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You are given a rooted tree consisting of $$$n$$$ vertices numbered from $$$1$$$ to $$$n$$$. The root is vertex $$$1$$$. There is also a string $$$s$$$ denoting the color of each vertex: if $$$s_i = \texttt{B}$$$, then vertex $$$i$$$ is black, and if $$$s_i = \texttt{W}$$$, then vertex $$$i$$$ is white.A subtree of the tree is called balanced if the number of white vertices equals the number of black vertices. Count the number of balanced subtrees.A tree is a connected undirected graph without cycles. A rooted tree is a tree with a selected vertex, which is called the root. In this problem, all trees have root $$$1$$$.The tree is specified by an array of parents $$$a_2, \dots, a_n$$$ containing $$$n-1$$$ numbers: $$$a_i$$$ is the parent of the vertex with the number $$$i$$$ for all $$$i = 2, \dots, n$$$. The parent of a vertex $$$u$$$ is a vertex that is the next vertex on a simple path from $$$u$$$ to the root.The subtree of a vertex $$$u$$$ is the set of all vertices that pass through $$$u$$$ on a simple path to the root. For example, in the picture below, $$$7$$$ is in the subtree of $$$3$$$ because the simple path $$$7 \to 5 \to 3 \to 1$$$ passes through $$$3$$$. Note that a vertex is included in its subtree, and the subtree of the root is the entire tree. The picture shows the tree for $$$n=7$$$, $$$a=[1,1,2,3,3,5]$$$, and $$$s=\texttt{WBBWWBW}$$$. The subtree at the vertex $$$3$$$ is balanced. | ['graphs', 'trees'] |
The $$$\text{$$$gcdSum$$$}$$$ of a positive integer is the $$$gcd$$$ of that integer with its sum of digits. Formally, $$$\text{$$$gcdSum$$$}(x) = gcd(x, \text{ sum of digits of } x)$$$ for a positive integer $$$x$$$. $$$gcd(a, b)$$$ denotes the greatest common divisor of $$$a$$$ and $$$b$$$ — the largest integer $$$d$$$ such that both integers $$$a$$$ and $$$b$$$ are divisible by $$$d$$$.For example: $$$\text{$$$gcdSum$$$}(762) = gcd(762, 7 + 6 + 2)=gcd(762,15) = 3$$$.Given an integer $$$n$$$, find the smallest integer $$$x \ge n$$$ such that $$$\text{$$$gcdSum$$$}(x) > 1$$$. | ['math'] |
We call a string good, if after merging all the consecutive equal characters, the resulting string is palindrome. For example, "aabba" is good, because after the merging step it will become "aba".Given a string, you have to find two values: the number of good substrings of even length; the number of good substrings of odd length. | ['math'] |
You are given a list of $$$n$$$ integers. You can perform the following operation: you choose an element $$$x$$$ from the list, erase $$$x$$$ from the list, and subtract the value of $$$x$$$ from all the remaining elements. Thus, in one operation, the length of the list is decreased by exactly $$$1$$$.Given an integer $$$k$$$ ($$$k>0$$$), find if there is some sequence of $$$n-1$$$ operations such that, after applying the operations, the only remaining element of the list is equal to $$$k$$$. | ['math'] |
Mahmoud wants to write a new dictionary that contains n words and relations between them. There are two types of relations: synonymy (i. e. the two words mean the same) and antonymy (i. e. the two words mean the opposite). From time to time he discovers a new relation between two words.He know that if two words have a relation between them, then each of them has relations with the words that has relations with the other. For example, if like means love and love is the opposite of hate, then like is also the opposite of hate. One more example: if love is the opposite of hate and hate is the opposite of like, then love means like, and so on.Sometimes Mahmoud discovers a wrong relation. A wrong relation is a relation that makes two words equal and opposite at the same time. For example if he knows that love means like and like is the opposite of hate, and then he figures out that hate means like, the last relation is absolutely wrong because it makes hate and like opposite and have the same meaning at the same time.After Mahmoud figured out many relations, he was worried that some of them were wrong so that they will make other relations also wrong, so he decided to tell every relation he figured out to his coder friend Ehab and for every relation he wanted to know is it correct or wrong, basing on the previously discovered relations. If it is wrong he ignores it, and doesn't check with following relations.After adding all relations, Mahmoud asked Ehab about relations between some words based on the information he had given to him. Ehab is busy making a Codeforces round so he asked you for help. | ['graphs'] |
You are given three strings $$$s$$$, $$$t$$$ and $$$p$$$ consisting of lowercase Latin letters. You may perform any number (possibly, zero) operations on these strings.During each operation you choose any character from $$$p$$$, erase it from $$$p$$$ and insert it into string $$$s$$$ (you may insert this character anywhere you want: in the beginning of $$$s$$$, in the end or between any two consecutive characters). For example, if $$$p$$$ is aba, and $$$s$$$ is de, then the following outcomes are possible (the character we erase from $$$p$$$ and insert into $$$s$$$ is highlighted): aba $$$\rightarrow$$$ ba, de $$$\rightarrow$$$ ade; aba $$$\rightarrow$$$ ba, de $$$\rightarrow$$$ dae; aba $$$\rightarrow$$$ ba, de $$$\rightarrow$$$ dea; aba $$$\rightarrow$$$ aa, de $$$\rightarrow$$$ bde; aba $$$\rightarrow$$$ aa, de $$$\rightarrow$$$ dbe; aba $$$\rightarrow$$$ aa, de $$$\rightarrow$$$ deb; aba $$$\rightarrow$$$ ab, de $$$\rightarrow$$$ ade; aba $$$\rightarrow$$$ ab, de $$$\rightarrow$$$ dae; aba $$$\rightarrow$$$ ab, de $$$\rightarrow$$$ dea; Your goal is to perform several (maybe zero) operations so that $$$s$$$ becomes equal to $$$t$$$. Please determine whether it is possible.Note that you have to answer $$$q$$$ independent queries. | ['strings'] |
You are given an array a1, a2, ..., an consisting of n integers, and an integer k. You have to split the array into exactly k non-empty subsegments. You'll then compute the minimum integer on each subsegment, and take the maximum integer over the k obtained minimums. What is the maximum possible integer you can get?Definitions of subsegment and array splitting are given in notes. | [] |
So, the Berland is at war with its eternal enemy Flatland again, and Vasya, an accountant, was assigned to fulfil his duty to the nation. Right now the situation in Berland is dismal — their both cities are surrounded! The armies of flatlanders stand on the borders of circles, the circles' centers are in the surrounded cities. At any moment all points of the flatland ring can begin to move quickly in the direction of the city — that's the strategy the flatlanders usually follow when they besiege cities.The berlanders are sure that they can repel the enemy's attack if they learn the exact time the attack starts. For that they need to construct a radar that would register any movement at the distance of at most r from it. Thus, we can install a radar at such point, that at least one point of the enemy ring will be in its detecting range (that is, at a distance of at most r). Then the radar can immediately inform about the enemy's attack. Due to the newest technologies, we can place a radar at any point without any problems. But the problem is that the berlanders have the time to make only one radar. Besides, the larger the detection radius (r) is, the more the radar costs.That's why Vasya's task (that is, your task) is to find the minimum possible detection radius for the radar. In other words, your task is to find the minimum radius r (r ≥ 0) such, that a radar with radius r can be installed at some point and it can register the start of the movements of both flatland rings from that point. In this problem you can consider the cities as material points, the attacking enemy rings - as circles with centers in the cities, the radar's detection range — as a disk (including the border) with the center at the point where the radar is placed. | ['geometry'] |
Let A = {a1, a2, ..., an} be any permutation of the first n natural numbers {1, 2, ..., n}. You are given a positive integer k and another sequence B = {b1, b2, ..., bn}, where bi is the number of elements aj in A to the left of the element at = i such that aj ≥ (i + k).For example, if n = 5, a possible A is {5, 1, 4, 2, 3}. For k = 2, B is given by {1, 2, 1, 0, 0}. But if k = 3, then B = {1, 1, 0, 0, 0}.For two sequences X = {x1, x2, ..., xn} and Y = {y1, y2, ..., yn}, let i-th elements be the first elements such that xi ≠ yi. If xi < yi, then X is lexicographically smaller than Y, while if xi > yi, then X is lexicographically greater than Y.Given n, k and B, you need to determine the lexicographically smallest A. | [] |
In ABBYY a wonderful Smart Beaver lives. This time, he began to study history. When he read about the Roman Empire, he became interested in the life of merchants.The Roman Empire consisted of n cities numbered from 1 to n. It also had m bidirectional roads numbered from 1 to m. Each road connected two different cities. Any two cities were connected by no more than one road.We say that there is a path between cities c1 and c2 if there exists a finite sequence of cities t1, t2, ..., tp (p ≥ 1) such that: t1 = c1 tp = c2 for any i (1 ≤ i < p), cities ti and ti + 1 are connected by a road We know that there existed a path between any two cities in the Roman Empire.In the Empire k merchants lived numbered from 1 to k. For each merchant we know a pair of numbers si and li, where si is the number of the city where this merchant's warehouse is, and li is the number of the city where his shop is. The shop and the warehouse could be located in different cities, so the merchants had to deliver goods from the warehouse to the shop.Let's call a road important for the merchant if its destruction threatens to ruin the merchant, that is, without this road there is no path from the merchant's warehouse to his shop. Merchants in the Roman Empire are very greedy, so each merchant pays a tax (1 dinar) only for those roads which are important for him. In other words, each merchant pays di dinars of tax, where di (di ≥ 0) is the number of roads important for the i-th merchant.The tax collection day came in the Empire. The Smart Beaver from ABBYY is very curious by nature, so he decided to count how many dinars each merchant had paid that day. And now he needs your help. | [] |
There are n incoming messages for Vasya. The i-th message is going to be received after ti minutes. Each message has a cost, which equals to A initially. After being received, the cost of a message decreases by B each minute (it can become negative). Vasya can read any message after receiving it at any moment of time. After reading the message, Vasya's bank account receives the current cost of this message. Initially, Vasya's bank account is at 0.Also, each minute Vasya's bank account receives C·k, where k is the amount of received but unread messages.Vasya's messages are very important to him, and because of that he wants to have all messages read after T minutes.Determine the maximum amount of money Vasya's bank account can hold after T minutes. | ['math'] |
While looking at the kitchen fridge, the little boy Tyler noticed magnets with symbols, that can be aligned into a string $$$s$$$.Tyler likes strings, and especially those that are lexicographically smaller than another string, $$$t$$$. After playing with magnets on the fridge, he is wondering, how many distinct strings can be composed out of letters of string $$$s$$$ by rearranging them, so that the resulting string is lexicographically smaller than the string $$$t$$$? Tyler is too young, so he can't answer this question. The alphabet Tyler uses is very large, so for your convenience he has already replaced the same letters in $$$s$$$ and $$$t$$$ to the same integers, keeping that different letters have been replaced to different integers.We call a string $$$x$$$ lexicographically smaller than a string $$$y$$$ if one of the followings conditions is fulfilled: There exists such position of symbol $$$m$$$ that is presented in both strings, so that before $$$m$$$-th symbol the strings are equal, and the $$$m$$$-th symbol of string $$$x$$$ is smaller than $$$m$$$-th symbol of string $$$y$$$. String $$$x$$$ is the prefix of string $$$y$$$ and $$$x \neq y$$$. Because the answer can be too large, print it modulo $$$998\,244\,353$$$. | ['math'] |
Andrew loves the sea. That's why, at the height of the summer season, he decided to go to the beach, taking a sunbed with him to sunbathe.The beach is a rectangular field with $$$n$$$ rows and $$$m$$$ columns. Some cells of the beach are free, some have roads, stones, shops and other non-movable objects. Some of two adjacent along the side cells can have sunbeds located either horizontally or vertically.Andrew hopes to put his sunbed somewhere, but that's a bad luck, there may no longer be free places for him! That's why Andrew asked you to help him to find a free place for his sunbed. Andrew's sunbed also should be places on two adjacent cells. If there are no two adjacent free cells, then in order to free some place for a sunbed, you will have to disturb other tourists. You can do the following actions: Come to some sunbed and, after causing $$$p$$$ units of discomfort to its owner, lift the sunbed by one of its sides and rotate it by $$$90$$$ degrees. One half of the sunbed must remain in the same cell and another half of the sunbed must move to the free cell. At the same time, anything could be on the way of a sunbed during the rotation . Rotation of the sunbed by $$$90$$$ degrees around cell $$$(1, 2)$$$. Come to some sunbed and, after causing $$$q$$$ units of discomfort to its owner, shift the sunbed along its long side by one cell. One half of the sunbed must move to the place of another, and another — to the free cell. Shift of the sunbed by one cell to the right. In any moment each sunbed occupies two adjacent free cells. You cannot move more than one sunbed at a time.Help Andrew to free a space for his sunbed, causing the minimum possible number of units of discomfort to other tourists, or detect that it is impossible. | ['graphs'] |
Fox Ciel wants to write a task for a programming contest. The task is: "You are given a simple undirected graph with n vertexes. Each its edge has unit length. You should calculate the number of shortest paths between vertex 1 and vertex 2."Same with some writers, she wants to make an example with some certain output: for example, her birthday or the number of her boyfriend. Can you help her to make a test case with answer equal exactly to k? | ['math', 'graphs'] |
This is an interactive problem.In good old times dwarves tried to develop extrasensory abilities: Exactly n dwarves entered completely dark cave. Each dwarf received a hat — white or black. While in cave, none of the dwarves was able to see either his own hat or hats of other Dwarves. Dwarves went out of the cave to the meadow and sat at an arbitrary place one after the other. When a dwarf leaves the cave, he sees the colors of all hats of all dwarves that are seating on the meadow (i.e. left the cave before him). However, he is not able to see the color of his own hat and none of the dwarves can give him this information. The task for dwarves was to got diverged into two parts — one with dwarves with white hats and one with black hats. After many centuries, dwarves finally managed to select the right place on the meadow without error. Will you be able to repeat their success?You are asked to successively name n different integer points on the plane. After naming each new point you will be given its color — black or white. Your task is to ensure that the named points can be split by a line in such a way that all points of one color lie on the same side from the line and points of different colors lie on different sides. Moreover, no points can belong to the line. Also, you need to report any such line at the end of the process.In this problem, the interactor is adaptive — the colors of the points in the tests are not fixed beforehand and the jury program can select them arbitrarily, in particular, depending on your program output. | ['geometry'] |
A little boy Gerald entered a clothes shop and found out something very unpleasant: not all clothes turns out to match. For example, Gerald noticed that he looks rather ridiculous in a smoking suit and a baseball cap.Overall the shop sells n clothing items, and exactly m pairs of clothing items match. Each item has its price, represented by an integer number of rubles. Gerald wants to buy three clothing items so that they matched each other. Besides, he wants to spend as little money as possible. Find the least possible sum he can spend. | [] |
The territory of Berland is represented by a rectangular field n × m in size. The king of Berland lives in the capital, located on the upper left square (1, 1). The lower right square has coordinates (n, m). One day the king decided to travel through the whole country and return back to the capital, having visited every square (except the capital) exactly one time. The king must visit the capital exactly two times, at the very beginning and at the very end of his journey. The king can only move to the side-neighboring squares. However, the royal advise said that the King possibly will not be able to do it. But there is a way out — one can build the system of one way teleporters between some squares so that the king could fulfill his plan. No more than one teleporter can be installed on one square, every teleporter can be used any number of times, however every time it is used, it transports to the same given for any single teleporter square. When the king reaches a square with an installed teleporter he chooses himself whether he is or is not going to use the teleport. What minimum number of teleporters should be installed for the king to complete the journey? You should also compose the journey path route for the king. | [] |
Vasya goes to visit his classmate Petya. Vasya knows that Petya's apartment number is $$$n$$$. There is only one entrance in Petya's house and the distribution of apartments is the following: the first floor contains $$$2$$$ apartments, every other floor contains $$$x$$$ apartments each. Apartments are numbered starting from one, from the first floor. I.e. apartments on the first floor have numbers $$$1$$$ and $$$2$$$, apartments on the second floor have numbers from $$$3$$$ to $$$(x + 2)$$$, apartments on the third floor have numbers from $$$(x + 3)$$$ to $$$(2 \cdot x + 2)$$$, and so on.Your task is to find the number of floor on which Petya lives. Assume that the house is always high enough to fit at least $$$n$$$ apartments.You have to answer $$$t$$$ independent test cases. | ['math'] |
You are given a permutation $$$p$$$ of $$$n$$$ integers $$$1$$$, $$$2$$$, ..., $$$n$$$ (a permutation is an array where each element from $$$1$$$ to $$$n$$$ occurs exactly once).Let's call some subsegment $$$p[l, r]$$$ of this permutation special if $$$p_l + p_r = \max \limits_{i = l}^{r} p_i$$$. Please calculate the number of special subsegments. | [] |
You are given a special jigsaw puzzle consisting of $$$n\cdot m$$$ identical pieces. Every piece has three tabs and one blank, as pictured below. The jigsaw puzzle is considered solved if the following conditions hold: The pieces are arranged into a grid with $$$n$$$ rows and $$$m$$$ columns. For any two pieces that share an edge in the grid, a tab of one piece fits perfectly into a blank of the other piece. Through rotation and translation of the pieces, determine if it is possible to solve the jigsaw puzzle. | ['math'] |
Permutation p is an ordered set of integers p1, p2, ..., pn, consisting of n distinct positive integers, each of them doesn't exceed n. We'll denote the i-th element of permutation p as pi. We'll call number n the size or the length of permutation p1, p2, ..., pn.The decreasing coefficient of permutation p1, p2, ..., pn is the number of such i (1 ≤ i < n), that pi > pi + 1.You have numbers n and k. Your task is to print the permutation of length n with decreasing coefficient k. | [] |
You are given two huge binary integer numbers $$$a$$$ and $$$b$$$ of lengths $$$n$$$ and $$$m$$$ respectively. You will repeat the following process: if $$$b > 0$$$, then add to the answer the value $$$a~ \&~ b$$$ and divide $$$b$$$ by $$$2$$$ rounding down (i.e. remove the last digit of $$$b$$$), and repeat the process again, otherwise stop the process.The value $$$a~ \&~ b$$$ means bitwise AND of $$$a$$$ and $$$b$$$. Your task is to calculate the answer modulo $$$998244353$$$.Note that you should add the value $$$a~ \&~ b$$$ to the answer in decimal notation, not in binary. So your task is to calculate the answer in decimal notation. For example, if $$$a = 1010_2~ (10_{10})$$$ and $$$b = 1000_2~ (8_{10})$$$, then the value $$$a~ \&~ b$$$ will be equal to $$$8$$$, not to $$$1000$$$. | ['math'] |
An array is beautiful if both of the following two conditions meet: there are at least $$$l_1$$$ and at most $$$r_1$$$ elements in the array equal to its minimum; there are at least $$$l_2$$$ and at most $$$r_2$$$ elements in the array equal to its maximum. For example, the array $$$[2, 3, 2, 4, 4, 3, 2]$$$ has $$$3$$$ elements equal to its minimum ($$$1$$$-st, $$$3$$$-rd and $$$7$$$-th) and $$$2$$$ elements equal to its maximum ($$$4$$$-th and $$$5$$$-th).Another example: the array $$$[42, 42, 42]$$$ has $$$3$$$ elements equal to its minimum and $$$3$$$ elements equal to its maximum.Your task is to calculate the minimum possible number of elements in a beautiful array. | ['math'] |
Theofanis really likes sequences of positive integers, thus his teacher (Yeltsa Kcir) gave him a problem about a sequence that consists of only special numbers.Let's call a positive number special if it can be written as a sum of different non-negative powers of $$$n$$$. For example, for $$$n = 4$$$ number $$$17$$$ is special, because it can be written as $$$4^0 + 4^2 = 1 + 16 = 17$$$, but $$$9$$$ is not.Theofanis asks you to help him find the $$$k$$$-th special number if they are sorted in increasing order. Since this number may be too large, output it modulo $$$10^9+7$$$. | ['math'] |
You are given two positive integers $$$a$$$ and $$$b$$$.In one move, you can change $$$a$$$ in the following way: Choose any positive odd integer $$$x$$$ ($$$x > 0$$$) and replace $$$a$$$ with $$$a+x$$$; choose any positive even integer $$$y$$$ ($$$y > 0$$$) and replace $$$a$$$ with $$$a-y$$$. You can perform as many such operations as you want. You can choose the same numbers $$$x$$$ and $$$y$$$ in different moves.Your task is to find the minimum number of moves required to obtain $$$b$$$ from $$$a$$$. It is guaranteed that you can always obtain $$$b$$$ from $$$a$$$.You have to answer $$$t$$$ independent test cases. | ['math'] |
You are given a binary string $$$s$$$ (recall that a string is binary if each character is either $$$0$$$ or $$$1$$$).Let $$$f(t)$$$ be the decimal representation of integer $$$t$$$ written in binary form (possibly with leading zeroes). For example $$$f(011) = 3, f(00101) = 5, f(00001) = 1, f(10) = 2, f(000) = 0$$$ and $$$f(000100) = 4$$$.The substring $$$s_{l}, s_{l+1}, \dots , s_{r}$$$ is good if $$$r - l + 1 = f(s_l \dots s_r)$$$.For example string $$$s = 1011$$$ has $$$5$$$ good substrings: $$$s_1 \dots s_1 = 1$$$, $$$s_3 \dots s_3 = 1$$$, $$$s_4 \dots s_4 = 1$$$, $$$s_1 \dots s_2 = 10$$$ and $$$s_2 \dots s_4 = 011$$$. Your task is to calculate the number of good substrings of string $$$s$$$.You have to answer $$$t$$$ independent queries. | [] |
Ziota found a video game called "Monster Invaders".Similar to every other shooting RPG game, "Monster Invaders" involves killing monsters and bosses with guns.For the sake of simplicity, we only consider two different types of monsters and three different types of guns.Namely, the two types of monsters are: a normal monster with $$$1$$$ hp. a boss with $$$2$$$ hp. And the three types of guns are: Pistol, deals $$$1$$$ hp in damage to one monster, $$$r_1$$$ reloading time Laser gun, deals $$$1$$$ hp in damage to all the monsters in the current level (including the boss), $$$r_2$$$ reloading time AWP, instantly kills any monster, $$$r_3$$$ reloading time The guns are initially not loaded, and the Ziota can only reload 1 gun at a time.The levels of the game can be considered as an array $$$a_1, a_2, \ldots, a_n$$$, in which the $$$i$$$-th stage has $$$a_i$$$ normal monsters and 1 boss. Due to the nature of the game, Ziota cannot use the Pistol (the first type of gun) or AWP (the third type of gun) to shoot the boss before killing all of the $$$a_i$$$ normal monsters.If Ziota damages the boss but does not kill it immediately, he is forced to move out of the current level to an arbitrary adjacent level (adjacent levels of level $$$i$$$ $$$(1 < i < n)$$$ are levels $$$i - 1$$$ and $$$i + 1$$$, the only adjacent level of level $$$1$$$ is level $$$2$$$, the only adjacent level of level $$$n$$$ is level $$$n - 1$$$). Ziota can also choose to move to an adjacent level at any time. Each move between adjacent levels are managed by portals with $$$d$$$ teleportation time.In order not to disrupt the space-time continuum within the game, it is strictly forbidden to reload or shoot monsters during teleportation. Ziota starts the game at level 1. The objective of the game is rather simple, to kill all the bosses in all the levels. He is curious about the minimum time to finish the game (assuming it takes no time to shoot the monsters with a loaded gun and Ziota has infinite ammo on all the three guns). Please help him find this value. | [] |
Pak Chanek, a renowned scholar, invented a card puzzle using his knowledge. In the puzzle, you are given a board with $$$n$$$ rows and $$$m$$$ columns. Let $$$(r, c)$$$ represent the cell in the $$$r$$$-th row and the $$$c$$$-th column.Initially, there are $$$k$$$ cards stacked in cell $$$(1, 1)$$$. Each card has an integer from $$$1$$$ to $$$k$$$ written on it. More specifically, the $$$i$$$-th card from the top of the stack in cell $$$(1, 1)$$$ has the number $$$a_i$$$ written on it. It is known that no two cards have the same number written on them. In other words, the numbers written on the cards are a permutation of integers from $$$1$$$ to $$$k$$$. All other cells are empty.You need to move the $$$k$$$ cards to cell $$$(n, m)$$$ to create another stack of cards. Let $$$b_i$$$ be the number written on the $$$i$$$-th card from the top of the stack in cell $$$(n, m)$$$. You should create the stack in cell $$$(n, m)$$$ in such a way so that $$$b_i = i$$$ for all $$$1 \leq i \leq k$$$.In one move, you can remove the top card from a cell and place it onto an adjacent cell (a cell that shares a common side). If the target cell already contains one or more cards, you place your card on the top of the stack. You must do each operation while satisfying the following restrictions: Each cell other than $$$(1,1)$$$ and $$$(n,m)$$$ must not have more than one card on it. You cannot move a card onto cell $$$(1,1)$$$. You cannot move a card from cell $$$(n,m)$$$. Given the values of $$$n$$$, $$$m$$$, $$$k$$$ and the array $$$a$$$, determine if the puzzle is solvable. | [] |
There are many anime that are about "love triangles": Alice loves Bob, and Charlie loves Bob as well, but Alice hates Charlie. You are thinking about an anime which has n characters. The characters are labeled from 1 to n. Every pair of two characters can either mutually love each other or mutually hate each other (there is no neutral state).You hate love triangles (A-B are in love and B-C are in love, but A-C hate each other), and you also hate it when nobody is in love. So, considering any three characters, you will be happy if exactly one pair is in love (A and B love each other, and C hates both A and B), or if all three pairs are in love (A loves B, B loves C, C loves A).You are given a list of m known relationships in the anime. You know for sure that certain pairs love each other, and certain pairs hate each other. You're wondering how many ways you can fill in the remaining relationships so you are happy with every triangle. Two ways are considered different if two characters are in love in one way but hate each other in the other. Print this count modulo 1 000 000 007. | ['graphs'] |
It was decided in IT City to distinguish successes of local IT companies by awards in the form of stars covered with gold from one side. To order the stars it is necessary to estimate order cost that depends on the area of gold-plating. Write a program that can calculate the area of a star.A "star" figure having n ≥ 5 corners where n is a prime number is constructed the following way. On the circle of radius r n points are selected so that the distances between the adjacent ones are equal. Then every point is connected by a segment with two maximally distant points. All areas bounded by the segments parts are the figure parts. | ['geometry'] |
Let's look at the following process: initially you have an empty stack and an array $$$s$$$ of the length $$$l$$$. You are trying to push array elements to the stack in the order $$$s_1, s_2, s_3, \dots s_{l}$$$. Moreover, if the stack is empty or the element at the top of this stack is not equal to the current element, then you just push the current element to the top of the stack. Otherwise, you don't push the current element to the stack and, moreover, pop the top element of the stack. If after this process the stack remains empty, the array $$$s$$$ is considered stack exterminable.There are samples of stack exterminable arrays: $$$[1, 1]$$$; $$$[2, 1, 1, 2]$$$; $$$[1, 1, 2, 2]$$$; $$$[1, 3, 3, 1, 2, 2]$$$; $$$[3, 1, 3, 3, 1, 3]$$$; $$$[3, 3, 3, 3, 3, 3]$$$; $$$[5, 1, 2, 2, 1, 4, 4, 5]$$$; Let's consider the changing of stack more details if $$$s = [5, 1, 2, 2, 1, 4, 4, 5]$$$ (the top of stack is highlighted). after pushing $$$s_1 = 5$$$ the stack turn into $$$[\textbf{5}]$$$; after pushing $$$s_2 = 1$$$ the stack turn into $$$[5, \textbf{1}]$$$; after pushing $$$s_3 = 2$$$ the stack turn into $$$[5, 1, \textbf{2}]$$$; after pushing $$$s_4 = 2$$$ the stack turn into $$$[5, \textbf{1}]$$$; after pushing $$$s_5 = 1$$$ the stack turn into $$$[\textbf{5}]$$$; after pushing $$$s_6 = 4$$$ the stack turn into $$$[5, \textbf{4}]$$$; after pushing $$$s_7 = 4$$$ the stack turn into $$$[\textbf{5}]$$$; after pushing $$$s_8 = 5$$$ the stack is empty. You are given an array $$$a_1, a_2, \ldots, a_n$$$. You have to calculate the number of its subarrays which are stack exterminable.Note, that you have to answer $$$q$$$ independent queries. | [] |
On his trip to Luxor and Aswan, Sagheer went to a Nubian market to buy some souvenirs for his friends and relatives. The market has some strange rules. It contains n different items numbered from 1 to n. The i-th item has base cost ai Egyptian pounds. If Sagheer buys k items with indices x1, x2, ..., xk, then the cost of item xj is axj + xj·k for 1 ≤ j ≤ k. In other words, the cost of an item is equal to its base cost in addition to its index multiplied by the factor k.Sagheer wants to buy as many souvenirs as possible without paying more than S Egyptian pounds. Note that he cannot buy a souvenir more than once. If there are many ways to maximize the number of souvenirs, he will choose the way that will minimize the total cost. Can you help him with this task? | [] |
There was a big bank robbery in Tablecity. In order to catch the thief, the President called none other than Albert – Tablecity’s Chief of Police. Albert does not know where the thief is located, but he does know how he moves.Tablecity can be represented as 1000 × 2 grid, where every cell represents one district. Each district has its own unique name “(X, Y)”, where X and Y are the coordinates of the district in the grid. The thief’s movement is as Every hour the thief will leave the district (X, Y) he is currently hiding in, and move to one of the districts: (X - 1, Y), (X + 1, Y), (X - 1, Y - 1), (X - 1, Y + 1), (X + 1, Y - 1), (X + 1, Y + 1) as long as it exists in Tablecity. Below is an example of thief’s possible movements if he is located in district (7,1):Albert has enough people so that every hour he can pick any two districts in Tablecity and fully investigate them, making sure that if the thief is located in one of them, he will get caught. Albert promised the President that the thief will be caught in no more than 2015 hours and needs your help in order to achieve that. | [] |
Pashmak has fallen in love with an attractive girl called Parmida since one year ago...Today, Pashmak set up a meeting with his partner in a romantic garden. Unfortunately, Pashmak has forgotten where the garden is. But he remembers that the garden looks like a square with sides parallel to the coordinate axes. He also remembers that there is exactly one tree on each vertex of the square. Now, Pashmak knows the position of only two of the trees. Help him to find the position of two remaining ones. | [] |
Little walrus Fangy loves math very much. That's why when he is bored he plays with a number performing some operations.Fangy takes some positive integer x and wants to get a number one from it. While x is not equal to 1, Fangy repeats the following action: if x is odd, then he adds 1 to it, otherwise he divides x by 2. Fangy knows that for any positive integer number the process ends in finite time.How many actions should Fangy perform to get a number one from number x? | [] |
Screen resolution of Polycarp's monitor is $$$a \times b$$$ pixels. Unfortunately, there is one dead pixel at his screen. It has coordinates $$$(x, y)$$$ ($$$0 \le x < a, 0 \le y < b$$$). You can consider columns of pixels to be numbered from $$$0$$$ to $$$a-1$$$, and rows — from $$$0$$$ to $$$b-1$$$.Polycarp wants to open a rectangular window of maximal size, which doesn't contain the dead pixel. The boundaries of the window should be parallel to the sides of the screen.Print the maximal area (in pixels) of a window that doesn't contain the dead pixel inside itself. | [] |
Recently Duff has been a soldier in the army. Malek is her commander.Their country, Andarz Gu has n cities (numbered from 1 to n) and n - 1 bidirectional roads. Each road connects two different cities. There exist a unique path between any two cities.There are also m people living in Andarz Gu (numbered from 1 to m). Each person has and ID number. ID number of i - th person is i and he/she lives in city number ci. Note that there may be more than one person in a city, also there may be no people living in the city. Malek loves to order. That's why he asks Duff to answer to q queries. In each query, he gives her numbers v, u and a.To answer a query:Assume there are x people living in the cities lying on the path from city v to city u. Assume these people's IDs are p1, p2, ..., px in increasing order. If k = min(x, a), then Duff should tell Malek numbers k, p1, p2, ..., pk in this order. In the other words, Malek wants to know a minimums on that path (or less, if there are less than a people).Duff is very busy at the moment, so she asked you to help her and answer the queries. | ['trees'] |
You have a card deck of $$$n$$$ cards, numbered from top to bottom, i. e. the top card has index $$$1$$$ and bottom card — index $$$n$$$. Each card has its color: the $$$i$$$-th card has color $$$a_i$$$.You should process $$$q$$$ queries. The $$$j$$$-th query is described by integer $$$t_j$$$. For each query you should: find the highest card in the deck with color $$$t_j$$$, i. e. the card with minimum index; print the position of the card you found; take the card and place it on top of the deck. | ['trees'] |
You are given a set of $$$n$$$ ($$$n$$$ is always a power of $$$2$$$) elements containing all integers $$$0, 1, 2, \ldots, n-1$$$ exactly once.Find $$$\frac{n}{2}$$$ pairs of elements such that: Each element in the set is in exactly one pair. The sum over all pairs of the bitwise AND of its elements must be exactly equal to $$$k$$$. Formally, if $$$a_i$$$ and $$$b_i$$$ are the elements of the $$$i$$$-th pair, then the following must hold: $$$$$$\sum_{i=1}^{n/2}{a_i \& b_i} = k,$$$$$$ where $$$\&$$$ denotes the bitwise AND operation. If there are many solutions, print any of them, if there is no solution, print $$$-1$$$ instead. | [] |
A binary string is a string that consists of characters $$$0$$$ and $$$1$$$. A bi-table is a table that has exactly two rows of equal length, each being a binary string.Let $$$\operatorname{MEX}$$$ of a bi-table be the smallest digit among $$$0$$$, $$$1$$$, or $$$2$$$ that does not occur in the bi-table. For example, $$$\operatorname{MEX}$$$ for $$$\begin{bmatrix} 0011\\ 1010 \end{bmatrix}$$$ is $$$2$$$, because $$$0$$$ and $$$1$$$ occur in the bi-table at least once. $$$\operatorname{MEX}$$$ for $$$\begin{bmatrix} 111\\ 111 \end{bmatrix}$$$ is $$$0$$$, because $$$0$$$ and $$$2$$$ do not occur in the bi-table, and $$$0 < 2$$$.You are given a bi-table with $$$n$$$ columns. You should cut it into any number of bi-tables (each consisting of consecutive columns) so that each column is in exactly one bi-table. It is possible to cut the bi-table into a single bi-table — the whole bi-table.What is the maximal sum of $$$\operatorname{MEX}$$$ of all resulting bi-tables can be? | [] |
Misha and Grisha are funny boys, so they like to use new underground. The underground has n stations connected with n - 1 routes so that each route connects two stations, and it is possible to reach every station from any other.The boys decided to have fun and came up with a plan. Namely, in some day in the morning Misha will ride the underground from station s to station f by the shortest path, and will draw with aerosol an ugly text "Misha was here" on every station he will pass through (including s and f). After that on the same day at evening Grisha will ride from station t to station f by the shortest path and will count stations with Misha's text. After that at night the underground workers will wash the texts out, because the underground should be clean. The boys have already chosen three stations a, b and c for each of several following days, one of them should be station s on that day, another should be station f, and the remaining should be station t. They became interested how they should choose these stations s, f, t so that the number Grisha will count is as large as possible. They asked you for help. | ['graphs', 'trees'] |
Polycarp has recently got himself a new job. He now earns so much that his old wallet can't even store all the money he has.Berland bills somehow come in lots of different sizes. However, all of them are shaped as rectangles (possibly squares). All wallets are also produced in form of rectangles (possibly squares).A bill $$$x \times y$$$ fits into some wallet $$$h \times w$$$ if either $$$x \le h$$$ and $$$y \le w$$$ or $$$y \le h$$$ and $$$x \le w$$$. Bills can overlap with each other in a wallet and an infinite amount of bills can fit into a wallet. That implies that all the bills Polycarp currently have fit into a wallet if every single one of them fits into it independently of the others.Now you are asked to perform the queries of two types: $$$+~x~y$$$ — Polycarp earns a bill of size $$$x \times y$$$; $$$?~h~w$$$ — Polycarp wants to check if all the bills he has earned to this moment fit into a wallet of size $$$h \times w$$$. It is guaranteed that there is at least one query of type $$$1$$$ before the first query of type $$$2$$$ and that there is at least one query of type $$$2$$$ in the input data.For each query of type $$$2$$$ print "YES" if all the bills he has earned to this moment fit into a wallet of given size. Print "NO" otherwise. | [] |
Lunar New Year is approaching, and Bob is planning to go for a famous restaurant — "Alice's".The restaurant "Alice's" serves $$$n$$$ kinds of food. The cost for the $$$i$$$-th kind is always $$$c_i$$$. Initially, the restaurant has enough ingredients for serving exactly $$$a_i$$$ dishes of the $$$i$$$-th kind. In the New Year's Eve, $$$m$$$ customers will visit Alice's one after another and the $$$j$$$-th customer will order $$$d_j$$$ dishes of the $$$t_j$$$-th kind of food. The $$$(i + 1)$$$-st customer will only come after the $$$i$$$-th customer is completely served.Suppose there are $$$r_i$$$ dishes of the $$$i$$$-th kind remaining (initially $$$r_i = a_i$$$). When a customer orders $$$1$$$ dish of the $$$i$$$-th kind, the following principles will be processed. If $$$r_i > 0$$$, the customer will be served exactly $$$1$$$ dish of the $$$i$$$-th kind. The cost for the dish is $$$c_i$$$. Meanwhile, $$$r_i$$$ will be reduced by $$$1$$$. Otherwise, the customer will be served $$$1$$$ dish of the cheapest available kind of food if there are any. If there are multiple cheapest kinds of food, the one with the smallest index among the cheapest will be served. The cost will be the cost for the dish served and the remain for the corresponding dish will be reduced by $$$1$$$. If there are no more dishes at all, the customer will leave angrily. Therefore, no matter how many dishes are served previously, the cost for the customer is $$$0$$$.If the customer doesn't leave after the $$$d_j$$$ dishes are served, the cost for the customer will be the sum of the cost for these $$$d_j$$$ dishes.Please determine the total cost for each of the $$$m$$$ customers. | [] |
During their New Year holidays, Alice and Bob play the following game using an array $$$a$$$ of $$$n$$$ integers: Players take turns, Alice moves first. Each turn a player chooses any element and removes it from the array. If Alice chooses even value, then she adds it to her score. If the chosen value is odd, Alice's score does not change. Similarly, if Bob chooses odd value, then he adds it to his score. If the chosen value is even, then Bob's score does not change. If there are no numbers left in the array, then the game ends. The player with the highest score wins. If the scores of the players are equal, then a draw is declared.For example, if $$$n = 4$$$ and $$$a = [5, 2, 7, 3]$$$, then the game could go as follows (there are other options): On the first move, Alice chooses $$$2$$$ and get two points. Her score is now $$$2$$$. The array $$$a$$$ is now $$$[5, 7, 3]$$$. On the second move, Bob chooses $$$5$$$ and get five points. His score is now $$$5$$$. The array $$$a$$$ is now $$$[7, 3]$$$. On the third move, Alice chooses $$$7$$$ and get no points. Her score is now $$$2$$$. The array $$$a$$$ is now $$$[3]$$$. On the last move, Bob chooses $$$3$$$ and get three points. His score is now $$$8$$$. The array $$$a$$$ is empty now. Since Bob has more points at the end of the game, he is the winner. You want to find out who will win if both players play optimally. Note that there may be duplicate numbers in the array. | ['games'] |
You are given an undirected graph without self-loops or multiple edges which consists of $$$n$$$ vertices and $$$m$$$ edges. Also you are given three integers $$$n_1$$$, $$$n_2$$$ and $$$n_3$$$.Can you label each vertex with one of three numbers 1, 2 or 3 in such way, that: Each vertex should be labeled by exactly one number 1, 2 or 3; The total number of vertices with label 1 should be equal to $$$n_1$$$; The total number of vertices with label 2 should be equal to $$$n_2$$$; The total number of vertices with label 3 should be equal to $$$n_3$$$; $$$|col_u - col_v| = 1$$$ for each edge $$$(u, v)$$$, where $$$col_x$$$ is the label of vertex $$$x$$$. If there are multiple valid labelings, print any of them. | ['graphs'] |
Two players decided to play one interesting card game.There is a deck of $$$n$$$ cards, with values from $$$1$$$ to $$$n$$$. The values of cards are pairwise different (this means that no two different cards have equal values). At the beginning of the game, the deck is completely distributed between players such that each player has at least one card. The game goes as follows: on each turn, each player chooses one of their cards (whichever they want) and puts on the table, so that the other player doesn't see which card they chose. After that, both cards are revealed, and the player, value of whose card was larger, takes both cards in his hand. Note that as all cards have different values, one of the cards will be strictly larger than the other one. Every card may be played any amount of times. The player loses if he doesn't have any cards.For example, suppose that $$$n = 5$$$, the first player has cards with values $$$2$$$ and $$$3$$$, and the second player has cards with values $$$1$$$, $$$4$$$, $$$5$$$. Then one possible flow of the game is:The first player chooses the card $$$3$$$. The second player chooses the card $$$1$$$. As $$$3>1$$$, the first player gets both cards. Now the first player has cards $$$1$$$, $$$2$$$, $$$3$$$, the second player has cards $$$4$$$, $$$5$$$.The first player chooses the card $$$3$$$. The second player chooses the card $$$4$$$. As $$$3<4$$$, the second player gets both cards. Now the first player has cards $$$1$$$, $$$2$$$. The second player has cards $$$3$$$, $$$4$$$, $$$5$$$.The first player chooses the card $$$1$$$. The second player chooses the card $$$3$$$. As $$$1<3$$$, the second player gets both cards. Now the first player has only the card $$$2$$$. The second player has cards $$$1$$$, $$$3$$$, $$$4$$$, $$$5$$$.The first player chooses the card $$$2$$$. The second player chooses the card $$$4$$$. As $$$2<4$$$, the second player gets both cards. Now the first player is out of cards and loses. Therefore, the second player wins.Who will win if both players are playing optimally? It can be shown that one of the players has a winning strategy. | ['math', 'games'] |
Mayor of city S just hates trees and lawns. They take so much space and there could be a road on the place they occupy!The Mayor thinks that one of the main city streets could be considerably widened on account of lawn nobody needs anyway. Moreover, that might help reduce the car jams which happen from time to time on the street.The street is split into n equal length parts from left to right, the i-th part is characterized by two integers: width of road si and width of lawn gi. For each of n parts the Mayor should decide the size of lawn to demolish. For the i-th part he can reduce lawn width by integer xi (0 ≤ xi ≤ gi). After it new road width of the i-th part will be equal to s'i = si + xi and new lawn width will be equal to g'i = gi - xi.On the one hand, the Mayor wants to demolish as much lawn as possible (and replace it with road). On the other hand, he does not want to create a rapid widening or narrowing of the road, which would lead to car accidents. To avoid that, the Mayor decided that width of the road for consecutive parts should differ by at most 1, i.e. for each i (1 ≤ i < n) the inequation |s'i + 1 - s'i| ≤ 1 should hold. Initially this condition might not be true.You need to find the the total width of lawns the Mayor will destroy according to his plan. | [] |
Arya has n opponents in the school. Each day he will fight with all opponents who are present this day. His opponents have some fighting plan that guarantees they will win, but implementing this plan requires presence of them all. That means if one day at least one of Arya's opponents is absent at the school, then Arya will beat all present opponents. Otherwise, if all opponents are present, then they will beat Arya.For each opponent Arya knows his schedule — whether or not he is going to present on each particular day. Tell him the maximum number of consecutive days that he will beat all present opponents.Note, that if some day there are no opponents present, Arya still considers he beats all the present opponents. | [] |
Vasya plays the Need For Brake. He plays because he was presented with a new computer wheel for birthday! Now he is sure that he will win the first place in the championship in his favourite racing computer game! n racers take part in the championship, which consists of a number of races. After each race racers are arranged from place first to n-th (no two racers share the same place) and first m places are awarded. Racer gains bi points for i-th awarded place, which are added to total points, obtained by him for previous races. It is known that current summary score of racer i is ai points. In the final standings of championship all the racers will be sorted in descending order of points. Racers with an equal amount of points are sorted by increasing of the name in lexicographical order.Unfortunately, the championship has come to an end, and there is only one race left. Vasya decided to find out what the highest and lowest place he can take up as a result of the championship. | [] |
Dmitry has $$$n$$$ segments of different colors on the coordinate axis $$$Ox$$$. Each segment is characterized by three integers $$$l_i$$$, $$$r_i$$$ and $$$c_i$$$ ($$$1 \le l_i \le r_i \le 10^9, 1 \le c_i \le n$$$), where $$$l_i$$$ and $$$r_i$$$ are are the coordinates of the ends of the $$$i$$$-th segment, and $$$c_i$$$ is its color.Dmitry likes to find the minimum distances between segments. However, he considers pairs of segments of the same color uninteresting. Therefore, he wants to know for each segment the distance from this segment to the nearest differently colored segment.The distance between two segments is the minimum of the distances between a point of the first segment and a point of the second segment. In particular, if the segments intersect, then the distance between them is equal to $$$0$$$.For example, Dmitry has $$$5$$$ segments: The first segment intersects with the second (and these are segments of different colors), so the answers for them are equal to $$$0$$$. For the $$$3$$$-rd segment, the nearest segment of a different color is the $$$2$$$-nd segment, the distance to which is equal to $$$2$$$. For the $$$4$$$-th segment, the nearest segment of a different color is the $$$5$$$-th segment, the distance to which is equal to $$$1$$$. The $$$5$$$-th segment lies inside the $$$2$$$-nd segment (and these are segments of different colors), so the answers for them are equal to $$$0$$$. | ['math'] |
Lolek and Bolek are about to travel abroad by plane. The local airport has a special "Choose Your Plane" offer. The offer's conditions are as follows: it is up to a passenger to choose a plane to fly on; if the chosen plane has x (x > 0) empty seats at the given moment, then the ticket for such a plane costs x zlotys (units of Polish currency). The only ticket office of the airport already has a queue of n passengers in front of it. Lolek and Bolek have not stood in the queue yet, but they are already wondering what is the maximum and the minimum number of zlotys the airport administration can earn if all n passengers buy tickets according to the conditions of this offer?The passengers buy tickets in turn, the first person in the queue goes first, then goes the second one, and so on up to n-th person. | [] |
Leo Jr. draws pictures in his notebook with checkered sheets (that is, each sheet has a regular square grid printed on it). We can assume that the sheets are infinitely large in any direction.To draw a picture, Leo Jr. colors some of the cells on a sheet gray. He considers the resulting picture beautiful if the following conditions are satisfied: The picture is connected, that is, it is possible to get from any gray cell to any other by following a chain of gray cells, with each pair of adjacent cells in the path being neighbours (that is, sharing a side). Each gray cell has an even number of gray neighbours. There are exactly $$$n$$$ gray cells with all gray neighbours. The number of other gray cells can be arbitrary (but reasonable, so that they can all be listed).Leo Jr. is now struggling to draw a beautiful picture with a particular choice of $$$n$$$. Help him, and provide any example of a beautiful picture.To output cell coordinates in your answer, assume that the sheet is provided with a Cartesian coordinate system such that one of the cells is chosen to be the origin $$$(0, 0)$$$, axes $$$0x$$$ and $$$0y$$$ are orthogonal and parallel to grid lines, and a unit step along any axis in any direction takes you to a neighbouring cell. | [] |
Long time ago there was a symmetric array $$$a_1,a_2,\ldots,a_{2n}$$$ consisting of $$$2n$$$ distinct integers. Array $$$a_1,a_2,\ldots,a_{2n}$$$ is called symmetric if for each integer $$$1 \le i \le 2n$$$, there exists an integer $$$1 \le j \le 2n$$$ such that $$$a_i = -a_j$$$.For each integer $$$1 \le i \le 2n$$$, Nezzar wrote down an integer $$$d_i$$$ equal to the sum of absolute differences from $$$a_i$$$ to all integers in $$$a$$$, i. e. $$$d_i = \sum_{j = 1}^{2n} {|a_i - a_j|}$$$.Now a million years has passed and Nezzar can barely remember the array $$$d$$$ and totally forget $$$a$$$. Nezzar wonders if there exists any symmetric array $$$a$$$ consisting of $$$2n$$$ distinct integers that generates the array $$$d$$$. | ['math'] |
A revolution took place on the Buka Island. New government replaced the old one. The new government includes n parties and each of them is entitled to some part of the island according to their contribution to the revolution. However, they can't divide the island.The island can be conventionally represented as two rectangles a × b and c × d unit squares in size correspondingly. The rectangles are located close to each other. At that, one of the sides with the length of a and one of the sides with the length of c lie on one line. You can see this in more details on the picture. The i-th party is entitled to a part of the island equal to xi unit squares. Every such part should fully cover several squares of the island (it is not allowed to cover the squares partially) and be a connected figure. A "connected figure" presupposes that from any square of this party one can move to any other square of the same party moving through edge-adjacent squares also belonging to that party.Your task is to divide the island between parties. | [] |
Pasha got a very beautiful string s for his birthday, the string consists of lowercase Latin letters. The letters in the string are numbered from 1 to |s| from left to right, where |s| is the length of the given string.Pasha didn't like his present very much so he decided to change it. After his birthday Pasha spent m days performing the following transformations on his string — each day he chose integer ai and reversed a piece of string (a segment) from position ai to position |s| - ai + 1. It is guaranteed that 2·ai ≤ |s|.You face the following task: determine what Pasha's string will look like after m days. | ['math', 'strings'] |
For a vector $$$\vec{v} = (x, y)$$$, define $$$|v| = \sqrt{x^2 + y^2}$$$.Allen had a bit too much to drink at the bar, which is at the origin. There are $$$n$$$ vectors $$$\vec{v_1}, \vec{v_2}, \cdots, \vec{v_n}$$$. Allen will make $$$n$$$ moves. As Allen's sense of direction is impaired, during the $$$i$$$-th move he will either move in the direction $$$\vec{v_i}$$$ or $$$-\vec{v_i}$$$. In other words, if his position is currently $$$p = (x, y)$$$, he will either move to $$$p + \vec{v_i}$$$ or $$$p - \vec{v_i}$$$.Allen doesn't want to wander too far from home (which happens to also be the bar). You need to help him figure out a sequence of moves (a sequence of signs for the vectors) such that his final position $$$p$$$ satisfies $$$|p| \le 1.5 \cdot 10^6$$$ so that he can stay safe. | ['math', 'geometry'] |
Arkady needs your help again! This time he decided to build his own high-speed Internet exchange point. It should consist of n nodes connected with minimum possible number of wires into one network (a wire directly connects two nodes). Exactly k of the nodes should be exit-nodes, that means that each of them should be connected to exactly one other node of the network, while all other nodes should be connected to at least two nodes in order to increase the system stability.Arkady wants to make the system as fast as possible, so he wants to minimize the maximum distance between two exit-nodes. The distance between two nodes is the number of wires a package needs to go through between those two nodes.Help Arkady to find such a way to build the network that the distance between the two most distant exit-nodes is as small as possible. | ['trees'] |
Mihai plans to watch a movie. He only likes palindromic movies, so he wants to skip some (possibly zero) scenes to make the remaining parts of the movie palindromic.You are given a list $$$s$$$ of $$$n$$$ non-empty strings of length at most $$$3$$$, representing the scenes of Mihai's movie.A subsequence of $$$s$$$ is called awesome if it is non-empty and the concatenation of the strings in the subsequence, in order, is a palindrome.Can you help Mihai check if there is at least one awesome subsequence of $$$s$$$?A palindrome is a string that reads the same backward as forward, for example strings "z", "aaa", "aba", "abccba" are palindromes, but strings "codeforces", "reality", "ab" are not.A sequence $$$a$$$ is a non-empty subsequence of a non-empty sequence $$$b$$$ if $$$a$$$ can be obtained from $$$b$$$ by deletion of several (possibly zero, but not all) elements. | ['strings'] |
Stepan likes to repeat vowel letters when he writes words. For example, instead of the word "pobeda" he can write "pobeeeedaaaaa".Sergey does not like such behavior, so he wants to write a program to format the words written by Stepan. This program must combine all consecutive equal vowels to a single vowel. The vowel letters are "a", "e", "i", "o", "u" and "y".There are exceptions: if letters "e" or "o" repeat in a row exactly 2 times, like in words "feet" and "foot", the program must skip them and do not transform in one vowel. For example, the word "iiiimpleeemeentatiioon" must be converted to the word "implemeentatioon".Sergey is very busy and asks you to help him and write the required program. | ['strings'] |
There is a prison that can be represented as a rectangular matrix with $$$n$$$ rows and $$$m$$$ columns. Therefore, there are $$$n \cdot m$$$ prison cells. There are also $$$n \cdot m$$$ prisoners, one in each prison cell. Let's denote the cell in the $$$i$$$-th row and the $$$j$$$-th column as $$$(i, j)$$$.There's a secret tunnel in the cell $$$(r, c)$$$, that the prisoners will use to escape! However, to avoid the risk of getting caught, they will escape at night.Before the night, every prisoner is in his own cell. When night comes, they can start moving to adjacent cells. Formally, in one second, a prisoner located in cell $$$(i, j)$$$ can move to cells $$$( i - 1 , j )$$$ , $$$( i + 1 , j )$$$ , $$$( i , j - 1 )$$$ , or $$$( i , j + 1 )$$$, as long as the target cell is inside the prison. They can also choose to stay in cell $$$(i, j)$$$.The prisoners want to know the minimum number of seconds needed so that every prisoner can arrive to cell $$$( r , c )$$$ if they move optimally. Note that there can be any number of prisoners in the same cell at the same time. | ['math'] |
You have a deck of $$$n$$$ cards, and you'd like to reorder it to a new one.Each card has a value between $$$1$$$ and $$$n$$$ equal to $$$p_i$$$. All $$$p_i$$$ are pairwise distinct. Cards in a deck are numbered from bottom to top, i. e. $$$p_1$$$ stands for the bottom card, $$$p_n$$$ is the top card. In each step you pick some integer $$$k > 0$$$, take the top $$$k$$$ cards from the original deck and place them, in the order they are now, on top of the new deck. You perform this operation until the original deck is empty. (Refer to the notes section for the better understanding.)Let's define an order of a deck as $$$\sum\limits_{i = 1}^{n}{n^{n - i} \cdot p_i}$$$.Given the original deck, output the deck with maximum possible order you can make using the operation above. | ['math'] |
The Super Duper Secret Meeting of the Super Duper Secret Military Squad takes place in a Super Duper Secret Place. The place is an infinite plane with introduced Cartesian coordinate system. The meeting table is represented as a rectangle whose sides are parallel to the coordinate axes and whose vertexes are located at the integer points of the plane. At each integer point which belongs to the table perimeter there is a chair in which a general sits.Some points on the plane contain radiators for the generals not to freeze in winter. Each radiator is characterized by the number ri — the radius of the area this radiator can heat. That is, if the distance between some general and the given radiator is less than or equal to ri, than the general feels comfortable and warm. Here distance is defined as Euclidean distance, so the distance between points (x1, y1) and (x2, y2) is Each general who is located outside the radiators' heating area can get sick. Thus, you should bring him a warm blanket. Your task is to count the number of warm blankets you should bring to the Super Duper Secret Place.The generals who are already comfortable do not need a blanket. Also the generals never overheat, ever if they are located in the heating area of several radiators. The radiators can be located at any integer points on the plane, even inside the rectangle (under the table) or on the perimeter (directly under some general). Even in this case their radius does not change. | [] |
The Bytelandian Institute for Biological Research (BIBR) is investigating the properties of two species of bacteria, named simply 0 and 1. Even under a microscope, bacteria of those two species are very difficult to distinguish. In fact, the only thing the scientists possess that is able to differentiate between them is a plant called Formurosa.If the scientists place a sample of colonies of bacteria on each on Formurosa's leaves, it will activate a complicated nutrition process. During that process color of Formurosa changes to reflect the result of a — possibly very complicated — logical formula on the species of bacteria, involving constants and the operators | (OR), & (AND) and ^ (XOR). If it is 0, the plant will turn red, otherwise — it will turn blue.For example, if the nutrition process of Formurosa is described by the formula: (((?^?)|?)&(1^?)); then Formurosa has four leaves (the "?" signs denote the leaves). If we place 0, 1, 0, 0 on the respective leaves, the result of the nutrition process will be (((0^1)|0)&(1^0)) = 1, therefore the plant will turn blue.The scientists have n colonies of bacteria. They do not know their types; the only thing they know for sure is that not all colonies are of the same type. They want to attempt to determine the bacteria's species by repeated evaluations with Formurosa. During each evaluation they must place exactly one sample on every leaf of the plant. However, they may use multiple samples of one colony during a single evaluation; they can even cover the whole plant with bacteria from one colony!Is it possible for them to always determine the species of each colony, no matter what they are (assuming they are not all the same)? | [] |
Little Dormi received a histogram with $$$n$$$ bars of height $$$a_1, a_2, \ldots, a_n$$$ for Christmas. However, the more he played with his new histogram, the more he realized its imperfections, so today he wanted to modify it to his liking.To modify the histogram, Little Dormi is able to perform the following operation an arbitrary number of times: Select an index $$$i$$$ ($$$1 \le i \le n$$$) where $$$a_i>0$$$, and assign $$$a_i := a_i-1$$$.Little Dormi defines the ugliness score of his histogram (after performing some number of operations) as the sum of the vertical length of its outline and the number of operations he performed on it. And to make the histogram as perfect as possible, he would like to minimize the ugliness score after modifying it with some number of operations.However, as his histogram is very large, Little Dormi is having trouble minimizing the ugliness score, so as Little Dormi's older brother, help him find the minimal ugliness.Consider the following example where the histogram has $$$4$$$ columns of heights $$$4,8,9,6$$$: The blue region represents the histogram, and the red lines represent the vertical portion of the outline. Currently, the vertical length of the outline is $$$4+4+1+3+6 = 18$$$, so if Little Dormi does not modify the histogram at all, the ugliness would be $$$18$$$.However, Little Dormi can apply the operation once on column $$$2$$$ and twice on column $$$3$$$, resulting in a histogram with heights $$$4,7,7,6$$$: Now, as the total vertical length of the outline (red lines) is $$$4+3+1+6=14$$$, the ugliness is $$$14+3=17$$$ dollars. It can be proven that this is optimal. | ['math'] |
You have two strings $$$a$$$ and $$$b$$$ of equal even length $$$n$$$ consisting of characters 0 and 1.We're in the endgame now. To finally make the universe perfectly balanced, you need to make strings $$$a$$$ and $$$b$$$ equal.In one step, you can choose any prefix of $$$a$$$ of even length and reverse it. Formally, if $$$a = a_1 a_2 \ldots a_n$$$, you can choose a positive even integer $$$p \le n$$$ and set $$$a$$$ to $$$a_p a_{p-1} \ldots a_1 a_{p+1} a_{p+2} \ldots a_n$$$.Find a way to make $$$a$$$ equal to $$$b$$$ using at most $$$n + 1$$$ reversals of the above kind, or determine that such a way doesn't exist. The number of reversals doesn't have to be minimized. | [] |
A has a string consisting of some number of lowercase English letters 'a'. He gives it to his friend B who appends some number of letters 'b' to the end of this string. Since both A and B like the characters 'a' and 'b', they have made sure that at this point, at least one 'a' and one 'b' exist in the string.B now gives this string to C and he appends some number of letters 'c' to the end of the string. However, since C is a good friend of A and B, the number of letters 'c' he appends is equal to the number of 'a' or to the number of 'b' in the string. It is also possible that the number of letters 'c' equals both to the number of letters 'a' and to the number of letters 'b' at the same time.You have a string in your hands, and you want to check if it is possible to obtain the string in this way or not. If it is possible to obtain the string, print "YES", otherwise print "NO" (without the quotes). | [] |
This problem is given in two versions that differ only by constraints. If you can solve this problem in large constraints, then you can just write a single solution to the both versions. If you find the problem too difficult in large constraints, you can write solution to the simplified version only.Waking up in the morning, Apollinaria decided to bake cookies. To bake one cookie, she needs n ingredients, and for each ingredient she knows the value ai — how many grams of this ingredient one needs to bake a cookie. To prepare one cookie Apollinaria needs to use all n ingredients.Apollinaria has bi gram of the i-th ingredient. Also she has k grams of a magic powder. Each gram of magic powder can be turned to exactly 1 gram of any of the n ingredients and can be used for baking cookies.Your task is to determine the maximum number of cookies, which Apollinaria is able to bake using the ingredients that she has and the magic powder. | [] |
This problem uses a simplified network topology model, please read the problem statement carefully and use it as a formal document as you develop the solution.Polycarpus continues working as a system administrator in a large corporation. The computer network of this corporation consists of n computers, some of them are connected by a cable. The computers are indexed by integers from 1 to n. It's known that any two computers connected by cable directly or through other computersPolycarpus decided to find out the network's topology. A network topology is the way of describing the network configuration, the scheme that shows the location and the connections of network devices.Polycarpus knows three main network topologies: bus, ring and star. A bus is the topology that represents a shared cable with all computers connected with it. In the ring topology the cable connects each computer only with two other ones. A star is the topology where all computers of a network are connected to the single central node.Let's represent each of these network topologies as a connected non-directed graph. A bus is a connected graph that is the only path, that is, the graph where all nodes are connected with two other ones except for some two nodes that are the beginning and the end of the path. A ring is a connected graph, where all nodes are connected with two other ones. A star is a connected graph, where a single central node is singled out and connected with all other nodes. For clarifications, see the picture. (1) — bus, (2) — ring, (3) — star You've got a connected non-directed graph that characterizes the computer network in Polycarpus' corporation. Help him find out, which topology type the given network is. If that is impossible to do, say that the network's topology is unknown. | ['graphs'] |
International Women's Day is coming soon! Polycarp is preparing for the holiday.There are $$$n$$$ candy boxes in the shop for sale. The $$$i$$$-th box contains $$$d_i$$$ candies.Polycarp wants to prepare the maximum number of gifts for $$$k$$$ girls. Each gift will consist of exactly two boxes. The girls should be able to share each gift equally, so the total amount of candies in a gift (in a pair of boxes) should be divisible by $$$k$$$. In other words, two boxes $$$i$$$ and $$$j$$$ ($$$i \ne j$$$) can be combined as a gift if $$$d_i + d_j$$$ is divisible by $$$k$$$.How many boxes will Polycarp be able to give? Of course, each box can be a part of no more than one gift. Polycarp cannot use boxes "partially" or redistribute candies between them. | ['math', 'number theory'] |
Mikhail walks on a Cartesian plane. He starts at the point $$$(0, 0)$$$, and in one move he can go to any of eight adjacent points. For example, if Mikhail is currently at the point $$$(0, 0)$$$, he can go to any of the following points in one move: $$$(1, 0)$$$; $$$(1, 1)$$$; $$$(0, 1)$$$; $$$(-1, 1)$$$; $$$(-1, 0)$$$; $$$(-1, -1)$$$; $$$(0, -1)$$$; $$$(1, -1)$$$. If Mikhail goes from the point $$$(x1, y1)$$$ to the point $$$(x2, y2)$$$ in one move, and $$$x1 \ne x2$$$ and $$$y1 \ne y2$$$, then such a move is called a diagonal move.Mikhail has $$$q$$$ queries. For the $$$i$$$-th query Mikhail's target is to go to the point $$$(n_i, m_i)$$$ from the point $$$(0, 0)$$$ in exactly $$$k_i$$$ moves. Among all possible movements he want to choose one with the maximum number of diagonal moves. Your task is to find the maximum number of diagonal moves or find that it is impossible to go from the point $$$(0, 0)$$$ to the point $$$(n_i, m_i)$$$ in $$$k_i$$$ moves.Note that Mikhail can visit any point any number of times (even the destination point!). | ['math'] |
Artem is building a new robot. He has a matrix $$$a$$$ consisting of $$$n$$$ rows and $$$m$$$ columns. The cell located on the $$$i$$$-th row from the top and the $$$j$$$-th column from the left has a value $$$a_{i,j}$$$ written in it. If two adjacent cells contain the same value, the robot will break. A matrix is called good if no two adjacent cells contain the same value, where two cells are called adjacent if they share a side. Artem wants to increment the values in some cells by one to make $$$a$$$ good.More formally, find a good matrix $$$b$$$ that satisfies the following condition — For all valid ($$$i,j$$$), either $$$b_{i,j} = a_{i,j}$$$ or $$$b_{i,j} = a_{i,j}+1$$$. For the constraints of this problem, it can be shown that such a matrix $$$b$$$ always exists. If there are several such tables, you can output any of them. Please note that you do not have to minimize the number of increments. | [] |
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 3 × 5 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 3 × 5 table is 15 + 8 + 3 = 26. | ['math'] |
Perhaps many have heard that the World Biathlon Championship has finished. Although our hero Valera was not present at this spectacular event himself and only watched it on TV, it excited him so much that he decided to enroll in a biathlon section.Of course, biathlon as any sport, proved very difficult in practice. It takes much time and effort. Workouts, workouts, and workouts, — that's what awaited Valera on his way to great achievements in biathlon.As for the workouts, you all probably know that every professional biathlete should ski fast and shoot precisely at the shooting range. Only in this case you can hope to be successful, because running and shooting are the two main components of biathlon. Valera has been diligent in his ski trainings, which is why he runs really fast, however, his shooting accuracy is nothing to write home about.On a biathlon base where Valera is preparing for the competition, there is a huge rifle range with n targets. Each target have shape of a circle, and the center of each circle is located on the Ox axis. At the last training session Valera made the total of m shots. To make monitoring of his own results easier for him, one rather well-known programmer (of course it is you) was commissioned to write a program that would reveal how many and which targets Valera hit. More specifically, for each target the program must print the number of the first successful shot (in the target), or "-1" if this was not hit. The target is considered hit if the shot is inside the circle or on its boundary. Valera is counting on you and perhaps, thanks to you he will one day win international competitions. | [] |
There are $$$n$$$ stones arranged on an axis. Initially the $$$i$$$-th stone is located at the coordinate $$$s_i$$$. There may be more than one stone in a single place.You can perform zero or more operations of the following type: take two stones with indices $$$i$$$ and $$$j$$$ so that $$$s_i \leq s_j$$$, choose an integer $$$d$$$ ($$$0 \leq 2 \cdot d \leq s_j - s_i$$$), and replace the coordinate $$$s_i$$$ with $$$(s_i + d)$$$ and replace coordinate $$$s_j$$$ with $$$(s_j - d)$$$. In other words, draw stones closer to each other. You want to move the stones so that they are located at positions $$$t_1, t_2, \ldots, t_n$$$. The order of the stones is not important — you just want for the multiset of the stones resulting positions to be the same as the multiset of $$$t_1, t_2, \ldots, t_n$$$.Detect whether it is possible to move the stones this way, and if yes, construct a way to do so. You don't need to minimize the number of moves. | ['math'] |
Leha plays a computer game, where is on each level is given a connected graph with n vertices and m edges. Graph can contain multiple edges, but can not contain self loops. Each vertex has an integer di, which can be equal to 0, 1 or - 1. To pass the level, he needs to find a «good» subset of edges of the graph or say, that it doesn't exist. Subset is called «good», if by by leaving only edges from this subset in the original graph, we obtain the following: for every vertex i, di = - 1 or it's degree modulo 2 is equal to di. Leha wants to pass the game as soon as possible and ask you to help him. In case of multiple correct answers, print any of them. | ['graphs'] |
You are given a number $$$n$$$ (divisible by $$$3$$$) and an array $$$a[1 \dots n]$$$. In one move, you can increase any of the array elements by one. Formally, you choose the index $$$i$$$ ($$$1 \le i \le n$$$) and replace $$$a_i$$$ with $$$a_i + 1$$$. You can choose the same index $$$i$$$ multiple times for different moves.Let's denote by $$$c_0$$$, $$$c_1$$$ and $$$c_2$$$ the number of numbers from the array $$$a$$$ that have remainders $$$0$$$, $$$1$$$ and $$$2$$$ when divided by the number $$$3$$$, respectively. Let's say that the array $$$a$$$ has balanced remainders if $$$c_0$$$, $$$c_1$$$ and $$$c_2$$$ are equal.For example, if $$$n = 6$$$ and $$$a = [0, 2, 5, 5, 4, 8]$$$, then the following sequence of moves is possible: initially $$$c_0 = 1$$$, $$$c_1 = 1$$$ and $$$c_2 = 4$$$, these values are not equal to each other. Let's increase $$$a_3$$$, now the array $$$a = [0, 2, 6, 5, 4, 8]$$$; $$$c_0 = 2$$$, $$$c_1 = 1$$$ and $$$c_2 = 3$$$, these values are not equal. Let's increase $$$a_6$$$, now the array $$$a = [0, 2, 6, 5, 4, 9]$$$; $$$c_0 = 3$$$, $$$c_1 = 1$$$ and $$$c_2 = 2$$$, these values are not equal. Let's increase $$$a_1$$$, now the array $$$a = [1, 2, 6, 5, 4, 9]$$$; $$$c_0 = 2$$$, $$$c_1 = 2$$$ and $$$c_2 = 2$$$, these values are equal to each other, which means that the array $$$a$$$ has balanced remainders. Find the minimum number of moves needed to make the array $$$a$$$ have balanced remainders. | ['math'] |
Medicine faculty of Berland State University has just finished their admission campaign. As usual, about $$$80\%$$$ of applicants are girls and majority of them are going to live in the university dormitory for the next $$$4$$$ (hopefully) years.The dormitory consists of $$$n$$$ rooms and a single mouse! Girls decided to set mouse traps in some rooms to get rid of the horrible monster. Setting a trap in room number $$$i$$$ costs $$$c_i$$$ burles. Rooms are numbered from $$$1$$$ to $$$n$$$.Mouse doesn't sit in place all the time, it constantly runs. If it is in room $$$i$$$ in second $$$t$$$ then it will run to room $$$a_i$$$ in second $$$t + 1$$$ without visiting any other rooms inbetween ($$$i = a_i$$$ means that mouse won't leave room $$$i$$$). It's second $$$0$$$ in the start. If the mouse is in some room with a mouse trap in it, then the mouse get caught into this trap.That would have been so easy if the girls actually knew where the mouse at. Unfortunately, that's not the case, mouse can be in any room from $$$1$$$ to $$$n$$$ at second $$$0$$$.What it the minimal total amount of burles girls can spend to set the traps in order to guarantee that the mouse will eventually be caught no matter the room it started from? | ['graphs'] |
Polycarp came up with a new programming language. There are only two types of statements in it: "x := s": assign the variable named x the value s (where s is a string). For example, the statement var := hello assigns the variable named var the value hello. Note that s is the value of a string, not the name of a variable. Between the variable name, the := operator and the string contains exactly one space each. "x = a + b": assign the variable named x the concatenation of values of two variables a and b. For example, if the program consists of three statements a := hello, b := world, c = a + b, then the variable c will contain the string helloworld. It is guaranteed that the program is correct and the variables a and b were previously defined. There is exactly one space between the variable names and the = and + operators. All variable names and strings only consist of lowercase letters of the English alphabet and do not exceed $$$5$$$ characters.The result of the program is the number of occurrences of string haha in the string that was written to the variable in the last statement.Polycarp was very tired while inventing that language. He asks you to implement it. Your task is — for given program statements calculate the number of occurrences of string haha in the last assigned variable. | ['strings'] |
Valery is very interested in magic. Magic attracts him so much that he sees it everywhere. He explains any strange and weird phenomenon through intervention of supernatural forces. But who would have thought that even in a regular array of numbers Valera manages to see something beautiful and magical.Valera absolutely accidentally got a piece of ancient parchment on which an array of numbers was written. He immediately thought that the numbers in this array were not random. As a result of extensive research Valera worked out a wonderful property that a magical array should have: an array is defined as magic if its minimum and maximum coincide.He decided to share this outstanding discovery with you, but he asks you for help in return. Despite the tremendous intelligence and wit, Valera counts very badly and so you will have to complete his work. All you have to do is count the number of magical subarrays of the original array of numbers, written on the parchment. Subarray is defined as non-empty sequence of consecutive elements. | ['math'] |
An IPv6-address is a 128-bit number. For convenience, this number is recorded in blocks of 16 bits in hexadecimal record, the blocks are separated by colons — 8 blocks in total, each block has four hexadecimal digits. Here is an example of the correct record of a IPv6 address: "0124:5678:90ab:cdef:0124:5678:90ab:cdef". We'll call such format of recording an IPv6-address full.Besides the full record of an IPv6 address there is a short record format. The record of an IPv6 address can be shortened by removing one or more leading zeroes at the beginning of each block. However, each block should contain at least one digit in the short format. For example, the leading zeroes can be removed like that: "a56f:00d3:0000:0124:0001:f19a:1000:0000" → "a56f:d3:0:0124:01:f19a:1000:00". There are more ways to shorten zeroes in this IPv6 address.Some IPv6 addresses contain long sequences of zeroes. Continuous sequences of 16-bit zero blocks can be shortened to "::". A sequence can consist of one or several consecutive blocks, with all 16 bits equal to 0. You can see examples of zero block shortenings below: "a56f:00d3:0000:0124:0001:0000:0000:0000" → "a56f:00d3:0000:0124:0001::"; "a56f:0000:0000:0124:0001:0000:1234:0ff0" → "a56f::0124:0001:0000:1234:0ff0"; "a56f:0000:0000:0000:0001:0000:1234:0ff0" → "a56f:0000::0000:0001:0000:1234:0ff0"; "a56f:00d3:0000:0124:0001:0000:0000:0000" → "a56f:00d3:0000:0124:0001::0000"; "0000:0000:0000:0000:0000:0000:0000:0000" → "::". It is not allowed to shorten zero blocks in the address more than once. This means that the short record can't contain the sequence of characters "::" more than once. Otherwise, it will sometimes be impossible to determine the number of zero blocks, each represented by a double colon.The format of the record of the IPv6 address after removing the leading zeroes and shortening the zero blocks is called short.You've got several short records of IPv6 addresses. Restore their full record. | ['strings'] |
Imp is really pleased that you helped him. But it you solve the last problem, his gladness would raise even more. Let's define for some set of integers as the number of pairs a, b in , such that: a is strictly less than b; a divides b without a remainder. You are to find such a set , which is a subset of {1, 2, ..., n} (the set that contains all positive integers not greater than n), that . | ['number theory'] |
In an ICPC contest, balloons are distributed as follows: Whenever a team solves a problem, that team gets a balloon. The first team to solve a problem gets an additional balloon. A contest has 26 problems, labelled $$$\textsf{A}$$$, $$$\textsf{B}$$$, $$$\textsf{C}$$$, ..., $$$\textsf{Z}$$$. You are given the order of solved problems in the contest, denoted as a string $$$s$$$, where the $$$i$$$-th character indicates that the problem $$$s_i$$$ has been solved by some team. No team will solve the same problem twice.Determine the total number of balloons that the teams received. Note that some problems may be solved by none of the teams. | [] |
You are given one integer $$$n$$$ ($$$n > 1$$$).Recall that a permutation of length $$$n$$$ is an array consisting of $$$n$$$ distinct integers from $$$1$$$ to $$$n$$$ in arbitrary order. For example, $$$[2, 3, 1, 5, 4]$$$ is a permutation of length $$$5$$$, but $$$[1, 2, 2]$$$ is not a permutation ($$$2$$$ appears twice in the array) and $$$[1, 3, 4]$$$ is also not a permutation ($$$n = 3$$$ but there is $$$4$$$ in the array).Your task is to find a permutation $$$p$$$ of length $$$n$$$ that there is no index $$$i$$$ ($$$1 \le i \le n$$$) such that $$$p_i = i$$$ (so, for all $$$i$$$ from $$$1$$$ to $$$n$$$ the condition $$$p_i \ne i$$$ should be satisfied).You have to answer $$$t$$$ independent test cases.If there are several answers, you can print any. It can be proven that the answer exists for each $$$n > 1$$$. | ['probabilities'] |
A film festival is coming up in the city N. The festival will last for exactly n days and each day will have a premiere of exactly one film. Each film has a genre — an integer from 1 to k.On the i-th day the festival will show a movie of genre ai. We know that a movie of each of k genres occurs in the festival programme at least once. In other words, each integer from 1 to k occurs in the sequence a1, a2, ..., an at least once.Valentine is a movie critic. He wants to watch some movies of the festival and then describe his impressions on his site.As any creative person, Valentine is very susceptive. After he watched the movie of a certain genre, Valentine forms the mood he preserves until he watches the next movie. If the genre of the next movie is the same, it does not change Valentine's mood. If the genres are different, Valentine's mood changes according to the new genre and Valentine has a stress.Valentine can't watch all n movies, so he decided to exclude from his to-watch list movies of one of the genres. In other words, Valentine is going to choose exactly one of the k genres and will skip all the movies of this genre. He is sure to visit other movies.Valentine wants to choose such genre x (1 ≤ x ≤ k), that the total number of after-movie stresses (after all movies of genre x are excluded) were minimum. | [] |
A sequence $$$a = [a_1, a_2, \ldots, a_l]$$$ of length $$$l$$$ has an ascent if there exists a pair of indices $$$(i, j)$$$ such that $$$1 \le i < j \le l$$$ and $$$a_i < a_j$$$. For example, the sequence $$$[0, 2, 0, 2, 0]$$$ has an ascent because of the pair $$$(1, 4)$$$, but the sequence $$$[4, 3, 3, 3, 1]$$$ doesn't have an ascent.Let's call a concatenation of sequences $$$p$$$ and $$$q$$$ the sequence that is obtained by writing down sequences $$$p$$$ and $$$q$$$ one right after another without changing the order. For example, the concatenation of the $$$[0, 2, 0, 2, 0]$$$ and $$$[4, 3, 3, 3, 1]$$$ is the sequence $$$[0, 2, 0, 2, 0, 4, 3, 3, 3, 1]$$$. The concatenation of sequences $$$p$$$ and $$$q$$$ is denoted as $$$p+q$$$.Gyeonggeun thinks that sequences with ascents bring luck. Therefore, he wants to make many such sequences for the new year. Gyeonggeun has $$$n$$$ sequences $$$s_1, s_2, \ldots, s_n$$$ which may have different lengths. Gyeonggeun will consider all $$$n^2$$$ pairs of sequences $$$s_x$$$ and $$$s_y$$$ ($$$1 \le x, y \le n$$$), and will check if its concatenation $$$s_x + s_y$$$ has an ascent. Note that he may select the same sequence twice, and the order of selection matters.Please count the number of pairs ($$$x, y$$$) of sequences $$$s_1, s_2, \ldots, s_n$$$ whose concatenation $$$s_x + s_y$$$ contains an ascent. | [] |
You're given Q queries of the form (L, R). For each query you have to find the number of such x that L ≤ x ≤ R and there exist integer numbers a > 0, p > 1 such that x = ap. | ['math', 'number theory'] |
You are given an array $$$a$$$ of length $$$n$$$ that has a special condition: every element in this array has at most 7 divisors. Find the length of the shortest non-empty subsequence of this array product of whose elements is a perfect square.A sequence $$$a$$$ is a subsequence of an array $$$b$$$ if $$$a$$$ can be obtained from $$$b$$$ by deletion of several (possibly, zero or all) elements. | ['graphs', 'number theory'] |
Xenia has a set of weights and pan scales. Each weight has an integer weight from 1 to 10 kilos. Xenia is going to play with scales and weights a little. For this, she puts weights on the scalepans, one by one. The first weight goes on the left scalepan, the second weight goes on the right scalepan, the third one goes on the left scalepan, the fourth one goes on the right scalepan and so on. Xenia wants to put the total of m weights on the scalepans.Simply putting weights on the scales is not interesting, so Xenia has set some rules. First, she does not put on the scales two consecutive weights of the same weight. That is, the weight that goes i-th should be different from the (i + 1)-th weight for any i (1 ≤ i < m). Second, every time Xenia puts a weight on some scalepan, she wants this scalepan to outweigh the other one. That is, the sum of the weights on the corresponding scalepan must be strictly greater than the sum on the other pan.You are given all types of weights available for Xenia. You can assume that the girl has an infinite number of weights of each specified type. Your task is to help Xenia lay m weights on the scales or to say that it can't be done. | ['graphs'] |
There are two main kinds of events in the life of top-model: fashion shows and photo shoots. Participating in any of these events affects the rating of appropriate top-model. After each photo shoot model's rating increases by a and after each fashion show decreases by b (designers do too many experiments nowadays). Moreover, sometimes top-models participates in talk shows. After participating in talk show model becomes more popular and increasing of her rating after photo shoots become c and decreasing of her rating after fashion show becomes d.Izabella wants to participate in a talk show, but she wants to do it in such a way that her rating will never become negative. Help her to find a suitable moment for participating in the talk show. Let's assume that model's career begins in moment 0. At that moment Izabella's rating was equal to start. If talk show happens in moment t if will affect all events in model's life in interval of time [t..t + len) (including t and not including t + len), where len is duration of influence.Izabella wants to participate in a talk show, but she wants to do it in such a way that her rating will not become become negative before talk show or during period of influence of talk show. Help her to find a suitable moment for participating in the talk show. | [] |
Vasya had a strictly increasing sequence of positive integers a1, ..., an. Vasya used it to build a new sequence b1, ..., bn, where bi is the sum of digits of ai's decimal representation. Then sequence ai got lost and all that remained is sequence bi.Vasya wonders what the numbers ai could be like. Of all the possible options he likes the one sequence with the minimum possible last number an. Help Vasya restore the initial sequence.It is guaranteed that such a sequence always exists. | [] |
You have $$$n$$$ sticks of the given lengths.Your task is to choose exactly four of them in such a way that they can form a rectangle. No sticks can be cut to pieces, each side of the rectangle must be formed by a single stick. No stick can be chosen multiple times. It is guaranteed that it is always possible to choose such sticks.Let $$$S$$$ be the area of the rectangle and $$$P$$$ be the perimeter of the rectangle. The chosen rectangle should have the value $$$\frac{P^2}{S}$$$ minimal possible. The value is taken without any rounding.If there are multiple answers, print any of them.Each testcase contains several lists of sticks, for each of them you are required to solve the problem separately. | [] |
A bracket sequence is a string containing only characters "(" and ")". A regular bracket sequence (or, shortly, an RBS) 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: bracket sequences "()()" and "(())" are regular (the resulting expressions are: "(1)+(1)" and "((1+1)+1)"); bracket sequences ")(", "(" and ")" are not. There was an RBS. Some brackets have been replaced with question marks. Is it true that there is a unique way to replace question marks with brackets, so that the resulting sequence is an RBS? | ['strings'] |
In 2013, the writers of Berland State University should prepare problems for n Olympiads. We will assume that the Olympiads are numbered with consecutive integers from 1 to n. For each Olympiad we know how many members of the jury must be involved in its preparation, as well as the time required to prepare the problems for her. Namely, the Olympiad number i should be prepared by pi people for ti days, the preparation for the Olympiad should be a continuous period of time and end exactly one day before the Olympiad. On the day of the Olympiad the juries who have prepared it, already do not work on it.For example, if the Olympiad is held on December 9th and the preparation takes 7 people and 6 days, all seven members of the jury will work on the problems of the Olympiad from December, 3rd to December, 8th (the jury members won't be working on the problems of this Olympiad on December 9th, that is, some of them can start preparing problems for some other Olympiad). And if the Olympiad is held on November 3rd and requires 5 days of training, the members of the jury will work from October 29th to November 2nd.In order not to overload the jury the following rule was introduced: one member of the jury can not work on the same day on the tasks for different Olympiads. Write a program that determines what the minimum number of people must be part of the jury so that all Olympiads could be prepared in time. | [] |
You've got an undirected graph, consisting of n vertices and m edges. We will consider the graph's vertices numbered with integers from 1 to n. Each vertex of the graph has a color. The color of the i-th vertex is an integer ci.Let's consider all vertices of the graph, that are painted some color k. Let's denote a set of such as V(k). Let's denote the value of the neighbouring color diversity for color k as the cardinality of the set Q(k) = {cu : cu ≠ k and there is vertex v belonging to set V(k) such that nodes v and u are connected by an edge of the graph}.Your task is to find such color k, which makes the cardinality of set Q(k) maximum. In other words, you want to find the color that has the most diverse neighbours. Please note, that you want to find such color k, that the graph has at least one vertex with such color. | ['graphs'] |
One popular website developed an unusual username editing procedure. One can change the username only by deleting some characters from it: to change the current name s, a user can pick number p and character c and delete the p-th occurrence of character c from the name. After the user changed his name, he can't undo the change.For example, one can change name "arca" by removing the second occurrence of character "a" to get "arc". Polycarpus learned that some user initially registered under nickname t, where t is a concatenation of k copies of string s. Also, Polycarpus knows the sequence of this user's name changes. Help Polycarpus figure out the user's final name. | ['strings'] |
Once upon a time Mike and Mike decided to come up with an outstanding problem for some stage of ROI (rare olympiad in informatics). One of them came up with a problem prototype but another stole the idea and proposed that problem for another stage of the same olympiad. Since then the first Mike has been waiting for an opportunity to propose the original idea for some other contest... Mike waited until this moment!You are given an array $$$a$$$ of $$$n$$$ integers. You are also given $$$q$$$ queries of two types: Replace $$$i$$$-th element in the array with integer $$$x$$$. Replace each element in the array with integer $$$x$$$. After performing each query you have to calculate the sum of all elements in the array. | [] |
PolandBall is playing a game with EnemyBall. The rules are simple. Players have to say words in turns. You cannot say a word which was already said. PolandBall starts. The Ball which can't say a new word loses.You're given two lists of words familiar to PolandBall and EnemyBall. Can you determine who wins the game, if both play optimally? | ['strings', 'games'] |
Petya has got 2n cards, each card contains some integer. The numbers on the cards can be the same. Let's index all cards by consecutive integers from 1 to 2n. We'll denote the number that is written on a card with number i, as ai. In order to play one entertaining game with his friends, Petya needs to split the cards into pairs so that each pair had equal numbers on the cards. Help Petya do that. | [] |
You are given n points on Cartesian plane. Every point is a lattice point (i. e. both of its coordinates are integers), and all points are distinct.You may draw two straight lines (not necessarily distinct). Is it possible to do this in such a way that every point lies on at least one of these lines? | ['geometry'] |
You are given a rectangular matrix of size $$$n \times m$$$ consisting of integers from $$$1$$$ to $$$2 \cdot 10^5$$$.In one move, you can: choose any element of the matrix and change its value to any integer between $$$1$$$ and $$$n \cdot m$$$, inclusive; take any column and shift it one cell up cyclically (see the example of such cyclic shift below). A cyclic shift is an operation such that you choose some $$$j$$$ ($$$1 \le j \le m$$$) and set $$$a_{1, j} := a_{2, j}, a_{2, j} := a_{3, j}, \dots, a_{n, j} := a_{1, j}$$$ simultaneously. Example of cyclic shift of the first column You want to perform the minimum number of moves to make this matrix look like this: In other words, the goal is to obtain the matrix, where $$$a_{1, 1} = 1, a_{1, 2} = 2, \dots, a_{1, m} = m, a_{2, 1} = m + 1, a_{2, 2} = m + 2, \dots, a_{n, m} = n \cdot m$$$ (i.e. $$$a_{i, j} = (i - 1) \cdot m + j$$$) with the minimum number of moves performed. | ['math'] |
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