 1 and 9, inclusive.

Partial Scores

In the test set worth 100 points, |S| ≦ 2.

In the test set worth another 100 points, |S| ≦ 16.

In the test set worth another 200 points, |S| ≦ 100.

In the test set worth another 200 points, |S| ≦ 2,000.

Input

The input is given from Standard Input in the following format:

K
S

Output

Print the minimum possible value.

Sample Input 1

2
15267315

Sample Output 1

315

When the string is separated into 152, 67 and 315, the maximum among these is 315, which is the minimum possible value.

Sample Input 2

0
12456174517653111

Sample Output 2

12456174517653111

12456174517653111 itself is the answer.

Sample Input 3

8
127356176351764127645176543176531763517635176531278461856198765816581726586715987216581

Sample Output 3

5317635176
Score : 1500 points

Problem Statement

There is an undirected connected graph with N vertices numbered 1 through N.
The lengths of all edges in this graph are 1.
It is known that for each i (1≦i≦N), the distance between vertex 1 and vertex i is A_i, and the distance between vertex 2 and vertex i is B_i.
Determine whether there exists such a graph.
If it exists, find the minimum possible number of edges in it.

Constraints

2≦N≦10^5

0≦A_i,B_i≦N-1

Input

The input is given from Standard Input in the following format:

N
A_1 B_1
A_2 B_2
:
A_N B_N

Output

If there exists a graph satisfying the conditions, print the minimum possible number of edges in such a graph. Otherwise, print -1.

Sample Input 1

4
0 1
1 0
1 1
2 1

Sample Output 1

4

The figure below shows two possible graphs. The graph on the right has fewer edges.

Sample Input 2

3
0 1
1 0
2 2

Sample Output 2

-1

Such a graph does not exist.
Score : 1500 points

Problem Statement

The currency used in Takahashi Kingdom is Myon.
There are 1-, 10-, 100-, 1000- and 10000-Myon coins, and so forth. Formally, there are 10^n-Myon coins for any non-negative integer n.

There are N items being sold at Ex Store.
The price of the i-th (1≦i≦N) item is A_i Myon.

Takahashi is going to buy some, at least one, possibly all, of these N items.
He hates receiving change, so he wants to bring coins to the store so that he can pay the total price without receiving change, no matter what items he chooses to buy.
Also, since coins are heavy, he wants to bring as few coins as possible.

Find the minimum number of coins he must bring to the store. It can be assumed that he has an infinite supply of coins.

Constraints

1≦N≦20,000

1≦A_i≦10^{12}

Input

The input is given from Standard Input in the following format:

N
A_1 A_2 ... A_N

Output

Print the minimum number of coins Takahashi must bring to the store, so that he can pay the total price without receiving change, no matter what items he chooses to buy.

Sample Input 1

3
43 24 37

Sample Output 1

16

There are seven possible total prices: 24, 37, 43, 61, 67, 80, and 104.
With seven 1-Myon coins, eight 10-Myon coins and one 100-Myon coin, Takahashi can pay any of these without receiving change.

Sample Input 2

5
49735011221 970534221705 411566391637 760836201000 563515091165

Sample Output 2

105
Score : 100 points

Problem Statement

There is a grid with H rows and W columns.

The square at the i-th row and j-th column contains a string S_{i,j} of length 5.

The rows are labeled with the numbers from 1 through H, and the columns are labeled with the uppercase English letters from A through the W-th letter of the alphabet.

Exactly one of the squares in the grid contains the string snuke. Find this square and report its location.

For example, the square at the 6-th row and 8-th column should be reported as H6.

Constraints

1≦H, W≦26

The length of S_{i,j} is 5.

S_{i,j} consists of lowercase English letters (a-z).

Exactly one of the given strings is equal to snuke.

Input

The input is given from Standard Input in the following format:

H W
S_{1,1} S_{1,2} ... S_{1,W}
S_{2,1} S_{2,2} ... S_{2,W}
:
S_{H,1} S_{H,2} ... S_{H,W}

Output

Print the labels of the row and the column of the square containing the string snuke, with no space inbetween.

Sample Input 1

15 10
snake snake snake snake snake snake snake snake snake snake
snake snake snake snake snake snake snake snake snake snake
snake snake snake snake snake snake snake snake snake snake
snake snake snake snake snake snake snake snake snake snake
snake snake snake snake snake snake snake snake snake snake
snake snake snake snake snake snake snake snuke snake snake
snake snake snake snake snake snake snake snake snake snake
snake snake snake snake snake snake snake snake snake snake
snake snake snake snake snake snake snake snake snake snake
snake snake snake snake snake snake snake snake snake snake
snake snake snake snake snake snake snake snake snake snake
snake snake snake snake snake snake snake snake snake snake
snake snake snake snake snake snake snake snake snake snake
snake snake snake snake snake snake snake snake snake snake
snake snake snake snake snake snake snake snake snake snake

Sample Output 1

H6

Sample Input 2

1 1
snuke

Sample Output 2

A1
Score : 300 points

Problem Statement

The problem set at CODE FESTIVAL 20XX Finals consists of N problems.

The score allocated to the i-th (1≦i≦N) problem is i points.

Takahashi, a contestant, is trying to score exactly N points. For that, he is deciding which problems to solve.

As problems with higher scores are harder, he wants to minimize the highest score of a problem among the ones solved by him.

Determine the set of problems that should be solved.

Constraints

1≦N≦10^7

Partial Score

200 points will be awarded for passing the test set satisfying 1≦N≦1000.

Additional 100 points will be awarded for passing the test set without additional constraints.

Input

The input is given from Standard Input in the following format:

N

Output

Among the sets of problems with the total score of N, find a set in which the highest score of a problem is minimum, then print the indices of the problems in the set in any order, one per line.

If there exists more than one such set, any of them will be accepted.

Sample Input 1

4

Sample Output 1

1
3

Solving only the 4-th problem will also result in the total score of 4 points, but solving the 1-st and 3-rd problems will lower the highest score of a solved problem.

Sample Input 2

7

Sample Output 2

1
2
4

The set \{3,4\} will also be accepted.

Sample Input 3

1

Sample Output 3

1
Score : 400 points

Problem Statement

On a planet far, far away, M languages are spoken. They are conveniently numbered 1 through M.

For CODE FESTIVAL 20XX held on this planet, N participants gathered from all over the planet.

The i-th (1≦i≦N) participant can speak K_i languages numbered L_{i,1}, L_{i,2}, ..., L_{i,{}K_i}.

Two participants A and B can communicate with each other if and only if one of the following conditions is satisfied:

There exists a language that both A and B can speak.

There exists a participant X that both A and B can communicate with.

Determine whether all N participants can communicate with all other participants.

Constraints

2≦N≦10^5

1≦M≦10^5

1≦K_i≦M

(The sum of all K_i)≦10^5

1≦L_{i,j}≦M

L_{i,1}, L_{i,2}, ..., L_{i,{}K_i} are pairwise distinct.

Partial Score

200 points will be awarded for passing the test set satisfying the following: N≦1000, M≦1000 and (The sum of all K_i)≦1000.

Additional 200 points will be awarded for passing the test set without additional constraints.

Input

The input is given from Standard Input in the following format:

N M
K_1 L_{1,1} L_{1,2} ... L_{1,{}K_1}
K_2 L_{2,1} L_{2,2} ... L_{2,{}K_2}
:
K_N L_{N,1} L_{N,2} ... L_{N,{}K_N}

Output

If all N participants can communicate with all other participants, print YES. Otherwise, print NO.

Sample Input 1

4 6
3 1 2 3
2 4 2
2 4 6
1 6

Sample Output 1

YES

Any two participants can communicate with each other, as follows:

Participants 1 and 2: both can speak language 2.

Participants 2 and 3: both can speak language 4.

Participants 1 and 3: both can communicate with participant 2.

Participants 3 and 4: both can speak language 6.

Participants 2 and 4: both can communicate with participant 3.

Participants 1 and 4: both can communicate with participant 2.

Note that there can be languages spoken by no participant.

Sample Input 2

4 4
2 1 2
2 1 2
1 3
2 4 3

Sample Output 2

NO

For example, participants 1 and 3 cannot communicate with each other.
Score : 700 points

Problem Statement

Takahashi is playing with N cards.

The i-th card has an integer X_i on it.

Takahashi is trying to create as many pairs of cards as possible satisfying one of the following conditions:

The integers on the two cards are the same.

The sum of the integers on the two cards is a multiple of M.

Find the maximum number of pairs that can be created.

Note that a card cannot be used in more than one pair.

Constraints

2≦N≦10^5

1≦M≦10^5

1≦X_i≦10^5

Input

The input is given from Standard Input in the following format:

N M
X_1 X_2 ... X_N

Output

Print the maximum number of pairs that can be created.

Sample Input 1

7 5
3 1 4 1 5 9 2

Sample Output 1

3

Three pairs (3,2), (1,4) and (1,9) can be created.

It is possible to create pairs (3,2) and (1,1), but the number of pairs is not maximized with this.

Sample Input 2

15 10
1 5 6 10 11 11 11 20 21 25 25 26 99 99 99

Sample Output 2

6
Score : 1000 points

Problem Statement

Rng is baking cookies.

Initially, he can bake one cookie per second.

He can also eat the cookies baked by himself.
When there are x cookies not yet eaten, he can choose to eat all those cookies. After he finishes eating those cookies, the number of cookies he can bake per second becomes x. Note that a cookie always needs to be baked for 1 second, that is, he cannot bake a cookie in 1/x seconds when x > 1.
When he choose to eat the cookies, he must eat all of them; he cannot choose to eat only part of them.
It takes him A seconds to eat the cookies regardless of how many, during which no cookies can be baked.

He wants to give N cookies to Grandma.
Find the shortest time needed to produce at least N cookies not yet eaten.

Constraints

1≦N≦10^{12}

0≦A≦10^{12}

A is an integer.

Partial Score

500 points will be awarded for passing the test set satisfying N≦10^6 and A≦10^6.

Additional 500 points will be awarded for passing the test set without additional constraints.

Input

The input is given from Standard Input in the following format:

N A

Output

Print the shortest time needed to produce at least N cookies not yet eaten.

Sample Input 1

8 1

Sample Output 1

7

It is possible to produce 8 cookies in 7 seconds, as follows:

After 1 second: 1 cookie is done.

After 2 seconds: 1 more cookie is done, totaling 2. Now, Rng starts eating those 2 cookies.

After 3 seconds: He finishes eating the cookies, and he can now bake 2 cookies per second.

After 4 seconds: 2 cookies are done.

After 5 seconds: 2 more cookies are done, totaling 4.

After 6 seconds: 2 more cookies are done, totaling 6.

After 7 seconds: 2 more cookies are done, totaling 8.

Sample Input 2

1000000000000 1000000000000

Sample Output 2

1000000000000
Score : 1000 points

Problem Statement

There are N towns in Takahashi Kingdom. They are conveniently numbered 1 through N.

Takahashi the king is planning to go on a tour of inspection for M days. He will determine a sequence of towns c, and visit town c_i on the i-th day. That is, on the i-th day, he will travel from his current location to town c_i. If he is already at town c_i, he will stay at that town. His location just before the beginning of the tour is town 1, the capital. The tour ends at town c_M, without getting back to the capital.

The problem is that there is no paved road in this kingdom. He decided to resolve this issue by paving the road himself while traveling. When he travels from town a to town b, there will be a newly paved one-way road from town a to town b.

Since he cares for his people, he wants the following condition to be satisfied after his tour is over: "it is possible to travel from any town to any other town by traversing roads paved by him". How many sequences of towns c satisfy this condition?

Constraints

2≦N≦300

1≦M≦300

Input

The input is given from Standard Input in the following format:

N M

Output

Print the number of sequences of towns satisfying the condition, modulo 1000000007 (=10^9+7).

Sample Input 1

3 3

Sample Output 1

2

As shown below, the condition is satisfied only when c = (2,3,1) or c = (3,2,1). Sequences such as c = (2,3,2), c = (2,1,3), c = (1,2,2) do not satisfy the condition.

Sample Input 2

150 300

Sample Output 2

734286322

Sample Input 3

300 150

Sample Output 3

0
Score : 1300 points

Problem Statement

We have a graph with N vertices, numbered 0 through N-1. Edges are yet to be added.

We will process Q queries to add edges.
In the i-th (1≦i≦Q) query, three integers A_i, B_i and C_i will be given, and we will add infinitely many edges to the graph as follows:

The two vertices numbered A_i and B_i will be connected by an edge with a weight of C_i.

The two vertices numbered B_i and A_i+1 will be connected by an edge with a weight of C_i+1.

The two vertices numbered A_i+1 and B_i+1 will be connected by an edge with a weight of C_i+2.

The two vertices numbered B_i+1 and A_i+2 will be connected by an edge with a weight of C_i+3.

The two vertices numbered A_i+2 and B_i+2 will be connected by an edge with a weight of C_i+4.

The two vertices numbered B_i+2 and A_i+3 will be connected by an edge with a weight of C_i+5.

The two vertices numbered A_i+3 and B_i+3 will be connected by an edge with a weight of C_i+6.

...

Here, consider the indices of the vertices modulo N.
For example, the vertice numbered N is the one numbered 0, and the vertice numbered 2N-1 is the one numbered N-1.

The figure below shows the first seven edges added when N=16, A_i=7, B_i=14, C_i=1:

After processing all the queries, find the total weight of the edges contained in a minimum spanning tree of the graph.

Constraints

2≦N≦200,000

1≦Q≦200,000

0≦A_i,B_i≦N-1

1≦C_i≦10^9

Input

The input is given from Standard Input in the following format:

N Q
A_1 B_1 C_1
A_2 B_2 C_2
:
A_Q B_Q C_Q

Output

Print the total weight of the edges contained in a minimum spanning tree of the graph.

Sample Input 1

7 1
5 2 1

Sample Output 1

21

The figure below shows the minimum spanning tree of the graph:

Note that there can be multiple edges connecting the same pair of vertices.

Sample Input 2

2 1
0 0 1000000000

Sample Output 2

1000000001

Also note that there can be self-loops.

Sample Input 3

5 3
0 1 10
0 2 10
0 4 10

Sample Output 3

42