question_text stringlengths 2 3.82k | input_outputs stringlengths 23 941 | algo_tags sequence |
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This is the hard version of the problem. The only difference between the simple and hard versions is that in this version u can take any possible value.As is known, Omsk is the capital of Berland. Like any capital, Omsk has a well-developed metro system. The Omsk metro consists of a certain number of stations connected... | Input: ['18+ 1 -1? 1 1 2? 1 2 1+ 1 1? 1 3 -1? 1 1 1? 1 3 2? 1 1 0'] Output:['NO', 'YES', 'NO', 'YES', 'YES', 'YES', ''] | [
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This is the simple version of the problem. The only difference between the simple and hard versions is that in this version u = 1.As is known, Omsk is the capital of Berland. Like any capital, Omsk has a well-developed metro system. The Omsk metro consists of a certain number of stations connected by tunnels, and betwe... | Input: ['18+ 1 -1? 1 1 2? 1 2 1+ 1 1? 1 3 -1? 1 1 1? 1 3 2? 1 1 0'] Output:['NO', 'YES', 'NO', 'YES', 'YES', 'YES', ''] | [
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You are given an array a consisting of n zeros. You are also given a set of m not necessarily different segments. Each segment is defined by two numbers l_i and r_i (1 <= l_i <= r_i <= n) and represents a subarray a_{l_i}, a_{l_i+1}, ..., a_{r_i} of the array a.Let's call the segment l_i, r_i beautiful if the number of... | Input: ['65 51 24 51 51 32 45531244 21 14 42235 21 51 5421345 21 51 35412355 51 51 51 51 51 431433 22 21 33231'] Output:['3', '-1', '3', '3', '3', '1', ''] | [
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Timofey has an apple tree growing in his garden; it is a rooted tree of n vertices with the root in vertex 1 (the vertices are numbered from 1 to n). A tree is a connected graph without loops and multiple edges.This tree is very unusual — it grows with its root upwards. However, it's quite normal for programmer's trees... | Input: ['251 23 45 33 243 45 14 41 331 21 331 12 33 1'] Output:['2', '2', '1', '4', '4', '1', '2', ''] | [
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Vanya really likes math. One day when he was solving another math problem, he came up with an interesting tree. This tree is built as follows.Initially, the tree has only one vertex with the number 1 — the root of the tree. Then, Vanya adds two children to it, assigning them consecutive numbers — 2 and 3, respectively.... | Input: ['63103711000000000000000015'] Output:['4', '18', '71', '1', '19999999999999980', '26', ''] | [
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Today Alex was brought array a_1, a_2, ..., a_n of length n. He can apply as many operations as he wants (including zero operations) to change the array elements.In 1 operation Alex can choose any l and r such that 1 <=q l <=q r <=q n, and multiply all elements of the array from l to r inclusive by -1. In other words, ... | Input: ['56-1 7 -4 -2 5 -88-1 0 0 -2 1 0 -3 052 -1 0 -3 -750 -17 0 1 04-1 0 -2 -1'] Output:['27 3', '7 2', '13 1', '18 1', '4 1', ''] | [
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Sasha found an array a consisting of n integers and asked you to paint elements.You have to paint each element of the array. You can use as many colors as you want, but each element should be painted into exactly one color, and for each color, there should be at least one element of that color.The cost of one color is ... | Input: ['651 5 6 3 41541 6 3 961 13 9 3 7 242 2 2 254 5 2 2 3'] Output:['7', '0', '11', '23', '0', '5', ''] | [
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bright, sunny and innocent......Tenzing has a beautiful necklace. The necklace consists of n pearls numbered from 1 to n with a string connecting pearls i and (i \text{ mod } n)+1 for all 1 <=q i <=q n.One day Tenzing wants to cut the necklace into several parts by cutting some strings. But for each connected part of t... | Input: ['45 21 1 1 1 15 21 2 3 4 56 34 2 5 1 3 310 32 5 6 5 2 1 7 9 7 2'] Output:['3', '7', '5', '15', ''] | [
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There are n uniform random real variables between 0 and 1, inclusive, which are denoted as x_1, x_2, ..., x_n.Tenzing has m conditions. Each condition has the form of x_i+x_j<= 1 or x_i+x_j>= 1.Tenzing wants to know the probability that all the conditions are satisfied, modulo 998~244~353.Formally, let M = 998~244~353.... | Input: ['3 2', '0 1 2', '1 3 3', ''] Output:['748683265', ''] | [
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Yet another random problem.Tenzing has an array a of length n and an integer v.Tenzing will perform the following operation m times: Choose an integer i such that 1 <=q i <=q n uniformly at random. For all j such that i <=q j <=q n, set a_j := a_j + v. Tenzing wants to know the expected value of \prod_{i=1}^n a_i aft... | Input: ['2 2 5', '2 2', ''] Output:['84', ''] | [
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Tenzing has an undirected tree of n vertices.Define the value of a tree with black and white vertices in the following way. The value of an edge is the absolute difference between the number of black nodes in the two components of the tree after deleting the edge. The value of the tree is the sum of values over all edg... | Input: ['4', '1 2', '3 2', '2 4', ''] Output:['0 3 4 5 6 ', ''] | [
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There are n pairwise-distinct points and a line x+y=k on a two-dimensional plane. The i-th point is at (x_i,y_i). All points have non-negative coordinates and are strictly below the line. Alternatively, 0 <=q x_i,y_i, x_i+y_i < k.Tenzing wants to erase all the points. He can perform the following two operations: Draw ... | Input: ['4 6 1', '1 2 1', '2 1 1', '1 1 1', '3 2 6', ''] Output:['4', ''] | [
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Tell a story about me and my animal friends.Tenzing has n animal friends. He numbers them from 1 to n.One day Tenzing wants to play with his animal friends. To do so, Tenzing will host several games.In one game, he will choose a set S which is a subset of \{1,2,3,...,n\} and choose an integer t. Then, he will play the ... | Input: ['5 4', '1 3 2', '1 4 2', '2 3 1', '2 5 1', ''] Output:['4 4', '10000 1', '10010 1', '10100 1', '11110 1', ''] | [
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Tenzing received 3n books from his fans. The books are arranged in 3 stacks with n books in each stack. Each book has a non-negative integer difficulty rating.Tenzing wants to read some (possibly zero) books. At first, his knowledge is 0.To read the books, Tenzing will choose a non-empty stack, read the book on the top... | Input: ['35 71 2 3 4 55 4 3 2 11 3 5 7 95 23 2 3 4 55 4 3 2 13 3 5 7 93 01 2 33 2 12 2 2'] Output:['Yes', 'No', 'Yes', ''] | [
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Tsondu always runs first! ! !Tsondu and Tenzing are playing a card game. Tsondu has n monsters with ability values a_1, a_2, ..., a_n while Tenzing has m monsters with ability values b_1, b_2, ..., b_m.Tsondu and Tenzing take turns making moves, with Tsondu going first. In each move, the current player chooses two mons... | Input: ['61 391 2 32 31 21 1 13 21 2 31 13 31 1 12 2 210 101 2 3 3 2 2 1 1 2 23 3 3 3 2 1 1 1 1 110 101 2 3 4 5 6 7 8 9 106 7 8 9 10 11 1 1 1 1'] Output:['Tsondu', 'Draw', 'Tsondu', 'Tenzing', 'Draw', 'Draw', ''] | [
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There is a square matrix, consisting of n rows and n columns of cells, both numbered from 1 to n. The cells are colored white or black. Cells from 1 to a_i are black, and cells from a_i+1 to n are white, in the i-th column.You want to place m integers in the matrix, from 1 to m. There are two rules: each cell should ... | Input: ['630 0 0942 0 3 1542 0 3 1642 0 3 110100 2 2 1 5 10 3 4 1 120110'] Output:['6', '3', '4', '4', '16', '0', ''] | [
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Two segments [l_1, r_1] and [l_2, r_2] intersect if there exists at least one x such that l_1 <= x <= r_1 and l_2 <= x <= r_2.An array of segments [[l_1, r_1], [l_2, r_2], ..., [l_k, r_k]] is called beautiful if k is even, and is possible to split the elements of this array into \frac{k}{2} pairs in such a way that: e... | Input: ['372 49 122 47 74 810 136 852 22 80 101 25 641 12 23 34 4'] Output:['1', '3', '4', ''] | [
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No, not "random" numbers.Ranom digits are denoted by uppercase Latin letters from A to E. Moreover, the value of the letter A is 1, B is 10, C is 100, D is 1000, E is 10000.A Ranom number is a sequence of Ranom digits. The value of the Ranom number is calculated as follows: the values of all digits are summed up, but s... | Input: ['4DAAABDCAABABCDEEDCBADDDDAAADDABECD'] Output:['11088', '10010', '31000', '15886', ''] | [
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The only difference between easy and hard versions is the maximum number of queries. In this version, you are allowed to ask at most 1000 queries.This is an interactive problem.You are playing a game. The circle is divided into n sectors, sectors are numbered from 1 to n in some order. You are in the adjacent room and ... | Input: ['1', '', '5', '', '6', '', '7', '', '2', '', '10', '', '9', '', '8', '', '4', '', '3', '', '1'] Output:['+ 1', '', '+ 1', '', '+ 1', '', '+ 1', '', '+ 1', '', '+ 1', '', '+ 1', '', '+ 1', '', '+ 1', '', '+ 1', '', '! 10'] | [
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The only difference between easy and hard versions is the maximum number of queries. In this version, you are allowed to ask at most 2023 queries.This is an interactive problem.You are playing a game. The circle is divided into n sectors, sectors are numbered from 1 to n in some order. You are in the adjacent room and ... | Input: ['1', '', '5', '', '6', '', '7', '', '2', '', '10', '', '9', '', '8', '', '4', '', '3', '', '1'] Output:['+ 1', '', '+ 1', '', '+ 1', '', '+ 1', '', '+ 1', '', '+ 1', '', '+ 1', '', '+ 1', '', '+ 1', '', '+ 1', '', '! 10'] | [
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Tema is playing a very interesting computer game.During the next mission, Tema's character found himself on an unfamiliar planet. Unlike Earth, this planet is flat and can be represented as an n * m rectangle.Tema's character is located at the point with coordinates (0, 0). In order to successfully complete the mission... | Input: ['51 341 2 02 2 13 2 24 1 13 362 1 02 1 12 1 22 2 02 2 12 2 22 137 1 22 1 17 2 12 259 1 23 2 05 1 24 2 27 1 04 676 1 212 1 34 1 017 2 31 2 616 2 63 2 4'] Output:['5', '-1', '3', '6', '10', ''] | [
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In a small town, there is a workshop specializing in woodwork. Since the town is small, only three carvers work there.Soon, a wooden toy festival is planned in the town. The workshop employees want to prepare for it.They know that n people will come to the workshop with a request to make a wooden toy. People are differ... | Input: ['561 7 7 9 9 965 4 2 1 30 60914 19 37 59 1 4 4 98 731263 10 1 17 15 11'] Output:['0', '2', '13', '0', '1', ''] | [
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Dima Vatrushin is a math teacher at school. He was sent on vacation for n days for his good work. Dima has long dreamed of going to a ski resort, so he wants to allocate several consecutive days and go skiing. Since the vacation requires careful preparation, he will only go for at least k days.You are given an array a ... | Input: ['73 1 15-5 0 -105 3 -338 12 9 0 54 3 1212 12 10 154 1 -50 -1 2 55 5 03 -1 4 -5 -31 1 556 1 30 3 -2 5 -4 -4'] Output:['6', '0', '1', '0', '0', '1', '9', ''] | [
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Once upon a time, Toma found himself in a binary cafe. It is a very popular and unusual place.The cafe offers visitors k different delicious desserts. The desserts are numbered from 0 to k-1. The cost of the i-th dessert is 2^i coins, because it is a binary cafe! Toma is willing to spend no more than n coins on tasting... | Input: ['51 22 12 210 2179 100'] Output:['2', '2', '3', '4', '180', ''] | [
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This is an interactive problem.Consider the following game for two players: Initially, an array of integers a_1, a_2, ..., a_n of length n is written on blackboard. Game consists of rounds. On each round, the following happens: The first player selects any i such that a_i \gt 0. If there is no such i, the first play... | Input: ['4', '10 4 6 3', '', '', '3', '', '1', '', '0'] Output:['', '', 'First', '1', '', '2', '', '4'] | [
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You have n lamps, numbered by integers from 1 to n. Each lamp i has two integer parameters a_i and b_i.At each moment each lamp is in one of three states: it may be turned on, turned off, or broken.Initially all lamps are turned off. In one operation you can select one lamp that is turned off and turn it on (you can't ... | Input: ['442 21 61 101 1353 43 12 53 23 361 23 41 43 43 52 311 1'] Output:['15', '14', '20', '1', ''] | [
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You are given two integers n and k.An array a_1, a_2, ..., a_n of length n, consisting of zeroes and ones is good if for all integers i from 1 to n both of the following conditions are satisfied: at least \lceil \frac{i}{k} \rceil of the first i elements of a are equal to 1, at least \lceil \frac{i}{k} \rceil of the ... | Input: ['73 25 29 37 110 49 58 8'] Output:['2', '3', '4', '7', '4', '3', '2', ''] | [
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This is an interactive problem.There is an n by n grid of conveyor belts, in positions (1, 1) through (n, n) of a coordinate plane. Every other square in the plane is empty. Each conveyor belt can be configured to move boxes up ('^'), down ('v'), left ('<'), or right ('>'). If a box moves onto an empty square, it stops... | Input: ['3', '', '', '', '', '-1 -1', '', '', '', '', '0 2'] Output:['', '? 2 2', '>><', '>>v', '^<<', '', '? 1 1', '>><', '>>v', '^<<', '', '! 1 2 ^'] | [
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You are given an array a of n integers, where all elements a_i lie in the range [1, k]. How many different arrays b of m integers, where all elements b_i lie in the range [1, k], contain a as a subsequence? Two arrays are considered different if they differ in at least one position.A sequence x is a subsequence of a se... | Input: ['71 1000000 113 4 31 2 25 7 81 2 3 4 16 6 1818 2 2 5 2 161 10 218 10 12345671 1 2 1 2 2 2 15 1000000000 1000000000525785549 816356460 108064697 194447117 725595511'] Output:['1', '9', '1079', '1', '1023', '906241579', '232432822', ''] | [
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There is a string s of length n consisting of the characters '(' and ')'. You are walking on this string. You start by standing on top of the first character of s, and you want to make a sequence of moves such that you end on the n-th character. In one step, you can move one space to the left (if you are not standing o... | Input: ['10 9', '(())()()))', '9', '7', '2', '6', '3', '6', '7', '4', '8', ''] Output:['YES', 'YES', 'NO', 'NO', 'YES', 'NO', 'YES', 'NO', 'NO', ''] | [
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You are given integers n and m. Fill an n by m grid with the integers 1 through n\cdot m, in such a way that for any two adjacent cells in the grid, the absolute difference of the values in those cells is not a prime number. Two cells in the grid are considered adjacent if they share a side. It can be shown that under... | Input: ['34 45 76 4'] Output:['16 7 1 9', '12 8 2 3', '13 4 10 11', '14 5 6 15', '', '29 23 17 9 5 6 2', '33 27 21 15 11 7 1', '32 31 25 19 20 16 10', '26 30 24 18 14 8 4', '35 34 28 22 13 12 3', '', ' 2 3 7 11', ' 8 9 1 10', '17 13 5 4', '18 14 6 12', '19 23 15 21', '20 24 16 22', ''] | [
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You are given a permutation p of size n. You want to minimize the number of subarrays of p that are permutations. In order to do so, you must perform the following operation exactly once: Select integers i, j, where 1 <= i, j <= n, then Swap p_i and p_j. For example, if p = [5, 1, 4, 2, 3] and we choose i = 2, j = 3,... | Input: ['831 2 331 3 251 3 2 5 464 5 6 1 2 398 7 6 3 2 1 4 5 9107 10 5 1 9 8 3 2 6 4108 5 10 9 2 1 3 4 6 7102 3 5 7 10 1 8 6 4 9'] Output:['2 3', '1 1', '5 2', '1 4', '9 5', '8 8', '6 10', '5 4', ''] | [
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Two integers were written on a blackboard. After that, the following step was carried out n-2 times: Select any two integers on the board, and write the absolute value of their difference on the board. After this process was complete, the list of n integers was shuffled. You are given the final list. Recover one of th... | Input: ['939 2 7315 -4 114-9 1 11 -1053 0 0 0 378 16 8 0 8 16 840 0 0 01027 1 24 28 2 -1 26 25 28 276600000000 800000000 0 -200000000 1000000000 80000000030 -1000000000 1000000000'] Output:['9', '11', '-9', '3', '8', '0', '-1', '600000000', '0', ''] | [
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Berland Intercollegiate Contest has just finished. Monocarp and Polycarp, as the jury, are going to conduct an editorial. Unfortunately, the time is limited, since they have to finish before the closing ceremony.There were n problems in the contest. The problems are numbered from 1 to n. The editorial for the i-th prob... | Input: ['65 41 10 1 1 15 31 20 5 15 35 31 20 3 15 510 610 8 20 14 3 8 6 4 16 1110 59 9 2 13 15 19 4 9 13 121 11'] Output:['2', '6', '5', '21', '18', '1', ''] | [
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A regular bracket sequence is a bracket sequence that can be transformed into a correct arithmetic expression by inserting characters "1" and "+" between the original characters of the sequence. For example: the bracket sequences "()()" and "(())" are regular (the resulting expressions are: "(1)+(1)" and "((1+1)+1)");... | Input: ['48((())))(4(())4))((3(()'] Output:['2', '2 2 2 1 2 2 2 1', '1', '1 1 1 1', '1', '1 1 1 1', '-1', ''] | [
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You are given a string s consisting of the characters 0, 1 and/or ?. Let's call it a pattern.Let's say that the binary string (a string where each character is either 0 or 1) matches the pattern if you can replace each character ? with 0 or 1 (for each character, the choice is independent) so that the strings become eq... | Input: ['4??01?101001??10?0?1?10?10'] Output:['00011', '10100', '111101', '011110010', ''] | [
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You are given a string s of length n, where each character is either < or >.An array a consisting of n+1 elements is compatible with the string s if, for every i from 1 to n, the character s_i represents the result of comparing a_i and a_{i+1}, i. e.: s_i is < if and only if a_i < a_{i+1}; s_i is > if and only if a_i... | Input: ['44<<>>4>><<5>>>>>7<><><><'] Output:['3', '3', '6', '2', ''] | [
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You are given two integers x and k. Grasshopper starts in a point 0 on an OX axis. In one move, it can jump some integer distance, that is not divisible by k, to the left or to the right.What's the smallest number of moves it takes the grasshopper to reach point x? What are these moves? If there are multiple answers, p... | Input: ['310 210 33 4'] Output:['2', '7 3', '1', '10', '1', '3', ''] | [
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In many, many years, far, far away, there will be a launch of the first flight to Mars. To celebrate the success, n astrophysicists working on the project will be given bonuses of a total value of k gold coins.You have to distribute the money among the astrophysicists, and to make it easier, you have to assign bonuses ... | Input: ['53 3 1002 1 1491 2 1336 16 673 8 22'] Output:['100', '0', '26', '72', '176', ''] | [
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The rebels have been crushed in the most recent battle with the imperial forces, but there is a ray of new hope.Meanwhile, on one of the conquered planets, Luke was getting ready for an illegal street race (which should come as no surprise, given his family history). Luke arrived at the finish line with 88 miles per ho... | Input: ['7 8 346', '1 2', '1 3', '2 4', '3 4', '4 5', '5 1', '6 7', '7 6', ''] Output:['5', ''] | [
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Famous worldwide astrophysicist Mleil waGrasse Tysok recently read about the existence of twin galaxy clusters. Before he shares this knowledge with the broader audience in his podcast called S.tarT-ok, he wants to prove their presence on his own. Mleil is aware that the vastness of the universe is astounding (almost a... | Input: ['424 15 0 7 11 8 3 210 1 2 300 0315 63 57 39 61 25 42 61 50 41 27 41 56 23 17 27'] Output:['2 4 6 6', '2 2 3 4', '1 1 2 2', '1 1 4 10', ''] | [
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n people indexed with integers from 1 to n came to take part in a lottery. Each received a ticket with an integer from 0 to m.In a lottery, one integer called target is drawn uniformly from 0 to m. k tickets (or less, if there are not enough participants) with the closest numbers to the target are declared the winners.... | Input: ['3 6 2', '1 4 5', ''] Output:['4 2', ''] | [
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Consider all equalities of form a + b = c, where a has A digits, b has B digits, and c has C digits. All the numbers are positive integers and are written without leading zeroes. Find the k-th lexicographically smallest equality when written as a string like above or determine that it does not exist.For example, the fi... | Input: ['71 1 1 92 2 3 12 2 1 11 5 6 421 6 6 100000005 5 6 30315688156 6 6 1000000000000'] Output:['2 + 1 = 3', '10 + 90 = 100', '-1', '9 + 99996 = 100005', '-1', '78506 + 28543 = 107049', '-1', ''] | [
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Recently, Polycarp was given an unusual typewriter as a gift! Unfortunately, the typewriter was defective and had a rather strange design.The typewriter consists of n cells numbered from left to right from 1 to n, and a carriage that moves over them. The typewriter cells contain n distinct integers from 1 to n, and eac... | Input: ['3', '2 3 1', '0', ''] Output:['1', ''] | [
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You are given an array a of length n. A positive integer x is called good if it is impossible to find a subsegment of the array such that the least common multiple of all its elements is equal to x.You need to find the smallest good integer.A subsegment of the array a is a set of elements a_l, a_{l + 1}, ..., a_r for s... | Input: ['631 2 351 2 3 4 522 311000000000121 8 4 2 3 5 7 2 9 10 11 13127 2 5 4 2 1 1 2 3 11 8 9'] Output:['4', '7', '1', '1', '16', '13', ''] | [
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Zinaida Viktorovna has n students in her history class. The homework for today included m topics, but the students had little time to prepare, so i-th student learned only topics from l_i to r_i inclusive.At the beginning of the lesson, each student holds their hand at 0. The teacher wants to ask some topics. It goes l... | Input: ['64 82 64 82 71 53 31 32 32 23 51 51 51 53 51 13 35 54 71 71 33 34 52 41 32 4'] Output:['6', '4', '0', '2', '12', '2', ''] | [
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Alice and Bob are playing a game. They have two strings S and T of the same length n consisting of lowercase latin letters. Players take turns alternately, with Alice going first.On her turn, Alice chooses an integer i from 1 to n, one of the strings S or T, and any lowercase latin letter c, and replaces the i-th symbo... | Input: ['75abcdeabxde5helloolleo2abcd7aaaaaaaabbbbba1qq6yoyoyooyoyoy8abcdefghhguedfbh'] Output:['1', '2', '3', '9', '0', '2', '6', ''] | [
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Fedya is playing a new game called "The Legend of Link", in which one of the character's abilities is to combine two materials into one weapon. Each material has its own strength, which can be represented by a positive integer x. The strength of the resulting weapon is determined as the sum of the absolute differences ... | Input: ['653 57179 23913 37132228 13222854943329752812629795 5515758193968886336688 1914'] Output:['4', '19', '11', '0', '163', '28', ''] | [
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Given an array a of length n, which elements are equal to -1 and 1. Let's call the array a good if the following conditions are held at the same time: a_1 + a_2 + ... + a_n >= 0; a_1 \cdot a_2 \cdot ... \cdot a_n = 1. In one operation, you can select an arbitrary element of the array a_i and change its value to the o... | Input: ['74-1 -1 1 -15-1 -1 -1 1 14-1 1 -1 13-1 -1 -151 1 1 1 11-12-1 -1'] Output:['1', '1', '0', '3', '0', '1', '2', ''] | [
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Ksyusha has a pet chinchilla, a tree on n vertices and huge scissors. A tree is a connected graph without cycles. During a boring physics lesson Ksyusha thought about how to entertain her pet.Chinchillas like to play with branches. A branch is a tree of 3 vertices. The branch looks like this. A cut is the removal of s... | Input: ['491 24 37 95 44 63 28 71 761 21 34 31 56 161 23 23 44 56 551 35 35 23 4'] Output:['2', '2 8 ', '-1', '1', '3 ', '-1', ''] | [
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Ira loves Spanish flamenco dance very much. She decided to start her own dance studio and found n students, ith of whom has level a_i.Ira can choose several of her students and set a dance with them. So she can set a huge number of dances, but she is only interested in magnificent dances. The dance is called magnificen... | Input: ['97 48 10 10 9 6 11 75 34 2 2 3 68 21 5 2 2 3 1 3 33 33 3 35 13 4 3 10 712 35 2 1 1 4 3 5 5 5 2 7 51 113 21 2 32 21 2'] Output:['5', '2', '10', '0', '5', '11', '1', '2', '1', ''] | [
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You are given a permutation p of length n.A permutation is an array consisting of n distinct integers from 1 to n in any order. For example, \{2,3,1,5,4\} is a permutation, while \{1,2,2\} is not (since 2 appears twice), and \{1,3,4\} is also not a permutation (as n=3, but the array contains 4).To the permutation p, yo... | Input: ['952 3 1 5 494 1 6 7 2 8 5 3 944 3 2 122 163 2 4 1 5 673 2 1 5 7 6 41010 2 5 6 1 9 3 8 4 744 2 1 311'] Output:['5 4 1 3 2 ', '9 4 1 6 7 2 8 5 3 ', '3 2 1 4 ', '1 2 ', '6 5 3 2 4 1 ', '7 6 4 5 3 2 1 ', '9 3 8 4 7 1 10 2 5 6 ', '3 4 2 1 ', '1 ', ''] | [
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Vlad was given an array a of n positive integers. Now he wants to build a beautiful array b of length n from it.Vlad considers an array beautiful if all the numbers in it are positive and have the same parity. That is, all numbers in the beautiful array are greater than zero and are either all even or all odd.To build ... | Input: ['752 6 8 4 351 4 7 6 942 6 4 1075 29 13 9 10000001 11 352 1 2 4 252 4 5 4 342 5 5 4'] Output:['NO', 'YES', 'YES', 'YES', 'YES', 'NO', 'NO', ''] | [
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You are given an array a containing the weather forecast for Berlandia for the last n days. That is, a_i — is the estimated air temperature on day i (1 <= i <= n).You are also given an array b — the air temperature that was actually present on each of the days. However, all the values in array b are mixed up. Determine... | Input: ['35 21 3 5 3 92 5 11 2 46 1-1 3 -2 0 -5 -1-4 0 -1 4 0 03 37 7 79 4 8'] Output:['2 2 5 4 11', '0 4 -1 0 -4 0', '8 4 9'] | [
2
] |
Polycarp plays a computer game in a post-apocalyptic setting. The zombies have taken over the world, and Polycarp with a small team of survivors is defending against hordes trying to invade their base. The zombies are invading for x minutes starting from minute 0. There are n entrances to the base, and every minute one... | Input: ['3 3 10 3', '0 2', '1 7', '4 7', ''] Output:['18', ''] | [
4
] |
Recall that the binomial coefficient \binom{x}{y} is calculated as follows (x and y are non-negative integers): if x < y, then \binom{x}{y} = 0; otherwise, \binom{x}{y} = \frac{x!}{y! \cdot (x-y)!}. You are given an array a_1, a_2, ..., a_n and an integer k. You have to calculate a new array b_1, b_2, ..., b_n, where... | Input: ['5 8 2 3 100 2', ''] Output:['1283', ''] | [
0
] |
The only difference between easy and hard versions is the maximum values of n and q.You are given an array, consisting of n integers. Initially, all elements are red.You can apply the following operation to the array multiple times. During the i-th operation, you select an element of the array; then: if the element i... | Input: ['4 10', '5 2 8 4', '1 2 3 4 5 6 7 8 9 10', ''] Output:['3 4 5 6 7 8 8 10 8 12', ''] | [
2,
3,
4
] |
The only difference between easy and hard versions is the maximum values of n and q.You are given an array, consisting of n integers. Initially, all elements are red.You can apply the following operation to the array multiple times. During the i-th operation, you select an element of the array; then: if the element i... | Input: ['4 10', '5 2 8 4', '1 2 3 4 5 6 7 8 9 10', ''] Output:['3 4 5 6 7 8 8 10 8 12', ''] | [
2,
3,
4
] |
For an array of integers [a_1, a_2, ..., a_n], let's call the value |a_1-a_2|+|a_2-a_3|+\cdots+|a_{n-1}-a_n| the contrast of the array. Note that the contrast of an array of size 1 is equal to 0.You are given an array of integers a. Your task is to build an array of b in such a way that all the following conditions are... | Input: ['451 3 3 3 724 241 1 1 175 4 2 1 0 0 4'] Output:['2', '2', '1', '3', ''] | [
2
] |
You are given an array a_1, a_2, ..., a_n, where all elements are different.You have to perform exactly k operations with it. During each operation, you do exactly one of the following two actions (you choose which to do yourself): find two minimum elements in the array, and delete them; find the maximum element in t... | Input: ['65 12 5 1 10 65 22 5 1 10 63 11 2 36 115 22 12 10 13 116 215 22 12 10 13 115 1999999996 999999999 999999997 999999998 999999995'] Output:['21', '11', '3', '62', '46', '3999999986', ''] | [
0
] |
You are given two arrays a and b both of length n.You will merge^ these arrays forming another array c of length 2 \cdot n. You have to find the maximum length of a subarray consisting of equal values across all arrays c that could be obtained.^ A merge of two arrays results in an array c composed by successively takin... | Input: ['412231 2 34 5 621 22 151 2 2 2 22 1 1 1 1'] Output:['2', '1', '2', '5', ''] | [
2
] |
On a permutation p of length n, we define a bully swap as follows: Let i be the index of the largest element p_i such that p_i \neq i. Let j be the index of the smallest element p_j such that i < j. Swap p_i and p_j. We define f(p) as the number of bully swaps we need to perform until p becomes sorted. Note that if ... | Input: ['8 5', '6 2 1 5 3 4 7 8', '1 8', '2 3', '4 7', '7 8', '3 6', ''] Output:['5', '6', '9', '8', '7', ''] | [
3
] |
You are given a tree with n nodes. For each node, you either color it in 0 or 1.The value of a path (u,v) is equal to the MEX^ of the colors of the nodes from the shortest path between u and v.The value of a coloring is equal to the sum of values of all paths (u,v) such that 1 <=q u <=q v <=q n.What is the maximum poss... | Input: ['431 22 341 21 31 4101 21 33 43 51 65 72 86 96 101'] Output:['8', '15', '96', '1', ''] | [
0
] |
You are given an integer n and k intervals. The i-th interval is [l_i,r_i] where 1 <=q l_i <=q r_i <=q n.Let us call a regular bracket sequence^{,\ddagger} of length n hyperregular if for each i such that 1 <=q i <=q k, the substring \overline{s_{l_i} s_{l_{i}+1} ... s_{r_i}} is also a regular bracket sequence.Your tas... | Input: ['76 05 08 11 310 23 46 91000 3100 701200 801300 90128 51 123 2011 144 918 194 31 41 41 4'] Output:['5', '0', '0', '4', '839415253', '140', '2', ''] | [
2,
3
] |
You are given two arrays a and b, both of length n.Your task is to count the number of pairs of integers (i,j) such that 1 <=q i < j <=q n and a_i \cdot a_j = b_i+b_j. | Input: ['332 3 23 3 184 2 8 2 1 2 7 53 5 8 8 1 1 6 584 4 8 8 8 8 8 88 8 8 8 8 8 8 8'] Output:['2', '7', '1', ''] | [
0,
3
] |
Sadly, the problem setter couldn't think of an interesting story, thus he just asks you to solve the following problem.Given an array a consisting of n positive integers, count the number of non-empty subsequences for which the bitwise \mathsf{AND} of the elements in the subsequence has exactly k set bits in its binary... | Input: ['65 11 1 1 1 14 00 1 2 35 15 5 7 4 21 2312 00 2 0 2 0 2 0 2 0 2 0 210 663 0 63 5 5 63 63 4 12 13'] Output:['31', '10', '10', '1', '4032', '15', ''] | [
3
] |
In a carnival game, there is a huge pyramid of cans with 2023 rows, numbered in a regular pattern as shown. If can 9^2 is hit initially, then all cans colored red in the picture above would fall. You throw a ball at the pyramid, and it hits a single can with number n^2. This causes all cans that are stacked on top of t... | Input: ['1091234561014341000000'] Output:['156', '1', '5', '10', '21', '39', '46', '146', '63145186', '58116199242129511', ''] | [
3
] |
A snowflake graph is generated from two integers x and y, both greater than 1, as follows: Start with one central vertex. Connect x new vertices to this central vertex. Connect y new vertices to each of these x vertices. For example, below is a snowflake graph for x=5 and y=3. The snowflake graph above has a cent... | Input: ['321 2021 205 2013 201 311 310 34 819 814 89 712 717 718 616 62 66 157 158 1520 153 157 61 21 32 42 53 63 79 89 33 66 22 15 22 74 33 8'] Output:['5 3', '2 2', '2 3', ''] | [
3
] |
Initially you have a single pile with n gold nuggets. In an operation you can do the following: Take any pile and split it into two piles, so that one of the resulting piles has exactly twice as many gold nuggets as the other. (All piles should have an integer number of nuggets.) One possible move is to take a pile... | Input: ['116 49 44 218 2727 427 227 101 13 15 1746001 2984004'] Output:['YES', 'YES', 'NO', 'NO', 'YES', 'YES', 'NO', 'YES', 'YES', 'NO', 'NO', ''] | [
0
] |
Victor wants to become "Mr. Perfectly Fine". For that, he needs to acquire a certain set of skills. More precisely, he has 2 skills he needs to acquire. Victor has n books. Reading book i takes him m_i minutes and will give him some (possibly none) of the required two skills, represented by a binary string of length 2.... | Input: ['642 003 104 014 0053 013 015 012 109 1015 1139 118 017 1064 016 017 018 009 011 0048 009 109 118 11'] Output:['7', '5', '5', '9', '-1', '8', ''] | [
2
] |
You are given an unsorted permutation p_1, p_2, ..., p_n. To sort the permutation, you choose a constant k (k >= 1) and do some operations on the permutation. In one operation, you can choose two integers i, j (1 <= j < i <= n) such that i - j = k, then swap p_i and p_j.What is the maximum value of k that you can choos... | Input: ['733 1 243 4 1 274 2 6 7 5 3 191 6 7 4 9 2 3 8 561 5 3 4 2 6103 10 5 2 9 6 7 8 1 4111 11 6 4 8 3 7 5 9 10 2'] Output:['1', '2', '3', '4', '3', '2', '3', ''] | [
3
] |
You are given a positive integer n. Please find an array a_1, a_2, ..., a_n that is perfect.A perfect array a_1, a_2, ..., a_n satisfies the following criteria: 1 <= a_i <= 1000 for all 1 <= i <= n. a_i is divisible by i for all 1 <= i <= n. a_1 + a_2 + ... + a_n is divisible by n. | Input: ['71234567'] Output:['1', '2 4', '1 2 3', '2 8 6 4', '3 4 9 4 5', '1 10 18 8 5 36', '3 6 21 24 10 6 14', ''] | [
3
] |
You are given a permutation a_1,a_2,...,a_n of the first n positive integers. A subarray [l,r] is called copium if we can rearrange it so that it becomes a sequence of consecutive integers, or more formally, if \max(a_l,a_{l+1},...,a_r)-\min(a_l,a_{l+1},...,a_r)=r-l For each k in the range [0,n], print out the maximum ... | Input: ['555 2 1 4 342 1 4 31187 5 8 1 4 2 6 3101 4 5 3 7 8 9 2 10 6'] Output:['15 15 11 10 9 9 ', '10 8 8 7 7 ', '1 1 ', '36 30 25 19 15 13 12 9 9 ', '55 55 41 35 35 25 22 22 19 17 17 ', ''] | [
2
] |
There is a country consisting of n cities and n - 1 bidirectional roads connecting them such that we can travel between any two cities using these roads. In other words, these cities and roads form a tree.There are m bus routes connecting the cities together. A bus route between city x and city y allows you to travel b... | Input: ['45 21 22 33 42 51 45 25 11 22 33 42 51 52 01 26 31 22 33 44 55 61 32 54 6'] Output:['YES', 'NO', '1 3', 'NO', '1 2', 'NO', '1 6'] | [
2,
4
] |
You are given a tree (an undirected connected acyclic graph) which initially only contains vertex 1. There will be several queries to the given tree. In the i-th query, vertex i + 1 will appear and be connected to vertex p_i (1 <= p_i <= i). After each query, please find out the least number of operations required to m... | Input: ['52131 141 2 371 2 3 2 5 2101 2 2 4 5 5 7 8 9'] Output:['0', '0 1', '0 1 0', '0 1 0 1 2 3', '0 1 2 1 0 1 0 1 2'] | [
2
] |
A substring is a continuous and non-empty segment of letters from a given string, without any reorders.An even palindrome is a string that reads the same backward as forward and has an even length. For example, strings "zz", "abba", "abccba" are even palindromes, but strings "codeforces", "reality", "aba", "c" are not.... | Input: ['66abaaba1a2aa6abcdef12accabccbacca6abbaaa'] Output:['3', '0', '1', '0', '14', '6', ''] | [
0,
4
] |
The only difference between this problem and the easy version is the constraints on t and n.You are given an array a, consisting of n distinct integers a_1, a_2, ..., a_n. Define the beauty of an array p_1, p_2, ... p_k as the minimum amount of time needed to sort this array using an arbitrary number of range-sort oper... | Input: ['526 433 10 644 8 7 259 8 2 4 6122 6 13 3 15 5 10 8 16 9 11 18'] Output:['1', '2', '8', '16', '232', ''] | [
2,
4
] |
The only difference between this problem and the hard version is the constraints on t and n.You are given an array a, consisting of n distinct integers a_1, a_2, ..., a_n. Define the beauty of an array p_1, p_2, ... p_k as the minimum amount of time needed to sort this array using an arbitrary number of range-sort oper... | Input: ['526 433 10 644 8 7 259 8 2 4 6122 6 13 3 15 5 10 8 16 9 11 18'] Output:['1', '2', '8', '16', '232', ''] | [
2,
4
] |
You are given two arrays a and b each consisting of n integers. All elements of a are pairwise distinct.Find the number of ways to reorder a such that a_i > b_i for all 1 <= i <= n, modulo 10^9 + 7.Two ways of reordering are considered different if the resulting arrays are different. | Input: ['569 6 8 4 5 24 1 5 6 3 134 3 23 4 912132 3 41 3 3122 3 7 10 23 28 29 50 69 135 420 10001 1 2 3 5 8 13 21 34 55 89 144'] Output:['32', '0', '1', '0', '13824', ''] | [
3
] |
This is an interactive problem.There are n distinct hidden points with real coordinates on a two-dimensional Euclidean plane. In one query, you can ask some line ax + by + c = 0 and get the projections of all n points to this line in some order. The given projections are not exact, please read the interaction section f... | Input: ['1', '2', '', '1 1 2.5 1', '', '1.500000001 1.500000000 2 2', '', ''] Output:['', '', '? 0 1 -1', '', '? 0.2 -0.2 0', '', '! 1 3 2.5 0.500000001'] | [
3
] |
A fashion tour consists of m identical runway shows in different cities. There are n models willing to participate in the tour, numbered from 1 to n. People in different cities have different views on the fashion industry, so they rate each model differently. In particular, people in city i rate model j with rating r_{... | Input: ['3 5', '10 10 10 10 10', '1 2 3 4 5', '1 5 2 3 4', '2 3 4 5 1', ''] Output:['30', ''] | [
0
] |
There is a street with n sights, with sight number i being i miles from the beginning of the street. Sight number i has beauty b_i. You want to start your morning jog l miles and end it r miles from the beginning of the street. By the time you run, you will see sights you run by (including sights at l and r miles from ... | Input: ['455 1 4 2 341 1 1 169 8 7 6 5 47100000000 1 100000000 1 100000000 1 100000000'] Output:['8', '1', '22', '299999996', ''] | [
0,
2
] |
Because to take away a man's freedom of choice, even his freedom to make the wrong choice, is to manipulate him as though he were a puppet and not a person.— Madeleine L'EngleThere are n programmers choosing their favorite algorithm amongst m different choice options. Before the first round, all m options are available... | Input: ['53 24 25 31000000 10000001 1000000'] Output:['YES', 'NO', 'YES', 'NO', 'YES', ''] | [
2,
3
] |
You have an array a of n non-negative integers. Let's define f(a, x) = [a_1 \bmod x, a_2 \bmod x, ..., a_n \bmod x] for some positive integer x. Find the biggest x, such that f(a, x) is a palindrome.Here, a \bmod x is the remainder of the integer division of a by x.An array is a palindrome if it reads the same backward... | Input: ['421 283 0 1 2 0 3 2 1103100 1 1000000000'] Output:['1', '2', '0', '999999900', ''] | [
3
] |
There is a group of n people. Some of them might be liars, who always tell lies. Other people always tell the truth. The i-th person says "There are at least l_i liars amongst us". Determine if what people are saying is contradictory, or if it is possible. If it is possible, output the number of liars in the group. If ... | Input: ['721 222 220 0111055 5 3 3 565 3 6 6 3 5'] Output:['1', '-1', '0', '-1', '0', '3', '4', ''] | [
0,
2
] |
LuoTianyi gave an array b of n \cdot m integers. She asks you to construct a table a of size n * m, filled with these n \cdot m numbers, and each element of the array must be used exactly once. Also she asked you to maximize the following value: \sum\limits_{i=1}^{n}\sum\limits_{j=1}^{m}<=ft(\max\limits_{1 <= x <= i, 1... | Input: ['52 21 3 1 42 2-1 -1 -1 -12 37 8 9 -3 10 83 24 8 -3 0 -7 14 3-32030 59554 16854 -85927 68060 -64460 -79547 90932 85063 82703 -12001 38762'] Output:['9', '0', '64', '71', '1933711', ''] | [
2,
3
] |
LuoTianyi gives you a palindrome string s, and she wants you to find out the length of the longest non-empty subsequence^{\ddagger} of s which is not a palindrome string. If there is no such subsequence, output -1 instead. A palindrome is a string that reads the same backward as forward. For example, strings "z", "aaa"... | Input: ['4abacabaaaacodeforcesecrofedoclol'] Output:['6', '-1', '18', '2', ''] | [
2
] |
LuoTianyi gives you a tree with values in its vertices, and the root of the tree is vertex 1.In one operation, you can change the value in one vertex to any non-negative integer.Now you need to find the minimum number of operations you need to perform to make each path from the root to leaf has a bitwise XOR value of z... | Input: ['6', '3 5 7 5 8 4', '1 2', '1 3', '1 4', '3 5', '4 6', ''] Output:['3'] | [
2
] |
This is the hard version of the problem. The only difference is that in this version k<= n. You can make hacks only if both versions of the problem are solved. Chtholly and the floating islands. LuoTianyi now lives in a world with n floating islands. The floating islands are connected by n-1 undirected air routes, an... | Input: ['4 2', '1 2', '2 3', '3 4', ''] Output:['666666674', ''] | [
3
] |
This is the easy version of the problem. The only difference is that in this version k<=\min(n,3). You can make hacks only if both versions of the problem are solved. Chtholly and the floating islands. LuoTianyi now lives in a world with n floating islands. The floating islands are connected by n-1 undirected air rout... | Input: ['4 2', '1 2', '2 3', '3 4', ''] Output:['666666674', ''] | [
3
] |
There are n people taking part in a show about VOCALOID. They will sit in the row of seats, numbered 1 to m from left to right.The n people come and sit in order. Each person occupies a seat in one of three ways: Sit in the seat next to the left of the leftmost person who is already sitting, or if seat 1 is taken, the... | Input: ['103 105 5 54 61 -2 -2 15 7-1 -1 4 -2 -26 75 -2 -2 -2 -2 -26 6-1 1 4 5 -1 46 8-1 -1 -1 3 -1 -26 75 -1 -2 -2 -2 -23 1-2 -2 12 55 -21 2-1'] Output:['1', '3', '5', '6', '5', '5', '5', '1', '2', '1', ''] | [
2
] |
You are given a tree consisting of n vertices and n - 1 edges, and each vertex v has a counter c(v) assigned to it.Initially, there is a chip placed at vertex s and all counters, except c(s), are set to 0; c(s) is set to 1.Your goal is to place the chip at vertex t. You can achieve it by a series of moves. Suppose righ... | Input: ['3 1 3', '1 2', '2 3', ''] Output:['2 2 1 ', ''] | [
3
] |
Alice and Bob are playing a game on a graph. They have an undirected graph without self-loops and multiple edges. All vertices of the graph have degree equal to 2. The graph may consist of several components. Note that if such graph has n vertices, it will have exactly n edges.Alice and Bob take turn. Alice goes first.... | Input: ['6 2 3', '1 2', '2 3', '3 1', '4 5', '5 6', '6 4', ''] Output:['Bob', ''] | [
0,
3
] |
A palindrome is a string that reads the same backwards as forwards. For example, the string abcba is palindrome, while the string abca is not.Let p(t) be the number of unique palindromic substrings of string t, i. e. the number of substrings t[l ... r] that are palindromes themselves. Even if some substring occurs in t... | Input: ['710 25 105 63 1334 23 43 34 23 43 44 14510 34 6 104 5 810 44 6 7 104 5 7 8'] Output:['YES', 'abcbbcabcb', 'YES', 'foo', 'YES', 'ayda', 'YES', 'wada', 'NO', 'YES', 'abcbcacbab', 'NO', ''] | [
3
] |
A prime number is an integer greater than 1, which has exactly two divisors. For example, 7 is a prime, since it has two divisors \{1, 7\}. A composite number is an integer greater than 1, which has more than two different divisors.Note that the integer 1 is neither prime nor composite.Let's look at some composite numb... | Input: ['823 633 4 522 333 10 14225 301108093 3 3 5 5 5 7 7 72012 15 2 2 2 2 2 3 3 3 17 21 21 21 30 6 6 33 31 39'] Output:['1', '1', '0', '2', '2', '3', '4', '15', ''] | [
2,
3
] |
Let's define a permutation of length n as an array p of length n, which contains every number from 1 to n exactly once.You are given a permutation p_1, p_2, ..., p_n and a number k. You need to sort this permutation in the ascending order. In order to do it, you can repeat the following operation any number of times (p... | Input: ['64 13 1 2 44 23 4 1 24 23 1 4 210 34 5 9 1 8 6 10 2 3 710 34 6 9 1 8 5 10 2 3 710 34 6 9 1 8 5 10 3 2 7'] Output:['0', '0', '1', '0', '1', '-1', ''] | [
0,
3
] |
Consider an array a_1, a_2, ..., a_n consisting of numbers 1 and -1. Define A-characteristic of this array as a number of pairs of indices 1 <= i < j <= n, such that a_i \cdot a_j = 1.Find any array a with given length n with A-characteristic equal to the given value k. | Input: ['72 02 13 13 23 35 45 5'] Output:['YES', '1 -1 ', 'YES', '1 1 ', 'YES', '1 -1 1 ', 'NO', 'YES', '1 1 1 ', 'YES', '-1 1 -1 1 1 ', 'NO'] | [
3
] |
This is the hard version of the problem. The only difference is that in this version, a_i <= 10^9.For a given sequence of n integers a, a triple (i, j, k) is called magic if: 1 <= i, j, k <= n. i, j, k are pairwise distinct. there exists a positive integer b such that a_i \cdot b = a_j and a_j \cdot b = a_k. Kolya ... | Input: ['751 7 7 2 736 2 1891 2 3 4 5 6 7 8 941000 993 986 17971 10 100 1000 10000 100000 100000081 1 2 2 4 4 8 891 1 1 2 2 2 4 4 4'] Output:['6', '1', '3', '0', '9', '16', '45', ''] | [
0,
3
] |
This is the easy version of the problem. The only difference is that in this version, a_i <= 10^6.For a given sequence of n integers a, a triple (i, j, k) is called magic if: 1 <= i, j, k <= n. i, j, k are pairwise distinct. there exists a positive integer b such that a_i \cdot b = a_j and a_j \cdot b = a_k. Kolya ... | Input: ['751 7 7 2 736 2 1891 2 3 4 5 6 7 8 941000 993 986 17971 10 100 1000 10000 100000 100000081 1 2 2 4 4 8 891 1 1 2 2 2 4 4 4'] Output:['6', '1', '3', '0', '9', '16', '45', ''] | [
0,
3
] |
Two friends, Alisa and Yuki, planted a tree with n vertices in their garden. A tree is an undirected graph without cycles, loops, or multiple edges. Each edge in this tree has a length of k. Initially, vertex 1 is the root of the tree.Alisa and Yuki are growing the tree not just for fun, they want to sell it. The cost ... | Input: ['43 2 32 13 15 4 12 14 25 43 46 5 34 16 12 65 13 210 6 41 31 99 77 66 49 22 88 55 10'] Output:['2', '12', '17', '32', ''] | [
0
] |
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