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Peter got a new snow blower as a New Year present. Of course, Peter decided to try it immediately. After reading the instructions he realized that it does not work like regular snow blowing machines. In order to make it work, you need to tie it to some point that it does not cover, and then switch it on. As a result it will go along a circle around this point and will remove all the snow from its path.Formally, we assume that Peter's machine is a polygon on a plane. Then, after the machine is switched on, it will make a circle around the point to which Peter tied it (this point lies strictly outside the polygon). That is, each of the points lying within or on the border of the polygon will move along the circular trajectory, with the center of the circle at the point to which Peter tied his machine.Peter decided to tie his car to point P and now he is wondering what is the area of ββthe region that will be cleared from snow. Help him. | Input: ['3 0 00 1-1 21 2'] Output:['12.566370614359172464'] | [
4,
4
] |
A permutation of length n is an array containing each integer from 1 to n exactly once. For example, qβ=β[4,β5,β1,β2,β3] is a permutation. For the permutation q the square of permutation is the permutation p that p[i]β=βq[q[i]] for each iβ=β1... n. For example, the square of qβ=β[4,β5,β1,β2,β3] is pβ=βq2β=β[2,β3,β4,β5,β1].This problem is about the inverse operation: given the permutation p you task is to find such permutation q that q2β=βp. If there are several such q find any of them. | Input: ['42 1 4 3'] Output:['3 4 2 1'] | [
3
] |
You are given n segments on the coordinate axis Ox and the number k. The point is satisfied if it belongs to at least k segments. Find the smallest (by the number of segments) set of segments on the coordinate axis Ox which contains all satisfied points and no others. | Input: ['3 20 5-3 23 8'] Output:['20 23 5'] | [
2
] |
You are given string s consists of opening and closing brackets of four kinds <>, {}, [], (). There are two types of brackets: opening and closing. You can replace any bracket by another of the same type. For example, you can replace < by the bracket {, but you can't replace it by ) or >.The following definition of a regular bracket sequence is well-known, so you can be familiar with it.Let's define a regular bracket sequence (RBS). Empty string is RBS. Let s1 and s2 be a RBS then the strings <s1>s2, {s1}s2, [s1]s2, (s1)s2 are also RBS.For example the string "[[(){}]<>]" is RBS, but the strings "[)()" and "][()()" are not.Determine the least number of replaces to make the string s RBS. | Input: ['[<}){}'] Output:['2'] | [
3
] |
HDD hard drives group data by sectors. All files are split to fragments and each of them are written in some sector of hard drive. Note the fragments can be written in sectors in arbitrary order.One of the problems of HDD hard drives is the following: the magnetic head should move from one sector to another to read some file.Find the time need to read file split to n fragments. The i-th sector contains the fi-th fragment of the file (1ββ€βfiββ€βn). Note different sectors contains the different fragments. At the start the magnetic head is in the position that contains the first fragment. The file are reading in the following manner: at first the first fragment is read, then the magnetic head moves to the sector that contains the second fragment, then the second fragment is read and so on until the n-th fragment is read. The fragments are read in the order from the first to the n-th.It takes |aβ-βb| time units to move the magnetic head from the sector a to the sector b. Reading a fragment takes no time. | Input: ['33 1 2'] Output:['3'] | [
3
] |
You are given the string s of length n and the numbers p,βq. Split the string s to pieces of length p and q.For example, the string "Hello" for pβ=β2, qβ=β3 can be split to the two strings "Hel" and "lo" or to the two strings "He" and "llo".Note it is allowed to split the string s to the strings only of length p or to the strings only of length q (see the second sample test). | Input: ['5 2 3Hello'] Output:['2Hello'] | [
0
] |
Limak is a little polar bear. His parents told him to clean a house before the New Year's Eve. Their house is a rectangular grid with h rows and w columns. Each cell is an empty square.He is a little bear and thus he can't clean a house by himself. Instead, he is going to use a cleaning robot.A cleaning robot has a built-in pattern of n moves, defined by a string of the length n. A single move (character) moves a robot to one of four adjacent cells. Each character is one of the following four: 'U' (up), 'D' (down), 'L' (left), 'R' (right). One move takes one minute.A cleaning robot must be placed and started in some cell. Then it repeats its pattern of moves till it hits a wall (one of four borders of a house). After hitting a wall it can be placed and used again.Limak isn't sure if placing a cleaning robot in one cell will be enough. Thus, he is going to start it wΒ·h times, one time in each cell. Maybe some cells will be cleaned more than once but who cares?Limak asks you one question. How much time will it take to clean a house? Find and print the number of minutes modulo 109β+β7. It's also possible that a cleaning robot will never stop β then print "-1" (without the quotes) instead.Placing and starting a robot takes no time, however, you must count a move when robot hits a wall. Take a look into samples for further clarification. | Input: ['1 10 2R'] Output:['30'] | [
4
] |
Do you know the story about the three musketeers? Anyway, you must help them now.Richelimakieu is a cardinal in the city of Bearis. He found three brave warriors and called them the three musketeers. Athos has strength a, Borthos strength b, and Caramis has strength c.The year 2015 is almost over and there are still n criminals to be defeated. The i-th criminal has strength ti. It's hard to defeat strong criminals β maybe musketeers will have to fight together to achieve it.Richelimakieu will coordinate musketeers' actions. In each hour each musketeer can either do nothing or be assigned to one criminal. Two or three musketeers can be assigned to the same criminal and then their strengths are summed up. A criminal can be defeated in exactly one hour (also if two or three musketeers fight him). Richelimakieu can't allow the situation where a criminal has strength bigger than the sum of strengths of musketeers fighting him β a criminal would win then!In other words, there are three ways to defeat a criminal. A musketeer of the strength x in one hour can defeat a criminal of the strength not greater than x. So, for example Athos in one hour can defeat criminal i only if tiββ€βa. Two musketeers can fight together and in one hour defeat a criminal of the strength not greater than the sum of strengths of these two musketeers. So, for example Athos and Caramis in one hour can defeat criminal i only if tiββ€βaβ+βc. Note that the third remaining musketeer can either do nothing or fight some other criminal. Similarly, all three musketeers can fight together and in one hour defeat a criminal of the strength not greater than the sum of musketeers' strengths, i.e. tiββ€βaβ+βbβ+βc. Richelimakieu doesn't want musketeers to fight during the New Year's Eve. Thus, he must coordinate their actions in order to minimize the number of hours till all criminals will be defeated.Find the minimum number of hours to defeat all criminals. If musketeers can't defeat them all then print "-1" (without the quotes) instead. | Input: ['510 20 301 1 1 1 50'] Output:['2'] | [
2
] |
The year 2015 is almost over.Limak is a little polar bear. He has recently learnt about the binary system. He noticed that the passing year has exactly one zero in its representation in the binary system β 201510β=β111110111112. Note that he doesn't care about the number of zeros in the decimal representation.Limak chose some interval of years. He is going to count all years from this interval that have exactly one zero in the binary representation. Can you do it faster?Assume that all positive integers are always written without leading zeros. | Input: ['5 10'] Output:['2'] | [
0
] |
Pasha has a wooden stick of some positive integer length n. He wants to perform exactly three cuts to get four parts of the stick. Each part must have some positive integer length and the sum of these lengths will obviously be n. Pasha likes rectangles but hates squares, so he wonders, how many ways are there to split a stick into four parts so that it's possible to form a rectangle using these parts, but is impossible to form a square.Your task is to help Pasha and count the number of such ways. Two ways to cut the stick are considered distinct if there exists some integer x, such that the number of parts of length x in the first way differ from the number of parts of length x in the second way. | Input: ['6'] Output:['1'] | [
3
] |
There are n frogs sitting on the coordinate axis Ox. For each frog two values xi,βti are known β the position and the initial length of the tongue of the i-th frog (it is guaranteed that all positions xi are different). m mosquitoes one by one are landing to the coordinate axis. For each mosquito two values are known pj β the coordinate of the position where the j-th mosquito lands and bj β the size of the j-th mosquito. Frogs and mosquitoes are represented as points on the coordinate axis.The frog can eat mosquito if mosquito is in the same position with the frog or to the right, and the distance between them is not greater than the length of the tongue of the frog.If at some moment several frogs can eat a mosquito the leftmost frog will eat it (with minimal xi). After eating a mosquito the length of the tongue of a frog increases with the value of the size of eaten mosquito. It's possible that after it the frog will be able to eat some other mosquitoes (the frog should eat them in this case).For each frog print two values β the number of eaten mosquitoes and the length of the tongue after landing all mosquitoes and after eating all possible mosquitoes by frogs.Each mosquito is landing to the coordinate axis only after frogs eat all possible mosquitoes landed before. Mosquitoes are given in order of their landing to the coordinate axis. | Input: ['4 610 215 06 10 1110 101 16 015 1014 10012 2'] Output:['3 1141 101 11 2'] | [
2
] |
Nura wants to buy k gadgets. She has only s burles for that. She can buy each gadget for dollars or for pounds. So each gadget is selling only for some type of currency. The type of currency and the cost in that currency are not changing.Nura can buy gadgets for n days. For each day you know the exchange rates of dollar and pound, so you know the cost of conversion burles to dollars or to pounds.Each day (from 1 to n) Nura can buy some gadgets by current exchange rate. Each day she can buy any gadgets she wants, but each gadget can be bought no more than once during n days.Help Nura to find the minimum day index when she will have k gadgets. Nura always pays with burles, which are converted according to the exchange rate of the purchase day. Nura can't buy dollars or pounds, she always stores only burles. Gadgets are numbered with integers from 1 to m in order of their appearing in input. | Input: ['5 4 2 21 2 3 2 13 2 1 2 31 12 11 22 2'] Output:['31 12 3'] | [
2,
4
] |
In the school computer room there are n servers which are responsible for processing several computing tasks. You know the number of scheduled tasks for each server: there are mi tasks assigned to the i-th server.In order to balance the load for each server, you want to reassign some tasks to make the difference between the most loaded server and the least loaded server as small as possible. In other words you want to minimize expression maβ-βmb, where a is the most loaded server and b is the least loaded one.In one second you can reassign a single task. Thus in one second you can choose any pair of servers and move a single task from one server to another.Write a program to find the minimum number of seconds needed to balance the load of servers. | Input: ['21 6'] Output:['2'] | [
3
] |
Sean is trying to save a large file to a USB flash drive. He has n USB flash drives with capacities equal to a1,βa2,β...,βan megabytes. The file size is equal to m megabytes. Find the minimum number of USB flash drives needed to write Sean's file, if he can split the file between drives. | Input: ['35213'] Output:['2'] | [
2
] |
Saitama accidentally destroyed a hotel again. To repay the hotel company, Genos has volunteered to operate an elevator in one of its other hotels. The elevator is special β it starts on the top floor, can only move down, and has infinite capacity. Floors are numbered from 0 to s and elevator initially starts on floor s at time 0.The elevator takes exactly 1 second to move down exactly 1 floor and negligible time to pick up passengers. Genos is given a list detailing when and on which floor passengers arrive. Please determine how long in seconds it will take Genos to bring all passengers to floor 0. | Input: ['3 72 13 85 2'] Output:['11'] | [
3
] |
Genos has been given n distinct lines on the Cartesian plane. Let be a list of intersection points of these lines. A single point might appear multiple times in this list if it is the intersection of multiple pairs of lines. The order of the list does not matter.Given a query point (p,βq), let be the corresponding list of distances of all points in to the query point. Distance here refers to euclidean distance. As a refresher, the euclidean distance between two points (x1,βy1) and (x2,βy2) is .Genos is given a point (p,βq) and a positive integer m. He is asked to find the sum of the m smallest elements in . Duplicate elements in are treated as separate elements. Genos is intimidated by Div1 E problems so he asked for your help. | Input: ['41000 1000 31000 0-1000 00 50000 -5000'] Output:['14.282170363'] | [
4
] |
There are n beacons located at distinct positions on a number line. The i-th beacon has position ai and power level bi. When the i-th beacon is activated, it destroys all beacons to its left (direction of decreasing coordinates) within distance bi inclusive. The beacon itself is not destroyed however. Saitama will activate the beacons one at a time from right to left. If a beacon is destroyed, it cannot be activated.Saitama wants Genos to add a beacon strictly to the right of all the existing beacons, with any position and any power level, such that the least possible number of beacons are destroyed. Note that Genos's placement of the beacon means it will be the first beacon activated. Help Genos by finding the minimum number of beacons that could be destroyed. | Input: ['41 93 16 17 4'] Output:['1'] | [
4
] |
An infinitely long railway has a train consisting of n cars, numbered from 1 to n (the numbers of all the cars are distinct) and positioned in arbitrary order. David Blaine wants to sort the railway cars in the order of increasing numbers. In one move he can make one of the cars disappear from its place and teleport it either to the beginning of the train, or to the end of the train, at his desire. What is the minimum number of actions David Blaine needs to perform in order to sort the train? | Input: ['54 1 2 5 3'] Output:['2'] | [
2
] |
Kevin Sun wants to move his precious collection of n cowbells from Naperthrill to Exeter, where there is actually grass instead of corn. Before moving, he must pack his cowbells into k boxes of a fixed size. In order to keep his collection safe during transportation, he won't place more than two cowbells into a single box. Since Kevin wishes to minimize expenses, he is curious about the smallest size box he can use to pack his entire collection. Kevin is a meticulous cowbell collector and knows that the size of his i-th (1ββ€βiββ€βn) cowbell is an integer si. In fact, he keeps his cowbells sorted by size, so siβ-β1ββ€βsi for any iβ>β1. Also an expert packer, Kevin can fit one or two cowbells into a box of size s if and only if the sum of their sizes does not exceed s. Given this information, help Kevin determine the smallest s for which it is possible to put all of his cowbells into k boxes of size s. | Input: ['2 12 5'] Output:['7'] | [
2,
4
] |
In the land of Bovinia there are n pastures, but no paths connecting the pastures. Of course, this is a terrible situation, so Kevin Sun is planning to rectify it by constructing m undirected paths connecting pairs of distinct pastures. To make transportation more efficient, he also plans to pave some of these new paths.Kevin is very particular about certain aspects of path-paving. Since he loves odd numbers, he wants each pasture to have an odd number of paved paths connected to it. Thus we call a paving sunny if each pasture is incident to an odd number of paved paths. He also enjoys short paths more than long paths, so he would like the longest paved path to be as short as possible. After adding each path, Kevin wants to know if a sunny paving exists for the paths of Bovinia, and if at least one does, the minimum possible length of the longest path in such a paving. Note that "longest path" here means maximum-weight edge. | Input: ['4 41 3 42 4 81 2 23 4 3'] Output:['-1883'] | [
3
] |
Kevin Sun is ruminating on the origin of cows while standing at the origin of the Cartesian plane. He notices n lines on the plane, each representable by an equation of the form axβ+βbyβ=βc. He also observes that no two lines are parallel and that no three lines pass through the same point.For each triple (i,βj,βk) such that 1ββ€βiβ<βjβ<βkββ€βn, Kevin considers the triangle formed by the three lines . He calls a triangle original if the circumcircle of that triangle passes through the origin. Since Kevin believes that the circles of bovine life are tied directly to such triangles, he wants to know the number of original triangles formed by unordered triples of distinct lines. Recall that the circumcircle of a triangle is the circle which passes through all the vertices of that triangle. | Input: ['41 0 00 1 01 1 -11 -1 2'] Output:['2'] | [
3
] |
Kevin and Nicky Sun have invented a new game called Lieges of Legendre. In this game, two players take turns modifying the game state with Kevin moving first. Initially, the game is set up so that there are n piles of cows, with the i-th pile containing ai cows. During each player's turn, that player calls upon the power of Sunlight, and uses it to either: Remove a single cow from a chosen non-empty pile. Choose a pile of cows with even size 2Β·x (xβ>β0), and replace it with k piles of x cows each. The player who removes the last cow wins. Given n, k, and a sequence a1,βa2,β...,βan, help Kevin and Nicky find the winner, given that both sides play in optimal way. | Input: ['2 13 4'] Output:['Kevin'] | [
3
] |
As behooves any intelligent schoolboy, Kevin Sun is studying psycowlogy, cowculus, and cryptcowgraphy at the Bovinia State University (BGU) under Farmer Ivan. During his Mathematics of Olympiads (MoO) class, Kevin was confronted with a weird functional equation and needs your help. For two fixed integers k and p, where p is an odd prime number, the functional equation states that for some function . (This equation should hold for any integer x in the range 0 to pβ-β1, inclusive.)It turns out that f can actually be many different functions. Instead of finding a solution, Kevin wants you to count the number of distinct functions f that satisfy this equation. Since the answer may be very large, you should print your result modulo 109β+β7. | Input: ['3 2'] Output:['3'] | [
3
] |
Kevin has just recevied his disappointing results on the USA Identification of Cows Olympiad (USAICO) in the form of a binary string of length n. Each character of Kevin's string represents Kevin's score on one of the n questions of the olympiadβ'1' for a correctly identified cow and '0' otherwise.However, all is not lost. Kevin is a big proponent of alternative thinking and believes that his score, instead of being the sum of his points, should be the length of the longest alternating subsequence of his string. Here, we define an alternating subsequence of a string as a not-necessarily contiguous subsequence where no two consecutive elements are equal. For example, {0,β1,β0,β1}, {1,β0,β1}, and {1,β0,β1,β0} are alternating sequences, while {1,β0,β0} and {0,β1,β0,β1,β1} are not.Kevin, being the sneaky little puffball that he is, is willing to hack into the USAICO databases to improve his score. In order to be subtle, he decides that he will flip exactly one substringβthat is, take a contiguous non-empty substring of his score and change all '0's in that substring to '1's and vice versa. After such an operation, Kevin wants to know the length of the longest possible alternating subsequence that his string could have. | Input: ['810000011'] Output:['5'] | [
2,
3
] |
After seeing the "ALL YOUR BASE ARE BELONG TO US" meme for the first time, numbers X and Y realised that they have different bases, which complicated their relations.You're given a number X represented in base bx and a number Y represented in base by. Compare those two numbers. | Input: ['6 21 0 1 1 1 12 104 7'] Output:['='] | [
0
] |
KleofΓ‘Ε‘ is participating in an n-thlon - a tournament consisting of n different competitions in n different disciplines (numbered 1 through n). There are m participants in the n-thlon and each of them participates in all competitions.In each of these n competitions, the participants are given ranks from 1 to m in such a way that no two participants are given the same rank - in other words, the ranks in each competition form a permutation of numbers from 1 to m. The score of a participant in a competition is equal to his/her rank in it.The overall score of each participant is computed as the sum of that participant's scores in all competitions.The overall rank of each participant is equal to 1β+βk, where k is the number of participants with strictly smaller overall score.The n-thlon is over now, but the results haven't been published yet. KleofΓ‘Ε‘ still remembers his ranks in each particular competition; however, he doesn't remember anything about how well the other participants did. Therefore, KleofΓ‘Ε‘ would like to know his expected overall rank.All competitors are equally good at each discipline, so all rankings (permutations of ranks of everyone except KleofΓ‘Ε‘) in each competition are equiprobable. | Input: ['4 102121'] Output:['1.0000000000000000'] | [
3
] |
A function is called Lipschitz continuous if there is a real constant K such that the inequality |f(x)β-βf(y)|ββ€βKΒ·|xβ-βy| holds for all . We'll deal with a more... discrete version of this term.For an array , we define it's Lipschitz constant as follows: if nβ<β2, if nββ₯β2, over all 1ββ€βiβ<βjββ€βn In other words, is the smallest non-negative integer such that |h[i]β-βh[j]|ββ€βLΒ·|iβ-βj| holds for all 1ββ€βi,βjββ€βn.You are given an array of size n and q queries of the form [l,βr]. For each query, consider the subarray ; determine the sum of Lipschitz constants of all subarrays of . | Input: ['10 41 5 2 9 1 3 4 2 1 72 43 87 101 9'] Output:['178223210'] | [
3
] |
A string is called palindrome if it reads the same from left to right and from right to left. For example "kazak", "oo", "r" and "mikhailrubinchikkihcniburliahkim" are palindroms, but strings "abb" and "ij" are not.You are given string s consisting of lowercase Latin letters. At once you can choose any position in the string and change letter in that position to any other lowercase letter. So after each changing the length of the string doesn't change. At first you can change some letters in s. Then you can permute the order of letters as you want. Permutation doesn't count as changes. You should obtain palindrome with the minimal number of changes. If there are several ways to do that you should get the lexicographically (alphabetically) smallest palindrome. So firstly you should minimize the number of changes and then minimize the palindrome lexicographically. | Input: ['aabc'] Output:['abba'] | [
2
] |
You are given two arrays of integers a and b. For each element of the second array bj you should find the number of elements in array a that are less than or equal to the value bj. | Input: ['5 41 3 5 7 96 4 2 8'] Output:['3 2 1 4'] | [
4
] |
Spongebob is already tired trying to reason his weird actions and calculations, so he simply asked you to find all pairs of n and m, such that there are exactly x distinct squares in the table consisting of n rows and m columns. For example, in a 3βΓβ5 table there are 15 squares with side one, 8 squares with side two and 3 squares with side three. The total number of distinct squares in a 3βΓβ5 table is 15β+β8β+β3β=β26. | Input: ['26'] Output:['61 262 93 55 39 226 1'] | [
0,
3
] |
You have a rectangular chocolate bar consisting of nβΓβm single squares. You want to eat exactly k squares, so you may need to break the chocolate bar. In one move you can break any single rectangular piece of chocolate in two rectangular pieces. You can break only by lines between squares: horizontally or vertically. The cost of breaking is equal to square of the break length.For example, if you have a chocolate bar consisting of 2βΓβ3 unit squares then you can break it horizontally and get two 1βΓβ3 pieces (the cost of such breaking is 32β=β9), or you can break it vertically in two ways and get two pieces: 2βΓβ1 and 2βΓβ2 (the cost of such breaking is 22β=β4).For several given values n, m and k find the minimum total cost of breaking. You can eat exactly k squares of chocolate if after all operations of breaking there is a set of rectangular pieces of chocolate with the total size equal to k squares. The remaining nΒ·mβ-βk squares are not necessarily form a single rectangular piece. | Input: ['42 2 12 2 32 2 22 2 4'] Output:['5540'] | [
0
] |
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 nβ=β4 the sum is equal to β-β1β-β2β+β3β-β4β=ββ-β4, because 1, 2 and 4 are 20, 21 and 22 respectively.Calculate the answer for t values of n. | Input: ['241000000000'] Output:['-4499999998352516354'] | [
3
] |
A restaurant received n orders for the rental. Each rental order reserve the restaurant for a continuous period of time, the i-th order is characterized by two time values β the start time li and the finish time ri (liββ€βri).Restaurant management can accept and reject orders. What is the maximal number of orders the restaurant can accept?No two accepted orders can intersect, i.e. they can't share even a moment of time. If one order ends in the moment other starts, they can't be accepted both. | Input: ['27 114 7'] Output:['1'] | [
2
] |
Find the number of k-divisible numbers on the segment [a,βb]. In other words you need to find the number of such integer values x that aββ€βxββ€βb and x is divisible by k. | Input: ['1 1 10'] Output:['10'] | [
3
] |
Wilbur the pig really wants to be a beaver, so he decided today to pretend he is a beaver and bite at trees to cut them down.There are n trees located at various positions on a line. Tree i is located at position xi. All the given positions of the trees are distinct.The trees are equal, i.e. each tree has height h. Due to the wind, when a tree is cut down, it either falls left with probability p, or falls right with probability 1β-βp. If a tree hits another tree while falling, that tree will fall in the same direction as the tree that hit it. A tree can hit another tree only if the distance between them is strictly less than h. For example, imagine there are 4 trees located at positions 1, 3, 5 and 8, while hβ=β3 and the tree at position 1 falls right. It hits the tree at position 3 and it starts to fall too. In it's turn it hits the tree at position 5 and it also starts to fall. The distance between 8 and 5 is exactly 3, so the tree at position 8 will not fall.As long as there are still trees standing, Wilbur will select either the leftmost standing tree with probability 0.5 or the rightmost standing tree with probability 0.5. Selected tree is then cut down. If there is only one tree remaining, Wilbur always selects it. As the ground is covered with grass, Wilbur wants to know the expected total length of the ground covered with fallen trees after he cuts them all down because he is concerned about his grass-eating cow friends. Please help Wilbur. | Input: ['2 2 0.5000001 2'] Output:['3.250000000'] | [
3
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Wilbur is playing with a set of n points on the coordinate plane. All points have non-negative integer coordinates. Moreover, if some point (x, y) belongs to the set, then all points (x', y'), such that 0ββ€βx'ββ€βx and 0ββ€βy'ββ€βy also belong to this set.Now Wilbur wants to number the points in the set he has, that is assign them distinct integer numbers from 1 to n. In order to make the numbering aesthetically pleasing, Wilbur imposes the condition that if some point (x, y) gets number i, then all (x',y') from the set, such that x'ββ₯βx and y'ββ₯βy must be assigned a number not less than i. For example, for a set of four points (0, 0), (0, 1), (1, 0) and (1, 1), there are two aesthetically pleasing numberings. One is 1, 2, 3, 4 and another one is 1, 3, 2, 4.Wilbur's friend comes along and challenges Wilbur. For any point he defines it's special value as s(x,βy)β=βyβ-βx. Now he gives Wilbur some w1, w2,..., wn, and asks him to find an aesthetically pleasing numbering of the points in the set, such that the point that gets number i has it's special value equal to wi, that is s(xi,βyi)β=βyiβ-βxiβ=βwi.Now Wilbur asks you to help him with this challenge. | Input: ['52 00 01 01 10 10 -1 -2 1 0'] Output:['YES0 01 02 00 11 1'] | [
2
] |
Wilbur the pig is tinkering with arrays again. He has the array a1,βa2,β...,βan initially consisting of n zeros. At one step, he can choose any index i and either add 1 to all elements ai,βaiβ+β1,β... ,βan or subtract 1 from all elements ai,βaiβ+β1,β...,βan. His goal is to end up with the array b1,βb2,β...,βbn. Of course, Wilbur wants to achieve this goal in the minimum number of steps and asks you to compute this value. | Input: ['51 2 3 4 5'] Output:['5'] | [
2
] |
Pasha has recently bought a new phone jPager and started adding his friends' phone numbers there. Each phone number consists of exactly n digits.Also Pasha has a number k and two sequences of length nβ/βk (n is divisible by k) a1,βa2,β...,βanβ/βk and b1,βb2,β...,βbnβ/βk. Let's split the phone number into blocks of length k. The first block will be formed by digits from the phone number that are on positions 1, 2,..., k, the second block will be formed by digits from the phone number that are on positions kβ+β1, kβ+β2, ..., 2Β·k and so on. Pasha considers a phone number good, if the i-th block doesn't start from the digit bi and is divisible by ai if represented as an integer. To represent the block of length k as an integer, let's write it out as a sequence c1, c2,...,ck. Then the integer is calculated as the result of the expression c1Β·10kβ-β1β+βc2Β·10kβ-β2β+β...β+βck.Pasha asks you to calculate the number of good phone numbers of length n, for the given k, ai and bi. As this number can be too big, print it modulo 109β+β7. | Input: ['6 238 56 497 3 4'] Output:['8'] | [
3,
4
] |
Edo has got a collection of n refrigerator magnets!He decided to buy a refrigerator and hang the magnets on the door. The shop can make the refrigerator with any size of the door that meets the following restrictions: the refrigerator door must be rectangle, and both the length and the width of the door must be positive integers.Edo figured out how he wants to place the magnets on the refrigerator. He introduced a system of coordinates on the plane, where each magnet is represented as a rectangle with sides parallel to the coordinate axes.Now he wants to remove no more than k magnets (he may choose to keep all of them) and attach all remaining magnets to the refrigerator door, and the area of ββthe door should be as small as possible. A magnet is considered to be attached to the refrigerator door if its center lies on the door or on its boundary. The relative positions of all the remaining magnets must correspond to the plan.Let us explain the last two sentences. Let's suppose we want to hang two magnets on the refrigerator. If the magnet in the plan has coordinates of the lower left corner (x1, y1) and the upper right corner (x2, y2), then its center is located at (, ) (may not be integers). By saying the relative position should correspond to the plan we mean that the only available operation is translation, i.e. the vector connecting the centers of two magnets in the original plan, must be equal to the vector connecting the centers of these two magnets on the refrigerator.The sides of the refrigerator door must also be parallel to coordinate axes. | Input: ['3 11 1 2 22 2 3 33 3 4 4'] Output:['1'] | [
0,
2
] |
For months Maxim has been coming to work on his favorite bicycle. And quite recently he decided that he is ready to take part in a cyclists' competitions.He knows that this year n competitions will take place. During the i-th competition the participant must as quickly as possible complete a ride along a straight line from point si to point fi (siβ<βfi).Measuring time is a complex process related to usage of a special sensor and a time counter. Think of the front wheel of a bicycle as a circle of radius r. Let's neglect the thickness of a tire, the size of the sensor, and all physical effects. The sensor is placed on the rim of the wheel, that is, on some fixed point on a circle of radius r. After that the counter moves just like the chosen point of the circle, i.e. moves forward and rotates around the center of the circle.At the beginning each participant can choose any point bi, such that his bike is fully behind the starting line, that is, biβ<βsiβ-βr. After that, he starts the movement, instantly accelerates to his maximum speed and at time tsi, when the coordinate of the sensor is equal to the coordinate of the start, the time counter starts. The cyclist makes a complete ride, moving with his maximum speed and at the moment the sensor's coordinate is equal to the coordinate of the finish (moment of time tfi), the time counter deactivates and records the final time. Thus, the counter records that the participant made a complete ride in time tfiβ-βtsi. Maxim is good at math and he suspects that the total result doesn't only depend on his maximum speed v, but also on his choice of the initial point bi. Now Maxim is asking you to calculate for each of n competitions the minimum possible time that can be measured by the time counter. The radius of the wheel of his bike is equal to r. | Input: ['2 1 21 105 9'] Output:['3.8496447105021.106060157705'] | [
4
] |
Bogdan has a birthday today and mom gave him a tree consisting of n vertecies. For every edge of the tree i, some number xi was written on it. In case you forget, a tree is a connected non-directed graph without cycles. After the present was granted, m guests consecutively come to Bogdan's party. When the i-th guest comes, he performs exactly one of the two possible operations: Chooses some number yi, and two vertecies ai and bi. After that, he moves along the edges of the tree from vertex ai to vertex bi using the shortest path (of course, such a path is unique in the tree). Every time he moves along some edge j, he replaces his current number yi by , that is, by the result of integer division yi div xj. Chooses some edge pi and replaces the value written in it xpi by some positive integer ciβ<βxpi. As Bogdan cares about his guests, he decided to ease the process. Write a program that performs all the operations requested by guests and outputs the resulting value yi for each i of the first type. | Input: ['6 61 2 11 3 71 4 42 5 52 6 21 4 6 172 3 21 4 6 171 5 5 202 4 11 5 1 3'] Output:['24203'] | [
3
] |
Every day Ruslan tried to count sheep to fall asleep, but this didn't help. Now he has found a more interesting thing to do. First, he thinks of some set of circles on a plane, and then tries to choose a beautiful set of points, such that there is at least one point from the set inside or on the border of each of the imagined circles.Yesterday Ruslan tried to solve this problem for the case when the set of points is considered beautiful if it is given as (xtβ=βf(t),βytβ=βg(t)), where argument t takes all integer values from 0 to 50. Moreover, f(t) and g(t) should be correct functions.Assume that w(t) and h(t) are some correct functions, and c is an integer ranging from 0 to 50. The function s(t) is correct if it's obtained by one of the following rules: s(t)β=βabs(w(t)), where abs(x) means taking the absolute value of a number x, i.e. |x|; s(t)β=β(w(t)β+βh(t)); s(t)β=β(w(t)β-βh(t)); s(t)β=β(w(t)β*βh(t)), where β*β means multiplication, i.e. (w(t)Β·h(t)); s(t)β=βc; s(t)β=βt;Yesterday Ruslan thought on and on, but he could not cope with the task. Now he asks you to write a program that computes the appropriate f(t) and g(t) for any set of at most 50 circles.In each of the functions f(t) and g(t) you are allowed to use no more than 50 multiplications. The length of any function should not exceed 100Β·n characters. The function should not contain spaces.Ruslan can't keep big numbers in his memory, so you should choose f(t) and g(t), such that for all integer t from 0 to 50 value of f(t) and g(t) and all the intermediate calculations won't exceed 109 by their absolute value. | Input: ['30 10 410 0 420 10 4'] Output:['t abs((t-10))'] | [
3
] |
Andrew often reads articles in his favorite magazine 2Char. The main feature of these articles is that each of them uses at most two distinct letters. Andrew decided to send an article to the magazine, but as he hasn't written any article, he just decided to take a random one from magazine 26Char. However, before sending it to the magazine 2Char, he needs to adapt the text to the format of the journal. To do so, he removes some words from the chosen article, in such a way that the remaining text can be written using no more than two distinct letters.Since the payment depends from the number of non-space characters in the article, Andrew wants to keep the words with the maximum total length. | Input: ['4abbcaccaaabbb'] Output:['9'] | [
0
] |
BCPC stands for Byteforces Collegiate Programming Contest, and is the most famous competition in Byteforces.BCPC is a team competition. Each team is composed by a coach and three contestants. Blenda is the coach of the Bit State University(BSU), and she is very strict selecting the members of her team. In BSU there are n students numbered from 1 to n. Since all BSU students are infinitely smart, the only important parameters for Blenda are their reading and writing speed. After a careful measuring, Blenda have found that the i-th student have a reading speed equal to ri (words per minute), and a writing speed of wi (symbols per minute). Since BSU students are very smart, the measured speeds are sometimes very big and Blenda have decided to subtract some constant value c from all the values of reading speed and some value d from all the values of writing speed. Therefore she considers ri'β=βriβ-βc and wi'β=βwiβ-βd. The student i is said to overwhelm the student j if and only if ri'Β·wj'β>βrj'Β·wi'. Blenda doesnβt like fights in teams, so she thinks that a team consisting of three distinct students i,βj and k is good if i overwhelms j, j overwhelms k, and k overwhelms i. Yes, the relation of overwhelming is not transitive as it often happens in real life.Since Blenda is busy preparing a training camp in Codeforces, you are given a task to calculate the number of different good teams in BSU. Two teams are considered to be different if there is at least one student that is present in one team but is not present in the other. In other words, two teams are different if the sets of students that form these teams are different. | Input: ['5 2 21 14 12 33 23 4'] Output:['4'] | [
4
] |
Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today. Willman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. While watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). Note that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.Since the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today? | Input: ['10 3 2'] Output:['3/10'] | [
3
] |
Ari the monster always wakes up very early with the first ray of the sun and the first thing she does is feeding her squirrel.Ari draws a regular convex polygon on the floor and numbers it's vertices 1,β2,β...,βn in clockwise order. Then starting from the vertex 1 she draws a ray in the direction of each other vertex. The ray stops when it reaches a vertex or intersects with another ray drawn before. Ari repeats this process for vertex 2,β3,β...,βn (in this particular order). And then she puts a walnut in each region inside the polygon. Ada the squirrel wants to collect all the walnuts, but she is not allowed to step on the lines drawn by Ari. That means Ada have to perform a small jump if she wants to go from one region to another. Ada can jump from one region P to another region Q if and only if P and Q share a side or a corner.Assuming that Ada starts from outside of the picture, what is the minimum number of jumps she has to perform in order to collect all the walnuts? | Input: ['5'] Output:['9'] | [
3
] |
Harry Potter and He-Who-Must-Not-Be-Named engaged in a fight to the death once again. This time they are located at opposite ends of the corridor of length l. Two opponents simultaneously charge a deadly spell in the enemy. We know that the impulse of Harry's magic spell flies at a speed of p meters per second, and the impulse of You-Know-Who's magic spell flies at a speed of q meters per second.The impulses are moving through the corridor toward each other, and at the time of the collision they turn round and fly back to those who cast them without changing their original speeds. Then, as soon as the impulse gets back to it's caster, the wizard reflects it and sends again towards the enemy, without changing the original speed of the impulse.Since Harry has perfectly mastered the basics of magic, he knows that after the second collision both impulses will disappear, and a powerful explosion will occur exactly in the place of their collision. However, the young wizard isn't good at math, so he asks you to calculate the distance from his position to the place of the second meeting of the spell impulses, provided that the opponents do not change positions during the whole fight. | Input: ['1005050'] Output:['50'] | [
3
] |
A team of furry rescue rangers was sitting idle in their hollow tree when suddenly they received a signal of distress. In a few moments they were ready, and the dirigible of the rescue chipmunks hit the road.We assume that the action takes place on a Cartesian plane. The headquarters of the rescuers is located at point (x1,βy1), and the distress signal came from the point (x2,βy2).Due to Gadget's engineering talent, the rescuers' dirigible can instantly change its current velocity and direction of movement at any moment and as many times as needed. The only limitation is: the speed of the aircraft relative to the air can not exceed meters per second.Of course, Gadget is a true rescuer and wants to reach the destination as soon as possible. The matter is complicated by the fact that the wind is blowing in the air and it affects the movement of the dirigible. According to the weather forecast, the wind will be defined by the vector (vx,βvy) for the nearest t seconds, and then will change to (wx,βwy). These vectors give both the direction and velocity of the wind. Formally, if a dirigible is located at the point (x,βy), while its own velocity relative to the air is equal to zero and the wind (ux,βuy) is blowing, then after seconds the new position of the dirigible will be .Gadget is busy piloting the aircraft, so she asked Chip to calculate how long will it take them to reach the destination if they fly optimally. He coped with the task easily, but Dale is convinced that Chip has given the random value, aiming only not to lose the face in front of Gadget. Dale has asked you to find the right answer.It is guaranteed that the speed of the wind at any moment of time is strictly less than the maximum possible speed of the airship relative to the air. | Input: ['0 0 5 53 2-1 -1-1 0'] Output:['3.729935587093555327'] | [
3,
4
] |
Duff is in love with lovely numbers! A positive integer x is called lovely if and only if there is no such positive integer aβ>β1 such that a2 is a divisor of x. Malek has a number store! In his store, he has only divisors of positive integer n (and he has all of them). As a birthday present, Malek wants to give her a lovely number from his store. He wants this number to be as big as possible.Malek always had issues in math, so he asked for your help. Please tell him what is the biggest lovely number in his store. | Input: ['10'] Output:['10'] | [
3
] |
Duff is addicted to meat! Malek wants to keep her happy for n days. In order to be happy in i-th day, she needs to eat exactly ai kilograms of meat. There is a big shop uptown and Malek wants to buy meat for her from there. In i-th day, they sell meat for pi dollars per kilogram. Malek knows all numbers a1,β...,βan and p1,β...,βpn. In each day, he can buy arbitrary amount of meat, also he can keep some meat he has for the future.Malek is a little tired from cooking meat, so he asked for your help. Help him to minimize the total money he spends to keep Duff happy for n days. | Input: ['31 32 23 1'] Output:['10'] | [
2
] |
Duff is one if the heads of Mafia in her country, Andarz Gu. Andarz Gu has n cities (numbered from 1 to n) connected by m bidirectional roads (numbered by 1 to m).Each road has a destructing time, and a color. i-th road connects cities vi and ui and its color is ci and its destructing time is ti.Mafia wants to destruct a matching in Andarz Gu. A matching is a subset of roads such that no two roads in this subset has common endpoint. They can destruct these roads in parallel, i. e. the total destruction time is a maximum over destruction times of all selected roads. They want two conditions to be satisfied: The remaining roads form a proper coloring. Destructing time of this matching is minimized. The remaining roads after destructing this matching form a proper coloring if and only if no two roads of the same color have same endpoint, or, in the other words, edges of each color should form a matching.There is no programmer in Mafia. That's why Duff asked for your help. Please help her and determine which matching to destruct in order to satisfied those conditions (or state that this is not possible). | Input: ['5 72 1 3 73 1 1 65 4 1 84 5 1 13 2 2 34 5 2 52 3 2 4'] Output:['Yes3 24 5'] | [
4
] |
Recently, Duff has been practicing weight lifting. As a hard practice, Malek gave her a task. He gave her a sequence of weights. Weight of i-th of them is 2wi pounds. In each step, Duff can lift some of the remaining weights and throw them away. She does this until there's no more weight left. Malek asked her to minimize the number of steps. Duff is a competitive programming fan. That's why in each step, she can only lift and throw away a sequence of weights 2a1,β...,β2ak if and only if there exists a non-negative integer x such that 2a1β+β2a2β+β...β+β2akβ=β2x, i. e. the sum of those numbers is a power of two.Duff is a competitive programming fan, but not a programmer. That's why she asked for your help. Help her minimize the number of steps. | Input: ['51 1 2 3 3'] Output:['2'] | [
2
] |
Vitalik the philatelist has a birthday today!As he is a regular customer in a stamp store called 'Robin Bobin', the store management decided to make him a gift.Vitalik wants to buy one stamp and the store will give him a non-empty set of the remaining stamps, such that the greatest common divisor (GCD) of the price of the stamps they give to him is more than one. If the GCD of prices of the purchased stamp and prices of present stamps set will be equal to 1, then Vitalik will leave the store completely happy.The store management asks you to count the number of different situations in which Vitalik will leave the store completely happy. Since the required number of situations can be very large, you need to find the remainder of this number modulo 109β+β7. The situations are different if the stamps purchased by Vitalik are different, or if one of the present sets contains a stamp that the other present does not contain. | Input: ['32 3 2'] Output:['5'] | [
3
] |
Gennady is one of the best child dentists in Berland. Today n children got an appointment with him, they lined up in front of his office.All children love to cry loudly at the reception at the dentist. We enumerate the children with integers from 1 to n in the order they go in the line. Every child is associated with the value of his cofidence pi. The children take turns one after another to come into the office; each time the child that is the first in the line goes to the doctor.While Gennady treats the teeth of the i-th child, the child is crying with the volume of vi. At that the confidence of the first child in the line is reduced by the amount of vi, the second one β by value viβ-β1, and so on. The children in the queue after the vi-th child almost do not hear the crying, so their confidence remains unchanged.If at any point in time the confidence of the j-th child is less than zero, he begins to cry with the volume of dj and leaves the line, running towards the exit, without going to the doctor's office. At this the confidence of all the children after the j-th one in the line is reduced by the amount of dj.All these events occur immediately one after the other in some order. Some cries may lead to other cries, causing a chain reaction. Once in the hallway it is quiet, the child, who is first in the line, goes into the doctor's office.Help Gennady the Dentist to determine the numbers of kids, whose teeth he will cure. Print their numbers in the chronological order. | Input: ['54 2 24 1 25 2 43 3 55 1 2'] Output:['21 3 '] | [
0
] |
Anton loves transforming one permutation into another one by swapping elements for money, and Ira doesn't like paying for stupid games. Help them obtain the required permutation by paying as little money as possible.More formally, we have two permutations, p and s of numbers from 1 to n. We can swap pi and pj, by paying |iβ-βj| coins for it. Find and print the smallest number of coins required to obtain permutation s from permutation p. Also print the sequence of swap operations at which we obtain a solution. | Input: ['44 2 1 33 2 4 1'] Output:['324 33 1'] | [
2,
3
] |
Dima loves representing an odd number as the sum of multiple primes, and Lisa loves it when there are at most three primes. Help them to represent the given number as the sum of at most than three primes.More formally, you are given an odd numer n. Find a set of numbers pi (1ββ€βiββ€βk), such that 1ββ€βkββ€β3 pi is a prime The numbers pi do not necessarily have to be distinct. It is guaranteed that at least one possible solution exists. | Input: ['27'] Output:['35 11 11'] | [
0,
3
] |
Marina loves strings of the same length and Vasya loves when there is a third string, different from them in exactly t characters. Help Vasya find at least one such string.More formally, you are given two strings s1, s2 of length n and number t. Let's denote as f(a,βb) the number of characters in which strings a and b are different. Then your task will be to find any string s3 of length n, such that f(s1,βs3)β=βf(s2,βs3)β=βt. If there is no such string, print β-β1. | Input: ['3 2abcxyc'] Output:['ayd'] | [
2
] |
Olesya loves numbers consisting of n digits, and Rodion only likes numbers that are divisible by t. Find some number that satisfies both of them.Your task is: given the n and t print an integer strictly larger than zero consisting of n digits that is divisible by t. If such number doesn't exist, print β-β1. | Input: ['3 2'] Output:['712'] | [
3
] |
Robot Doc is located in the hall, with n computers stand in a line, numbered from left to right from 1 to n. Each computer contains exactly one piece of information, each of which Doc wants to get eventually. The computers are equipped with a security system, so to crack the i-th of them, the robot needs to collect at least ai any pieces of information from the other computers. Doc can hack the computer only if he is right next to it.The robot is assembled using modern technologies and can move along the line of computers in either of the two possible directions, but the change of direction requires a large amount of resources from Doc. Tell the minimum number of changes of direction, which the robot will have to make to collect all n parts of information if initially it is next to computer with number 1.It is guaranteed that there exists at least one sequence of the robot's actions, which leads to the collection of all information. Initially Doc doesn't have any pieces of information. | Input: ['30 2 0'] Output:['1'] | [
2
] |
For a given prime integer p and integers Ξ±,βA calculate the number of pairs of integers (n,βk), such that 0ββ€βkββ€βnββ€βA and is divisible by pΞ±. As the answer can be rather large, print the remainder of the answer moduly 109β+β7.Let us remind you that is the number of ways k objects can be chosen from the set of n objects. | Input: ['2 27'] Output:['3'] | [
3
] |
The GCD table G of size nβΓβn for an array of positive integers a of length n is defined by formula Let us remind you that the greatest common divisor (GCD) of two positive integers x and y is the greatest integer that is divisor of both x and y, it is denoted as . For example, for array aβ=β{4,β3,β6,β2} of length 4 the GCD table will look as follows: Given all the numbers of the GCD table G, restore array a. | Input: ['42 1 2 3 4 3 2 6 1 1 2 2 1 2 3 2'] Output:['4 3 6 2'] | [
2
] |
Motorist Kojiro spent 10 years saving up for his favorite car brand, Furrari. Finally Kojiro's dream came true! Kojiro now wants to get to his girlfriend Johanna to show off his car to her.Kojiro wants to get to his girlfriend, so he will go to her along a coordinate line. For simplicity, we can assume that Kojiro is at the point f of a coordinate line, and Johanna is at point e. Some points of the coordinate line have gas stations. Every gas station fills with only one type of fuel: Regular-92, Premium-95 or Super-98. Thus, each gas station is characterized by a pair of integers ti and xi β the number of the gas type and its position.One liter of fuel is enough to drive for exactly 1 km (this value does not depend on the type of fuel). Fuels of three types differ only in quality, according to the research, that affects the lifetime of the vehicle motor. A Furrari tank holds exactly s liters of fuel (regardless of the type of fuel). At the moment of departure from point f Kojiro's tank is completely filled with fuel Super-98. At each gas station Kojiro can fill the tank with any amount of fuel, but of course, at no point in time, the amount of fuel in the tank can be more than s liters. Note that the tank can simultaneously have different types of fuel. The car can moves both left and right.To extend the lifetime of the engine Kojiro seeks primarily to minimize the amount of fuel of type Regular-92. If there are several strategies to go from f to e, using the minimum amount of fuel of type Regular-92, it is necessary to travel so as to minimize the amount of used fuel of type Premium-95.Write a program that can for the m possible positions of the start fi minimize firstly, the amount of used fuel of type Regular-92 and secondly, the amount of used fuel of type Premium-95. | Input: ['8 4 1 12 40'] Output:['0 4'] | [
2
] |
Three companies decided to order a billboard with pictures of their logos. A billboard is a big square board. A logo of each company is a rectangle of a non-zero area. Advertisers will put up the ad only if it is possible to place all three logos on the billboard so that they do not overlap and the billboard has no empty space left. When you put a logo on the billboard, you should rotate it so that the sides were parallel to the sides of the billboard.Your task is to determine if it is possible to put the logos of all the three companies on some square billboard without breaking any of the described rules. | Input: ['5 1 2 5 5 2'] Output:['5AAAAABBBBBBBBBBCCCCCCCCCC'] | [
0,
3
] |
Petya loves computer games. Finally a game that he's been waiting for so long came out!The main character of this game has n different skills, each of which is characterized by an integer ai from 0 to 100. The higher the number ai is, the higher is the i-th skill of the character. The total rating of the character is calculated as the sum of the values ββof for all i from 1 to n. The expression β xβ denotes the result of rounding the number x down to the nearest integer.At the beginning of the game Petya got k improvement units as a bonus that he can use to increase the skills of his character and his total rating. One improvement unit can increase any skill of Petya's character by exactly one. For example, if a4β=β46, after using one imporvement unit to this skill, it becomes equal to 47. A hero's skill cannot rise higher more than 100. Thus, it is permissible that some of the units will remain unused.Your task is to determine the optimal way of using the improvement units so as to maximize the overall rating of the character. It is not necessary to use all the improvement units. | Input: ['2 47 9'] Output:['2'] | [
3
] |
The capital of Berland has n multifloor buildings. The architect who built up the capital was very creative, so all the houses were built in one row.Let's enumerate all the houses from left to right, starting with one. A house is considered to be luxurious if the number of floors in it is strictly greater than in all the houses with larger numbers. In other words, a house is luxurious if the number of floors in it is strictly greater than in all the houses, which are located to the right from it. In this task it is assumed that the heights of floors in the houses are the same.The new architect is interested in n questions, i-th of them is about the following: "how many floors should be added to the i-th house to make it luxurious?" (for all i from 1 to n, inclusive). You need to help him cope with this task.Note that all these questions are independent from each other β the answer to the question for house i does not affect other answers (i.e., the floors to the houses are not actually added). | Input: ['51 2 3 1 2'] Output:['3 2 0 2 0 '] | [
3
] |
One day Vasya the Hipster decided to count how many socks he had. It turned out that he had a red socks and b blue socks.According to the latest fashion, hipsters should wear the socks of different colors: a red one on the left foot, a blue one on the right foot.Every day Vasya puts on new socks in the morning and throws them away before going to bed as he doesn't want to wash them.Vasya wonders, what is the maximum number of days when he can dress fashionable and wear different socks, and after that, for how many days he can then wear the same socks until he either runs out of socks or cannot make a single pair from the socks he's got.Can you help him? | Input: ['3 1'] Output:['1 1'] | [
3
] |
Kefa wants to celebrate his first big salary by going to restaurant. However, he needs company. Kefa has n friends, each friend will agree to go to the restaurant if Kefa asks. Each friend is characterized by the amount of money he has and the friendship factor in respect to Kefa. The parrot doesn't want any friend to feel poor compared to somebody else in the company (Kefa doesn't count). A friend feels poor if in the company there is someone who has at least d units of money more than he does. Also, Kefa wants the total friendship factor of the members of the company to be maximum. Help him invite an optimal company! | Input: ['4 575 50 100150 2075 1'] Output:['100'] | [
4
] |
Kefa decided to make some money doing business on the Internet for exactly n days. He knows that on the i-th day (1ββ€βiββ€βn) he makes ai money. Kefa loves progress, that's why he wants to know the length of the maximum non-decreasing subsegment in sequence ai. Let us remind you that the subsegment of the sequence is its continuous fragment. A subsegment of numbers is called non-decreasing if all numbers in it follow in the non-decreasing order.Help Kefa cope with this task! | Input: ['62 2 1 3 4 1'] Output:['3'] | [
0
] |
There is a programing contest named SnakeUp, 2n people want to compete for it. In order to attend this contest, people need to form teams of exactly two people. You are given the strength of each possible combination of two people. All the values of the strengths are distinct.Every contestant hopes that he can find a teammate so that their teamβs strength is as high as possible. That is, a contestant will form a team with highest strength possible by choosing a teammate from ones who are willing to be a teammate with him/her. More formally, two people A and B may form a team if each of them is the best possible teammate (among the contestants that remain unpaired) for the other one. Can you determine who will be each personβs teammate? | Input: ['261 23 4 5'] Output:['2 1 4 3'] | [
0
] |
There is a sand trail in front of Alice's home.In daytime, people walk over it and leave a footprint on the trail for their every single step. Alice cannot distinguish the order of the footprints, but she can tell whether each footprint is made by left foot or right foot. Also she's certain that all people are walking by alternating left foot and right foot.For example, suppose that one person walked through the trail and left some footprints. The footprints are RRLRL in order along the trail ('R' means right foot and 'L' means left foot). You might think the outcome of the footprints is strange. But in fact, some steps are resulting from walking backwards!There are some possible order of steps that produce these footprints such as 1βββ3βββ2βββ5βββ4 or 2βββ3βββ4βββ5βββ1 (we suppose that the distance between two consecutive steps can be arbitrarily long). The number of backward steps from above two examples are 2 and 1 separately.Alice is interested in these footprints. Whenever there is a person walking trough the trail, she takes a picture of all these footprints along the trail and erase all of them so that next person will leave a new set of footprints. We know that people walk by alternating right foot and left foot, but we don't know if the first step is made by left foot or right foot.Alice wants to know the minimum possible number of backward steps made by a person. But it's a little hard. Please help Alice to calculate it. You also need to construct one possible history of these footprints. | Input: ['RRLRL'] Output:['12 5 1 3 4'] | [
2
] |
You are given a string S of length n with each character being one of the first m lowercase English letters. Calculate how many different strings T of length n composed from the first m lowercase English letters exist such that the length of LCS (longest common subsequence) between S and T is nβ-β1.Recall that LCS of two strings S and T is the longest string C such that C both in S and T as a subsequence. | Input: ['3 3aaa'] Output:['6'] | [
2
] |
You are given a sequence of n integers a1,βa2,β...,βan. Determine a real number x such that the weakness of the sequence a1β-βx,βa2β-βx,β...,βanβ-βx is as small as possible.The weakness of a sequence is defined as the maximum value of the poorness over all segments (contiguous subsequences) of a sequence.The poorness of a segment is defined as the absolute value of sum of the elements of segment. | Input: ['31 2 3'] Output:['1.000000000000000'] | [
4
] |
You are given n numbers a1,βa2,β...,βan. You can perform at most k operations. For each operation you can multiply one of the numbers by x. We want to make as large as possible, where denotes the bitwise OR. Find the maximum possible value of after performing at most k operations optimally. | Input: ['3 1 21 1 1'] Output:['3'] | [
0,
2
] |
There is a polyline going through points (0,β0)βββ(x,βx)βββ(2x,β0)βββ(3x,βx)βββ(4x,β0)βββ...β-β(2kx,β0)βββ(2kxβ+βx,βx)βββ.... We know that the polyline passes through the point (a,βb). Find minimum positive value x such that it is true or determine that there is no such x. | Input: ['3 1'] Output:['1.000000000000'] | [
3
] |
Note the unusual memory limit for this problem.You are given an undirected graph consisting of n vertices and m edges. The vertices are numbered with integers from 1 to n, the edges are numbered with integers from 1 to m. Each edge can be unpainted or be painted in one of the k colors, which are numbered with integers from 1 to k. Initially, none of the edges is painted in any of the colors.You get queries of the form "Repaint edge ei to color ci". At any time the graph formed by the edges of the same color must be bipartite. If after the repaint this condition is violated, then the query is considered to be invalid and edge ei keeps its color. Otherwise, edge ei is repainted in color ci, and the query is considered to valid.Recall that the graph is called bipartite if the set of its vertices can be divided into two parts so that no edge connected vertices of the same parts.For example, suppose you are given a triangle graph, that is a graph with three vertices and edges (1,β2), (2,β3) and (3,β1). Suppose that the first two edges are painted color 1, and the third one is painted color 2. Then the query of "repaint the third edge in color 1" will be incorrect because after its execution the graph formed by the edges of color 1 will not be bipartite. On the other hand, it is possible to repaint the second edge in color 2.You receive q queries. For each query, you should either apply it, and report that the query is valid, or report that the query is invalid. | Input: ['3 3 2 51 22 31 31 12 13 23 12 2'] Output:['YESYESYESNOYES'] | [
4
] |
On a plane are n points (xi, yi) with integer coordinates between 0 and 106. The distance between the two points with numbers a and b is said to be the following value: (the distance calculated by such formula is called Manhattan distance).We call a hamiltonian path to be some permutation pi of numbers from 1 to n. We say that the length of this path is value .Find some hamiltonian path with a length of no more than 25βΓβ108. Note that you do not have to minimize the path length. | Input: ['50 78 103 45 09 12'] Output:['4 3 1 2 5 '] | [
2
] |
A tree of size n is an undirected connected graph consisting of n vertices without cycles.Consider some tree with n vertices. We call a tree invariant relative to permutation pβ=βp1p2... pn, if for any two vertices of the tree u and v the condition holds: "vertices u and v are connected by an edge if and only if vertices pu and pv are connected by an edge".You are given permutation p of size n. Find some tree size n, invariant relative to the given permutation. | Input: ['44 3 2 1'] Output:['YES4 14 21 3'] | [
2
] |
Vasya and Petya are playing a simple game. Vasya thought of number x between 1 and n, and Petya tries to guess the number.Petya can ask questions like: "Is the unknown number divisible by number y?".The game is played by the following rules: first Petya asks all the questions that interest him (also, he can ask no questions), and then Vasya responds to each question with a 'yes' or a 'no'. After receiving all the answers Petya should determine the number that Vasya thought of.Unfortunately, Petya is not familiar with the number theory. Help him find the minimum number of questions he should ask to make a guaranteed guess of Vasya's number, and the numbers yi, he should ask the questions about. | Input: ['4'] Output:['32 4 3 '] | [
3
] |
Sasha and Ira are two best friends. But they arenβt just friends, they are software engineers and experts in artificial intelligence. They are developing an algorithm for two bots playing a two-player game. The game is cooperative and turn based. In each turn, one of the players makes a move (it doesnβt matter which player, it's possible that players turns do not alternate). Algorithm for bots that Sasha and Ira are developing works by keeping track of the state the game is in. Each time either bot makes a move, the state changes. And, since the game is very dynamic, it will never go back to the state it was already in at any point in the past.Sasha and Ira are perfectionists and want their algorithm to have an optimal winning strategy. They have noticed that in the optimal winning strategy, both bots make exactly N moves each. But, in order to find the optimal strategy, their algorithm needs to analyze all possible states of the game (they havenβt learned about alpha-beta pruning yet) and pick the best sequence of moves.They are worried about the efficiency of their algorithm and are wondering what is the total number of states of the game that need to be analyzed? | Input: ['2'] Output:['19'] | [
3
] |
Bananistan is a beautiful banana republic. Beautiful women in beautiful dresses. Beautiful statues of beautiful warlords. Beautiful stars in beautiful nights.In Bananistan people play this crazy game β Bulbo. Thereβs an array of bulbs and player at the position, which represents one of the bulbs. The distance between two neighboring bulbs is 1. Before each turn player can change his position with cost |posnewβ-βposold|. After that, a contiguous set of bulbs lights-up and player pays the cost thatβs equal to the distance to the closest shining bulb. Then, all bulbs go dark again. The goal is to minimize your summed cost. I tell you, Bananistanians are spending their nights playing with bulbs.Banana day is approaching, and you are hired to play the most beautiful Bulbo game ever. A huge array of bulbs is installed, and you know your initial position and all the light-ups in advance. You need to play the ideal game and impress Bananistanians, and their families. | Input: ['5 42 79 168 109 171 6'] Output:['8'] | [
2
] |
Note the unusual memory limit for the problem.People working in MDCS (Microsoft Development Center Serbia) like partying. They usually go to night clubs on Friday and Saturday.There are N people working in MDCS and there are N clubs in the city. Unfortunately, if there is more than one Microsoft employee in night club, level of coolness goes infinitely high and party is over, so club owners will never let more than one Microsoft employee enter their club in the same week (just to be sure).You are organizing night life for Microsoft employees and you have statistics about how much every employee likes Friday and Saturday parties for all clubs.You need to match people with clubs maximizing overall sum of their happiness (they are happy as much as they like the club), while half of people should go clubbing on Friday and the other half on Saturday. | Input: ['41 2 3 42 3 4 13 4 1 24 1 2 35 8 7 16 9 81 355 78 1 61 1 1 1'] Output:['167'] | [
0
] |
Fibonotci sequence is an integer recursive sequence defined by the recurrence relation Fnβ=βsnβ-β1Β·Fnβ-β1β+βsnβ-β2Β·Fnβ-β2 with F0β=β0,βF1β=β1 Sequence s is an infinite and almost cyclic sequence with a cycle of length N. A sequence s is called almost cyclic with a cycle of length N if , for iββ₯βN, except for a finite number of values si, for which (iββ₯βN).Following is an example of an almost cyclic sequence with a cycle of length 4: s = (5,3,8,11,5,3,7,11,5,3,8,11,β¦) Notice that the only value of s for which the equality does not hold is s6 (s6β=β7 and s2β=β8). You are given s0,βs1,β...sNβ-β1 and all the values of sequence s for which (iββ₯βN).Find . | Input: ['10 831 2 127 35 4'] Output:['4'] | [
3
] |
Do you know a story about the three musketeers? Anyway, you will learn about its origins now.Richelimakieu is a cardinal in the city of Bearis. He is tired of dealing with crime by himself. He needs three brave warriors to help him to fight against bad guys.There are n warriors. Richelimakieu wants to choose three of them to become musketeers but it's not that easy. The most important condition is that musketeers must know each other to cooperate efficiently. And they shouldn't be too well known because they could be betrayed by old friends. For each musketeer his recognition is the number of warriors he knows, excluding other two musketeers.Help Richelimakieu! Find if it is possible to choose three musketeers knowing each other, and what is minimum possible sum of their recognitions. | Input: ['5 61 21 32 32 43 44 5'] Output:['2'] | [
0
] |
Limak is a grizzly bear who desires power and adoration. He wants to win in upcoming elections and rule over the Bearland.There are n candidates, including Limak. We know how many citizens are going to vote for each candidate. Now i-th candidate would get ai votes. Limak is candidate number 1. To win in elections, he must get strictly more votes than any other candidate.Victory is more important than everything else so Limak decided to cheat. He will steal votes from his opponents by bribing some citizens. To bribe a citizen, Limak must give him or her one candy - citizens are bears and bears like candies. Limak doesn't have many candies and wonders - how many citizens does he have to bribe? | Input: ['55 1 11 2 8'] Output:['4'] | [
2
] |
Limak is an old brown bear. He often goes bowling with his friends. Today he feels really good and tries to beat his own record!For rolling a ball one gets a score β an integer (maybe negative) number of points. Score for i-th roll is multiplied by i and scores are summed up. So, for k rolls with scores s1,βs2,β...,βsk, total score is . Total score is 0 if there were no rolls.Limak made n rolls and got score ai for i-th of them. He wants to maximize his total score and he came up with an interesting idea. He will cancel some rolls, saying that something distracted him or there was a strong wind.Limak is able to cancel any number of rolls, maybe even all or none of them. Total score is calculated as if there were only non-canceled rolls. Look at the sample tests for clarification. What maximum total score can Limak get? | Input: ['5-2 -8 0 5 -3'] Output:['13'] | [
2
] |
Limak is a little bear who loves to play. Today he is playing by destroying block towers. He built n towers in a row. The i-th tower is made of hi identical blocks. For clarification see picture for the first sample.Limak will repeat the following operation till everything is destroyed.Block is called internal if it has all four neighbors, i.e. it has each side (top, left, down and right) adjacent to other block or to the floor. Otherwise, block is boundary. In one operation Limak destroys all boundary blocks. His paws are very fast and he destroys all those blocks at the same time.Limak is ready to start. You task is to count how many operations will it take him to destroy all towers. | Input: ['62 1 4 6 2 2'] Output:['3'] | [
3,
4
] |
Limak is an old brown bear. He often plays poker with his friends. Today they went to a casino. There are n players (including Limak himself) and right now all of them have bids on the table. i-th of them has bid with size ai dollars.Each player can double his bid any number of times and triple his bid any number of times. The casino has a great jackpot for making all bids equal. Is it possible that Limak and his friends will win a jackpot? | Input: ['475 150 75 50'] Output:['Yes'] | [
3
] |
In this task you need to process a set of stock exchange orders and use them to create order book.An order is an instruction of some participant to buy or sell stocks on stock exchange. The order number i has price pi, direction di β buy or sell, and integer qi. This means that the participant is ready to buy or sell qi stocks at price pi for one stock. A value qi is also known as a volume of an order.All orders with the same price p and direction d are merged into one aggregated order with price p and direction d. The volume of such order is a sum of volumes of the initial orders.An order book is a list of aggregated orders, the first part of which contains sell orders sorted by price in descending order, the second contains buy orders also sorted by price in descending order.An order book of depth s contains s best aggregated orders for each direction. A buy order is better if it has higher price and a sell order is better if it has lower price. If there are less than s aggregated orders for some direction then all of them will be in the final order book.You are given n stock exhange orders. Your task is to print order book of depth s for these orders. | Input: ['6 2B 10 3S 50 2S 40 1S 50 6B 20 4B 25 10'] Output:['S 50 8S 40 1B 25 10B 20 4'] | [
2
] |
Geometric progression with the first element a and common ratio b is a sequence of numbers a,βab,βab2,βab3,β....You are given n integer geometric progressions. Your task is to find the smallest integer x, that is the element of all the given progressions, or else state that such integer does not exist. | Input: ['22 24 1'] Output:['4'] | [
3
] |
Oscolcovo city has a campus consisting of n student dormitories, n universities and n military offices. Initially, the i-th dormitory belongs to the i-th university and is assigned to the i-th military office.Life goes on and the campus is continuously going through some changes. The changes can be of four types: University aj merges with university bj. After that all the dormitories that belonged to university bj are assigned to to university aj, and university bj disappears. Military office cj merges with military office dj. After that all the dormitories that were assigned to military office dj, are assigned to military office cj, and military office dj disappears. Students of university xj move in dormitories. Lets kxj is the number of dormitories that belong to this university at the time when the students move in. Then the number of students in each dormitory of university xj increases by kxj (note that the more dormitories belong to the university, the more students move in each dormitory of the university). Military office number yj conducts raids on all the dormitories assigned to it and takes all students from there. Thus, at each moment of time each dormitory is assigned to exactly one university and one military office. Initially, all the dormitory are empty.Your task is to process the changes that take place in the campus and answer the queries, how many people currently live in dormitory qj. | Input: ['2 7A 1Q 1U 1 2A 1Z 1Q 1Q 2'] Output:['102'] | [
4
] |
'In Boolean logic, a formula is in conjunctive normal form (CNF) or clausal normal form if it is a conjunction of clauses, where a clause is a disjunction of literals' (cited from https://en.wikipedia.org/wiki/Conjunctive_normal_form)In the other words, CNF is a formula of type , where & represents a logical "AND" (conjunction), represents a logical "OR" (disjunction), and vij are some boolean variables or their negations. Each statement in brackets is called a clause, and vij are called literals.You are given a CNF containing variables x1,β...,βxm and their negations. We know that each variable occurs in at most two clauses (with negation and without negation in total). Your task is to determine whether this CNF is satisfiable, that is, whether there are such values of variables where the CNF value is true. If CNF is satisfiable, then you also need to determine the values of the variables at which the CNF is true. It is guaranteed that each variable occurs at most once in each clause. | Input: ['2 22 1 -22 2 -1'] Output:['YES11'] | [
2
] |
You've got array A, consisting of n integers and a positive integer k. Array A is indexed by integers from 1 to n.You need to permute the array elements so that value became minimal possible. In particular, it is allowed not to change order of elements at all. | Input: ['3 21 2 4'] Output:['1'] | [
2
] |
You are given three sticks with positive integer lengths of a,βb, and c centimeters. You can increase length of some of them by some positive integer number of centimeters (different sticks can be increased by a different length), but in total by at most l centimeters. In particular, it is allowed not to increase the length of any stick.Determine the number of ways to increase the lengths of some sticks so that you can form from them a non-degenerate (that is, having a positive area) triangle. Two ways are considered different, if the length of some stick is increased by different number of centimeters in them. | Input: ['1 1 1 2'] Output:['4'] | [
3
] |
Roman planted a tree consisting of n vertices. Each vertex contains a lowercase English letter. Vertex 1 is the root of the tree, each of the nβ-β1 remaining vertices has a parent in the tree. Vertex is connected with its parent by an edge. The parent of vertex i is vertex pi, the parent index is always less than the index of the vertex (i.e., piβ<βi).The depth of the vertex is the number of nodes on the path from the root to v along the edges. In particular, the depth of the root is equal to 1.We say that vertex u is in the subtree of vertex v, if we can get from u to v, moving from the vertex to the parent. In particular, vertex v is in its subtree.Roma gives you m queries, the i-th of which consists of two numbers vi, hi. Let's consider the vertices in the subtree vi located at depth hi. Determine whether you can use the letters written at these vertices to make a string that is a palindrome. The letters that are written in the vertexes, can be rearranged in any order to make a palindrome, but all letters should be used. | Input: ['6 51 1 1 3 3zacccd1 13 34 16 11 2'] Output:['YesNoYesYesYes'] | [
4
] |
One day Misha and Andrew were playing a very simple game. First, each player chooses an integer in the range from 1 to n. Let's assume that Misha chose number m, and Andrew chose number a.Then, by using a random generator they choose a random integer c in the range between 1 and n (any integer from 1 to n is chosen with the same probability), after which the winner is the player, whose number was closer to c. The boys agreed that if m and a are located on the same distance from c, Misha wins.Andrew wants to win very much, so he asks you to help him. You know the number selected by Misha, and number n. You need to determine which value of a Andrew must choose, so that the probability of his victory is the highest possible.More formally, you need to find such integer a (1ββ€βaββ€βn), that the probability that is maximal, where c is the equiprobably chosen integer from 1 to n (inclusive). | Input: ['3 1'] Output:['2'] | [
2,
3
] |
Companies always have a lot of equipment, furniture and other things. All of them should be tracked. To do this, there is an inventory number assigned with each item. It is much easier to create a database by using those numbers and keep the track of everything.During an audit, you were surprised to find out that the items are not numbered sequentially, and some items even share the same inventory number! There is an urgent need to fix it. You have chosen to make the numbers of the items sequential, starting with 1. Changing a number is quite a time-consuming process, and you would like to make maximum use of the current numbering.You have been given information on current inventory numbers for n items in the company. Renumber items so that their inventory numbers form a permutation of numbers from 1 to n by changing the number of as few items as possible. Let us remind you that a set of n numbers forms a permutation if all the numbers are in the range from 1 to n, and no two numbers are equal. | Input: ['31 3 2'] Output:['1 3 2 '] | [
2,
3
] |
Little Lesha loves listening to music via his smartphone. But the smartphone doesn't have much memory, so Lesha listens to his favorite songs in a well-known social network InTalk.Unfortunately, internet is not that fast in the city of Ekaterinozavodsk and the song takes a lot of time to download. But Lesha is quite impatient. The song's duration is T seconds. Lesha downloads the first S seconds of the song and plays it. When the playback reaches the point that has not yet been downloaded, Lesha immediately plays the song from the start (the loaded part of the song stays in his phone, and the download is continued from the same place), and it happens until the song is downloaded completely and Lesha listens to it to the end. For q seconds of real time the Internet allows you to download qβ-β1 seconds of the track.Tell Lesha, for how many times he will start the song, including the very first start. | Input: ['5 2 2'] Output:['2'] | [
3
] |
One Khanate had a lot of roads and very little wood. Riding along the roads was inconvenient, because the roads did not have road signs indicating the direction to important cities.The Han decided that it's time to fix the issue, and ordered to put signs on every road. The Minister of Transport has to do that, but he has only k signs. Help the minister to solve his problem, otherwise the poor guy can lose not only his position, but also his head.More formally, every road in the Khanate is a line on the Oxy plane, given by an equation of the form Axβ+βByβ+βCβ=β0 (A and B are not equal to 0 at the same time). You are required to determine whether you can put signs in at most k points so that each road had at least one sign installed. | Input: ['3 11 0 00 -1 07 -93 0'] Output:['YES11 2'] | [
0,
3
] |
Living in Byteland was good enough to begin with, but the good king decided to please his subjects and to introduce a national language. He gathered the best of wise men, and sent an expedition to faraway countries, so that they would find out all about how a language should be designed.After some time, the wise men returned from the trip even wiser. They locked up for six months in the dining room, after which they said to the king: "there are a lot of different languages, but almost all of them have letters that are divided into vowels and consonants; in a word, vowels and consonants must be combined correctly."There are very many rules, all of them have exceptions, but our language will be deprived of such defects! We propose to introduce a set of formal rules of combining vowels and consonants, and include in the language all the words that satisfy them.The rules of composing words are: The letters are divided into vowels and consonants in some certain way; All words have a length of exactly n; There are m rules of the form (pos1,βt1,βpos2,βt2). Each rule is: if the position pos1 has a letter of type t1, then the position pos2 has a letter of type t2.You are given some string s of length n, it is not necessarily a correct word of the new language. Among all the words of the language that lexicographically not smaller than the string s, find the minimal one in lexicographic order. | Input: ['VC2 11 V 2 Caa'] Output:['ab'] | [
2
] |
Little Johnny has recently learned about set theory. Now he is studying binary relations. You've probably heard the term "equivalence relation". These relations are very important in many areas of mathematics. For example, the equality of the two numbers is an equivalence relation.A set Ο of pairs (a,βb) of elements of some set A is called a binary relation on set A. For two elements a and b of the set A we say that they are in relation Ο, if pair , in this case we use a notation .Binary relation is equivalence relation, if: It is reflexive (for any a it is true that ); It is symmetric (for any a, b it is true that if , then ); It is transitive (if and , than ).Little Johnny is not completely a fool and he noticed that the first condition is not necessary! Here is his "proof":Take any two elements, a and b. If , then (according to property (2)), which means (according to property (3)).It's very simple, isn't it? However, you noticed that Johnny's "proof" is wrong, and decided to show him a lot of examples that prove him wrong.Here's your task: count the number of binary relations over a set of size n such that they are symmetric, transitive, but not an equivalence relations (i.e. they are not reflexive).Since their number may be very large (not 0, according to Little Johnny), print the remainder of integer division of this number by 109β+β7. | Input: ['1'] Output:['1'] | [
3
] |
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