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Solve the programming task below in a Python markdown code block.
Sonya decided that having her own hotel business is the best way of earning money because she can profit and rest wherever she wants.
The country where Sonya lives is an endless line. There is a city in each integer coordinate on this line. She has n hotels, where the i-th hotel is located in the city with coordinate x_i. Sonya is a smart girl, so she does not open two or more hotels in the same city.
Sonya understands that her business needs to be expanded by opening new hotels, so she decides to build one more. She wants to make the minimum distance from this hotel to all others to be equal to d. The girl understands that there are many possible locations to construct such a hotel. Thus she wants to know the number of possible coordinates of the cities where she can build a new hotel.
Because Sonya is lounging in a jacuzzi in one of her hotels, she is asking you to find the number of cities where she can build a new hotel so that the minimum distance from the original n hotels to the new one is equal to d.
Input
The first line contains two integers n and d (1β€ nβ€ 100, 1β€ dβ€ 10^9) β the number of Sonya's hotels and the needed minimum distance from a new hotel to all others.
The second line contains n different integers in strictly increasing order x_1, x_2, β¦, x_n (-10^9β€ x_iβ€ 10^9) β coordinates of Sonya's hotels.
Output
Print the number of cities where Sonya can build a new hotel so that the minimum distance from this hotel to all others is equal to d.
Examples
Input
4 3
-3 2 9 16
Output
6
Input
5 2
4 8 11 18 19
Output
5
Note
In the first example, there are 6 possible cities where Sonya can build a hotel. These cities have coordinates -6, 5, 6, 12, 13, and 19.
In the second example, there are 5 possible cities where Sonya can build a hotel. These cities have coordinates 2, 6, 13, 16, and 21.
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
Consider a table of size n Γ m, initially fully white. Rows are numbered 1 through n from top to bottom, columns 1 through m from left to right. Some square inside the table with odd side length was painted black. Find the center of this square.
Input
The first line contains two integers n and m (1 β€ n, m β€ 115) β the number of rows and the number of columns in the table.
The i-th of the next n lines contains a string of m characters s_{i1} s_{i2} β¦ s_{im} (s_{ij} is 'W' for white cells and 'B' for black cells), describing the i-th row of the table.
Output
Output two integers r and c (1 β€ r β€ n, 1 β€ c β€ m) separated by a space β the row and column numbers of the center of the black square.
Examples
Input
5 6
WWBBBW
WWBBBW
WWBBBW
WWWWWW
WWWWWW
Output
2 4
Input
3 3
WWW
BWW
WWW
Output
2 1
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
Everybody seems to think that the Martians are green, but it turns out they are metallic pink and fat. Ajs has two bags of distinct nonnegative integers. The bags are disjoint, and the union of the sets of numbers in the bags is \{0,1,β¦,M-1\}, for some positive integer M. Ajs draws a number from the first bag and a number from the second bag, and then sums them modulo M.
What are the residues modulo M that Ajs cannot obtain with this action?
Input
The first line contains two positive integer N (1 β€ N β€ 200 000) and M (N+1 β€ M β€ 10^{9}), denoting the number of the elements in the first bag and the modulus, respectively.
The second line contains N nonnegative integers a_1,a_2,β¦,a_N (0 β€ a_1<a_2< β¦< a_N<M), the contents of the first bag.
Output
In the first line, output the cardinality K of the set of residues modulo M which Ajs cannot obtain.
In the second line of the output, print K space-separated integers greater or equal than zero and less than M, which represent the residues Ajs cannot obtain. The outputs should be sorted in increasing order of magnitude. If K=0, do not output the second line.
Examples
Input
2 5
3 4
Output
1
2
Input
4 1000000000
5 25 125 625
Output
0
Input
2 4
1 3
Output
2
0 2
Note
In the first sample, the first bag and the second bag contain \{3,4\} and \{0,1,2\}, respectively. Ajs can obtain every residue modulo 5 except the residue 2: 4+1 β‘ 0, 4+2 β‘ 1, 3+0 β‘ 3, 3+1 β‘ 4 modulo 5. One can check that there is no choice of elements from the first and the second bag which sum to 2 modulo 5.
In the second sample, the contents of the first bag are \{5,25,125,625\}, while the second bag contains all other nonnegative integers with at most 9 decimal digits. Every residue modulo 1 000 000 000 can be obtained as a sum of an element in the first bag and an element in the second bag.
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
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Solve the programming task below in a Python markdown code block.
Elections in Berland are coming. There are only two candidates β Alice and Bob.
The main Berland TV channel plans to show political debates. There are n people who want to take part in the debate as a spectator. Each person is described by their influence and political views. There are four kinds of political views:
* supporting none of candidates (this kind is denoted as "00"),
* supporting Alice but not Bob (this kind is denoted as "10"),
* supporting Bob but not Alice (this kind is denoted as "01"),
* supporting both candidates (this kind is denoted as "11").
The direction of the TV channel wants to invite some of these people to the debate. The set of invited spectators should satisfy three conditions:
* at least half of spectators support Alice (i.e. 2 β
a β₯ m, where a is number of spectators supporting Alice and m is the total number of spectators),
* at least half of spectators support Bob (i.e. 2 β
b β₯ m, where b is number of spectators supporting Bob and m is the total number of spectators),
* the total influence of spectators is maximal possible.
Help the TV channel direction to select such non-empty set of spectators, or tell that this is impossible.
Input
The first line contains integer n (1 β€ n β€ 4β
10^5) β the number of people who want to take part in the debate as a spectator.
These people are described on the next n lines. Each line describes a single person and contains the string s_i and integer a_i separated by space (1 β€ a_i β€ 5000), where s_i denotes person's political views (possible values β "00", "10", "01", "11") and a_i β the influence of the i-th person.
Output
Print a single integer β maximal possible total influence of a set of spectators so that at least half of them support Alice and at least half of them support Bob. If it is impossible print 0 instead.
Examples
Input
6
11 6
10 4
01 3
00 3
00 7
00 9
Output
22
Input
5
11 1
01 1
00 100
10 1
01 1
Output
103
Input
6
11 19
10 22
00 18
00 29
11 29
10 28
Output
105
Input
3
00 5000
00 5000
00 5000
Output
0
Note
In the first example 4 spectators can be invited to maximize total influence: 1, 2, 3 and 6. Their political views are: "11", "10", "01" and "00". So in total 2 out of 4 spectators support Alice and 2 out of 4 spectators support Bob. The total influence is 6+4+3+9=22.
In the second example the direction can select all the people except the 5-th person.
In the third example the direction can select people with indices: 1, 4, 5 and 6.
In the fourth example it is impossible to select any non-empty set of spectators.
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
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Solve the programming task below in a Python markdown code block.
There are n students in a university. The number of students is even. The i-th student has programming skill equal to a_i.
The coach wants to form n/2 teams. Each team should consist of exactly two students, and each student should belong to exactly one team. Two students can form a team only if their skills are equal (otherwise they cannot understand each other and cannot form a team).
Students can solve problems to increase their skill. One solved problem increases the skill by one.
The coach wants to know the minimum total number of problems students should solve to form exactly n/2 teams (i.e. each pair of students should form a team). Your task is to find this number.
Input
The first line of the input contains one integer n (2 β€ n β€ 100) β the number of students. It is guaranteed that n is even.
The second line of the input contains n integers a_1, a_2, ..., a_n (1 β€ a_i β€ 100), where a_i is the skill of the i-th student.
Output
Print one number β the minimum total number of problems students should solve to form exactly n/2 teams.
Examples
Input
6
5 10 2 3 14 5
Output
5
Input
2
1 100
Output
99
Note
In the first example the optimal teams will be: (3, 4), (1, 6) and (2, 5), where numbers in brackets are indices of students. Then, to form the first team the third student should solve 1 problem, to form the second team nobody needs to solve problems and to form the third team the second student should solve 4 problems so the answer is 1 + 4 = 5.
In the second example the first student should solve 99 problems to form a team with the second one.
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
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Solve the programming task below in a Python markdown code block.
Let's denote that some array b is bad if it contains a subarray b_l, b_{l+1}, ..., b_{r} of odd length more than 1 (l < r and r - l + 1 is odd) such that β i β \{0, 1, ..., r - l\} b_{l + i} = b_{r - i}.
If an array is not bad, it is good.
Now you are given an array a_1, a_2, ..., a_n. Some elements are replaced by -1. Calculate the number of good arrays you can obtain by replacing each -1 with some integer from 1 to k.
Since the answer can be large, print it modulo 998244353.
Input
The first line contains two integers n and k (2 β€ n, k β€ 2 β
10^5) β the length of array a and the size of "alphabet", i. e., the upper bound on the numbers you may use to replace -1.
The second line contains n integers a_1, a_2, ..., a_n (a_i = -1 or 1 β€ a_i β€ k) β the array a.
Output
Print one integer β the number of good arrays you can get, modulo 998244353.
Examples
Input
2 3
-1 -1
Output
9
Input
5 2
1 -1 -1 1 2
Output
0
Input
5 3
1 -1 -1 1 2
Output
2
Input
4 200000
-1 -1 12345 -1
Output
735945883
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
Innokenty works at a flea market and sells some random stuff rare items. Recently he found an old rectangular blanket. It turned out that the blanket is split in n β
m colored pieces that form a rectangle with n rows and m columns.
The colored pieces attracted Innokenty's attention so he immediately came up with the following business plan. If he cuts out a subrectangle consisting of three colored stripes, he can sell it as a flag of some country. Innokenty decided that a subrectangle is similar enough to a flag of some country if it consists of three stripes of equal heights placed one above another, where each stripe consists of cells of equal color. Of course, the color of the top stripe must be different from the color of the middle stripe; and the color of the middle stripe must be different from the color of the bottom stripe.
Innokenty has not yet decided what part he will cut out, but he is sure that the flag's boundaries should go along grid lines. Also, Innokenty won't rotate the blanket. Please help Innokenty and count the number of different subrectangles Innokenty can cut out and sell as a flag. Two subrectangles located in different places but forming the same flag are still considered different.
<image> <image> <image>
These subrectangles are flags.
<image> <image> <image> <image> <image> <image>
These subrectangles are not flags.
Input
The first line contains two integers n and m (1 β€ n, m β€ 1 000) β the number of rows and the number of columns on the blanket.
Each of the next n lines contains m lowercase English letters from 'a' to 'z' and describes a row of the blanket. Equal letters correspond to equal colors, different letters correspond to different colors.
Output
In the only line print the number of subrectangles which form valid flags.
Examples
Input
4 3
aaa
bbb
ccb
ddd
Output
6
Input
6 1
a
a
b
b
c
c
Output
1
Note
<image> <image>
The selected subrectangles are flags in the first example.
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
Vasya is about to take his first university exam in about several minutes. And it's not just some ordinary exam, it's on mathematical analysis. Of course, right now Vasya can only think of one thing: what the result of his talk with the examiner will be...
To prepare for the exam, one has to study proofs of n theorems. It is known that there will be k examination cards on the exam and each card contains <image> distinct theorems. Besides, no theorem is mentioned in more than one card (that is, <image> theorems won't be mentioned in any card). During the exam several students may get the same card.
We do not know the exact way theorems are distributed by cards, however the students that took the exam before Vasya told him what theorems their cards contained. Vasya evaluates his level of proficiency in the i-th theorem by some number ai. The level of proficiency in some card is the average of the levels of proficiency in the theorems that are included in the card. Now Vasya wants to know the minimally and maximally possible levels of his proficiency in the card he gets on the exam. Vasya wants to determine it by the data he has collected from other students. Unfortunately, Vasya has no time left to do the math and he asked you to help him.
Input
The first line contains two integers n and k (1 β€ k β€ n β€ 100) β the number of theorems and the number of cards correspondingly. The second line contains n integers ai (0 β€ ai β€ 100), the i-th number (1 β€ i β€ n) corresponds to Vasya's proficiency in the i-th theorem.
The third line contains number q (0 β€ q β€ 100) β the number of people that have taken the exam before Vasya. Each of the following q lines contains the description of a student's card: <image> integers from 1 to n inclusive. They are the numbers of theorems included in the card in the order in which they are enumerated in the input data. The numbers are given in an arbitrary order. It is guaranteed that the given cards are valid (that is, that all theorems in one card are different and that different people get cards that either don't contain the same theorems or coincide up to the theorems' permutation).
Output
Print two real numbers, representing Vasya's minimum and maximum proficiency in the card he will get on the exam. The absolute or relative error should not exceed 10 - 6.
Examples
Input
7 3
7 15 0 19 10 5 12
2
1 6
7 4
Output
5.0000000000 15.5000000000
Input
4 2
10 8 1 17
2
2 3
3 2
Output
4.5000000000 13.5000000000
Note
Let's analyze the first sample. Vasya's proficiency in the cards whose content he already knows equals 6 and 15.5 correspondingly. The three theorems that are left are only enough to make one exam card. If we consider all possible variants of theorems included in the card we can see that in the best case scenario Vasya gets the card that contains theorems 4 and 7 (his proficiency would equal 15.5) and in the worst case scenario he gets theorems 3 and 5 (his proficiency would equal 5).
The β xβ operation denotes taking integer part of real number x (rounding down).
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
You work as a system administrator in a dormitory, which has n rooms one after another along a straight hallway. Rooms are numbered from 1 to n.
You have to connect all n rooms to the Internet.
You can connect each room to the Internet directly, the cost of such connection for the i-th room is i coins.
Some rooms also have a spot for a router. The cost of placing a router in the i-th room is also i coins. You cannot place a router in a room which does not have a spot for it. When you place a router in the room i, you connect all rooms with the numbers from max(1,~i - k) to min(n,~i + k) inclusive to the Internet, where k is the range of router. The value of k is the same for all routers.
Calculate the minimum total cost of connecting all n rooms to the Internet. You can assume that the number of rooms which have a spot for a router is not greater than the number of routers you have.
Input
The first line of the input contains two integers n and k (1 β€ n, k β€ 2 β
10^5) β the number of rooms and the range of each router.
The second line of the input contains one string s of length n, consisting only of zeros and ones. If the i-th character of the string equals to '1' then there is a spot for a router in the i-th room. If the i-th character of the string equals to '0' then you cannot place a router in the i-th room.
Output
Print one integer β the minimum total cost of connecting all n rooms to the Internet.
Examples
Input
5 2
00100
Output
3
Input
6 1
000000
Output
21
Input
4 1
0011
Output
4
Input
12 6
000010000100
Output
15
Note
In the first example it is enough to place the router in the room 3, then all rooms will be connected to the Internet. The total cost of connection is 3.
In the second example you can place routers nowhere, so you need to connect all rooms directly. Thus, the total cost of connection of all rooms is 1 + 2 + 3 + 4 + 5 + 6 = 21.
In the third example you need to connect the room 1 directly and place the router in the room 3. Thus, the total cost of connection of all rooms is 1 + 3 = 4.
In the fourth example you need to place routers in rooms 5 and 10. Then all rooms will be connected to the Internet. The total cost of connection is 5 + 10 = 15.
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
Welcome! Everything is fine.
You have arrived in The Medium Place, the place between The Good Place and The Bad Place. You are assigned a task that will either make people happier or torture them for eternity.
You have a list of k pairs of people who have arrived in a new inhabited neighborhood. You need to assign each of the 2k people into one of the 2k houses. Each person will be the resident of exactly one house, and each house will have exactly one resident.
Of course, in the neighborhood, it is possible to visit friends. There are 2k - 1 roads, each of which connects two houses. It takes some time to traverse a road. We will specify the amount of time it takes in the input. The neighborhood is designed in such a way that from anyone's house, there is exactly one sequence of distinct roads you can take to any other house. In other words, the graph with the houses as vertices and the roads as edges is a tree.
The truth is, these k pairs of people are actually soulmates. We index them from 1 to k. We denote by f(i) the amount of time it takes for the i-th pair of soulmates to go to each other's houses.
As we have said before, you will need to assign each of the 2k people into one of the 2k houses. You have two missions, one from the entities in The Good Place and one from the entities of The Bad Place. Here they are:
* The first mission, from The Good Place, is to assign the people into the houses such that the sum of f(i) over all pairs i is minimized. Let's define this minimized sum as G. This makes sure that soulmates can easily and efficiently visit each other;
* The second mission, from The Bad Place, is to assign the people into the houses such that the sum of f(i) over all pairs i is maximized. Let's define this maximized sum as B. This makes sure that soulmates will have a difficult time to visit each other.
What are the values of G and B?
Input
The first line of input contains a single integer t (1 β€ t β€ 500) denoting the number of test cases. The next lines contain descriptions of the test cases.
The first line of each test case contains a single integer k denoting the number of pairs of people (1 β€ k β€ 10^5). The next 2k - 1 lines describe the roads; the i-th of them contains three space-separated integers a_i, b_i, t_i which means that the i-th road connects the a_i-th and b_i-th houses with a road that takes t_i units of time to traverse (1 β€ a_i, b_i β€ 2k, a_i β b_i, 1 β€ t_i β€ 10^6). It is guaranteed that the given roads define a tree structure.
It is guaranteed that the sum of the k in a single file is at most 3 β
10^5.
Output
For each test case, output a single line containing two space-separated integers G and B.
Example
Input
2
3
1 2 3
3 2 4
2 4 3
4 5 6
5 6 5
2
1 2 1
1 3 2
1 4 3
Output
15 33
6 6
Note
For the sample test case, we have a minimum sum equal to G = 15. One way this can be achieved is with the following assignment:
* The first pair of people get assigned to houses 5 and 6, giving us f(1) = 5;
* The second pair of people get assigned to houses 1 and 4, giving us f(2) = 6;
* The third pair of people get assigned to houses 3 and 2, giving us f(3) = 4.
Note that the sum of the f(i) is 5 + 6 + 4 = 15.
We also have a maximum sum equal to B = 33. One way this can be achieved is with the following assignment:
* The first pair of people get assigned to houses 1 and 4, giving us f(1) = 6;
* The second pair of people get assigned to houses 6 and 2, giving us f(2) = 14;
* The third pair of people get assigned to houses 3 and 5, giving us f(3) = 13.
Note that the sum of the f(i) is 6 + 14 + 13 = 33.
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
Warawreh created a great company called Nanosoft. The only thing that Warawreh still has to do is to place a large picture containing its logo on top of the company's building.
The logo of Nanosoft can be described as four squares of the same size merged together into one large square. The top left square is colored with red, the top right square is colored with green, the bottom left square is colored with yellow and the bottom right square is colored with blue.
An Example of some correct logos:
<image>
An Example of some incorrect logos:
<image>
Warawreh went to Adhami's store in order to buy the needed picture. Although Adhami's store is very large he has only one picture that can be described as a grid of n rows and m columns. The color of every cell in the picture will be green (the symbol 'G'), red (the symbol 'R'), yellow (the symbol 'Y') or blue (the symbol 'B').
Adhami gave Warawreh q options, in every option he gave him a sub-rectangle from that picture and told him that he can cut that sub-rectangle for him. To choose the best option, Warawreh needs to know for every option the maximum area of sub-square inside the given sub-rectangle that can be a Nanosoft logo. If there are no such sub-squares, the answer is 0.
Warawreh couldn't find the best option himself so he asked you for help, can you help him?
Input
The first line of input contains three integers n, m and q (1 β€ n , m β€ 500, 1 β€ q β€ 3 β
10^{5}) β the number of row, the number columns and the number of options.
For the next n lines, every line will contain m characters. In the i-th line the j-th character will contain the color of the cell at the i-th row and j-th column of the Adhami's picture. The color of every cell will be one of these: {'G','Y','R','B'}.
For the next q lines, the input will contain four integers r_1, c_1, r_2 and c_2 (1 β€ r_1 β€ r_2 β€ n, 1 β€ c_1 β€ c_2 β€ m). In that option, Adhami gave to Warawreh a sub-rectangle of the picture with the upper-left corner in the cell (r_1, c_1) and with the bottom-right corner in the cell (r_2, c_2).
Output
For every option print the maximum area of sub-square inside the given sub-rectangle, which can be a NanoSoft Logo. If there are no such sub-squares, print 0.
Examples
Input
5 5 5
RRGGB
RRGGY
YYBBG
YYBBR
RBBRG
1 1 5 5
2 2 5 5
2 2 3 3
1 1 3 5
4 4 5 5
Output
16
4
4
4
0
Input
6 10 5
RRRGGGRRGG
RRRGGGRRGG
RRRGGGYYBB
YYYBBBYYBB
YYYBBBRGRG
YYYBBBYBYB
1 1 6 10
1 3 3 10
2 2 6 6
1 7 6 10
2 1 5 10
Output
36
4
16
16
16
Input
8 8 8
RRRRGGGG
RRRRGGGG
RRRRGGGG
RRRRGGGG
YYYYBBBB
YYYYBBBB
YYYYBBBB
YYYYBBBB
1 1 8 8
5 2 5 7
3 1 8 6
2 3 5 8
1 2 6 8
2 1 5 5
2 1 7 7
6 5 7 5
Output
64
0
16
4
16
4
36
0
Note
Picture for the first test:
<image>
The pictures from the left to the right corresponds to the options. The border of the sub-rectangle in the option is marked with black, the border of the sub-square with the maximal possible size, that can be cut is marked with gray.
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
You are given an array a of length n that has a special condition: every element in this array has at most 7 divisors. Find the length of the shortest non-empty subsequence of this array product of whose elements is a perfect square.
A sequence a is a subsequence of an array b if a can be obtained from b by deletion of several (possibly, zero or all) elements.
Input
The first line contains an integer n (1 β€ n β€ 10^5) β the length of a.
The second line contains n integers a_1, a_2, β¦, a_{n} (1 β€ a_i β€ 10^6) β the elements of the array a.
Output
Output the length of the shortest non-empty subsequence of a product of whose elements is a perfect square. If there are several shortest subsequences, you can find any of them. If there's no such subsequence, print "-1".
Examples
Input
3
1 4 6
Output
1
Input
4
2 3 6 6
Output
2
Input
3
6 15 10
Output
3
Input
4
2 3 5 7
Output
-1
Note
In the first sample, you can choose a subsequence [1].
In the second sample, you can choose a subsequence [6, 6].
In the third sample, you can choose a subsequence [6, 15, 10].
In the fourth sample, there is no such subsequence.
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
You are given n integers a_1, a_2, ..., a_n.
For each a_i find its two divisors d_1 > 1 and d_2 > 1 such that \gcd(d_1 + d_2, a_i) = 1 (where \gcd(a, b) is the greatest common divisor of a and b) or say that there is no such pair.
Input
The first line contains single integer n (1 β€ n β€ 5 β
10^5) β the size of the array a.
The second line contains n integers a_1, a_2, ..., a_n (2 β€ a_i β€ 10^7) β the array a.
Output
To speed up the output, print two lines with n integers in each line.
The i-th integers in the first and second lines should be corresponding divisors d_1 > 1 and d_2 > 1 such that \gcd(d_1 + d_2, a_i) = 1 or -1 and -1 if there is no such pair. If there are multiple answers, print any of them.
Example
Input
10
2 3 4 5 6 7 8 9 10 24
Output
-1 -1 -1 -1 3 -1 -1 -1 2 2
-1 -1 -1 -1 2 -1 -1 -1 5 3
Note
Let's look at a_7 = 8. It has 3 divisors greater than 1: 2, 4, 8. As you can see, the sum of any pair of divisors is divisible by 2 as well as a_7.
There are other valid pairs of d_1 and d_2 for a_{10}=24, like (3, 4) or (8, 3). You can print any of them.
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
You are given an undirected graph where each edge has one of two colors: black or red.
Your task is to assign a real number to each node so that:
* for each black edge the sum of values at its endpoints is 1;
* for each red edge the sum of values at its endpoints is 2;
* the sum of the absolute values of all assigned numbers is the smallest possible.
Otherwise, if it is not possible, report that there is no feasible assignment of the numbers.
Input
The first line contains two integers N (1 β€ N β€ 100 000) and M (0 β€ M β€ 200 000): the number of nodes and the number of edges, respectively. The nodes are numbered by consecutive integers: 1, 2, β¦, N.
The next M lines describe the edges. Each line contains three integers a, b and c denoting that there is an edge between nodes a and b (1 β€ a, b β€ N) with color c (1 denotes black, 2 denotes red).
Output
If there is a solution, the first line should contain the word "YES" and the second line should contain N space-separated numbers. For each i (1 β€ i β€ N), the i-th number should be the number assigned to the node i.
Output should be such that:
* the sum of the numbers at the endpoints of each edge differs from the precise value by less than 10^{-6};
* the sum of the absolute values of all assigned numbers differs from the smallest possible by less than 10^{-6}.
If there are several valid solutions, output any of them.
If there is no solution, the only line should contain the word "NO".
Scoring
Subtasks:
1. (5 points) N β€ 5, M β€ 14
2. (12 points) N β€ 100
3. (17 points) N β€ 1000
4. (24 points) N β€ 10 000
5. (42 points) No further constraints
Examples
Input
4 4
1 2 1
2 3 2
1 3 2
3 4 1
Output
YES
0.5 0.5 1.5 -0.5
Input
2 1
1 2 1
Output
YES
0.3 0.7
Input
3 2
1 2 2
2 3 2
Output
YES
0 2 0
Input
3 4
1 2 2
2 2 1
2 1 1
1 2 2
Output
NO
Note
Note that in the second example the solution is not unique.
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
There are n robbers at coordinates (a_1, b_1), (a_2, b_2), ..., (a_n, b_n) and m searchlight at coordinates (c_1, d_1), (c_2, d_2), ..., (c_m, d_m).
In one move you can move each robber to the right (increase a_i of each robber by one) or move each robber up (increase b_i of each robber by one). Note that you should either increase all a_i or all b_i, you can't increase a_i for some points and b_i for some other points.
Searchlight j can see a robber i if a_i β€ c_j and b_i β€ d_j.
A configuration of robbers is safe if no searchlight can see a robber (i.e. if there is no pair i,j such that searchlight j can see a robber i).
What is the minimum number of moves you need to perform to reach a safe configuration?
Input
The first line of input contains two integers n and m (1 β€ n, m β€ 2000): the number of robbers and the number of searchlight.
Each of the next n lines contains two integers a_i, b_i (0 β€ a_i, b_i β€ 10^6), coordinates of robbers.
Each of the next m lines contains two integers c_i, d_i (0 β€ c_i, d_i β€ 10^6), coordinates of searchlights.
Output
Print one integer: the minimum number of moves you need to perform to reach a safe configuration.
Examples
Input
1 1
0 0
2 3
Output
3
Input
2 3
1 6
6 1
10 1
1 10
7 7
Output
4
Input
1 2
0 0
0 0
0 0
Output
1
Input
7 3
0 8
3 8
2 7
0 10
5 5
7 0
3 5
6 6
3 11
11 5
Output
6
Note
In the first test, you can move each robber to the right three times. After that there will be one robber in the coordinates (3, 0).
The configuration of the robbers is safe, because the only searchlight can't see the robber, because it is in the coordinates (2, 3) and 3 > 2.
In the second test, you can move each robber to the right two times and two times up. After that robbers will be in the coordinates (3, 8), (8, 3).
It's easy the see that the configuration of the robbers is safe.
It can be proved that you can't reach a safe configuration using no more than 3 moves.
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
Polycarp found on the street an array a of n elements.
Polycarp invented his criterion for the beauty of an array. He calls an array a beautiful if at least one of the following conditions must be met for each different pair of indices i β j:
* a_i is divisible by a_j;
* or a_j is divisible by a_i.
For example, if:
* n=5 and a=[7, 9, 3, 14, 63], then the a array is not beautiful (for i=4 and j=2, none of the conditions above is met);
* n=3 and a=[2, 14, 42], then the a array is beautiful;
* n=4 and a=[45, 9, 3, 18], then the a array is not beautiful (for i=1 and j=4 none of the conditions above is met);
Ugly arrays upset Polycarp, so he wants to remove some elements from the array a so that it becomes beautiful. Help Polycarp determine the smallest number of elements to remove to make the array a beautiful.
Input
The first line contains one integer t (1 β€ t β€ 10) β the number of test cases. Then t test cases follow.
The first line of each test case contains one integer n (1 β€ n β€ 2 β
10^5) β the length of the array a.
The second line of each test case contains n numbers a_1, a_2, β¦, a_n (1 β€ a_i β€ 2 β
10^5) β elements of the array a.
Output
For each test case output one integer β the minimum number of elements that must be removed to make the array a beautiful.
Example
Input
4
5
7 9 3 14 63
3
2 14 42
4
45 9 3 18
3
2 2 8
Output
2
0
1
0
Note
In the first test case, removing 7 and 14 will make array a beautiful.
In the second test case, the array a is already beautiful.
In the third test case, removing one of the elements 45 or 18 will make the array a beautiful.
In the fourth test case, the array a is beautiful.
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
As Sherlock Holmes was investigating a crime, he identified n suspects. He knows for sure that exactly one of them committed the crime. To find out which one did it, the detective lines up the suspects and numbered them from 1 to n. After that, he asked each one: "Which one committed the crime?". Suspect number i answered either "The crime was committed by suspect number ai", or "Suspect number ai didn't commit the crime". Also, the suspect could say so about himself (ai = i).
Sherlock Holmes understood for sure that exactly m answers were the truth and all other answers were a lie. Now help him understand this: which suspect lied and which one told the truth?
Input
The first line contains two integers n and m (1 β€ n β€ 105, 0 β€ m β€ n) β the total number of suspects and the number of suspects who told the truth. Next n lines contain the suspects' answers. The i-th line contains either "+ai" (without the quotes), if the suspect number i says that the crime was committed by suspect number ai, or "-ai" (without the quotes), if the suspect number i says that the suspect number ai didn't commit the crime (ai is an integer, 1 β€ ai β€ n).
It is guaranteed that at least one suspect exists, such that if he committed the crime, then exactly m people told the truth.
Output
Print n lines. Line number i should contain "Truth" if suspect number i has told the truth for sure. Print "Lie" if the suspect number i lied for sure and print "Not defined" if he could lie and could tell the truth, too, depending on who committed the crime.
Examples
Input
1 1
+1
Output
Truth
Input
3 2
-1
-2
-3
Output
Not defined
Not defined
Not defined
Input
4 1
+2
-3
+4
-1
Output
Lie
Not defined
Lie
Not defined
Note
The first sample has the single person and he confesses to the crime, and Sherlock Holmes knows that one person is telling the truth. That means that this person is telling the truth.
In the second sample there are three suspects and each one denies his guilt. Sherlock Holmes knows that only two of them are telling the truth. Any one of them can be the criminal, so we don't know for any of them, whether this person is telling the truth or not.
In the third sample the second and the fourth suspect defend the first and the third one. But only one is telling the truth, thus, the first or the third one is the criminal. Both of them can be criminals, so the second and the fourth one can either be lying or telling the truth. The first and the third one are lying for sure as they are blaming the second and the fourth one.
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
You've got a rectangular parallelepiped with integer edge lengths. You know the areas of its three faces that have a common vertex. Your task is to find the sum of lengths of all 12 edges of this parallelepiped.
Input
The first and the single line contains three space-separated integers β the areas of the parallelepiped's faces. The area's values are positive ( > 0) and do not exceed 104. It is guaranteed that there exists at least one parallelepiped that satisfies the problem statement.
Output
Print a single number β the sum of all edges of the parallelepiped.
Examples
Input
1 1 1
Output
12
Input
4 6 6
Output
28
Note
In the first sample the parallelepiped has sizes 1 Γ 1 Γ 1, in the second one β 2 Γ 2 Γ 3.
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
For he knew every Who down in Whoville beneath, Was busy now, hanging a mistletoe wreath. "And they're hanging their stockings!" he snarled with a sneer, "Tomorrow is Christmas! It's practically here!"
Dr. Suess, How The Grinch Stole Christmas
Christmas celebrations are coming to Whoville. Cindy Lou Who and her parents Lou Lou Who and Betty Lou Who decided to give sweets to all people in their street. They decided to give the residents of each house on the street, one kilogram of sweets. So they need as many kilos of sweets as there are homes on their street.
The street, where the Lou Who family lives can be represented as n consecutive sections of equal length. You can go from any section to a neighbouring one in one unit of time. Each of the sections is one of three types: an empty piece of land, a house or a shop. Cindy Lou and her family can buy sweets in a shop, but no more than one kilogram of sweets in one shop (the vendors care about the residents of Whoville not to overeat on sweets).
After the Lou Who family leave their home, they will be on the first section of the road. To get to this section of the road, they also require one unit of time. We can assume that Cindy and her mom and dad can carry an unlimited number of kilograms of sweets. Every time they are on a house section, they can give a kilogram of sweets to the inhabitants of the house, or they can simply move to another section. If the family have already given sweets to the residents of a house, they can't do it again. Similarly, if they are on the shop section, they can either buy a kilo of sweets in it or skip this shop. If they've bought a kilo of sweets in a shop, the seller of the shop remembered them and the won't sell them a single candy if they come again. The time to buy and give sweets can be neglected. The Lou Whos do not want the people of any house to remain without food.
The Lou Whos want to spend no more than t time units of time to give out sweets, as they really want to have enough time to prepare for the Christmas celebration. In order to have time to give all the sweets, they may have to initially bring additional k kilos of sweets.
Cindy Lou wants to know the minimum number of k kilos of sweets they need to take with them, to have time to give sweets to the residents of each house in their street.
Your task is to write a program that will determine the minimum possible value of k.
Input
The first line of the input contains two space-separated integers n and t (2 β€ n β€ 5Β·105, 1 β€ t β€ 109). The second line of the input contains n characters, the i-th of them equals "H" (if the i-th segment contains a house), "S" (if the i-th segment contains a shop) or "." (if the i-th segment doesn't contain a house or a shop).
It is guaranteed that there is at least one segment with a house.
Output
If there isn't a single value of k that makes it possible to give sweets to everybody in at most t units of time, print in a single line "-1" (without the quotes). Otherwise, print on a single line the minimum possible value of k.
Examples
Input
6 6
HSHSHS
Output
1
Input
14 100
...HHHSSS...SH
Output
0
Input
23 50
HHSS.......SSHHHHHHHHHH
Output
8
Note
In the first example, there are as many stores, as houses. If the family do not take a single kilo of sweets from home, in order to treat the inhabitants of the first house, they will need to make at least one step back, and they have absolutely no time for it. If they take one kilogram of sweets, they won't need to go back.
In the second example, the number of shops is equal to the number of houses and plenty of time. Available at all stores passing out candy in one direction and give them when passing in the opposite direction.
In the third example, the shops on the street are fewer than houses. The Lou Whos have to take the missing number of kilograms of sweets with them from home.
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
Yaroslav thinks that two strings s and w, consisting of digits and having length n are non-comparable if there are two numbers, i and j (1 β€ i, j β€ n), such that si > wi and sj < wj. Here sign si represents the i-th digit of string s, similarly, wj represents the j-th digit of string w.
A string's template is a string that consists of digits and question marks ("?").
Yaroslav has two string templates, each of them has length n. Yaroslav wants to count the number of ways to replace all question marks by some integers in both templates, so as to make the resulting strings incomparable. Note that the obtained strings can contain leading zeroes and that distinct question marks can be replaced by distinct or the same integers.
Help Yaroslav, calculate the remainder after dividing the described number of ways by 1000000007 (109 + 7).
Input
The first line contains integer n (1 β€ n β€ 105) β the length of both templates. The second line contains the first template β a string that consists of digits and characters "?". The string's length equals n. The third line contains the second template in the same format.
Output
In a single line print the remainder after dividing the answer to the problem by number 1000000007 (109 + 7).
Examples
Input
2
90
09
Output
1
Input
2
11
55
Output
0
Input
5
?????
?????
Output
993531194
Note
The first test contains no question marks and both strings are incomparable, so the answer is 1.
The second test has no question marks, but the given strings are comparable, so the answer is 0.
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
As it has been found out recently, all the Berland's current economical state can be described using a simple table n Γ m in size. n β the number of days in each Berland month, m β the number of months. Thus, a table cell corresponds to a day and a month of the Berland's year. Each cell will contain either 1, or -1, which means the state's gains in a particular month, on a particular day. 1 corresponds to profits, -1 corresponds to losses. It turned out important for successful development to analyze the data on the state of the economy of the previous year, however when the treasurers referred to the archives to retrieve the data, it turned out that the table had been substantially damaged. In some table cells the number values had faded and were impossible to be deciphered. It is known that the number of cells in which the data had been preserved is strictly less than max(n, m). However, there is additional information β the product of the numbers in each line and column equaled -1. Your task is to find out how many different tables may conform to the preserved data. As the answer to the task can be quite large, you have to find it modulo p.
Input
The first line contains integers n and m (1 β€ n, m β€ 1000). The second line contains the integer k (0 β€ k < max(n, m)) β the number of cells in which the data had been preserved. The next k lines contain the data on the state of the table in the preserved cells. Each line is of the form "a b c", where a (1 β€ a β€ n) β the number of the table row, b (1 β€ b β€ m) β the number of the column, c β the value containing in the cell (1 or -1). They are numbered starting from 1. It is guaranteed that no two lines with same a and b values exist. The last line contains an integer p (2 β€ p β€ 109 + 7).
Output
Print the number of different tables that could conform to the preserved data modulo p.
Examples
Input
2 2
0
100
Output
2
Input
2 2
1
1 1 -1
100
Output
1
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
Vasya has n pairs of socks. In the morning of each day Vasya has to put on a pair of socks before he goes to school. When he comes home in the evening, Vasya takes off the used socks and throws them away. Every m-th day (at days with numbers m, 2m, 3m, ...) mom buys a pair of socks to Vasya. She does it late in the evening, so that Vasya cannot put on a new pair of socks before the next day. How many consecutive days pass until Vasya runs out of socks?
Input
The single line contains two integers n and m (1 β€ n β€ 100; 2 β€ m β€ 100), separated by a space.
Output
Print a single integer β the answer to the problem.
Examples
Input
2 2
Output
3
Input
9 3
Output
13
Note
In the first sample Vasya spends the first two days wearing the socks that he had initially. Then on day three he puts on the socks that were bought on day two.
In the second sample Vasya spends the first nine days wearing the socks that he had initially. Then he spends three days wearing the socks that were bought on the third, sixth and ninth days. Than he spends another day wearing the socks that were bought on the twelfth day.
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
Breaking Good is a new video game which a lot of gamers want to have. There is a certain level in the game that is really difficult even for experienced gamers.
Walter William, the main character of the game, wants to join a gang called Los Hermanos (The Brothers). The gang controls the whole country which consists of n cities with m bidirectional roads connecting them. There is no road is connecting a city to itself and for any two cities there is at most one road between them. The country is connected, in the other words, it is possible to reach any city from any other city using the given roads.
The roads aren't all working. There are some roads which need some more work to be performed to be completely functioning.
The gang is going to rob a bank! The bank is located in city 1. As usual, the hardest part is to escape to their headquarters where the police can't get them. The gang's headquarters is in city n. To gain the gang's trust, Walter is in charge of this operation, so he came up with a smart plan.
First of all the path which they are going to use on their way back from city 1 to their headquarters n must be as short as possible, since it is important to finish operation as fast as possible.
Then, gang has to blow up all other roads in country that don't lay on this path, in order to prevent any police reinforcements. In case of non-working road, they don't have to blow up it as it is already malfunctional.
If the chosen path has some roads that doesn't work they'll have to repair those roads before the operation.
Walter discovered that there was a lot of paths that satisfied the condition of being shortest possible so he decided to choose among them a path that minimizes the total number of affected roads (both roads that have to be blown up and roads to be repaired).
Can you help Walter complete his task and gain the gang's trust?
Input
The first line of input contains two integers n, m (2 β€ n β€ 105, <image>), the number of cities and number of roads respectively.
In following m lines there are descriptions of roads. Each description consists of three integers x, y, z (1 β€ x, y β€ n, <image>) meaning that there is a road connecting cities number x and y. If z = 1, this road is working, otherwise it is not.
Output
In the first line output one integer k, the minimum possible number of roads affected by gang.
In the following k lines output three integers describing roads that should be affected. Each line should contain three integers x, y, z (1 β€ x, y β€ n, <image>), cities connected by a road and the new state of a road. z = 1 indicates that the road between cities x and y should be repaired and z = 0 means that road should be blown up.
You may output roads in any order. Each affected road should appear exactly once. You may output cities connected by a single road in any order. If you output a road, it's original state should be different from z.
After performing all operations accroding to your plan, there should remain working only roads lying on some certain shortest past between city 1 and n.
If there are multiple optimal answers output any.
Examples
Input
2 1
1 2 0
Output
1
1 2 1
Input
4 4
1 2 1
1 3 0
2 3 1
3 4 1
Output
3
1 2 0
1 3 1
2 3 0
Input
8 9
1 2 0
8 3 0
2 3 1
1 4 1
8 7 0
1 5 1
4 6 1
5 7 0
6 8 0
Output
3
2 3 0
1 5 0
6 8 1
Note
In the first test the only path is 1 - 2
In the second test the only shortest path is 1 - 3 - 4
In the third test there are multiple shortest paths but the optimal is 1 - 4 - 6 - 8
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
Soon a school Olympiad in Informatics will be held in Berland, n schoolchildren will participate there.
At a meeting of the jury of the Olympiad it was decided that each of the n participants, depending on the results, will get a diploma of the first, second or third degree. Thus, each student will receive exactly one diploma.
They also decided that there must be given at least min1 and at most max1 diplomas of the first degree, at least min2 and at most max2 diplomas of the second degree, and at least min3 and at most max3 diplomas of the third degree.
After some discussion it was decided to choose from all the options of distributing diplomas satisfying these limitations the one that maximizes the number of participants who receive diplomas of the first degree. Of all these options they select the one which maximizes the number of the participants who receive diplomas of the second degree. If there are multiple of these options, they select the option that maximizes the number of diplomas of the third degree.
Choosing the best option of distributing certificates was entrusted to Ilya, one of the best programmers of Berland. However, he found more important things to do, so it is your task now to choose the best option of distributing of diplomas, based on the described limitations.
It is guaranteed that the described limitations are such that there is a way to choose such an option of distributing diplomas that all n participants of the Olympiad will receive a diploma of some degree.
Input
The first line of the input contains a single integer n (3 β€ n β€ 3Β·106) β the number of schoolchildren who will participate in the Olympiad.
The next line of the input contains two integers min1 and max1 (1 β€ min1 β€ max1 β€ 106) β the minimum and maximum limits on the number of diplomas of the first degree that can be distributed.
The third line of the input contains two integers min2 and max2 (1 β€ min2 β€ max2 β€ 106) β the minimum and maximum limits on the number of diplomas of the second degree that can be distributed.
The next line of the input contains two integers min3 and max3 (1 β€ min3 β€ max3 β€ 106) β the minimum and maximum limits on the number of diplomas of the third degree that can be distributed.
It is guaranteed that min1 + min2 + min3 β€ n β€ max1 + max2 + max3.
Output
In the first line of the output print three numbers, showing how many diplomas of the first, second and third degree will be given to students in the optimal variant of distributing diplomas.
The optimal variant of distributing diplomas is the one that maximizes the number of students who receive diplomas of the first degree. Of all the suitable options, the best one is the one which maximizes the number of participants who receive diplomas of the second degree. If there are several of these options, the best one is the one that maximizes the number of diplomas of the third degree.
Examples
Input
6
1 5
2 6
3 7
Output
1 2 3
Input
10
1 2
1 3
1 5
Output
2 3 5
Input
6
1 3
2 2
2 2
Output
2 2 2
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
High school student Vasya got a string of length n as a birthday present. This string consists of letters 'a' and 'b' only. Vasya denotes beauty of the string as the maximum length of a substring (consecutive subsequence) consisting of equal letters.
Vasya can change no more than k characters of the original string. What is the maximum beauty of the string he can achieve?
Input
The first line of the input contains two integers n and k (1 β€ n β€ 100 000, 0 β€ k β€ n) β the length of the string and the maximum number of characters to change.
The second line contains the string, consisting of letters 'a' and 'b' only.
Output
Print the only integer β the maximum beauty of the string Vasya can achieve by changing no more than k characters.
Examples
Input
4 2
abba
Output
4
Input
8 1
aabaabaa
Output
5
Note
In the first sample, Vasya can obtain both strings "aaaa" and "bbbb".
In the second sample, the optimal answer is obtained with the string "aaaaabaa" or with the string "aabaaaaa".
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
In one school with Vasya there is a student Kostya. Kostya does not like physics, he likes different online games. Every day, having come home, Kostya throws his bag in the farthest corner and sits down at his beloved computer. Kostya even eats glued to the game. A few days ago Kostya bought a new RPG game "HaresButtle", which differs from all other games in this genre. It has a huge number of artifacts. As we know, artifacts are divided into basic and composite ones. Only the basic artifacts are available on sale. More powerful composite artifacts are collected from some number of basic artifacts.
After the composing composite artifact, all the components disappear.
Kostya is the head of the alliance, so he has to remember, what artifacts has not only himself, but also his allies. You must identify by sequence of artifacts purchased by Kostya and his allies, how many and which artifacts has been collected by each of them. It is believed that initially no one has any artifacts.
Input
The first line has 4 natural numbers: k (1 β€ k β€ 100) β the number of Kostya's allies, n (1 β€ n β€ 50) β the number of basic artifacts, m (0 β€ m β€ 50) β the number of composite artifacts, q (1 β€ q β€ 500) β the number of his friends' purchases. The following n lines contain the names of basic artifacts. After them m lines contain the descriptions of composite artifacts in the following format:
<Art. Name>: <Art. β1> <Art. β1 Number>, <Art. β2> <Art. β2 Number>, ... <Art. βX> <Art. βΠ₯ Number>
All the numbers are natural numbers not exceeding 100 (1 β€ X β€ n).
The names of all artifacts are different, they are composed of lowercase Latin letters, and the length of each name is from 1 to 100 characters inclusive. All the words in the format of the description of a composite artifact are separated by exactly one space. It is guaranteed that all components of the new artifact are different and have already been met in the input data as the names of basic artifacts.
Next, each of the following q lines is characterized by the number ai, the number of a friend who has bought the artifact (1 β€ ai β€ k), and the name of the purchased basic artifact. Let's assume that the backpacks of the heroes are infinitely large and any artifact bought later can fit in there.
It is guaranteed that after the i-th purchase no more than one opportunity to collect the composite artifact appears. If such an opportunity arose, the hero must take advantage of it.
Output
The output file should consist of k blocks. The first line should contain number bi β the number of different artifacts the i-th ally has. Then the block should contain bi lines with the names of these artifacts and the number of these artifacts. At that the lines should be printed in accordance with the lexicographical order of the names of the artifacts. In each block all the artifacts must be different, and all the numbers except the bi should be positive.
Examples
Input
2 3 2 5
desolator
refresher
perseverance
vanguard: desolator 1, refresher 1
maelstorm: perseverance 2
1 desolator
2 perseverance
1 refresher
2 desolator
2 perseverance
Output
1
vanguard 1
2
desolator 1
maelstorm 1
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
As you have noticed, there are lovely girls in Arpaβs land.
People in Arpa's land are numbered from 1 to n. Everyone has exactly one crush, i-th person's crush is person with the number crushi.
<image>
Someday Arpa shouted Owf loudly from the top of the palace and a funny game started in Arpa's land. The rules are as follows.
The game consists of rounds. Assume person x wants to start a round, he calls crushx and says: "Oww...wwf" (the letter w is repeated t times) and cuts off the phone immediately. If t > 1 then crushx calls crushcrushx and says: "Oww...wwf" (the letter w is repeated t - 1 times) and cuts off the phone immediately. The round continues until some person receives an "Owf" (t = 1). This person is called the Joon-Joon of the round. There can't be two rounds at the same time.
Mehrdad has an evil plan to make the game more funny, he wants to find smallest t (t β₯ 1) such that for each person x, if x starts some round and y becomes the Joon-Joon of the round, then by starting from y, x would become the Joon-Joon of the round. Find such t for Mehrdad if it's possible.
Some strange fact in Arpa's land is that someone can be himself's crush (i.e. crushi = i).
Input
The first line of input contains integer n (1 β€ n β€ 100) β the number of people in Arpa's land.
The second line contains n integers, i-th of them is crushi (1 β€ crushi β€ n) β the number of i-th person's crush.
Output
If there is no t satisfying the condition, print -1. Otherwise print such smallest t.
Examples
Input
4
2 3 1 4
Output
3
Input
4
4 4 4 4
Output
-1
Input
4
2 1 4 3
Output
1
Note
In the first sample suppose t = 3.
If the first person starts some round:
The first person calls the second person and says "Owwwf", then the second person calls the third person and says "Owwf", then the third person calls the first person and says "Owf", so the first person becomes Joon-Joon of the round. So the condition is satisfied if x is 1.
The process is similar for the second and the third person.
If the fourth person starts some round:
The fourth person calls himself and says "Owwwf", then he calls himself again and says "Owwf", then he calls himself for another time and says "Owf", so the fourth person becomes Joon-Joon of the round. So the condition is satisfied when x is 4.
In the last example if the first person starts a round, then the second person becomes the Joon-Joon, and vice versa.
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
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Solve the programming task below in a Python markdown code block.
Noora is a student of one famous high school. It's her final year in school β she is going to study in university next year. However, she has to get an Β«AΒ» graduation certificate in order to apply to a prestigious one.
In school, where Noora is studying, teachers are putting down marks to the online class register, which are integers from 1 to k. The worst mark is 1, the best is k. Mark that is going to the certificate, is calculated as an average of all the marks, rounded to the closest integer. If several answers are possible, rounding up is produced. For example, 7.3 is rounded to 7, but 7.5 and 7.8784 β to 8.
For instance, if Noora has marks [8, 9], then the mark to the certificate is 9, because the average is equal to 8.5 and rounded to 9, but if the marks are [8, 8, 9], Noora will have graduation certificate with 8.
To graduate with Β«AΒ» certificate, Noora has to have mark k.
Noora got n marks in register this year. However, she is afraid that her marks are not enough to get final mark k. Noora decided to ask for help in the internet, where hacker Leha immediately responded to her request. He is ready to hack class register for Noora and to add Noora any number of additional marks from 1 to k. At the same time, Leha want his hack be unseen to everyone, so he decided to add as less as possible additional marks. Please help Leha to calculate the minimal number of marks he has to add, so that final Noora's mark will become equal to k.
Input
The first line contains two integers n and k (1 β€ n β€ 100, 1 β€ k β€ 100) denoting the number of marks, received by Noora and the value of highest possible mark.
The second line contains n integers a1, a2, ..., an (1 β€ ai β€ k) denoting marks received by Noora before Leha's hack.
Output
Print a single integer β minimal number of additional marks, that Leha has to add in order to change Noora's final mark to k.
Examples
Input
2 10
8 9
Output
4
Input
3 5
4 4 4
Output
3
Note
Consider the first example testcase.
Maximal mark is 10, Noora received two marks β 8 and 9, so current final mark is 9. To fix it, Leha can add marks [10, 10, 10, 10] (4 marks in total) to the registry, achieving Noora having average mark equal to <image>. Consequently, new final mark is 10. Less number of marks won't fix the situation.
In the second example Leha can add [5, 5, 5] to the registry, so that making average mark equal to 4.5, which is enough to have 5 in the certificate.
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
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Solve the programming task below in a Python markdown code block.
Palindromic characteristics of string s with length |s| is a sequence of |s| integers, where k-th number is the total number of non-empty substrings of s which are k-palindromes.
A string is 1-palindrome if and only if it reads the same backward as forward.
A string is k-palindrome (k > 1) if and only if:
1. Its left half equals to its right half.
2. Its left and right halfs are non-empty (k - 1)-palindromes.
The left half of string t is its prefix of length β|t| / 2β, and right half β the suffix of the same length. β|t| / 2β denotes the length of string t divided by 2, rounded down.
Note that each substring is counted as many times as it appears in the string. For example, in the string "aaa" the substring "a" appears 3 times.
Input
The first line contains the string s (1 β€ |s| β€ 5000) consisting of lowercase English letters.
Output
Print |s| integers β palindromic characteristics of string s.
Examples
Input
abba
Output
6 1 0 0
Input
abacaba
Output
12 4 1 0 0 0 0
Note
In the first example 1-palindromes are substring Β«aΒ», Β«bΒ», Β«bΒ», Β«aΒ», Β«bbΒ», Β«abbaΒ», the substring Β«bbΒ» is 2-palindrome. There are no 3- and 4-palindromes here.
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
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Solve the programming task below in a Python markdown code block.
There are n points on a straight line, and the i-th point among them is located at xi. All these coordinates are distinct.
Determine the number m β the smallest number of points you should add on the line to make the distances between all neighboring points equal.
Input
The first line contains a single integer n (3 β€ n β€ 100 000) β the number of points.
The second line contains a sequence of integers x1, x2, ..., xn ( - 109 β€ xi β€ 109) β the coordinates of the points. All these coordinates are distinct. The points can be given in an arbitrary order.
Output
Print a single integer m β the smallest number of points you should add on the line to make the distances between all neighboring points equal.
Examples
Input
3
-5 10 5
Output
1
Input
6
100 200 400 300 600 500
Output
0
Input
4
10 9 0 -1
Output
8
Note
In the first example you can add one point with coordinate 0.
In the second example the distances between all neighboring points are already equal, so you shouldn't add anything.
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
Two famous competing companies ChemForces and TopChemist decided to show their sets of recently discovered chemical elements on an exhibition. However they know that no element should be present in the sets of both companies.
In order to avoid this representatives of both companies decided to make an agreement on the sets the companies should present. The sets should be chosen in the way that maximizes the total income of the companies.
All elements are enumerated with integers. The ChemForces company has discovered n distinct chemical elements with indices a_1, a_2, β¦, a_n, and will get an income of x_i Berland rubles if the i-th element from this list is in the set of this company.
The TopChemist company discovered m distinct chemical elements with indices b_1, b_2, β¦, b_m, and it will get an income of y_j Berland rubles for including the j-th element from this list to its set.
In other words, the first company can present any subset of elements from \\{a_1, a_2, β¦, a_n\} (possibly empty subset), the second company can present any subset of elements from \\{b_1, b_2, β¦, b_m\} (possibly empty subset). There shouldn't be equal elements in the subsets.
Help the representatives select the sets in such a way that no element is presented in both sets and the total income is the maximum possible.
Input
The first line contains a single integer n (1 β€ n β€ 10^5) β the number of elements discovered by ChemForces.
The i-th of the next n lines contains two integers a_i and x_i (1 β€ a_i β€ 10^9, 1 β€ x_i β€ 10^9) β the index of the i-th element and the income of its usage on the exhibition. It is guaranteed that all a_i are distinct.
The next line contains a single integer m (1 β€ m β€ 10^5) β the number of chemicals invented by TopChemist.
The j-th of the next m lines contains two integers b_j and y_j, (1 β€ b_j β€ 10^9, 1 β€ y_j β€ 10^9) β the index of the j-th element and the income of its usage on the exhibition. It is guaranteed that all b_j are distinct.
Output
Print the maximum total income you can obtain by choosing the sets for both companies in such a way that no element is presented in both sets.
Examples
Input
3
1 2
7 2
3 10
4
1 4
2 4
3 4
4 4
Output
24
Input
1
1000000000 239
3
14 15
92 65
35 89
Output
408
Note
In the first example ChemForces can choose the set (3, 7), while TopChemist can choose (1, 2, 4). This way the total income is (10 + 2) + (4 + 4 + 4) = 24.
In the second example ChemForces can choose the only element 10^9, while TopChemist can choose (14, 92, 35). This way the total income is (239) + (15 + 65 + 89) = 408.
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
Problem Statement:
Dom and Brian were trying to evade the truck drivers of Verone's gang. They were too fast, too furious for them, and could get away from the truck drivers easily. They decided to race with each other, and divided the city into N checkpoints for the same. Mia was monitoring Dom and Brian throughout the race and she noted down their speeds at every checkpoint.
Roman was jobless, as usual, and told Mia that Dom would definitely have the maximum change in speed between any two consecutive checkpoints. Mia, disagreed with him and challenged him that Brian would have maximum change in speed.
Help Mia and Roman to find out who has the maximum change in speed between any two consecutive checkpoints - Dom or Brian.
Note: Both negative and positive changes have to be considered. Hence, absolute value of maximum change in speed is to be considered.
Input:
First line contains a positive integer N - total number of checkpoints
Second line contains N space-separated integers - ai - the speeds of Dom's car at each checkpoint.
Third line contains N space-separated integers - bi - the speeds of Brian's car at each checkpoint.
Output:
Print who had the maximum change in speed between 2 consecutive checkpoints - Dom or Brian.
If both have equal maximum change in speed, print "Tie".
Print a single integer on a new line - the maximum absolute change in speed between 2 consecutive checkpoints.
Constraints:
2 β€ N β€ 10^6
-10^9 β€ ai β€ 10^9
SAMPLE INPUT
3
1 2 3
1 2 4
SAMPLE OUTPUT
Brian
2
Explanation
Max absolute change of speed for Dom = 1
Max absolute change of speed for Brian = 2
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
Soumika has a string S and its starting index is 1. The string S consists of characters from 1-9. As she is very intelligent, she wants to test his brother Vinay Tendulkar. She asked her brother Vinay Tendulkar to count the number of even numbered characters ( i.e 2,4,6,8 ) for every index i (1 β€ i β€ | S|). For an index i, the result should be calculated from i to the end of the string. As Vinay doesn't know about programming, he wants you to help him find the solution.
Input:
First line contains a string S.
Output:
Print |S| space-separated integers,the result of every index.
Constraints:
1 β€ |S| β€ 10^4
SAMPLE INPUT
574674546476
SAMPLE OUTPUT
7 7 7 6 5 5 4 4 3 2 1 1
Explanation
Given string S is 574674546476.
for index 1
Number of even numbers from 5 to end of the string is 7 so the result of index 1 is 7.
for index 2
Number of even numbers from 7 to end of the string is 7 so the result of index 2 is 7.
for index 3
Number of even numbers from 4 to end of the string is 7 so the result of index 3 is 7.
for index 3
Number of even numbers from 6 to end of the string is 6 so the result of index 4 is 6.....
...
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
There is a grid with N rows and N columns of squares. Let (i, j) be the square at the i-th row from the top and the j-th column from the left.
Each of the central (N-2) \times (N-2) squares in the grid has a black stone on it. Each of the 2N - 1 squares on the bottom side and the right side has a white stone on it.
Q queries are given. We ask you to process them in order. There are two kinds of queries. Their input format and description are as follows:
* `1 x`: Place a white stone on (1, x). After that, for each black stone between (1, x) and the first white stone you hit if you go down from (1, x), replace it with a white stone.
* `2 x`: Place a white stone on (x, 1). After that, for each black stone between (x, 1) and the first white stone you hit if you go right from (x, 1), replace it with a white stone.
How many black stones are there on the grid after processing all Q queries?
Constraints
* 3 \leq N \leq 2\times 10^5
* 0 \leq Q \leq \min(2N-4,2\times 10^5)
* 2 \leq x \leq N-1
* Queries are pairwise distinct.
Input
Input is given from Standard Input in the following format:
N Q
Query_1
\vdots
Query_Q
Output
Print how many black stones there are on the grid after processing all Q queries.
Examples
Input
5 5
1 3
2 3
1 4
2 2
1 2
Output
1
Input
200000 0
Output
39999200004
Input
176527 15
1 81279
2 22308
2 133061
1 80744
2 44603
1 170938
2 139754
2 15220
1 172794
1 159290
2 156968
1 56426
2 77429
1 97459
2 71282
Output
31159505795
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
We have A cards, each of which has an integer 1 written on it. Similarly, we also have B cards with 0s and C cards with -1s.
We will pick up K among these cards. What is the maximum possible sum of the numbers written on the cards chosen?
Constraints
* All values in input are integers.
* 0 \leq A, B, C
* 1 \leq K \leq A + B + C \leq 2 \times 10^9
Input
Input is given from Standard Input in the following format:
A B C K
Output
Print the maximum possible sum of the numbers written on the cards chosen.
Examples
Input
2 1 1 3
Output
2
Input
1 2 3 4
Output
0
Input
2000000000 0 0 2000000000
Output
2000000000
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
Niwango bought a piece of land that can be represented as a half-open interval [0, X).
Niwango will lay out N vinyl sheets on this land. The sheets are numbered 1,2, \ldots, N, and they are distinguishable. For Sheet i, he can choose an integer j such that 0 \leq j \leq X - L_i and cover [j, j + L_i) with this sheet.
Find the number of ways to cover the land with the sheets such that no point in [0, X) remains uncovered, modulo (10^9+7). We consider two ways to cover the land different if and only if there is an integer i (1 \leq i \leq N) such that the region covered by Sheet i is different.
Constraints
* 1 \leq N \leq 100
* 1 \leq L_i \leq X \leq 500
* All values in input are integers.
Input
Input is given from Standard Input in the following format:
N X
L_1 L_2 \ldots L_N
Output
Print the answer.
Examples
Input
3 3
1 1 2
Output
10
Input
18 477
324 31 27 227 9 21 41 29 50 34 2 362 92 11 13 17 183 119
Output
134796357
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
We will call a string obtained by arranging the characters contained in a string a in some order, an anagram of a.
For example, `greenbin` is an anagram of `beginner`. As seen here, when the same character occurs multiple times, that character must be used that number of times.
Given are N strings s_1, s_2, \ldots, s_N. Each of these strings has a length of 10 and consists of lowercase English characters. Additionally, all of these strings are distinct. Find the number of pairs of integers i, j (1 \leq i < j \leq N) such that s_i is an anagram of s_j.
Constraints
* 2 \leq N \leq 10^5
* s_i is a string of length 10.
* Each character in s_i is a lowercase English letter.
* s_1, s_2, \ldots, s_N are all distinct.
Input
Input is given from Standard Input in the following format:
N
s_1
s_2
:
s_N
Output
Print the number of pairs of integers i, j (1 \leq i < j \leq N) such that s_i is an anagram of s_j.
Examples
Input
3
acornistnt
peanutbomb
constraint
Output
1
Input
2
oneplustwo
ninemodsix
Output
0
Input
5
abaaaaaaaa
oneplustwo
aaaaaaaaba
twoplusone
aaaabaaaaa
Output
4
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
In the beginning, Takahashi has A cookies, and Aoki has B cookies. They will perform the following operation alternately, starting from Takahashi:
* If the number of cookies in his hand is odd, eat one of those cookies; if the number is even, do nothing. Then, give one-half of the cookies in his hand to the other person.
Find the numbers of cookies Takahashi and Aoki respectively have after performing K operations in total.
Constraints
* 1 \leq A,B \leq 10^9
* 1 \leq K \leq 100
* A,B and K are integers.
Input
Input is given from Standard Input in the following format:
A B K
Output
Print the number of cookies Takahashi has, and the number of cookies Aoki has, in this order, after performing K operations in total.
Examples
Input
5 4 2
Output
5 3
Input
3 3 3
Output
1 3
Input
314159265 358979323 84
Output
448759046 224379523
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
Akaki, a patissier, can make N kinds of doughnut using only a certain powder called "Okashi no Moto" (literally "material of pastry", simply called Moto below) as ingredient. These doughnuts are called Doughnut 1, Doughnut 2, ..., Doughnut N. In order to make one Doughnut i (1 β€ i β€ N), she needs to consume m_i grams of Moto. She cannot make a non-integer number of doughnuts, such as 0.5 doughnuts.
The recipes of these doughnuts are developed by repeated modifications from the recipe of Doughnut 1. Specifically, the recipe of Doughnut i (2 β€ i β€ N) is a direct modification of the recipe of Doughnut p_i (1 β€ p_i < i).
Now, she has X grams of Moto. She decides to make as many doughnuts as possible for a party tonight. However, since the tastes of the guests differ, she will obey the following condition:
* Let c_i be the number of Doughnut i (1 β€ i β€ N) that she makes. For each integer i such that 2 β€ i β€ N, c_{p_i} β€ c_i β€ c_{p_i} + D must hold. Here, D is a predetermined value.
At most how many doughnuts can be made here? She does not necessarily need to consume all of her Moto.
Constraints
* 2 β€ N β€ 50
* 1 β€ X β€ 10^9
* 0 β€ D β€ 10^9
* 1 β€ m_i β€ 10^9 (1 β€ i β€ N)
* 1 β€ p_i < i (2 β€ i β€ N)
* All values in input are integers.
Input
Input is given from Standard Input in the following format:
N X D
m_1
m_2 p_2
:
m_N p_N
Output
Print the maximum number of doughnuts that can be made under the condition.
Examples
Input
3 100 1
15
10 1
20 1
Output
7
Input
3 100 10
15
10 1
20 1
Output
10
Input
5 1000000000 1000000
123
159 1
111 1
135 3
147 3
Output
7496296
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
We call a 4-digit integer with three or more consecutive same digits, such as 1118, good.
You are given a 4-digit integer N. Answer the question: Is N good?
Constraints
* 1000 β€ N β€ 9999
* N is an integer.
Input
Input is given from Standard Input in the following format:
N
Output
If N is good, print `Yes`; otherwise, print `No`.
Examples
Input
1118
Output
Yes
Input
7777
Output
Yes
Input
1234
Output
No
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
You are given two integers A and B as the input. Output the value of A + B.
However, if A + B is 10 or greater, output `error` instead.
Constraints
* A and B are integers.
* 1 β€ A, B β€ 9
Input
Input is given from Standard Input in the following format:
A B
Output
If A + B is 10 or greater, print the string `error` (case-sensitive); otherwise, print the value of A + B.
Examples
Input
6 3
Output
9
Input
6 4
Output
error
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
Given a lowercase English letter c, determine whether it is a vowel. Here, there are five vowels in the English alphabet: `a`, `e`, `i`, `o` and `u`.
Constraints
* c is a lowercase English letter.
Input
The input is given from Standard Input in the following format:
c
Output
If c is a vowel, print `vowel`. Otherwise, print `consonant`.
Examples
Input
a
Output
vowel
Input
z
Output
consonant
Input
s
Output
consonant
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
In the city, there are two pastry shops. One shop was very popular because its cakes are pretty tasty. However, there was a man who is displeased at the shop. He was an owner of another shop. Although cause of his shop's unpopularity is incredibly awful taste of its cakes, he never improved it. He was just growing hate, ill, envy, and jealousy.
Finally, he decided to vandalize the rival.
His vandalize is to mess up sales record of cakes. The rival shop sells K kinds of cakes and sales quantity is recorded for each kind. He calculates sum of sales quantities for all pairs of cakes. Getting K(K-1)/2 numbers, then he rearranges them randomly, and replace an original sales record with them.
An owner of the rival shop is bothered. Could you write, at least, a program that finds total sales quantity of all cakes for the pitiful owner?
Constraints
* Judge data contains at most 100 data sets.
* 2 β€ K β€ 100
* 0 β€ ci β€ 100
Input
Input file contains several data sets. A single data set has following format:
K
c1 c2 ... cKΓ(K-1)/2
K is an integer that denotes how many kinds of cakes are sold. ci is an integer that denotes a number written on the card.
The end of input is denoted by a case where K = 0. You should output nothing for this case.
Output
For each data set, output the total sales quantity in one line.
Example
Input
2
2
3
5 4 3
0
Output
2
6
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
You are given a sequence a0a1...aN-1 digits and a prime number Q. For each i β€ j with ai β 0, the subsequence aiai+1...aj can be read as a decimal representation of a positive integer. Subsequences with leading zeros are not considered. Your task is to count the number of pairs (i, j) such that the corresponding subsequence is a multiple of Q.
Input
The input consists of at most 50 datasets. Each dataset is represented by a line containing four integers N, S, W, and Q, separated by spaces, where 1 β€ N β€ 105, 1 β€ S β€ 109, 1 β€ W β€ 109, and Q is a prime number less than 108. The sequence a0...aN-1 of length N is generated by the following code, in which ai is written as a[i].
int g = S;
for(int i=0; i<N; i++) {
a[i] = (g/7) % 10;
if( g%2 == 0 ) { g = (g/2); }
else { g = (g/2) ^ W; }
}
Note: the operators /, %, and ^ are the integer division, the modulo, and the bitwise exclusiveor, respectively. The above code is meant to be a random number generator. The intended solution does not rely on the way how the sequence is generated.
The end of the input is indicated by a line containing four zeros separated by spaces.
Output
For each dataset, output the answer in a line. You may assume that the answer is less than 230.
Example
Input
3 32 64 7
4 35 89 5
5 555 442 3
5 777 465 11
100000 666 701622763 65537
0 0 0 0
Output
2
4
6
3
68530
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
You are a programmer who loves bishojo games (a sub-genre of dating simulation games). A game, which is titled "I * C * P * C!" and was released yesterday, has arrived to you just now. This game has multiple endings. When you complete all of those endings, you can get a special figure of the main heroine, Sakuya. So, you want to hurry and play the game! But, let's calm down a bit and think how to complete all of the endings in the shortest time first.
In fact, you have a special skill that allows you to know the structure of branching points of games. By using the skill, you have found out that all of the branching points in this game are to select two choices "Yes" or "No", and once a different choice is taken the branched stories flow to different endings; they won't converge any more, like a binary tree. You also noticed that it takes exactly one minute to proceed the game from a branching point to another branching point or to an ending. In addition, you can assume it only takes negligible time to return to the beginning of the game (``reset'') and to play from the beginning to the first branching point.
The game has an additional feature called "Quick Save", which can significantly reduce the playing time for completion. The feature allows you to record the point where you are currently playing and return there at any time later. You can record any number of times, but you can hold only the last recorded point. That is, when you use Quick Save, you overwrite the previous record. If you want to return to the overwritten point, you must play the game from the beginning once again.
Well, let's estimate how long it will take for completing all of the endings in the shortest time.
Input
A data set is given in the following format.
The first line of the data set contains one integer N (2 \leq N \leq 500{,}000), which denotes the number of the endings in this game. The following N-1 lines describe the branching points. The i-th line describes the branching point of ID number i and contains two integers Yes_i and No_i (i + 1 \leq Yes_i, No_i \leq N), which denote the ID numbers of the next branching points when you select Yes or No respectively. Yes_i = N means that you can reach an ending if you select Yes, and so for No_i = N. The branching point with ID 1 is the first branching point. The branching points with ID between 2 and N-1 (inclusive) appear exactly once in Yes_i's and No_i's.
Output
Print the shortest time in a line.
Examples
Input
4
2 3
4 4
4 4
Output
6
Input
5
5 2
3 5
5 4
5 5
Output
8
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
A great king of a certain country suddenly decided to visit the land of a friendly country. The country is famous for trains, and the king visits various stations.
There are 52 train stations, each with a single uppercase or lowercase alphabetic name (no overlapping names). The line of this train is circular, with station a next to station b, station b next to station c, then station z next to station A, then station B, and so on. Proceed in order, and after Z station, it becomes a station and returns to the original. It is a single track, and there are no trains running in the opposite direction.
One day, a newspaper reporter got a list of the stations the King would visit.
"DcdkIlkP ..."
It is said that they will visit d station first, then c station, then d station, and so on. With this, when I thought that I could follow up on the king of a great country, I discovered something unexpected. The list was encrypted for counter-terrorism! A fellow reporter reportedly obtained the key to break the code. The reporter immediately gave him the key and started to modify the list. The key consists of a sequence of numbers.
"3 1 4 5 3"
This number means that the first station you visit is the one three stations before the one on the list. The station you visit second is the station in front of the second station on the list, which indicates how many stations you actually visit are in front of the stations on the list. The reporter started to fix it, but when I asked my friends what to do because the number of keys was smaller than the number of stations to visit, if you use the last key, use the first key again. It seems to be good. And the reporter was finally able to revise the list.
"AbZfFijL ..."
With this, there would be nothing scary anymore, and as soon as I thought so, an unexpected situation was discovered. The Great King stayed for days, and there was a list and keys for each date. The reporter was instructed by his boss to decrypt the entire list, but the amount was not enough for him alone. Your job is to help him and create a program that will automatically decrypt this list.
Input
The input consists of multiple datasets. The format of each data set is as follows.
n
k1 k2 ... kn
s
n is an integer representing the number of keys and can be assumed to be between 1 and 100. The following line contains a list of keys. ki indicates the i-th key. It can be assumed that it is 1 or more and 52 or less. s is a character string consisting of uppercase and lowercase letters, and indicates a list of stations to visit. It may be assumed that it is 1 or more and 100 or less. n = 0 indicates the end of input. This is not included in the dataset.
Output
Print the decrypted list for each dataset on one line each.
Sample Input
2
1 2
bdd
3
3 2 1
DDDA
Five
3 1 4 5 3
dcdkIlkP
0
Output for Sample Input
abc
ABCx
abZfFijL
Example
Input
2
1 2
bdd
3
3 2 1
DDDA
5
3 1 4 5 3
dcdkIlkP
0
Output
abc
ABCx
abZfFijL
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
problem
AOR Ika likes rooted trees with a fractal (self-similar) structure. Consider using the weighted rooted tree $ T $ consisting of $ N $ vertices to represent the rooted tree $ T'$ with the following fractal structure.
* $ T'$ is the addition of a tree rooted at $ x $ and having a tree structure similar to $ T $ (same cost) for each vertex $ x $ of $ T $.
* The root of $ T'$ is the same as that of $ T $.
The tree represented in this way is, for example, as shown in the figure below.
<image>
AOR Ika is trying to do a depth-first search for $ T'$, but finds that it takes a lot of time to trace all the vertices. Therefore, we decided to skip some node visits by performing a depth-first search with a policy of transitioning with a probability of $ p $ and not transitioning with a probability of $ 1-p $ at the time of transition during a depth-first search. Given $ T $ and the probability $ p $, find the expected value of the sum of the costs of all edges to follow when performing a depth-first search for $ T'$. The information of $ T $ is given by the number of vertices $ N $ and the information of the edges of $ N-1 $, and the vertex $ 1 $ is the root. Each vertex is labeled $ 1,2, \ dots, N $, and the $ i \ (1 \ le i \ le N-1) $ th edge costs the vertices $ x_i $ and $ y_i $ $ c_i $ It is tied with. The non-deterministic algorithm of the depth-first search that transitions to a child with a probability of $ p $, which is dealt with in this problem, is expressed as follows. The sum of the costs of the edges that the output $ \ mathrm {answer} $ follows.
1. Prepare an empty stack $ S $.
2. Set $ ββ\ mathrm {answer} = 0 $
3. Push the root vertex of $ T'$ to $ S $.
4. Extract the first element of $ S $ and make it $ x $.
5. For each child $ c $ of $ x $, do the following with probability $ p $ and do nothing with probability $ 1-p $.
* Add vertex $ c $ to $ S $. Then add the weight of the edge connecting $ x $ to $ c $ to $ \ mathrm {answer} $.
6. If S is not empty, transition to 3.
7. Output $ \ mathrm {answer} $.
output
Print the answer in one line. If the relative or absolute error is less than $ 10 ^ {-6} $, then AC. Also, output a line break at the end.
Example
Input
0.75
4
1 2 1
2 3 3
3 4 10
Output
24.8569335938
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
You have N coins each of which has a value ai. Find the number of combinations that result when you choose K different coins in such a way that the total value of the coins is greater than or equal to L and less than or equal to R.
Constraints
* 1 β€ K β€ N β€ 40
* 1 β€ ai β€ 1016
* 1 β€ L β€ R β€ 1016
* All input values are given in integers
Input
The input is given in the following format.
N K L R
a1 a2 ... aN
Output
Print the number of combinations in a line.
Examples
Input
2 2 1 9
5 1
Output
1
Input
5 2 7 19
3 5 4 2 2
Output
5
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
Division of Big Integers
Given two integers $A$ and $B$, compute the quotient, $\frac{A}{B}$. Round down to the nearest decimal.
Input
Two integers $A$ and $B$ separated by a space character are given in a line.
Output
Print the quotient in a line.
Constraints
* $-1 \times 10^{1000} \leq A, B \leq 10^{1000}$
* $B \ne 0$
Sample Input 1
5 8
Sample Output 1
0
Sample Input 2
100 25
Sample Output 2
4
Sample Input 3
-1 3
Sample Output 3
0
Sample Input 4
12 -3
Sample Output 4
-4
Example
Input
5 8
Output
0
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
Chef talks a lot on his mobile phone. As a result he exhausts his talk-value (in Rokdas) very quickly. One day at a mobile recharge shop, he noticed that his service provider gives add-on plans which can lower his calling rates (Rokdas/minute). One of the plans said "Recharge for 28 Rokdas and enjoy call rates of 0.50 Rokdas/min for one month". Chef was very pleased. His normal calling rate is 1 Rokda/min. And he talked for 200 minutes in last month, which costed him 200 Rokdas. If he had this plan activated, it would have costed him: 28 + 0.5*200 = 128 Rokdas only! Chef could have saved 72 Rokdas. But if he pays for this add-on and talks for very little in the coming month, he may end up saving nothing or even wasting money. Now, Chef is a simple guy and he doesn't worry much about future. He decides to choose the plan based upon his last monthβs usage.
There are numerous plans. Some for 1 month, some for 15 months. Some reduce call rate to 0.75 Rokdas/min, some reduce it to 0.60 Rokdas/min. And of course each of them differ in their activation costs. Note - If a plan is valid for M months, then we must pay for (re)activation after every M months (once in M months). Naturally, Chef is confused, and you (as always) are given the task to help him choose the best plan.
Input
First line contains T- the number of test cases. In each test case, first line contains D- the default rate (Rokdas/minute, real number), U- the number of minutes Chef talked in last month and N- the number of add-on plans available. Then N lines follow, each containing M- the number of months the plan is valid for, R- the calling rate for the plan (Rokdas/minute, real number) and C- the cost of the plan.
Output
For each test case, output one integer- the number of the best plan (from 1 to N). Output '0' if no plan is advantageous for Chef. No two plans are equally advantageous.
Constraints
1 β€ T β€ 1000.5 β€ D β€ 10.0 (exactly 2 digits after the decimal point)1 β€ U β€ 100001 β€ N β€ 1001 β€ M β€ 360.05 β€ R < D (exactly 2 digits after the decimal point)1 β€ C β€ 1000
Example
Input:
4
1.00 200 1
1 0.50 28
1.00 200 2
1 0.75 40
3 0.60 100
1.00 50 2
1 0.75 40
3 0.60 100
1.00 100 2
3 0.50 10
2 0.10 20
Output:
1
2
0
2
Explanation
Test Case 1: This test case is same as the example in the problem statement.Test Case 2: This is for you to work out!Test Case 3: Chef's monthly usage is only 50 Rokdas and none of the 2 plans are advantageous, hence the answer is zero '0'.Test Case 4: Again solve it yourself, but NOTE - if Chef had chosen plan 1, he would have to pay 10 Rokdas (Activation cost), after every 3 months and NOT every month. Similarly had he chosen plan 2, he would have to pay 20 Rokdas (Activation cost), after every 2 months.
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
Polycarp loves ciphers. He has invented his own cipher called Right-Left.
Right-Left cipher is used for strings. To encrypt the string s=s_{1}s_{2} ... s_{n} Polycarp uses the following algorithm:
* he writes down s_1,
* he appends the current word with s_2 (i.e. writes down s_2 to the right of the current result),
* he prepends the current word with s_3 (i.e. writes down s_3 to the left of the current result),
* he appends the current word with s_4 (i.e. writes down s_4 to the right of the current result),
* he prepends the current word with s_5 (i.e. writes down s_5 to the left of the current result),
* and so on for each position until the end of s.
For example, if s="techno" the process is: "t" β "te" β "cte" β "cteh" β "ncteh" β "ncteho". So the encrypted s="techno" is "ncteho".
Given string t β the result of encryption of some string s. Your task is to decrypt it, i.e. find the string s.
Input
The only line of the input contains t β the result of encryption of some string s. It contains only lowercase Latin letters. The length of t is between 1 and 50, inclusive.
Output
Print such string s that after encryption it equals t.
Examples
Input
ncteho
Output
techno
Input
erfdcoeocs
Output
codeforces
Input
z
Output
z
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
This is an interactive problem.
Vasya and Petya are going to play the following game: Petya has some positive integer number a. After that Vasya should guess this number using the following questions. He can say a pair of non-negative integer numbers (x, y). Petya will answer him:
* "x", if (x mod a) β₯ (y mod a).
* "y", if (x mod a) < (y mod a).
We define (x mod a) as a remainder of division x by a.
Vasya should guess the number a using no more, than 60 questions.
It's guaranteed that Petya has a number, that satisfies the inequality 1 β€ a β€ 10^9.
Help Vasya playing this game and write a program, that will guess the number a.
Interaction
Your program should play several games.
Before the start of any game your program should read the string:
* "start" (without quotes) β the start of the new game.
* "mistake" (without quotes) β in the previous game, you found the wrong answer. Your program should terminate after reading this string and it will get verdict "Wrong answer".
* "end" (without quotes) β all games finished. Your program should terminate after reading this string.
After reading the string "start" (without quotes) the new game starts.
At the beginning, your program should ask several questions about pairs of non-negative integer numbers (x, y). You can only ask the numbers, that satisfy the inequalities 0 β€ x, y β€ 2 β
10^9. To ask a question print "? x y" (without quotes). As the answer, you should read one symbol:
* "x" (without quotes), if (x mod a) β₯ (y mod a).
* "y" (without quotes), if (x mod a) < (y mod a).
* "e" (without quotes) β you asked more than 60 questions. Your program should terminate after reading this string and it will get verdict "Wrong answer".
After your program asked several questions your program should print the answer in form "! a" (without quotes). You should print the number a satisfying the inequalities 1 β€ a β€ 10^9. It's guaranteed that Petya's number a satisfied this condition. After that, the current game will finish.
We recall that your program can't ask more than 60 questions during one game.
If your program doesn't terminate after reading "mistake" (without quotes), "end" (without quotes) or "e" (without quotes), it can get any verdict, because it will continue reading from closed input. Also, if your program prints answer or question in the incorrect format it can get any verdict, too. Be careful.
Don't forget to flush the output after printing questions and answers.
To flush the output, you can use:
* fflush(stdout) in C++.
* System.out.flush() in Java.
* stdout.flush() in Python.
* flush(output) in Pascal.
* See the documentation for other languages.
It's guaranteed that you should play at least 1 and no more than 100 games.
Hacks:
In hacks, you can use only one game. To hack a solution with Petya's number a (1 β€ a β€ 10^9) in the first line you should write a single number 1 and in the second line you should write a single number a.
Example
Input
start
x
x
start
x
x
y
start
x
x
y
y
end
Output
? 0 0
? 10 1
! 1
? 0 0
? 3 4
? 2 5
! 2
? 2 4
? 2 5
? 3 10
? 9 1
! 3
Note
In the first test, you should play 3 games with Petya's numbers 1, 2 and 3.
In the first game, Petya will answer "x" (without quotes) to any question, because (x mod 1) = 0 for any integer x.
In the second game, if you will ask pair (0, 0), the answer will be "x" (without quotes), because (0 mod 2) β₯ (0 mod 2). But if you will ask pair (2, 5), the answer will be "y" (without quotes), because (2 mod 2) < (5 mod 2), because (2 mod 2) = 0 and (5 mod 2) = 1.
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
You came to a local shop and want to buy some chocolate bars. There are n bars in the shop, i-th of them costs a_i coins (and you want to buy all of them).
You have m different coupons that allow you to buy chocolate bars. i-th coupon allows you to buy q_i chocolate bars while you have to pay only for the q_i - 1 most expensive ones (so, the cheapest bar of those q_i bars is for free).
You can use only one coupon; if you use coupon i, you have to choose q_i bars and buy them using the coupon, and buy all the remaining n - q_i bars without any discounts.
To decide which coupon to choose, you want to know what will be the minimum total amount of money you have to pay if you use one of the coupons optimally.
Input
The first line contains one integer n (2 β€ n β€ 3 β
10^5) β the number of chocolate bars in the shop.
The second line contains n integers a_1, a_2, ..., a_n (1 β€ a_i β€ 10^9), where a_i is the cost of i-th chocolate bar.
The third line contains one integer m (1 β€ m β€ n - 1) β the number of coupons you have.
The fourth line contains m integers q_1, q_2, ..., q_m (2 β€ q_i β€ n), where q_i is the number of chocolate bars you have to buy using i-th coupon so that the least expensive of them will be for free. All values of q_i are pairwise distinct.
Output
Print m integers, i-th of them should be the minimum amount of money you have to pay if you buy q_i bars with i-th coupon, and all the remaining bars one by one for their full price.
Example
Input
7
7 1 3 1 4 10 8
2
3 4
Output
27
30
Note
Consider the first example.
If we use the first coupon, we may choose chocolate bars having indices 1, 6 and 7, and we pay 18 coins for them and 9 coins for all other bars.
If we use the second coupon, we may choose chocolate bars having indices 1, 5, 6 and 7, and we pay 25 coins for them and 5 coins for all other bars.
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
Anadi has a set of dominoes. Every domino has two parts, and each part contains some dots. For every a and b such that 1 β€ a β€ b β€ 6, there is exactly one domino with a dots on one half and b dots on the other half. The set contains exactly 21 dominoes. Here is an exact illustration of his set:
<image>
Also, Anadi has an undirected graph without self-loops and multiple edges. He wants to choose some dominoes and place them on the edges of this graph. He can use at most one domino of each type. Each edge can fit at most one domino. It's not necessary to place a domino on each edge of the graph.
When placing a domino on an edge, he also chooses its direction. In other words, one half of any placed domino must be directed toward one of the endpoints of the edge and the other half must be directed toward the other endpoint. There's a catch: if there are multiple halves of dominoes directed toward the same vertex, each of these halves must contain the same number of dots.
How many dominoes at most can Anadi place on the edges of his graph?
Input
The first line contains two integers n and m (1 β€ n β€ 7, 0 β€ m β€ (nβ
(n-1))/(2)) β the number of vertices and the number of edges in the graph.
The next m lines contain two integers each. Integers in the i-th line are a_i and b_i (1 β€ a, b β€ n, a β b) and denote that there is an edge which connects vertices a_i and b_i.
The graph might be disconnected. It's however guaranteed that the graph doesn't contain any self-loops, and that there is at most one edge between any pair of vertices.
Output
Output one integer which denotes the maximum number of dominoes which Anadi can place on the edges of the graph.
Examples
Input
4 4
1 2
2 3
3 4
4 1
Output
4
Input
7 0
Output
0
Input
3 1
1 3
Output
1
Input
7 21
1 2
1 3
1 4
1 5
1 6
1 7
2 3
2 4
2 5
2 6
2 7
3 4
3 5
3 6
3 7
4 5
4 6
4 7
5 6
5 7
6 7
Output
16
Note
Here is an illustration of Anadi's graph from the first sample test:
<image>
And here is one of the ways to place a domino on each of its edges:
<image>
Note that each vertex is faced by the halves of dominoes with the same number of dots. For instance, all halves directed toward vertex 1 have three dots.
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
Kamil likes streaming the competitive programming videos. His MeTube channel has recently reached 100 million subscribers. In order to celebrate this, he posted a video with an interesting problem he couldn't solve yet. Can you help him?
You're given a tree β a connected undirected graph consisting of n vertices connected by n - 1 edges. The tree is rooted at vertex 1. A vertex u is called an ancestor of v if it lies on the shortest path between the root and v. In particular, a vertex is an ancestor of itself.
Each vertex v is assigned its beauty x_v β a non-negative integer not larger than 10^{12}. This allows us to define the beauty of a path. Let u be an ancestor of v. Then we define the beauty f(u, v) as the greatest common divisor of the beauties of all vertices on the shortest path between u and v. Formally, if u=t_1, t_2, t_3, ..., t_k=v are the vertices on the shortest path between u and v, then f(u, v) = \gcd(x_{t_1}, x_{t_2}, ..., x_{t_k}). Here, \gcd denotes the greatest common divisor of a set of numbers. In particular, f(u, u) = \gcd(x_u) = x_u.
Your task is to find the sum
$$$ β_{u is an ancestor of v} f(u, v). $$$
As the result might be too large, please output it modulo 10^9 + 7.
Note that for each y, \gcd(0, y) = \gcd(y, 0) = y. In particular, \gcd(0, 0) = 0.
Input
The first line contains a single integer n (2 β€ n β€ 100 000) β the number of vertices in the tree.
The following line contains n integers x_1, x_2, ..., x_n (0 β€ x_i β€ 10^{12}). The value x_v denotes the beauty of vertex v.
The following n - 1 lines describe the edges of the tree. Each of them contains two integers a, b (1 β€ a, b β€ n, a β b) β the vertices connected by a single edge.
Output
Output the sum of the beauties on all paths (u, v) such that u is ancestor of v. This sum should be printed modulo 10^9 + 7.
Examples
Input
5
4 5 6 0 8
1 2
1 3
1 4
4 5
Output
42
Input
7
0 2 3 0 0 0 0
1 2
1 3
2 4
2 5
3 6
3 7
Output
30
Note
The following figure shows all 10 possible paths for which one endpoint is an ancestor of another endpoint. The sum of beauties of all these paths is equal to 42:
<image>
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
Three friends are going to meet each other. Initially, the first friend stays at the position x = a, the second friend stays at the position x = b and the third friend stays at the position x = c on the coordinate axis Ox.
In one minute each friend independently from other friends can change the position x by 1 to the left or by 1 to the right (i.e. set x := x - 1 or x := x + 1) or even don't change it.
Let's introduce the total pairwise distance β the sum of distances between each pair of friends. Let a', b' and c' be the final positions of the first, the second and the third friend, correspondingly. Then the total pairwise distance is |a' - b'| + |a' - c'| + |b' - c'|, where |x| is the absolute value of x.
Friends are interested in the minimum total pairwise distance they can reach if they will move optimally. Each friend will move no more than once. So, more formally, they want to know the minimum total pairwise distance they can reach after one minute.
You have to answer q independent test cases.
Input
The first line of the input contains one integer q (1 β€ q β€ 1000) β the number of test cases.
The next q lines describe test cases. The i-th test case is given as three integers a, b and c (1 β€ a, b, c β€ 10^9) β initial positions of the first, second and third friend correspondingly. The positions of friends can be equal.
Output
For each test case print the answer on it β the minimum total pairwise distance (the minimum sum of distances between each pair of friends) if friends change their positions optimally. Each friend will move no more than once. So, more formally, you have to find the minimum total pairwise distance they can reach after one minute.
Example
Input
8
3 3 4
10 20 30
5 5 5
2 4 3
1 1000000000 1000000000
1 1000000000 999999999
3 2 5
3 2 6
Output
0
36
0
0
1999999994
1999999994
2
4
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
You are given string s of length n consisting of 0-s and 1-s. You build an infinite string t as a concatenation of an infinite number of strings s, or t = ssss ... For example, if s = 10010, then t = 100101001010010...
Calculate the number of prefixes of t with balance equal to x. The balance of some string q is equal to cnt_{0, q} - cnt_{1, q}, where cnt_{0, q} is the number of occurrences of 0 in q, and cnt_{1, q} is the number of occurrences of 1 in q. The number of such prefixes can be infinite; if it is so, you must say that.
A prefix is a string consisting of several first letters of a given string, without any reorders. An empty prefix is also a valid prefix. For example, the string "abcd" has 5 prefixes: empty string, "a", "ab", "abc" and "abcd".
Input
The first line contains the single integer T (1 β€ T β€ 100) β the number of test cases.
Next 2T lines contain descriptions of test cases β two lines per test case. The first line contains two integers n and x (1 β€ n β€ 10^5, -10^9 β€ x β€ 10^9) β the length of string s and the desired balance, respectively.
The second line contains the binary string s (|s| = n, s_i β \{0, 1\}).
It's guaranteed that the total sum of n doesn't exceed 10^5.
Output
Print T integers β one per test case. For each test case print the number of prefixes or -1 if there is an infinite number of such prefixes.
Example
Input
4
6 10
010010
5 3
10101
1 0
0
2 0
01
Output
3
0
1
-1
Note
In the first test case, there are 3 good prefixes of t: with length 28, 30 and 32.
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
Vasya has a string s of length n. He decides to make the following modification to the string:
1. Pick an integer k, (1 β€ k β€ n).
2. For i from 1 to n-k+1, reverse the substring s[i:i+k-1] of s. For example, if string s is qwer and k = 2, below is the series of transformations the string goes through:
* qwer (original string)
* wqer (after reversing the first substring of length 2)
* weqr (after reversing the second substring of length 2)
* werq (after reversing the last substring of length 2)
Hence, the resulting string after modifying s with k = 2 is werq.
Vasya wants to choose a k such that the string obtained after the above-mentioned modification is lexicographically smallest possible among all choices of k. Among all such k, he wants to choose the smallest one. Since he is busy attending Felicity 2020, he asks for your help.
A string a is lexicographically smaller than a string b if and only if one of the following holds:
* a is a prefix of b, but a β b;
* in the first position where a and b differ, the string a has a letter that appears earlier in the alphabet than the corresponding letter in b.
Input
Each test contains multiple test cases.
The first line contains the number of test cases t (1 β€ t β€ 5000). The description of the test cases follows.
The first line of each test case contains a single integer n (1 β€ n β€ 5000) β the length of the string s.
The second line of each test case contains the string s of n lowercase latin letters.
It is guaranteed that the sum of n over all test cases does not exceed 5000.
Output
For each testcase output two lines:
In the first line output the lexicographically smallest string s' achievable after the above-mentioned modification.
In the second line output the appropriate value of k (1 β€ k β€ n) that you chose for performing the modification. If there are multiple values of k that give the lexicographically smallest string, output the smallest value of k among them.
Example
Input
6
4
abab
6
qwerty
5
aaaaa
6
alaska
9
lfpbavjsm
1
p
Output
abab
1
ertyqw
3
aaaaa
1
aksala
6
avjsmbpfl
5
p
1
Note
In the first testcase of the first sample, the string modification results for the sample abab are as follows :
* for k = 1 : abab
* for k = 2 : baba
* for k = 3 : abab
* for k = 4 : baba
The lexicographically smallest string achievable through modification is abab for k = 1 and 3. Smallest value of k needed to achieve is hence 1.
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
Xenia is a girl being born a noble. Due to the inflexibility and harshness of her family, Xenia has to find some ways to amuse herself.
<image>
Recently Xenia has bought n_r red gems, n_g green gems and n_b blue gems. Each of the gems has a weight.
Now, she is going to pick three gems.
Xenia loves colorful things, so she will pick exactly one gem of each color.
Xenia loves balance, so she will try to pick gems with little difference in weight.
Specifically, supposing the weights of the picked gems are x, y and z, Xenia wants to find the minimum value of (x-y)^2+(y-z)^2+(z-x)^2. As her dear friend, can you help her?
Input
The first line contains a single integer t (1β€ t β€ 100) β the number of test cases. Then t test cases follow.
The first line of each test case contains three integers n_r,n_g,n_b (1β€ n_r,n_g,n_bβ€ 10^5) β the number of red gems, green gems and blue gems respectively.
The second line of each test case contains n_r integers r_1,r_2,β¦,r_{n_r} (1β€ r_i β€ 10^9) β r_i is the weight of the i-th red gem.
The third line of each test case contains n_g integers g_1,g_2,β¦,g_{n_g} (1β€ g_i β€ 10^9) β g_i is the weight of the i-th green gem.
The fourth line of each test case contains n_b integers b_1,b_2,β¦,b_{n_b} (1β€ b_i β€ 10^9) β b_i is the weight of the i-th blue gem.
It is guaranteed that β n_r β€ 10^5, β n_g β€ 10^5, β n_b β€ 10^5 (the sum for all test cases).
Output
For each test case, print a line contains one integer β the minimum value which Xenia wants to find.
Example
Input
5
2 2 3
7 8
6 3
3 1 4
1 1 1
1
1
1000000000
2 2 2
1 2
5 4
6 7
2 2 2
1 2
3 4
6 7
3 4 1
3 2 1
7 3 3 4
6
Output
14
1999999996000000002
24
24
14
Note
In the first test case, Xenia has the following gems:
<image>
If she picks the red gem with weight 7, the green gem with weight 6, and the blue gem with weight 4, she will achieve the most balanced selection with (x-y)^2+(y-z)^2+(z-x)^2=(7-6)^2+(6-4)^2+(4-7)^2=14.
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
Little Petya very much likes rectangles and especially squares. Recently he has received 8 points on the plane as a gift from his mother. The points are pairwise distinct. Petya decided to split them into two sets each containing 4 points so that the points from the first set lay at the vertexes of some square and the points from the second set lay at the vertexes of a rectangle. Each point of initial 8 should belong to exactly one set. It is acceptable for a rectangle from the second set was also a square. If there are several partitions, Petya will be satisfied by any of them. Help him find such partition. Note that the rectangle and the square from the partition should have non-zero areas. The sides of the figures do not have to be parallel to the coordinate axes, though it might be the case.
Input
You are given 8 pairs of integers, a pair per line β the coordinates of the points Petya has. The absolute value of all coordinates does not exceed 104. It is guaranteed that no two points coincide.
Output
Print in the first output line "YES" (without the quotes), if the desired partition exists. In the second line output 4 space-separated numbers β point indexes from the input, which lie at the vertexes of the square. The points are numbered starting from 1. The numbers can be printed in any order. In the third line print the indexes of points lying at the vertexes of a rectangle in the similar format. All printed numbers should be pairwise distinct.
If the required partition does not exist, the first line should contain the word "NO" (without the quotes), after which no output is needed.
Examples
Input
0 0
10 11
10 0
0 11
1 1
2 2
2 1
1 2
Output
YES
5 6 7 8
1 2 3 4
Input
0 0
1 1
2 2
3 3
4 4
5 5
6 6
7 7
Output
NO
Input
0 0
4 4
4 0
0 4
1 2
2 3
3 2
2 1
Output
YES
1 2 3 4
5 6 7 8
Note
Pay attention to the third example: the figures do not necessarily have to be parallel to the coordinate axes.
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
Polycarpus likes studying at school a lot and he is always diligent about his homework. Polycarpus has never had any problems with natural sciences as his great-great-grandfather was the great physicist Seinstein. On the other hand though, Polycarpus has never had an easy time with history.
Everybody knows that the World history encompasses exactly n events: the i-th event had continued from the year ai to the year bi inclusive (ai < bi). Polycarpus easily learned the dates when each of n events started and ended (Polycarpus inherited excellent memory from his great-great-granddad). But the teacher gave him a more complicated task: Polycaprus should know when all events began and ended and he should also find out for each event whether it includes another event. Polycarpus' teacher thinks that an event j includes an event i if aj < ai and bi < bj. Your task is simpler: find the number of events that are included in some other event.
Input
The first input line contains integer n (1 β€ n β€ 105) which represents the number of events. Next n lines contain descriptions of the historical events, one event per line. The i + 1 line contains two integers ai and bi (1 β€ ai < bi β€ 109) β the beginning and the end of the i-th event. No two events start or finish in the same year, that is, ai β aj, ai β bj, bi β aj, bi β bj for all i, j (where i β j). Events are given in arbitrary order.
Output
Print the only integer β the answer to the problem.
Examples
Input
5
1 10
2 9
3 8
4 7
5 6
Output
4
Input
5
1 100
2 50
51 99
52 98
10 60
Output
4
Input
1
1 1000000000
Output
0
Note
In the first example the fifth event is contained in the fourth. Similarly, the fourth event is contained in the third, the third β in the second and the second β in the first.
In the second example all events except the first one are contained in the first.
In the third example only one event, so the answer is 0.
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
Little Petya likes to play very much. And most of all he likes to play the following game:
He is given a sequence of N integer numbers. At each step it is allowed to increase the value of any number by 1 or to decrease it by 1. The goal of the game is to make the sequence non-decreasing with the smallest number of steps. Petya is not good at math, so he asks for your help.
The sequence a is called non-decreasing if a1 β€ a2 β€ ... β€ aN holds, where N is the length of the sequence.
Input
The first line of the input contains single integer N (1 β€ N β€ 5000) β the length of the initial sequence. The following N lines contain one integer each β elements of the sequence. These numbers do not exceed 109 by absolute value.
Output
Output one integer β minimum number of steps required to achieve the goal.
Examples
Input
5
3 2 -1 2 11
Output
4
Input
5
2 1 1 1 1
Output
1
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
A permutation is a sequence of integers from 1 to n of length n containing each number exactly once. For example, [1], [4, 3, 5, 1, 2], [3, 2, 1] β are permutations, and [1, 1], [4, 3, 1], [2, 3, 4] β no.
Permutation a is lexicographically smaller than permutation b (they have the same length n), if in the first index i in which they differ, a[i] < b[i]. For example, the permutation [1, 3, 2, 4] is lexicographically smaller than the permutation [1, 3, 4, 2], because the first two elements are equal, and the third element in the first permutation is smaller than in the second.
The next permutation for a permutation a of length n β is the lexicographically smallest permutation b of length n that lexicographically larger than a. For example:
* for permutation [2, 1, 4, 3] the next permutation is [2, 3, 1, 4];
* for permutation [1, 2, 3] the next permutation is [1, 3, 2];
* for permutation [2, 1] next permutation does not exist.
You are given the number n β the length of the initial permutation. The initial permutation has the form a = [1, 2, β¦, n]. In other words, a[i] = i (1 β€ i β€ n).
You need to process q queries of two types:
* 1 l r: query for the sum of all elements on the segment [l, r]. More formally, you need to find a[l] + a[l + 1] + β¦ + a[r].
* 2 x: x times replace the current permutation with the next permutation. For example, if x=2 and the current permutation has the form [1, 3, 4, 2], then we should perform such a chain of replacements [1, 3, 4, 2] β [1, 4, 2, 3] β [1, 4, 3, 2].
For each query of the 1-st type output the required sum.
Input
The first line contains two integers n (2 β€ n β€ 2 β
10^5) and q (1 β€ q β€ 2 β
10^5), where n β the length of the initial permutation, and q β the number of queries.
The next q lines contain a single query of the 1-st or 2-nd type. The 1-st type query consists of three integers 1, l and r (1 β€ l β€ r β€ n), the 2-nd type query consists of two integers 2 and x (1 β€ x β€ 10^5).
It is guaranteed that all requests of the 2-nd type are possible to process.
Output
For each query of the 1-st type, output on a separate line one integer β the required sum.
Example
Input
4 4
1 2 4
2 3
1 1 2
1 3 4
Output
9
4
6
Note
Initially, the permutation has the form [1, 2, 3, 4]. Queries processing is as follows:
1. 2 + 3 + 4 = 9;
2. [1, 2, 3, 4] β [1, 2, 4, 3] β [1, 3, 2, 4] β [1, 3, 4, 2];
3. 1 + 3 = 4;
4. 4 + 2 = 6
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
The progress is not standing still in Berland. Recently all garbage containers in Bertown, the capital of Berland, were replaced by differentiated recycling bins, each accepting some category of waste. While this will definitely improve the ecological situation, for some citizens it's difficult to get used to the habit of sorting waste.
Monocarp is one of those citizens who tries to get used to waste sorting. Today he has to take out the trash from his house. There are three containers near the Monocarp's house, the first one accepts paper waste, the second one accepts plastic waste, and the third one β all other types of waste. It is possible to fit c_1 items into the first container, c_2 items into the second container and c_3 items into the third container.
Monocarp has a lot of items to throw into containers. Some are made of paper, so Monocarp has to put them into the first container (he has a_1 such items), some are made of plastic, so he has to put them into the second container (he has a_2 such items), and some are neither paper nor plastic β so Monocarp has to put them into the third container (he has a_3 such items).
Unfortunately, there are also two categories of items that Monocarp is unsure of: he has a_4 items which are partially made of paper, so he will put each of these items either into the first container or into the third container. Similarly, he has a_5 items partially made of plastic, so he has to put each of them either into the second container or into the third container. Obviously, this choice is made separately for each item β for example, Monocarp can throw several partially-plastic items into the second container, and all other partially-plastic items β into the third one.
Now Monocarp wonders: is it possible to put each item into some container so that the first container will hold no more than c_1 items, the second one β no more than c_2 items, and the third one β no more than c_3 items?
Input
The first line contains one integer t (1 β€ t β€ 3 β
10^4) β the number of test cases.
Each test case consists of two lines. The first line of each test case contains three integers c_1, c_2, c_3 (0 β€ c_1, c_2, c_3 β€ 10^8) β the capacities of the containers.
The second line of each test case contains five integers a_1, a_2, a_3, a_4, a_5 (0 β€ a_i β€ 10^8), where a_i is the number of items of the i-th category Monocarp has to throw out (i = 1 is paper waste, i = 2 is plastic waste, i = 3 is general waste, i = 4 is partially-paper waste, i = 5 is partially-plastic waste).
Output
For each test case, print either YES if it is possible to fit all items into containers, or NO otherwise. You may print each letter in any case (for example, YES, Yes, yes, yEs will all be recognized as positive answer).
Example
Input
7
1 2 3
1 2 3 0 0
2 2 3
1 2 3 1 0
2 2 3
1 2 3 0 1
1 2 5
1 2 3 1 1
0 0 0
0 0 0 0 0
0 0 4
1 0 0 0 0
13 37 42
0 0 0 40 47
Output
YES
YES
NO
YES
YES
NO
YES
Note
Explanations for the example test cases:
1. Monocarp can put 1 item of paper waste into the first container, 2 items of plastic waste into the second container, and 3 items of general waste into the third container;
2. Monocarp can put 1 item of paper waste and 1 item of partially-paper waste into the first container, 2 items of plastic waste into the second container, and 3 items of general waste into the third container;
3. there is no answer since either the second container should contain 3 items, or the third container should contain 4 items;
4. Monocarp can put 1 item of paper waste into the first container, 2 items of plastic waste into the second container, and 3 items of general waste, 1 item of partially-paper waste and 1 item of partially-plastic waste into the third container;
5. there is no waste at all, so all containers can be left empty;
6. there's no answer since it's impossible to put a paper item into the third container;
7. Monocarp can put 10 items of partially-paper waste into the first container, 37 items of partially-plastic waste into the second container, and 30 items of partially-paper waste and 10 items of partially-plastic waste into the third container.
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
Berland crossword is a puzzle that is solved on a square grid with n rows and n columns. Initially all the cells are white.
To solve the puzzle one has to color some cells on the border of the grid black in such a way that:
* exactly U cells in the top row are black;
* exactly R cells in the rightmost column are black;
* exactly D cells in the bottom row are black;
* exactly L cells in the leftmost column are black.
Note that you can color zero cells black and leave every cell white.
Your task is to check if there exists a solution to the given puzzle.
Input
The first line contains a single integer t (1 β€ t β€ 1000) β the number of testcases.
Then the descriptions of t testcases follow.
The only line of each testcase contains 5 integers n, U, R, D, L (2 β€ n β€ 100; 0 β€ U, R, D, L β€ n).
Output
For each testcase print "YES" if the solution exists and "NO" otherwise.
You may print every letter in any case you want (so, for example, the strings yEs, yes, Yes and YES are all recognized as positive answer).
Example
Input
4
5 2 5 3 1
3 0 0 0 0
4 4 1 4 0
2 1 1 1 1
Output
YES
YES
NO
YES
Note
Here are possible solutions to testcases 1, 2 and 4:
<image>
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
A number is called 2050-number if it is 2050, 20500, ..., (2050 β
10^k for integer k β₯ 0).
Given a number n, you are asked to represent n as the sum of some (not necessarily distinct) 2050-numbers. Compute the minimum number of 2050-numbers required for that.
Input
The first line contains a single integer T (1β€ Tβ€ 1 000) denoting the number of test cases.
The only line of each test case contains a single integer n (1β€ nβ€ 10^{18}) denoting the number to be represented.
Output
For each test case, output the minimum number of 2050-numbers in one line.
If n cannot be represented as the sum of 2050-numbers, output -1 instead.
Example
Input
6
205
2050
4100
20500
22550
25308639900
Output
-1
1
2
1
2
36
Note
In the third case, 4100 = 2050 + 2050.
In the fifth case, 22550 = 20500 + 2050.
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
Finally, you have defeated Razor and now, you are the Most Wanted street racer. Sergeant Cross has sent the full police force after you in a deadly pursuit. Fortunately, you have found a hiding spot but you fear that Cross and his force will eventually find you. To increase your chances of survival, you want to tune and repaint your BMW M3 GTR.
The car can be imagined as a permuted n-dimensional hypercube. A simple n-dimensional hypercube is an undirected unweighted graph built recursively as follows:
* Take two simple (n-1)-dimensional hypercubes one having vertices numbered from 0 to 2^{n-1}-1 and the other having vertices numbered from 2^{n-1} to 2^{n}-1. A simple 0-dimensional Hypercube is just a single vertex.
* Add an edge between the vertices i and i+2^{n-1} for each 0β€ i < 2^{n-1}.
A permuted n-dimensional hypercube is formed by permuting the vertex numbers of a simple n-dimensional hypercube in any arbitrary manner.
Examples of a simple and permuted 3-dimensional hypercubes are given below:
<image>
Note that a permuted n-dimensional hypercube has the following properties:
* There are exactly 2^n vertices.
* There are exactly nβ
2^{n-1} edges.
* Each vertex is connected to exactly n other vertices.
* There are no self-loops or duplicate edges.
Let's denote the permutation used to generate the permuted n-dimensional hypercube, representing your car, from a simple n-dimensional hypercube by P. Before messing up the functionalities of the car, you want to find this permutation so that you can restore the car if anything goes wrong. But the job isn't done yet.
You have n different colours numbered from 0 to n-1. You want to colour the vertices of this permuted n-dimensional hypercube in such a way that for each and every vertex u satisfying 0β€ u < 2^n and for each and every colour c satisfying 0β€ c < n, there is at least one vertex v adjacent to u having a colour c. In other words, from each and every vertex, it must be possible to reach a vertex of any colour by just moving to an adjacent vertex.
Given the permuted n-dimensional hypercube, find any valid permutation P and colouring.
Input
The first line of input contains a single integer t (1β€ tβ€ 4096) β the number of test cases.
For each test case, the first line contains a single integer n (1β€ nβ€ 16).
Each of the next nβ
2^{n-1} lines contain two integers u and v (0β€ u, v < 2^n) denoting that there is an edge between the vertices numbered u and v.
It is guaranteed that the graph described in the input is a permuted n-dimensional hypercube.
Additionally, it is guaranteed that the sum of 2^n over all test cases does not exceed 2^{16}=65 536.
Output
For each test case, print two lines.
In the first line, output any permutation P of length 2^n that can be used to transform a simple n-dimensional hypercube to the permuted n-dimensional hypercube given in the input. Two permuted hypercubes are considered the same if they have the same set of edges. If there are multiple answers, output any of them.
In the second line, print the colouring. If there is no way to colour the vertices satisfying the conditions, output -1. Otherwise, output a single line containing 2^n space separated integers. The i-th integer must be the colour of the vertex numbered (i-1) in the permuted n-dimensional hypercube. If there are multiple answers, output any of them.
Example
Input
3
1
0 1
2
0 1
1 2
2 3
3 0
3
0 1
0 5
0 7
1 2
1 4
2 5
2 6
3 5
3 6
3 7
4 6
4 7
Output
0 1
0 0
0 1 3 2
0 0 1 1
5 3 0 7 2 6 1 4
-1
Note
The colouring and the permuted hypercube for the first test case is shown below:
<image>
The colouring and the permuted hypercube for the second test case is shown below:
<image>
The permuted hypercube for the third test case is given in the problem statement. However, it can be shown that there exists no way to colour that cube satifying all the conditions. Note that some other permutations like [0, 5, 7, 3, 1, 2, 4, 6] and [0, 1, 5, 2, 7, 4, 3, 6] will also give the same permuted hypercube.
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
Input
The only line of input contains three integers a1, a2, a3 (1 β€ a1, a2, a3 β€ 20), separated by spaces.
Output
Output a single integer.
Examples
Input
2 3 2
Output
5
Input
13 14 1
Output
14
Input
14 5 9
Output
464
Input
17 18 3
Output
53
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
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Solve the programming task below in a Python markdown code block.
A non-empty string s is called binary, if it consists only of characters "0" and "1". Let's number the characters of binary string s from 1 to the string's length and let's denote the i-th character in string s as si.
Binary string s with length n is periodical, if there is an integer 1 β€ k < n such that:
* k is a divisor of number n
* for all 1 β€ i β€ n - k, the following condition fulfills: si = si + k
For example, binary strings "101010" and "11" are periodical and "10" and "10010" are not.
A positive integer x is periodical, if its binary representation (without leading zeroes) is a periodic string.
Your task is to calculate, how many periodic numbers are in the interval from l to r (both ends are included).
Input
The single input line contains two integers l and r (1 β€ l β€ r β€ 1018). The numbers are separated by a space.
Please, do not use the %lld specifier to read or write 64-bit integers in Π‘++. It is preferred to use the cin, cout streams or the %I64d specifier.
Output
Print a single integer, showing how many periodic numbers are in the interval from l to r (both ends are included).
Examples
Input
1 10
Output
3
Input
25 38
Output
2
Note
In the first sample periodic numbers are 3, 7 and 10.
In the second sample periodic numbers are 31 and 36.
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
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Solve the programming task below in a Python markdown code block.
Ilya is a very good-natured lion. He likes maths. Of all mathematical objects, his favourite one is matrices. Now he's faced a complicated matrix problem he needs to solve.
He's got a square 2n Γ 2n-sized matrix and 4n integers. You need to arrange all these numbers in the matrix (put each number in a single individual cell) so that the beauty of the resulting matrix with numbers is maximum.
The beauty of a 2n Γ 2n-sized matrix is an integer, obtained by the following algorithm:
1. Find the maximum element in the matrix. Let's denote it as m.
2. If n = 0, then the beauty of the matrix equals m. Otherwise, a matrix can be split into 4 non-intersecting 2n - 1 Γ 2n - 1-sized submatrices, then the beauty of the matrix equals the sum of number m and other four beauties of the described submatrices.
As you can see, the algorithm is recursive.
Help Ilya, solve the problem and print the resulting maximum beauty of the matrix.
Input
The first line contains integer 4n (1 β€ 4n β€ 2Β·106). The next line contains 4n integers ai (1 β€ ai β€ 109) β the numbers you need to arrange in the 2n Γ 2n-sized matrix.
Output
On a single line print the maximum value of the beauty of the described matrix.
Please, do not use the %lld specifier to read or write 64-bit integers in Π‘++. It is preferred to use the cin, cout streams or the %I64d specifier.
Examples
Input
1
13
Output
13
Input
4
1 2 3 4
Output
14
Note
Consider the second sample. You need to arrange the numbers in the matrix as follows:
1 2
3 4
Then the beauty of the matrix will equal: 4 + 1 + 2 + 3 + 4 = 14.
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
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Solve the programming task below in a Python markdown code block.
A nearby pie shop is having a special sale. For each pie you pay full price for, you may select one pie of a strictly lesser value to get for free. Given the prices of all the pies you wish to acquire, determine the minimum total amount you must pay for all of the pies.
Input
Input will begin with an integer n (1 β€ n β€ 500000), the number of pies you wish to acquire. Following this is a line with n integers, each indicating the cost of a pie. All costs are positive integers not exceeding 109.
Output
Print the minimum cost to acquire all the pies.
Please, do not use the %lld specifier to read or write 64-bit integers in Π‘++. It is preferred to use the cin, cout streams or the %I64d specifier.
Examples
Input
6
3 4 5 3 4 5
Output
14
Input
5
5 5 5 5 5
Output
25
Input
4
309999 6000 2080 2080
Output
314159
Note
In the first test case you can pay for a pie with cost 5 and get a pie with cost 4 for free, then pay for a pie with cost 5 and get a pie with cost 3 for free, then pay for a pie with cost 4 and get a pie with cost 3 for free.
In the second test case you have to pay full price for every pie.
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
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Solve the programming task below in a Python markdown code block.
Simon has a rectangular table consisting of n rows and m columns. Simon numbered the rows of the table from top to bottom starting from one and the columns β from left to right starting from one. We'll represent the cell on the x-th row and the y-th column as a pair of numbers (x, y). The table corners are cells: (1, 1), (n, 1), (1, m), (n, m).
Simon thinks that some cells in this table are good. Besides, it's known that no good cell is the corner of the table.
Initially, all cells of the table are colorless. Simon wants to color all cells of his table. In one move, he can choose any good cell of table (x1, y1), an arbitrary corner of the table (x2, y2) and color all cells of the table (p, q), which meet both inequations: min(x1, x2) β€ p β€ max(x1, x2), min(y1, y2) β€ q β€ max(y1, y2).
Help Simon! Find the minimum number of operations needed to color all cells of the table. Note that you can color one cell multiple times.
Input
The first line contains exactly two integers n, m (3 β€ n, m β€ 50).
Next n lines contain the description of the table cells. Specifically, the i-th line contains m space-separated integers ai1, ai2, ..., aim. If aij equals zero, then cell (i, j) isn't good. Otherwise aij equals one. It is guaranteed that at least one cell is good. It is guaranteed that no good cell is a corner.
Output
Print a single number β the minimum number of operations Simon needs to carry out his idea.
Examples
Input
3 3
0 0 0
0 1 0
0 0 0
Output
4
Input
4 3
0 0 0
0 0 1
1 0 0
0 0 0
Output
2
Note
In the first sample, the sequence of operations can be like this:
<image>
* For the first time you need to choose cell (2, 2) and corner (1, 1).
* For the second time you need to choose cell (2, 2) and corner (3, 3).
* For the third time you need to choose cell (2, 2) and corner (3, 1).
* For the fourth time you need to choose cell (2, 2) and corner (1, 3).
In the second sample the sequence of operations can be like this:
<image>
* For the first time you need to choose cell (3, 1) and corner (4, 3).
* For the second time you need to choose cell (2, 3) and corner (1, 1).
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
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Solve the programming task below in a Python markdown code block.
You have an array of positive integers a[1], a[2], ..., a[n] and a set of bad prime numbers b1, b2, ..., bm. The prime numbers that do not occur in the set b are considered good. The beauty of array a is the sum <image>, where function f(s) is determined as follows:
* f(1) = 0;
* Let's assume that p is the minimum prime divisor of s. If p is a good prime, then <image>, otherwise <image>.
You are allowed to perform an arbitrary (probably zero) number of operations to improve array a. The operation of improvement is the following sequence of actions:
* Choose some number r (1 β€ r β€ n) and calculate the value g = GCD(a[1], a[2], ..., a[r]).
* Apply the assignments: <image>, <image>, ..., <image>.
What is the maximum beauty of the array you can get?
Input
The first line contains two integers n and m (1 β€ n, m β€ 5000) showing how many numbers are in the array and how many bad prime numbers there are.
The second line contains n space-separated integers a[1], a[2], ..., a[n] (1 β€ a[i] β€ 109) β array a. The third line contains m space-separated integers b1, b2, ..., bm (2 β€ b1 < b2 < ... < bm β€ 109) β the set of bad prime numbers.
Output
Print a single integer β the answer to the problem.
Examples
Input
5 2
4 20 34 10 10
2 5
Output
-2
Input
4 5
2 4 8 16
3 5 7 11 17
Output
10
Note
Note that the answer to the problem can be negative.
The GCD(x1, x2, ..., xk) is the maximum positive integer that divides each xi.
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
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Solve the programming task below in a Python markdown code block.
Alex enjoys performing magic tricks. He has a trick that requires a deck of n cards. He has m identical decks of n different cards each, which have been mixed together. When Alex wishes to perform the trick, he grabs n cards at random and performs the trick with those. The resulting deck looks like a normal deck, but may have duplicates of some cards.
The trick itself is performed as follows: first Alex allows you to choose a random card from the deck. You memorize the card and put it back in the deck. Then Alex shuffles the deck, and pulls out a card. If the card matches the one you memorized, the trick is successful.
You don't think Alex is a very good magician, and that he just pulls a card randomly from the deck. Determine the probability of the trick being successful if this is the case.
Input
First line of the input consists of two integers n and m (1 β€ n, m β€ 1000), separated by space β number of cards in each deck, and number of decks.
Output
On the only line of the output print one floating point number β probability of Alex successfully performing the trick. Relative or absolute error of your answer should not be higher than 10 - 6.
Examples
Input
2 2
Output
0.6666666666666666
Input
4 4
Output
0.4000000000000000
Input
1 2
Output
1.0000000000000000
Note
In the first sample, with probability <image> Alex will perform the trick with two cards with the same value from two different decks. In this case the trick is guaranteed to succeed.
With the remaining <image> probability he took two different cards, and the probability of pulling off the trick is <image>.
The resulting probability is <image>
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
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Solve the programming task below in a Python markdown code block.
You have decided to watch the best moments of some movie. There are two buttons on your player:
1. Watch the current minute of the movie. By pressing this button, you watch the current minute of the movie and the player automatically proceeds to the next minute of the movie.
2. Skip exactly x minutes of the movie (x is some fixed positive integer). If the player is now at the t-th minute of the movie, then as a result of pressing this button, it proceeds to the minute (t + x).
Initially the movie is turned on in the player on the first minute, and you want to watch exactly n best moments of the movie, the i-th best moment starts at the li-th minute and ends at the ri-th minute (more formally, the i-th best moment consists of minutes: li, li + 1, ..., ri).
Determine, what is the minimum number of minutes of the movie you have to watch if you want to watch all the best moments?
Input
The first line contains two space-separated integers n, x (1 β€ n β€ 50, 1 β€ x β€ 105) β the number of the best moments of the movie and the value of x for the second button.
The following n lines contain the descriptions of the best moments of the movie, the i-th line of the description contains two integers separated by a space li, ri (1 β€ li β€ ri β€ 105).
It is guaranteed that for all integers i from 2 to n the following condition holds: ri - 1 < li.
Output
Output a single number β the answer to the problem.
Examples
Input
2 3
5 6
10 12
Output
6
Input
1 1
1 100000
Output
100000
Note
In the first sample, the player was initially standing on the first minute. As the minutes from the 1-st to the 4-th one don't contain interesting moments, we press the second button. Now we can not press the second button and skip 3 more minutes, because some of them contain interesting moments. Therefore, we watch the movie from the 4-th to the 6-th minute, after that the current time is 7. Similarly, we again skip 3 minutes and then watch from the 10-th to the 12-th minute of the movie. In total, we watch 6 minutes of the movie.
In the second sample, the movie is very interesting, so you'll have to watch all 100000 minutes of the movie.
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
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Solve the programming task below in a Python markdown code block.
The first algorithm for detecting a face on the image working in realtime was developed by Paul Viola and Michael Jones in 2001. A part of the algorithm is a procedure that computes Haar features. As part of this task, we consider a simplified model of this concept.
Let's consider a rectangular image that is represented with a table of size n Γ m. The table elements are integers that specify the brightness of each pixel in the image.
A feature also is a rectangular table of size n Γ m. Each cell of a feature is painted black or white.
To calculate the value of the given feature at the given image, you must perform the following steps. First the table of the feature is put over the table of the image (without rotations or reflections), thus each pixel is entirely covered with either black or white cell. The value of a feature in the image is the value of W - B, where W is the total brightness of the pixels in the image, covered with white feature cells, and B is the total brightness of the pixels covered with black feature cells.
Some examples of the most popular Haar features are given below.
<image>
Your task is to determine the number of operations that are required to calculate the feature by using the so-called prefix rectangles.
A prefix rectangle is any rectangle on the image, the upper left corner of which coincides with the upper left corner of the image.
You have a variable value, whose value is initially zero. In one operation you can count the sum of pixel values ββat any prefix rectangle, multiply it by any integer and add to variable value.
You are given a feature. It is necessary to calculate the minimum number of operations required to calculate the values of this attribute at an arbitrary image. For a better understanding of the statement, read the explanation of the first sample.
Input
The first line contains two space-separated integers n and m (1 β€ n, m β€ 100) β the number of rows and columns in the feature.
Next n lines contain the description of the feature. Each line consists of m characters, the j-th character of the i-th line equals to "W", if this element of the feature is white and "B" if it is black.
Output
Print a single number β the minimum number of operations that you need to make to calculate the value of the feature.
Examples
Input
6 8
BBBBBBBB
BBBBBBBB
BBBBBBBB
WWWWWWWW
WWWWWWWW
WWWWWWWW
Output
2
Input
3 3
WBW
BWW
WWW
Output
4
Input
3 6
WWBBWW
WWBBWW
WWBBWW
Output
3
Input
4 4
BBBB
BBBB
BBBB
BBBW
Output
4
Note
The first sample corresponds to feature B, the one shown in the picture. The value of this feature in an image of size 6 Γ 8 equals to the difference of the total brightness of the pixels in the lower and upper half of the image. To calculate its value, perform the following two operations:
1. add the sum of pixels in the prefix rectangle with the lower right corner in the 6-th row and 8-th column with coefficient 1 to the variable value (the rectangle is indicated by a red frame); <image>
2. add the number of pixels in the prefix rectangle with the lower right corner in the 3-rd row and 8-th column with coefficient - 2 and variable value. <image>
Thus, all the pixels in the lower three rows of the image will be included with factor 1, and all pixels in the upper three rows of the image will be included with factor 1 - 2 = - 1, as required.
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
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Solve the programming task below in a Python markdown code block.
In this problem you are to calculate the sum of all integers from 1 to n, but you should take all powers of two with minus in the sum.
For example, for n = 4 the sum is equal to - 1 - 2 + 3 - 4 = - 4, because 1, 2 and 4 are 20, 21 and 22 respectively.
Calculate the answer for t values of n.
Input
The first line of the input contains a single integer t (1 β€ t β€ 100) β the number of values of n to be processed.
Each of next t lines contains a single integer n (1 β€ n β€ 109).
Output
Print the requested sum for each of t integers n given in the input.
Examples
Input
2
4
1000000000
Output
-4
499999998352516354
Note
The answer for the first sample is explained in the statement.
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
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Solve the programming task below in a Python markdown code block.
A lot of people in Berland hates rain, but you do not. Rain pacifies, puts your thoughts in order. By these years you have developed a good tradition β when it rains, you go on the street and stay silent for a moment, contemplate all around you, enjoy freshness, think about big deeds you have to do.
Today everything had changed quietly. You went on the street with a cup contained water, your favorite drink. In a moment when you were drinking a water you noticed that the process became quite long: the cup still contained water because of rain. You decided to make a formal model of what was happening and to find if it was possible to drink all water in that situation.
Thus, your cup is a cylinder with diameter equals d centimeters. Initial level of water in cup equals h centimeters from the bottom.
<image>
You drink a water with a speed equals v milliliters per second. But rain goes with such speed that if you do not drink a water from the cup, the level of water increases on e centimeters per second. The process of drinking water from the cup and the addition of rain to the cup goes evenly and continuously.
Find the time needed to make the cup empty or find that it will never happen. It is guaranteed that if it is possible to drink all water, it will happen not later than after 104 seconds.
Note one milliliter equals to one cubic centimeter.
Input
The only line of the input contains four integer numbers d, h, v, e (1 β€ d, h, v, e β€ 104), where:
* d β the diameter of your cylindrical cup,
* h β the initial level of water in the cup,
* v β the speed of drinking process from the cup in milliliters per second,
* e β the growth of water because of rain if you do not drink from the cup.
Output
If it is impossible to make the cup empty, print "NO" (without quotes).
Otherwise print "YES" (without quotes) in the first line. In the second line print a real number β time in seconds needed the cup will be empty. The answer will be considered correct if its relative or absolute error doesn't exceed 10 - 4. It is guaranteed that if the answer exists, it doesn't exceed 104.
Examples
Input
1 2 3 100
Output
NO
Input
1 1 1 1
Output
YES
3.659792366325
Note
In the first example the water fills the cup faster than you can drink from it.
In the second example area of the cup's bottom equals to <image>, thus we can conclude that you decrease the level of water by <image> centimeters per second. At the same time water level increases by 1 centimeter per second due to rain. Thus, cup will be empty in <image> seconds.
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
Sonya was unable to think of a story for this problem, so here comes the formal description.
You are given the array containing n positive integers. At one turn you can pick any element and increase or decrease it by 1. The goal is the make the array strictly increasing by making the minimum possible number of operations. You are allowed to change elements in any way, they can become negative or equal to 0.
Input
The first line of the input contains a single integer n (1 β€ n β€ 3000) β the length of the array.
Next line contains n integer ai (1 β€ ai β€ 109).
Output
Print the minimum number of operation required to make the array strictly increasing.
Examples
Input
7
2 1 5 11 5 9 11
Output
9
Input
5
5 4 3 2 1
Output
12
Note
In the first sample, the array is going to look as follows:
2 3 5 6 7 9 11
|2 - 2| + |1 - 3| + |5 - 5| + |11 - 6| + |5 - 7| + |9 - 9| + |11 - 11| = 9
And for the second sample:
1 2 3 4 5
|5 - 1| + |4 - 2| + |3 - 3| + |2 - 4| + |1 - 5| = 12
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
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Solve the programming task below in a Python markdown code block.
Anton likes to play chess, and so does his friend Danik.
Once they have played n games in a row. For each game it's known who was the winner β Anton or Danik. None of the games ended with a tie.
Now Anton wonders, who won more games, he or Danik? Help him determine this.
Input
The first line of the input contains a single integer n (1 β€ n β€ 100 000) β the number of games played.
The second line contains a string s, consisting of n uppercase English letters 'A' and 'D' β the outcome of each of the games. The i-th character of the string is equal to 'A' if the Anton won the i-th game and 'D' if Danik won the i-th game.
Output
If Anton won more games than Danik, print "Anton" (without quotes) in the only line of the output.
If Danik won more games than Anton, print "Danik" (without quotes) in the only line of the output.
If Anton and Danik won the same number of games, print "Friendship" (without quotes).
Examples
Input
6
ADAAAA
Output
Anton
Input
7
DDDAADA
Output
Danik
Input
6
DADADA
Output
Friendship
Note
In the first sample, Anton won 6 games, while Danik β only 1. Hence, the answer is "Anton".
In the second sample, Anton won 3 games and Danik won 4 games, so the answer is "Danik".
In the third sample, both Anton and Danik won 3 games and the answer is "Friendship".
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
In Berland it is the holiday of equality. In honor of the holiday the king decided to equalize the welfare of all citizens in Berland by the expense of the state treasury.
Totally in Berland there are n citizens, the welfare of each of them is estimated as the integer in ai burles (burle is the currency in Berland).
You are the royal treasurer, which needs to count the minimum charges of the kingdom on the king's present. The king can only give money, he hasn't a power to take away them.
Input
The first line contains the integer n (1 β€ n β€ 100) β the number of citizens in the kingdom.
The second line contains n integers a1, a2, ..., an, where ai (0 β€ ai β€ 106) β the welfare of the i-th citizen.
Output
In the only line print the integer S β the minimum number of burles which are had to spend.
Examples
Input
5
0 1 2 3 4
Output
10
Input
5
1 1 0 1 1
Output
1
Input
3
1 3 1
Output
4
Input
1
12
Output
0
Note
In the first example if we add to the first citizen 4 burles, to the second 3, to the third 2 and to the fourth 1, then the welfare of all citizens will equal 4.
In the second example it is enough to give one burle to the third citizen.
In the third example it is necessary to give two burles to the first and the third citizens to make the welfare of citizens equal 3.
In the fourth example it is possible to give nothing to everyone because all citizens have 12 burles.
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
Bob recently read about bitwise operations used in computers: AND, OR and XOR. He have studied their properties and invented a new game.
Initially, Bob chooses integer m, bit depth of the game, which means that all numbers in the game will consist of m bits. Then he asks Peter to choose some m-bit number. After that, Bob computes the values of n variables. Each variable is assigned either a constant m-bit number or result of bitwise operation. Operands of the operation may be either variables defined before, or the number, chosen by Peter. After that, Peter's score equals to the sum of all variable values.
Bob wants to know, what number Peter needs to choose to get the minimum possible score, and what number he needs to choose to get the maximum possible score. In both cases, if there are several ways to get the same score, find the minimum number, which he can choose.
Input
The first line contains two integers n and m, the number of variables and bit depth, respectively (1 β€ n β€ 5000; 1 β€ m β€ 1000).
The following n lines contain descriptions of the variables. Each line describes exactly one variable. Description has the following format: name of a new variable, space, sign ":=", space, followed by one of:
1. Binary number of exactly m bits.
2. The first operand, space, bitwise operation ("AND", "OR" or "XOR"), space, the second operand. Each operand is either the name of variable defined before or symbol '?', indicating the number chosen by Peter.
Variable names are strings consisting of lowercase Latin letters with length at most 10. All variable names are different.
Output
In the first line output the minimum number that should be chosen by Peter, to make the sum of all variable values minimum possible, in the second line output the minimum number that should be chosen by Peter, to make the sum of all variable values maximum possible. Both numbers should be printed as m-bit binary numbers.
Examples
Input
3 3
a := 101
b := 011
c := ? XOR b
Output
011
100
Input
5 1
a := 1
bb := 0
cx := ? OR a
d := ? XOR ?
e := d AND bb
Output
0
0
Note
In the first sample if Peter chooses a number 0112, then a = 1012, b = 0112, c = 0002, the sum of their values is 8. If he chooses the number 1002, then a = 1012, b = 0112, c = 1112, the sum of their values is 15.
For the second test, the minimum and maximum sum of variables a, bb, cx, d and e is 2, and this sum doesn't depend on the number chosen by Peter, so the minimum Peter can choose is 0.
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
The marmots have prepared a very easy problem for this year's HC2 β this one. It involves numbers n, k and a sequence of n positive integers a1, a2, ..., an. They also came up with a beautiful and riveting story for the problem statement. It explains what the input means, what the program should output, and it also reads like a good criminal.
However I, Heidi, will have none of that. As my joke for today, I am removing the story from the statement and replacing it with these two unhelpful paragraphs. Now solve the problem, fools!
Input
The first line of the input contains two space-separated integers n and k (1 β€ k β€ n β€ 2200). The second line contains n space-separated integers a1, ..., an (1 β€ ai β€ 104).
Output
Output one number.
Examples
Input
8 5
1 1 1 1 1 1 1 1
Output
5
Input
10 3
16 8 2 4 512 256 32 128 64 1
Output
7
Input
5 1
20 10 50 30 46
Output
10
Input
6 6
6 6 6 6 6 6
Output
36
Input
1 1
100
Output
100
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
Connect the countless points with lines, till we reach the faraway yonder.
There are n points on a coordinate plane, the i-th of which being (i, yi).
Determine whether it's possible to draw two parallel and non-overlapping lines, such that every point in the set lies on exactly one of them, and each of them passes through at least one point in the set.
Input
The first line of input contains a positive integer n (3 β€ n β€ 1 000) β the number of points.
The second line contains n space-separated integers y1, y2, ..., yn ( - 109 β€ yi β€ 109) β the vertical coordinates of each point.
Output
Output "Yes" (without quotes) if it's possible to fulfill the requirements, and "No" otherwise.
You can print each letter in any case (upper or lower).
Examples
Input
5
7 5 8 6 9
Output
Yes
Input
5
-1 -2 0 0 -5
Output
No
Input
5
5 4 3 2 1
Output
No
Input
5
1000000000 0 0 0 0
Output
Yes
Note
In the first example, there are five points: (1, 7), (2, 5), (3, 8), (4, 6) and (5, 9). It's possible to draw a line that passes through points 1, 3, 5, and another one that passes through points 2, 4 and is parallel to the first one.
In the second example, while it's possible to draw two lines that cover all points, they cannot be made parallel.
In the third example, it's impossible to satisfy both requirements at the same time.
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
You are given n distinct points on a plane with integral coordinates. For each point you can either draw a vertical line through it, draw a horizontal line through it, or do nothing.
You consider several coinciding straight lines as a single one. How many distinct pictures you can get? Print the answer modulo 109 + 7.
Input
The first line contains single integer n (1 β€ n β€ 105) β the number of points.
n lines follow. The (i + 1)-th of these lines contains two integers xi, yi ( - 109 β€ xi, yi β€ 109) β coordinates of the i-th point.
It is guaranteed that all points are distinct.
Output
Print the number of possible distinct pictures modulo 109 + 7.
Examples
Input
4
1 1
1 2
2 1
2 2
Output
16
Input
2
-1 -1
0 1
Output
9
Note
In the first example there are two vertical and two horizontal lines passing through the points. You can get pictures with any subset of these lines. For example, you can get the picture containing all four lines in two ways (each segment represents a line containing it).
The first way: <image> The second way: <image>
In the second example you can work with two points independently. The number of pictures is 32 = 9.
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
Since you are the best Wraith King, Nizhniy Magazin Β«MirΒ» at the centre of Vinnytsia is offering you a discount.
You are given an array a of length n and an integer c.
The value of some array b of length k is the sum of its elements except for the <image> smallest. For example, the value of the array [3, 1, 6, 5, 2] with c = 2 is 3 + 6 + 5 = 14.
Among all possible partitions of a into contiguous subarrays output the smallest possible sum of the values of these subarrays.
Input
The first line contains integers n and c (1 β€ n, c β€ 100 000).
The second line contains n integers ai (1 β€ ai β€ 109) β elements of a.
Output
Output a single integer β the smallest possible sum of values of these subarrays of some partition of a.
Examples
Input
3 5
1 2 3
Output
6
Input
12 10
1 1 10 10 10 10 10 10 9 10 10 10
Output
92
Input
7 2
2 3 6 4 5 7 1
Output
17
Input
8 4
1 3 4 5 5 3 4 1
Output
23
Note
In the first example any partition yields 6 as the sum.
In the second example one of the optimal partitions is [1, 1], [10, 10, 10, 10, 10, 10, 9, 10, 10, 10] with the values 2 and 90 respectively.
In the third example one of the optimal partitions is [2, 3], [6, 4, 5, 7], [1] with the values 3, 13 and 1 respectively.
In the fourth example one of the optimal partitions is [1], [3, 4, 5, 5, 3, 4], [1] with the values 1, 21 and 1 respectively.
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
Petya loves lucky numbers. Everybody knows that positive integers are lucky if their decimal representation doesn't contain digits other than 4 and 7. For example, numbers 47, 744, 4 are lucky and 5, 17, 467 are not.
Lucky number is super lucky if it's decimal representation contains equal amount of digits 4 and 7. For example, numbers 47, 7744, 474477 are super lucky and 4, 744, 467 are not.
One day Petya came across a positive integer n. Help him to find the least super lucky number which is not less than n.
Input
The only line contains a positive integer n (1 β€ n β€ 109). This number doesn't have leading zeroes.
Output
Output the least super lucky number that is more than or equal to n.
Please, do not use the %lld specificator to read or write 64-bit integers in C++. It is preferred to use the cin, cout streams or the %I64d specificator.
Examples
Input
4500
Output
4747
Input
47
Output
47
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
Unlike Knights of a Round Table, Knights of a Polygonal Table deprived of nobility and happy to kill each other. But each knight has some power and a knight can kill another knight if and only if his power is greater than the power of victim. However, even such a knight will torment his conscience, so he can kill no more than k other knights. Also, each knight has some number of coins. After a kill, a knight can pick up all victim's coins.
Now each knight ponders: how many coins he can have if only he kills other knights?
You should answer this question for each knight.
Input
The first line contains two integers n and k (1 β€ n β€ 10^5, 0 β€ k β€ min(n-1,10)) β the number of knights and the number k from the statement.
The second line contains n integers p_1, p_2 ,β¦,p_n (1 β€ p_i β€ 10^9) β powers of the knights. All p_i are distinct.
The third line contains n integers c_1, c_2 ,β¦,c_n (0 β€ c_i β€ 10^9) β the number of coins each knight has.
Output
Print n integers β the maximum number of coins each knight can have it only he kills other knights.
Examples
Input
4 2
4 5 9 7
1 2 11 33
Output
1 3 46 36
Input
5 1
1 2 3 4 5
1 2 3 4 5
Output
1 3 5 7 9
Input
1 0
2
3
Output
3
Note
Consider the first example.
* The first knight is the weakest, so he can't kill anyone. That leaves him with the only coin he initially has.
* The second knight can kill the first knight and add his coin to his own two.
* The third knight is the strongest, but he can't kill more than k = 2 other knights. It is optimal to kill the second and the fourth knights: 2+11+33 = 46.
* The fourth knight should kill the first and the second knights: 33+1+2 = 36.
In the second example the first knight can't kill anyone, while all the others should kill the one with the index less by one than their own.
In the third example there is only one knight, so he can't kill anyone.
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
Harry was contesting to be the most stylist person in his college. He had to collect maximum points from the judges to be able to win. However there was a problem. The judges were sitting in a line and each pair of adjacent judges had ego issues with each other. So if one judge gave X points to Harry then the next judge wonβt give him any points. Harry had a friend in the organizing team and through him he had found out the exact points he would get from each judge if he chose their score to be considered. Help him find out the maximum points he can score.
INPUT
The first line of input contains the number of test cases, T.
0 < T < = 10
Each test case starts with a number N, the number of judges.
0 β€ N < = 10^4.
The next line will have N numbers, number of points each judge gave Harry
0 < = X(i) < = 10^9.
The order of the judges does not change.
OUTPUT
For each test case print βCase T: Aβ without quotes in a single line.
T is the case number, starting with 1.
A is the maximum number of points Harry can collect.
SAMPLE INPUT
2
5
1 2 3 4 5
1
10
SAMPLE OUTPUT
Case 1: 9
Case 2: 10
Explanation
Case 1 :
Harry chooses judges 1,3 and 5 with points 1 ,3 and 5 respectively to add up to 9 points in total.
Case 2:
There is only one judge so Harry chooses his points alone.
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
PRAVAH, an annual techfest is being organised at SKIT. Naina comes up with an idea for an event Guess The Number. In this event, the computer thinks up a random number and the player is asked to guess that number and enter it in the system. if the number is correct, the player wins.
Rakesh, a student who is good at hacking finds a breach in the system and has got the access the modify any 1 digit of the guessed number, but this access is only granted after the number has been entered by a player.
So, at the event, the computer thinks of a number A, a friend of Rakesh enters a number B and Rakesh changes one digit of the number A, so that his friend wins. You are given two integers A and B. Print "wins" if there is a possibility that the friend will win by changing of a digit, otherwise, print "loses". If the numbers A and B are same print "wins".
Input
First line of input contains the number generated by system.
Second line of input contains the number entered by player.
Output
A single word "wins" or "loses"
Constraints
1 β€ A β€ 900000
1 β€ B β€ 900000
Number of digits of A = Number of digits of B
Sample Input 1
1111
1011
Sample Output 1
wins
Sample Input 2
108
110
Sample Output 2
loses
SAMPLE INPUT
SAMPLE OUTPUT
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
Let us see how search engines work. Consider the following simple auto complete feature. When you type some characters in the text bar, the engine automatically gives best matching options among it's database. Your job is simple. Given an incomplete search text, output the best search result.
Each entry in engine's database has a priority factor attached to it. We consider a result / search suggestion best if it has maximum weight and completes the given incomplete search query. For each query in the input, print the maximum weight of the string in the database, that completes the given incomplete search string. In case no such string exists, print -1.
INPUT
First line contains two integers n and q, which represent number of database entries and number of search queries need to be completed. Next n lines contain a string s and an integer weight, which are the database entry and it's corresponding priority.
Next q lines follow, each line having a string t, which needs to be completed.
OUTPUT
Output q lines, each line containing the maximum possible weight of the match for given query, else -1, in case no valid result is obtained.
CONSTRAINTS
1 β€ n, weight, len(s), len(t) β€ 10^6
1 β€ q β€ 10^5
total length of all strings in database entries β€ 10^6
total length of all query strings β€ 10^6
SAMPLE INPUT
2 1
hackerearth 10
hackerrank 9
hacker
SAMPLE OUTPUT
10
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
Takahashi is a member of a programming competition site, ButCoder.
Each member of ButCoder is assigned two values: Inner Rating and Displayed Rating.
The Displayed Rating of a member is equal to their Inner Rating if the member has participated in 10 or more contests. Otherwise, the Displayed Rating will be their Inner Rating minus 100 \times (10 - K) when the member has participated in K contests.
Takahashi has participated in N contests, and his Displayed Rating is R. Find his Inner Rating.
Constraints
* All values in input are integers.
* 1 \leq N \leq 100
* 0 \leq R \leq 4111
Input
Input is given from Standard Input in the following format:
N R
Output
Print his Inner Rating.
Examples
Input
2 2919
Output
3719
Input
22 3051
Output
3051
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
Takahashi, who is A years old, is riding a Ferris wheel.
It costs B yen (B is an even number) to ride the Ferris wheel if you are 13 years old or older, but children between 6 and 12 years old (inclusive) can ride it for half the cost, and children who are 5 years old or younger are free of charge. (Yen is the currency of Japan.)
Find the cost of the Ferris wheel for Takahashi.
Constraints
* 0 β€ A β€ 100
* 2 β€ B β€ 1000
* B is an even number.
Input
Input is given from Standard Input in the following format:
A B
Output
Print the cost of the Ferris wheel for Takahashi.
Examples
Input
30 100
Output
100
Input
12 100
Output
50
Input
0 100
Output
0
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
Find the sum of the integers between 1 and N (inclusive), whose sum of digits written in base 10 is between A and B (inclusive).
Constraints
* 1 \leq N \leq 10^4
* 1 \leq A \leq B \leq 36
* All input values are integers.
Input
Input is given from Standard Input in the following format:
N A B
Output
Print the sum of the integers between 1 and N (inclusive), whose sum of digits written in base 10 is between A and B (inclusive).
Examples
Input
20 2 5
Output
84
Input
10 1 2
Output
13
Input
100 4 16
Output
4554
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
This contest, AtCoder Beginner Contest, is abbreviated as ABC.
When we refer to a specific round of ABC, a three-digit number is appended after ABC. For example, ABC680 is the 680th round of ABC.
What is the abbreviation for the N-th round of ABC? Write a program to output the answer.
Constraints
* 100 β€ N β€ 999
Input
Input is given from Standard Input in the following format:
N
Output
Print the abbreviation for the N-th round of ABC.
Examples
Input
100
Output
ABC100
Input
425
Output
ABC425
Input
999
Output
ABC999
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
Snuke loves constructing integer sequences.
There are N piles of stones, numbered 1 through N. The pile numbered i consists of a_i stones.
Snuke will construct an integer sequence s of length Ξ£a_i, as follows:
1. Among the piles with the largest number of stones remaining, let x be the index of the pile with the smallest index. Append x to the end of s.
2. Select a pile with one or more stones remaining, and remove a stone from that pile.
3. If there is a pile with one or more stones remaining, go back to step 1. Otherwise, terminate the process.
We are interested in the lexicographically smallest sequence that can be constructed. For each of the integers 1,2,3,...,N, how many times does it occur in the lexicographically smallest sequence?
Constraints
* 1 β€ N β€ 10^{5}
* 1 β€ a_i β€ 10^{9}
Input
The input is given from Standard Input in the following format:
N
a_1 a_2 ... a_{N}
Output
Print N lines. The i-th line should contain the number of the occurrences of the integer i in the lexicographically smallest sequence that can be constructed.
Examples
Input
2
1 2
Output
2
1
Input
10
1 2 1 3 2 4 2 5 8 1
Output
10
7
0
4
0
3
0
2
3
0
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
Read the coordinates of four different points on the plane, $ A (x_A, y_A) $, $ B (x_B, y_B) $, $ C (x_C, y_C) $, $ D (x_D, y_D) $, and straight line $ Create a program that outputs YES if AB $ and $ CD $ are orthogonal, and NO if they are not orthogonal. Here, "straight line" does not mean a line segment. Please refer to the figure below.
<image>
Input
Given multiple datasets. The format of each dataset is as follows.
$ x_A $ $ y_A $ $ x_B $ $ y_B $ $ x_C $ $ y_C $ $ x_D $ $ y_D $
$ x_A $, $ y_A $, $ x_B $, $ y_B $, $ x_C $, $ y_C $, $ x_D $, $ y_D $ are each -100 or more and 100 or less, and each value has a maximum of 5 digits after the decimal point. It is given as a real number including the number of.
The number of datasets does not exceed 100.
Output
Print YES or NO on one line for each dataset.
Example
Input
1.0 1.0 2.0 2.0 0.0 0.0 1.0 -1.0
0.0 0.0 2.0 0.0 -1.0 2.0 2.0 2.0
10.0 6.0 3.4 5.2 6.8 9.5 4.3 2.1
2.5 3.5 2.5 4.5 -3.3 -2.3 6.8 -2.3
Output
YES
NO
NO
YES
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
Mr. A, who will graduate next spring, decided to move when he got a job. The company that finds a job has offices in several towns, and the offices that go to work differ depending on the day. So Mr. A decided to live in a town where he had a short time to go to any office.
So you decided to find the most convenient town to live in to help Mr. A.
<image>
Towns are numbered starting with 0 and there is a road between towns. Commuting time is fixed for each road. If Mr. A lives in a town, the commuting time to the office in his town is 0. At this time, consider the total commuting time to all towns. For example, if the layout of towns and roads is as shown in the figure above and Mr. A lives in town 1, the commuting time to each town will be
Town 0 to 80
0 to town 1
Up to town 2 20
Up to town 3 70
Town 4 to 90
And the sum is 260.
Enter the number of roads and information on all roads, calculate the total commuting time when living in each town, and output the number of the town that minimizes it and the total commuting time at that time. Create a program. However, if there are multiple towns with the smallest total commuting time, please output the number of the smallest town and the total commuting time at that time. The total number of towns is 10 or less, the total number of roads is 45 or less, all roads can move in both directions, and commuting time does not change depending on the direction. We also assume that there are routes from any town to all other towns.
Input
A sequence of multiple datasets is given as input. The end of the input is indicated by a single line of zeros. Each dataset is given in the following format:
n
a1 b1 c1
a2 b2 c2
::
an bn cn
The number of roads n (1 β€ n β€ 45) is given on the first line. The following n lines give information on the i-th way. ai, bi (0 β€ ai, bi β€ 9) is the number of the town to which the i-th road connects, and ci (0 β€ ci β€ 100) is the commute time for that road.
Output
For each data set, the town number that minimizes the total commuting time and the total commuting time at that time are output on one line separated by blanks.
Example
Input
6
0 1 80
1 2 20
0 2 60
2 3 50
3 4 60
1 4 90
2
0 1 1
1 2 1
0
Output
2 240
1 2
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
Yuki made a sugoroku so that everyone can play at the children's association event. In this sugoroku, squares are lined up in a ring, and each square has an integer of 1 or more written on it.
The player chooses a square as a starting point and places his piece. Advance the pieces clockwise by the number written on the square. Advance the pieces clockwise again by the number written on the stopped square. Repeat this, and when the piece stops on the square selected as the starting point, it is "Agari".
In reality, depending on how you choose the squares, it may never be "finished". Yuki is trying to count the number of squares that can reach the "Agari" in this sugoroku.
Create a program that inputs sugoroku information and reports the number of squares that can reach "Agari".
Input
The input is given in the following format.
N
a1 a2 ... aN
The number N (1 β€ N β€ 100000) of all the cells contained in the sugoroku is given in the first line. On the second line, the number ai (1 β€ ai β€ 109) written in each cell is given in turn clockwise.
Output
The number of squares that can reach "Agari" is output in one line.
Examples
Input
3
1 1 1
Output
3
Input
3
1 1 2
Output
2
Input
8
2 3 7 3 3 3 4 4
Output
6
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
Solve the programming task below in a Python markdown code block.
A Bingo game is played by one gamemaster and several players. At the beginning of a game, each player is given a card with M Γ M numbers in a matrix (See Figure 10).
<image>
As the game proceeds, the gamemaster announces a series of numbers one by one. Each player punches a hole in his card on the announced number, if any.
When at least one 'Bingo' is made on the card, the player wins and leaves the game. The 'Bingo' means that all the M numbers in a line are punched vertically, horizontally or diagonally (See Figure 11).
<image>
The gamemaster continues announcing numbers until all the players make a Bingo.
In the ordinary Bingo games, the gamemaster chooses numbers by a random process and has no control on them. But in this problem the gamemaster knows all the cards at the beginning of the game and controls the game by choosing the number sequence to be announced at his will.
Specifically, he controls the game to satisfy the following condition.
Cardi makes a Bingo no later than Cardj, for i < j. (*)
Figure 12 shows an example of how a game proceeds. The gamemaster cannot announce '5' before '16', because Card4 makes a Bingo before Card2 and Card3, violating the condition (*).
Your job is to write a program which finds the minimum length of such sequence of numbers for the given cards.
Input
The input consists of multiple datasets. The format of each dataset is as follows.
<image>
All data items are integers. P is the number of the cards, namely the number of the players. M is the number of rows and the number of columns of the matrix on each card. Nkij means the number written at the position (i, j) on the k-th card. If (i, j) β (p, q), then Nkij β Nkpq. The parameters P, M, and N satisfy the conditions 2 β€ P β€ 4, 3 β€ M β€ 4, and 0 β€ Nkij β€ 99.
The end of the input is indicated by a line containing two zeros separated by a space. It is not a dataset.
Output
For each dataset, output the minimum length of the sequence of numbers which satisfy the condition (*). Output a zero if there are no such sequences. Output for each dataset must be printed on a separate line.
Example
Input
4 3
10 25 11 20 6 2 1 15 23
5 21 3 12 23 17 7 26 2
8 18 4 22 13 27 16 5 11
19 9 24 2 11 5 14 28 16
4 3
12 13 20 24 28 32 15 16 17
12 13 21 25 29 33 16 17 18
12 13 22 26 30 34 17 18 15
12 13 23 27 31 35 18 15 16
4 3
11 12 13 14 15 16 17 18 19
21 22 23 24 25 26 27 28 29
31 32 33 34 35 36 37 38 39
41 42 43 44 45 46 47 48 49
4 4
2 6 9 21 15 23 17 31 33 12 25 4 8 24 13 36
22 18 27 26 35 28 3 7 11 20 38 16 5 32 14 29
26 7 16 29 27 3 38 14 18 28 20 32 22 35 11 5
36 13 24 8 4 25 12 33 31 17 23 15 21 9 6 2
0 0
Output
5
4
12
0
The input will be give
Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
|
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