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Polycarpus works as a programmer in a start-up social network. His boss gave his a task to develop a mechanism for determining suggested friends. Polycarpus thought much about the task and came to the folowing conclusion. Let's say that all friendship relationships in a social network are given as m username pairs ai, bi (ai ≠ bi). Each pair ai, bi means that users ai and bi are friends. Friendship is symmetric, that is, if ai is friends with bi, then bi is also friends with ai. User y is a suggested friend for user x, if the following conditions are met: x ≠ y; x and y aren't friends; among all network users who meet the first two conditions, user y has most of all common friends with user x. User z is a common friend of user x and user y (z ≠ x, z ≠ y), if x and z are friends, and y and z are also friends. Your task is to help Polycarpus to implement a mechanism for determining suggested friends. | ['graphs'] |
There are $$$n$$$ pieces of tangerine peel, the $$$i$$$-th of them has size $$$a_i$$$. In one step it is possible to divide one piece of size $$$x$$$ into two pieces of positive integer sizes $$$y$$$ and $$$z$$$ so that $$$y + z = x$$$.You want that for each pair of pieces, their sizes differ strictly less than twice. In other words, there should not be two pieces of size $$$x$$$ and $$$y$$$, such that $$$2x \le y$$$. What is the minimum possible number of steps needed to satisfy the condition? | ['math'] |
As we know, DZY loves playing games. One day DZY decided to play with a n × m matrix. To be more precise, he decided to modify the matrix with exactly k operations.Each modification is one of the following: Pick some row of the matrix and decrease each element of the row by p. This operation brings to DZY the value of pleasure equal to the sum of elements of the row before the decreasing. Pick some column of the matrix and decrease each element of the column by p. This operation brings to DZY the value of pleasure equal to the sum of elements of the column before the decreasing. DZY wants to know: what is the largest total value of pleasure he could get after performing exactly k modifications? Please, help him to calculate this value. | [] |
Petya works as a PR manager for a successful Berland company BerSoft. He needs to prepare a presentation on the company income growth since 2001 (the year of its founding) till now. Petya knows that in 2001 the company income amounted to a1 billion bourles, in 2002 — to a2 billion, ..., and in the current (2000 + n)-th year — an billion bourles. On the base of the information Petya decided to show in his presentation the linear progress history which is in his opinion perfect. According to a graph Petya has already made, in the first year BerSoft company income must amount to 1 billion bourles, in the second year — 2 billion bourles etc., each following year the income increases by 1 billion bourles. Unfortunately, the real numbers are different from the perfect ones. Among the numbers ai can even occur negative ones that are a sign of the company’s losses in some years. That is why Petya wants to ignore some data, in other words, cross some numbers ai from the sequence and leave only some subsequence that has perfect growth.Thus Petya has to choose a sequence of years y1, y2, ..., yk,so that in the year y1 the company income amounted to 1 billion bourles, in the year y2 — 2 billion bourles etc., in accordance with the perfect growth dynamics. Help him to choose the longest such sequence. | [] |
You are given an array $$$a_1, a_2, \ldots, a_n$$$ consisting of $$$n$$$ positive integers and a positive integer $$$m$$$.You should divide elements of this array into some arrays. You can order the elements in the new arrays as you want.Let's call an array $$$m$$$-divisible if for each two adjacent numbers in the array (two numbers on the positions $$$i$$$ and $$$i+1$$$ are called adjacent for each $$$i$$$) their sum is divisible by $$$m$$$. An array of one element is $$$m$$$-divisible.Find the smallest number of $$$m$$$-divisible arrays that $$$a_1, a_2, \ldots, a_n$$$ is possible to divide into. | ['math'] |
A new pack of n t-shirts came to a shop. Each of the t-shirts is characterized by three integers pi, ai and bi, where pi is the price of the i-th t-shirt, ai is front color of the i-th t-shirt and bi is back color of the i-th t-shirt. All values pi are distinct, and values ai and bi are integers from 1 to 3.m buyers will come to the shop. Each of them wants to buy exactly one t-shirt. For the j-th buyer we know his favorite color cj.A buyer agrees to buy a t-shirt, if at least one side (front or back) is painted in his favorite color. Among all t-shirts that have colors acceptable to this buyer he will choose the cheapest one. If there are no such t-shirts, the buyer won't buy anything. Assume that the buyers come one by one, and each buyer is served only after the previous one is served.You are to compute the prices each buyer will pay for t-shirts. | [] |
Monocarp plays a computer game. There are $$$n$$$ different sets of armor and $$$m$$$ different weapons in this game. If a character equips the $$$i$$$-th set of armor and wields the $$$j$$$-th weapon, their power is usually equal to $$$i + j$$$; but some combinations of armor and weapons synergize well. Formally, there is a list of $$$q$$$ ordered pairs, and if the pair $$$(i, j)$$$ belongs to this list, the power of the character equipped with the $$$i$$$-th set of armor and wielding the $$$j$$$-th weapon is not $$$i + j$$$, but $$$i + j + 1$$$.Initially, Monocarp's character has got only the $$$1$$$-st armor set and the $$$1$$$-st weapon. Monocarp can obtain a new weapon or a new set of armor in one hour. If he wants to obtain the $$$k$$$-th armor set or the $$$k$$$-th weapon, he must possess a combination of an armor set and a weapon that gets his power to $$$k$$$ or greater. Of course, after Monocarp obtains a weapon or an armor set, he can use it to obtain new armor sets or weapons, but he can go with any of the older armor sets and/or weapons as well.Monocarp wants to obtain the $$$n$$$-th armor set and the $$$m$$$-th weapon. What is the minimum number of hours he has to spend on it? | [] |
Xenia the mathematician has a sequence consisting of n (n is divisible by 3) positive integers, each of them is at most 7. She wants to split the sequence into groups of three so that for each group of three a, b, c the following conditions held: a < b < c; a divides b, b divides c. Naturally, Xenia wants each element of the sequence to belong to exactly one group of three. Thus, if the required partition exists, then it has groups of three.Help Xenia, find the required partition or else say that it doesn't exist. | [] |
Let's call an array consisting of n integer numbers a1, a2, ..., an, beautiful if it has the following property: consider all pairs of numbers x, y (x ≠ y), such that number x occurs in the array a and number y occurs in the array a; for each pair x, y must exist some position j (1 ≤ j < n), such that at least one of the two conditions are met, either aj = x, aj + 1 = y, or aj = y, aj + 1 = x. Sereja wants to build a beautiful array a, consisting of n integers. But not everything is so easy, Sereja's friend Dima has m coupons, each contains two integers qi, wi. Coupon i costs wi and allows you to use as many numbers qi as you want when constructing the array a. Values qi are distinct. Sereja has no coupons, so Dima and Sereja have made the following deal. Dima builds some beautiful array a of n elements. After that he takes wi rubles from Sereja for each qi, which occurs in the array a. Sereja believed his friend and agreed to the contract, and now he is wondering, what is the maximum amount of money he can pay.Help Sereja, find the maximum amount of money he can pay to Dima. | ['graphs'] |
You are given a string $$$s$$$ of length $$$n$$$ consisting only of the characters 0 and 1.You perform the following operation until the string becomes empty: choose some consecutive substring of equal characters, erase it from the string and glue the remaining two parts together (any of them can be empty) in the same order. For example, if you erase the substring 111 from the string 111110, you will get the string 110. When you delete a substring of length $$$l$$$, you get $$$a \cdot l + b$$$ points.Your task is to calculate the maximum number of points that you can score in total, if you have to make the given string empty. | ['math'] |
Ksusha the Squirrel is standing at the beginning of a straight road, divided into n sectors. The sectors are numbered 1 to n, from left to right. Initially, Ksusha stands in sector 1. Ksusha wants to walk to the end of the road, that is, get to sector n. Unfortunately, there are some rocks on the road. We know that Ksusha hates rocks, so she doesn't want to stand in sectors that have rocks.Ksusha the squirrel keeps fit. She can jump from sector i to any of the sectors i + 1, i + 2, ..., i + k. Help Ksusha! Given the road description, say if she can reach the end of the road (note, she cannot stand on a rock)? | [] |
There are $$$n$$$ blocks arranged in a row and numbered from left to right, starting from one. Each block is either black or white. You may perform the following operation zero or more times: choose two adjacent blocks and invert their colors (white block becomes black, and vice versa). You want to find a sequence of operations, such that they make all the blocks having the same color. You don't have to minimize the number of operations, but it should not exceed $$$3 \cdot n$$$. If it is impossible to find such a sequence of operations, you need to report it. | ['math'] |
The only difference with E2 is the question of the problem..Vlad built a maze out of $$$n$$$ rooms and $$$n-1$$$ bidirectional corridors. From any room $$$u$$$ any other room $$$v$$$ can be reached through a sequence of corridors. Thus, the room system forms an undirected tree.Vlad invited $$$k$$$ friends to play a game with them.Vlad starts the game in the room $$$1$$$ and wins if he reaches a room other than $$$1$$$, into which exactly one corridor leads.Friends are placed in the maze: the friend with number $$$i$$$ is in the room $$$x_i$$$, and no two friends are in the same room (that is, $$$x_i \neq x_j$$$ for all $$$i \neq j$$$). Friends win if one of them meets Vlad in any room or corridor before he wins.For one unit of time, each participant of the game can go through one corridor. All participants move at the same time. Participants may not move. Each room can fit all participants at the same time. Friends know the plan of a maze and intend to win. Vlad is a bit afraid of their ardor. Determine if he can guarantee victory (i.e. can he win in any way friends play).In other words, determine if there is such a sequence of Vlad's moves that lets Vlad win in any way friends play. | ['trees'] |
Petya has got an interesting flower. Petya is a busy person, so he sometimes forgets to water it. You are given $$$n$$$ days from Petya's live and you have to determine what happened with his flower in the end.The flower grows as follows: If the flower isn't watered for two days in a row, it dies. If the flower is watered in the $$$i$$$-th day, it grows by $$$1$$$ centimeter. If the flower is watered in the $$$i$$$-th and in the $$$(i-1)$$$-th day ($$$i > 1$$$), then it grows by $$$5$$$ centimeters instead of $$$1$$$. If the flower is not watered in the $$$i$$$-th day, it does not grow. At the beginning of the $$$1$$$-st day the flower is $$$1$$$ centimeter tall. What is its height after $$$n$$$ days? | [] |
Yeah, we failed to make up a New Year legend for this problem.A permutation of length $$$n$$$ is an array of $$$n$$$ integers such that every integer from $$$1$$$ to $$$n$$$ appears in it exactly once. An element $$$y$$$ of permutation $$$p$$$ is reachable from element $$$x$$$ if $$$x = y$$$, or $$$p_x = y$$$, or $$$p_{p_x} = y$$$, and so on. The decomposition of a permutation $$$p$$$ is defined as follows: firstly, we have a permutation $$$p$$$, all elements of which are not marked, and an empty list $$$l$$$. Then we do the following: while there is at least one not marked element in $$$p$$$, we find the leftmost such element, list all elements that are reachable from it in the order they appear in $$$p$$$, mark all of these elements, then cyclically shift the list of those elements so that the maximum appears at the first position, and add this list as an element of $$$l$$$. After all elements are marked, $$$l$$$ is the result of this decomposition.For example, if we want to build a decomposition of $$$p = [5, 4, 2, 3, 1, 7, 8, 6]$$$, we do the following: initially $$$p = [5, 4, 2, 3, 1, 7, 8, 6]$$$ (bold elements are marked), $$$l = []$$$; the leftmost unmarked element is $$$5$$$; $$$5$$$ and $$$1$$$ are reachable from it, so the list we want to shift is $$$[5, 1]$$$; there is no need to shift it, since maximum is already the first element; $$$p = [\textbf{5}, 4, 2, 3, \textbf{1}, 7, 8, 6]$$$, $$$l = [[5, 1]]$$$; the leftmost unmarked element is $$$4$$$, the list of reachable elements is $$$[4, 2, 3]$$$; the maximum is already the first element, so there's no need to shift it; $$$p = [\textbf{5}, \textbf{4}, \textbf{2}, \textbf{3}, \textbf{1}, 7, 8, 6]$$$, $$$l = [[5, 1], [4, 2, 3]]$$$; the leftmost unmarked element is $$$7$$$, the list of reachable elements is $$$[7, 8, 6]$$$; we have to shift it, so it becomes $$$[8, 6, 7]$$$; $$$p = [\textbf{5}, \textbf{4}, \textbf{2}, \textbf{3}, \textbf{1}, \textbf{7}, \textbf{8}, \textbf{6}]$$$, $$$l = [[5, 1], [4, 2, 3], [8, 6, 7]]$$$; all elements are marked, so $$$[[5, 1], [4, 2, 3], [8, 6, 7]]$$$ is the result. The New Year transformation of a permutation is defined as follows: we build the decomposition of this permutation; then we sort all lists in decomposition in ascending order of the first elements (we don't swap the elements in these lists, only the lists themselves); then we concatenate the lists into one list which becomes a new permutation. For example, the New Year transformation of $$$p = [5, 4, 2, 3, 1, 7, 8, 6]$$$ is built as follows: the decomposition is $$$[[5, 1], [4, 2, 3], [8, 6, 7]]$$$; after sorting the decomposition, it becomes $$$[[4, 2, 3], [5, 1], [8, 6, 7]]$$$; $$$[4, 2, 3, 5, 1, 8, 6, 7]$$$ is the result of the transformation. We call a permutation good if the result of its transformation is the same as the permutation itself. For example, $$$[4, 3, 1, 2, 8, 5, 6, 7]$$$ is a good permutation; and $$$[5, 4, 2, 3, 1, 7, 8, 6]$$$ is bad, since the result of transformation is $$$[4, 2, 3, 5, 1, 8, 6, 7]$$$.Your task is the following: given $$$n$$$ and $$$k$$$, find the $$$k$$$-th (lexicographically) good permutation of length $$$n$$$. | [] |
There are $$$n$$$ cities in Berland. The city numbered $$$1$$$ is the capital. Some pairs of cities are connected by a one-way road of length 1.Before the trip, Polycarp for each city found out the value of $$$d_i$$$ — the shortest distance from the capital (the $$$1$$$-st city) to the $$$i$$$-th city.Polycarp begins his journey in the city with number $$$s$$$ and, being in the $$$i$$$-th city, chooses one of the following actions: Travel from the $$$i$$$-th city to the $$$j$$$-th city if there is a road from the $$$i$$$-th city to the $$$j$$$-th and $$$d_i < d_j$$$; Travel from the $$$i$$$-th city to the $$$j$$$-th city if there is a road from the $$$i$$$-th city to the $$$j$$$-th and $$$d_i \geq d_j$$$; Stop traveling. Since the government of Berland does not want all people to come to the capital, so Polycarp no more than once can take the second action from the list. in other words, he can perform the second action $$$0$$$ or $$$1$$$ time during his journey. Polycarp, on the other hand, wants to be as close to the capital as possible. For example, if $$$n = 6$$$ and the cities are connected, as in the picture above, then Polycarp could have made the following travels (not all possible options): $$$2 \rightarrow 5 \rightarrow 1 \rightarrow 2 \rightarrow 5$$$; $$$3 \rightarrow 6 \rightarrow 2$$$; $$$1 \rightarrow 3 \rightarrow 6 \rightarrow 2 \rightarrow 5$$$. Polycarp wants for each starting city $$$i$$$ to find out how close he can get to the capital. More formally: he wants to find the minimal value of $$$d_j$$$ that Polycarp can get from the city $$$i$$$ to the city $$$j$$$ according to the rules described above. | ['graphs'] |
You are given a keyboard that consists of $$$26$$$ keys. The keys are arranged sequentially in one row in a certain order. Each key corresponds to a unique lowercase Latin letter.You have to type the word $$$s$$$ on this keyboard. It also consists only of lowercase Latin letters.To type a word, you need to type all its letters consecutively one by one. To type each letter you must position your hand exactly over the corresponding key and press it.Moving the hand between the keys takes time which is equal to the absolute value of the difference between positions of these keys (the keys are numbered from left to right). No time is spent on pressing the keys and on placing your hand over the first letter of the word.For example, consider a keyboard where the letters from 'a' to 'z' are arranged in consecutive alphabetical order. The letters 'h', 'e', 'l' and 'o' then are on the positions $$$8$$$, $$$5$$$, $$$12$$$ and $$$15$$$, respectively. Therefore, it will take $$$|5 - 8| + |12 - 5| + |12 - 12| + |15 - 12| = 13$$$ units of time to type the word "hello". Determine how long it will take to print the word $$$s$$$. | ['strings'] |
You're given an array $$$a$$$. You should repeat the following operation $$$k$$$ times: find the minimum non-zero element in the array, print it, and then subtract it from all the non-zero elements of the array. If all the elements are 0s, just print 0. | [] |
Today at the lesson Vitya learned a very interesting function — mex. Mex of a sequence of numbers is the minimum non-negative number that is not present in the sequence as element. For example, mex([4, 33, 0, 1, 1, 5]) = 2 and mex([1, 2, 3]) = 0.Vitya quickly understood all tasks of the teacher, but can you do the same?You are given an array consisting of n non-negative integers, and m queries. Each query is characterized by one number x and consists of the following consecutive steps: Perform the bitwise addition operation modulo 2 (xor) of each array element with the number x. Find mex of the resulting array. Note that after each query the array changes. | [] |
You are given a string $$$s$$$. You have to reverse it — that is, the first letter should become equal to the last letter before the reversal, the second letter should become equal to the second-to-last letter before the reversal — and so on. For example, if your goal is to reverse the string "abddea", you should get the string "aeddba". To accomplish your goal, you can swap the neighboring elements of the string. Your task is to calculate the minimum number of swaps you have to perform to reverse the given string. | ['strings'] |
A motorcade of n trucks, driving from city «Z» to city «З», has approached a tunnel, known as Tunnel of Horror. Among truck drivers there were rumours about monster DravDe, who hunts for drivers in that tunnel. Some drivers fear to go first, others - to be the last, but let's consider the general case. Each truck is described with four numbers: v — value of the truck, of its passangers and cargo c — amount of passanger on the truck, the driver included l — total amount of people that should go into the tunnel before this truck, so that the driver can overcome his fear («if the monster appears in front of the motorcade, he'll eat them first») r — total amount of people that should follow this truck, so that the driver can overcome his fear («if the monster appears behind the motorcade, he'll eat them first»). Since the road is narrow, it's impossible to escape DravDe, if he appears from one side. Moreover, the motorcade can't be rearranged. The order of the trucks can't be changed, but it's possible to take any truck out of the motorcade, and leave it near the tunnel for an indefinite period. You, as the head of the motorcade, should remove some of the trucks so, that the rest of the motorcade can move into the tunnel and the total amount of the left trucks' values is maximal. | [] |
In Berland it is the holiday of equality. In honor of the holiday the king decided to equalize the welfare of all citizens in Berland by the expense of the state treasury. Totally in Berland there are n citizens, the welfare of each of them is estimated as the integer in ai burles (burle is the currency in Berland).You are the royal treasurer, which needs to count the minimum charges of the kingdom on the king's present. The king can only give money, he hasn't a power to take away them. | ['math'] |
You are given a string $$$s$$$ consisting only of first $$$20$$$ lowercase Latin letters ('a', 'b', ..., 't').Recall that the substring $$$s[l; r]$$$ of the string $$$s$$$ is the string $$$s_l s_{l + 1} \dots s_r$$$. For example, the substrings of "codeforces" are "code", "force", "f", "for", but not "coder" and "top".You can perform the following operation no more than once: choose some substring $$$s[l; r]$$$ and reverse it (i.e. the string $$$s_l s_{l + 1} \dots s_r$$$ becomes $$$s_r s_{r - 1} \dots s_l$$$).Your goal is to maximize the length of the maximum substring of $$$s$$$ consisting of distinct (i.e. unique) characters.The string consists of distinct characters if no character in this string appears more than once. For example, strings "abcde", "arctg" and "minecraft" consist of distinct characters but strings "codeforces", "abacaba" do not consist of distinct characters. | [] |
You are given a string $$$s$$$ consisting of $$$n$$$ lowercase Latin letters.Let's define a substring as a contiguous subsegment of a string. For example, "acab" is a substring of "abacaba" (it starts in position $$$3$$$ and ends in position $$$6$$$), but "aa" or "d" aren't substrings of this string. So the substring of the string $$$s$$$ from position $$$l$$$ to position $$$r$$$ is $$$s[l; r] = s_l s_{l + 1} \dots s_r$$$.You have to choose exactly one of the substrings of the given string and reverse it (i. e. make $$$s[l; r] = s_r s_{r - 1} \dots s_l$$$) to obtain a string that is less lexicographically. Note that it is not necessary to obtain the minimum possible string.If it is impossible to reverse some substring of the given string to obtain a string that is less, print "NO". Otherwise print "YES" and any suitable substring.String $$$x$$$ is lexicographically less than string $$$y$$$, if either $$$x$$$ is a prefix of $$$y$$$ (and $$$x \ne y$$$), or there exists such $$$i$$$ ($$$1 \le i \le min(|x|, |y|)$$$), that $$$x_i < y_i$$$, and for any $$$j$$$ ($$$1 \le j < i$$$) $$$x_j = y_j$$$. Here $$$|a|$$$ denotes the length of the string $$$a$$$. The lexicographic comparison of strings is implemented by operator < in modern programming languages. | ['strings'] |
You are asked to build an array $$$a$$$, consisting of $$$n$$$ integers, each element should be from $$$1$$$ to $$$k$$$.The array should be non-decreasing ($$$a_i \le a_{i+1}$$$ for all $$$i$$$ from $$$1$$$ to $$$n-1$$$). You are also given additional constraints on it. Each constraint is of one of three following types: $$$1~i~x$$$: $$$a_i$$$ should not be equal to $$$x$$$; $$$2~i~j~x$$$: $$$a_i + a_j$$$ should be less than or equal to $$$x$$$; $$$3~i~j~x$$$: $$$a_i + a_j$$$ should be greater than or equal to $$$x$$$. Build any non-decreasing array that satisfies all constraints or report that no such array exists. | ['graphs'] |
Monocarp is a little boy who lives in Byteland and he loves programming.Recently, he found a permutation of length $$$n$$$. He has to come up with a mystic permutation. It has to be a new permutation such that it differs from the old one in each position.More formally, if the old permutation is $$$p_1,p_2,\ldots,p_n$$$ and the new one is $$$q_1,q_2,\ldots,q_n$$$ it must hold that $$$$$$p_1\neq q_1, p_2\neq q_2, \ldots ,p_n\neq q_n.$$$$$$Monocarp is afraid of lexicographically large permutations. Can you please help him to find the lexicographically minimal mystic permutation? | [] |
It is the easy version of the problem. The only difference is that in this version $$$k = 3$$$.You are given a positive integer $$$n$$$. Find $$$k$$$ positive integers $$$a_1, a_2, \ldots, a_k$$$, such that: $$$a_1 + a_2 + \ldots + a_k = n$$$ $$$LCM(a_1, a_2, \ldots, a_k) \le \frac{n}{2}$$$ Here $$$LCM$$$ is the least common multiple of numbers $$$a_1, a_2, \ldots, a_k$$$.We can show that for given constraints the answer always exists. | ['math'] |
Creatnx has $$$n$$$ mirrors, numbered from $$$1$$$ to $$$n$$$. Every day, Creatnx asks exactly one mirror "Am I beautiful?". The $$$i$$$-th mirror will tell Creatnx that he is beautiful with probability $$$\frac{p_i}{100}$$$ for all $$$1 \le i \le n$$$.Creatnx asks the mirrors one by one, starting from the $$$1$$$-st mirror. Every day, if he asks $$$i$$$-th mirror, there are two possibilities: The $$$i$$$-th mirror tells Creatnx that he is beautiful. In this case, if $$$i = n$$$ Creatnx will stop and become happy, otherwise he will continue asking the $$$i+1$$$-th mirror next day; In the other case, Creatnx will feel upset. The next day, Creatnx will start asking from the $$$1$$$-st mirror again. You need to calculate the expected number of days until Creatnx becomes happy.This number should be found by modulo $$$998244353$$$. Formally, let $$$M = 998244353$$$. It can be shown that the answer can be expressed as an irreducible fraction $$$\frac{p}{q}$$$, where $$$p$$$ and $$$q$$$ are integers and $$$q \not \equiv 0 \pmod{M}$$$. Output the integer equal to $$$p \cdot q^{-1} \bmod M$$$. In other words, output such an integer $$$x$$$ that $$$0 \le x < M$$$ and $$$x \cdot q \equiv p \pmod{M}$$$. | ['math', 'probabilities'] |
You are given n × m table. Each cell of the table is colored white or black. Find the number of non-empty sets of cells such that: All cells in a set have the same color. Every two cells in a set share row or column. | ['math'] |
You have $$$r$$$ red and $$$b$$$ blue beans. You'd like to distribute them among several (maybe, one) packets in such a way that each packet: has at least one red bean (or the number of red beans $$$r_i \ge 1$$$); has at least one blue bean (or the number of blue beans $$$b_i \ge 1$$$); the number of red and blue beans should differ in no more than $$$d$$$ (or $$$|r_i - b_i| \le d$$$) Can you distribute all beans? | ['math'] |
You are given an array of positive integers $$$a = [a_0, a_1, \dots, a_{n - 1}]$$$ ($$$n \ge 2$$$).In one step, the array $$$a$$$ is replaced with another array of length $$$n$$$, in which each element is the greatest common divisor (GCD) of two neighboring elements (the element itself and its right neighbor; consider that the right neighbor of the $$$(n - 1)$$$-th element is the $$$0$$$-th element).Formally speaking, a new array $$$b = [b_0, b_1, \dots, b_{n - 1}]$$$ is being built from array $$$a = [a_0, a_1, \dots, a_{n - 1}]$$$ such that $$$b_i$$$ $$$= \gcd(a_i, a_{(i + 1) \mod n})$$$, where $$$\gcd(x, y)$$$ is the greatest common divisor of $$$x$$$ and $$$y$$$, and $$$x \mod y$$$ is the remainder of $$$x$$$ dividing by $$$y$$$. In one step the array $$$b$$$ is built and then the array $$$a$$$ is replaced with $$$b$$$ (that is, the assignment $$$a$$$ := $$$b$$$ is taking place).For example, if $$$a = [16, 24, 10, 5]$$$ then $$$b = [\gcd(16, 24)$$$, $$$\gcd(24, 10)$$$, $$$\gcd(10, 5)$$$, $$$\gcd(5, 16)]$$$ $$$= [8, 2, 5, 1]$$$. Thus, after one step the array $$$a = [16, 24, 10, 5]$$$ will be equal to $$$[8, 2, 5, 1]$$$.For a given array $$$a$$$, find the minimum number of steps after which all values $$$a_i$$$ become equal (that is, $$$a_0 = a_1 = \dots = a_{n - 1}$$$). If the original array $$$a$$$ consists of identical elements then consider the number of steps is equal to $$$0$$$. | ['number theory'] |
The Romans have attacked again. This time they are much more than the Persians but Shapur is ready to defeat them. He says: "A lion is never afraid of a hundred sheep". Nevertheless Shapur has to find weaknesses in the Roman army to defeat them. So he gives the army a weakness number.In Shapur's opinion the weakness of an army is equal to the number of triplets i, j, k such that i < j < k and ai > aj > ak where ax is the power of man standing at position x. The Roman army has one special trait — powers of all the people in it are distinct.Help Shapur find out how weak the Romans are. | ['trees'] |
Bill is a famous mathematician in BubbleLand. Thanks to his revolutionary math discoveries he was able to make enough money to build a beautiful house. Unfortunately, for not paying property tax on time, court decided to punish Bill by making him lose a part of his property.Bill’s property can be observed as a convex regular 2n-sided polygon A0 A1... A2n - 1 A2n, A2n = A0, with sides of the exactly 1 meter in length. Court rules for removing part of his property are as follows: Split every edge Ak Ak + 1, k = 0... 2n - 1 in n equal parts of size 1 / n with points P0, P1, ..., Pn - 1 On every edge A2k A2k + 1, k = 0... n - 1 court will choose one point B2k = Pi for some i = 0, ..., n - 1 such that On every edge A2k + 1A2k + 2, k = 0...n - 1 Bill will choose one point B2k + 1 = Pi for some i = 0, ..., n - 1 such that Bill gets to keep property inside of 2n-sided polygon B0 B1... B2n - 1 Luckily, Bill found out which B2k points the court chose. Even though he is a great mathematician, his house is very big and he has a hard time calculating. Therefore, he is asking you to help him choose points so he maximizes area of property he can keep. | [] |
You are given an array $$$A$$$ of length $$$N$$$ weights of masses $$$A_1$$$, $$$A_2$$$...$$$A_N$$$. No two weights have the same mass. You can put every weight on one side of the balance (left or right). You don't have to put weights in order $$$A_1$$$,...,$$$A_N$$$. There is also a string $$$S$$$ consisting of characters "L" and "R", meaning that after putting the $$$i-th$$$ weight (not $$$A_i$$$, but $$$i-th$$$ weight of your choice) left or right side of the balance should be heavier. Find the order of putting the weights on the balance such that rules of string $$$S$$$ are satisfied. | [] |
Bajtek, known for his unusual gifts, recently got an integer array $$$x_0, x_1, \ldots, x_{k-1}$$$.Unfortunately, after a huge array-party with his extraordinary friends, he realized that he'd lost it. After hours spent on searching for a new toy, Bajtek found on the arrays producer's website another array $$$a$$$ of length $$$n + 1$$$. As a formal description of $$$a$$$ says, $$$a_0 = 0$$$ and for all other $$$i$$$ ($$$1 \le i \le n$$$) $$$a_i = x_{(i-1)\bmod k} + a_{i-1}$$$, where $$$p \bmod q$$$ denotes the remainder of division $$$p$$$ by $$$q$$$.For example, if the $$$x = [1, 2, 3]$$$ and $$$n = 5$$$, then: $$$a_0 = 0$$$, $$$a_1 = x_{0\bmod 3}+a_0=x_0+0=1$$$, $$$a_2 = x_{1\bmod 3}+a_1=x_1+1=3$$$, $$$a_3 = x_{2\bmod 3}+a_2=x_2+3=6$$$, $$$a_4 = x_{3\bmod 3}+a_3=x_0+6=7$$$, $$$a_5 = x_{4\bmod 3}+a_4=x_1+7=9$$$. So, if the $$$x = [1, 2, 3]$$$ and $$$n = 5$$$, then $$$a = [0, 1, 3, 6, 7, 9]$$$.Now the boy hopes that he will be able to restore $$$x$$$ from $$$a$$$! Knowing that $$$1 \le k \le n$$$, help him and find all possible values of $$$k$$$ — possible lengths of the lost array. | [] |
You found a useless array $$$a$$$ of $$$2n$$$ positive integers. You have realized that you actually don't need this array, so you decided to throw out all elements of $$$a$$$.It could have been an easy task, but it turned out that you should follow some rules: In the beginning, you select any positive integer $$$x$$$. Then you do the following operation $$$n$$$ times: select two elements of array with sum equals $$$x$$$; remove them from $$$a$$$ and replace $$$x$$$ with maximum of that two numbers. For example, if initially $$$a = [3, 5, 1, 2]$$$, you can select $$$x = 6$$$. Then you can select the second and the third elements of $$$a$$$ with sum $$$5 + 1 = 6$$$ and throw them out. After this operation, $$$x$$$ equals $$$5$$$ and there are two elements in array: $$$3$$$ and $$$2$$$. You can throw them out on the next operation.Note, that you choose $$$x$$$ before the start and can't change it as you want between the operations.Determine how should you behave to throw out all elements of $$$a$$$. | [] |
The Winter holiday will be here soon. Mr. Chanek wants to decorate his house's wall with ornaments. The wall can be represented as a binary string $$$a$$$ of length $$$n$$$. His favorite nephew has another binary string $$$b$$$ of length $$$m$$$ ($$$m \leq n$$$).Mr. Chanek's nephew loves the non-negative integer $$$k$$$. His nephew wants exactly $$$k$$$ occurrences of $$$b$$$ as substrings in $$$a$$$. However, Mr. Chanek does not know the value of $$$k$$$. So, for each $$$k$$$ ($$$0 \leq k \leq n - m + 1$$$), find the minimum number of elements in $$$a$$$ that have to be changed such that there are exactly $$$k$$$ occurrences of $$$b$$$ in $$$a$$$.A string $$$s$$$ occurs exactly $$$k$$$ times in $$$t$$$ if there are exactly $$$k$$$ different pairs $$$(p,q)$$$ such that we can obtain $$$s$$$ by deleting $$$p$$$ characters from the beginning and $$$q$$$ characters from the end of $$$t$$$. | ['strings'] |
The King of Berland Polycarp LXXXIV has $$$n$$$ daughters. To establish his power to the neighbouring kingdoms he wants to marry his daughters to the princes of these kingdoms. As a lucky coincidence there are $$$n$$$ other kingdoms as well.So Polycarp LXXXIV has enumerated his daughters from $$$1$$$ to $$$n$$$ and the kingdoms from $$$1$$$ to $$$n$$$. For each daughter he has compiled a list of kingdoms princes of which she wanted to marry.Polycarp LXXXIV is very busy, so he finds a couple for his daughters greedily one after another.For the first daughter he takes the kingdom with the lowest number from her list and marries the daughter to their prince. For the second daughter he takes the kingdom with the lowest number from her list, prince of which hasn't been taken already. If there are no free princes in the list then the daughter marries nobody and Polycarp LXXXIV proceeds to the next daughter. The process ends after the $$$n$$$-th daughter.For example, let there be $$$4$$$ daughters and kingdoms, the lists daughters have are $$$[2, 3]$$$, $$$[1, 2]$$$, $$$[3, 4]$$$, $$$[3]$$$, respectively. In that case daughter $$$1$$$ marries the prince of kingdom $$$2$$$, daughter $$$2$$$ marries the prince of kingdom $$$1$$$, daughter $$$3$$$ marries the prince of kingdom $$$3$$$, leaving daughter $$$4$$$ nobody to marry to.Actually, before starting the marriage process Polycarp LXXXIV has the time to convince one of his daughters that some prince is also worth marrying to. Effectively, that means that he can add exactly one kingdom to exactly one of his daughter's list. Note that this kingdom should not be present in the daughter's list.Polycarp LXXXIV wants to increase the number of married couples.Unfortunately, what he doesn't have the time for is determining what entry to add. If there is no way to increase the total number of married couples then output that the marriages are already optimal. Otherwise, find such an entry that the total number of married couples increases if Polycarp LXXXIV adds it.If there are multiple ways to add an entry so that the total number of married couples increases then print any of them.For your and our convenience you are asked to answer $$$t$$$ independent test cases. | ['graphs'] |
Geometric progression with the first element a and common ratio b is a sequence of numbers a, ab, ab2, ab3, ....You are given n integer geometric progressions. Your task is to find the smallest integer x, that is the element of all the given progressions, or else state that such integer does not exist. | ['math'] |
Nastya likes reading and even spends whole days in a library sometimes. Today she found a chronicle of Byteland in the library, and it stated that there lived shamans long time ago. It is known that at every moment there was exactly one shaman in Byteland, and there were n shamans in total enumerated with integers from 1 to n in the order they lived. Also, each shaman had a magic power which can now be expressed as an integer.The chronicle includes a list of powers of the n shamans. Also, some shamans can be king-shamans, if they gathered all the power of their predecessors, i.e. their power is exactly the sum of powers of all previous shamans. Nastya is interested in whether there was at least one king-shaman in Byteland.Unfortunately many of the powers are unreadable in the list, so Nastya is doing the following: Initially she supposes some power for each shaman. After that she changes the power of some shaman q times (the shamans can differ) and after that wants to check if there is at least one king-shaman in the list. If yes, she wants to know the index of any king-shaman. Unfortunately the list is too large and Nastya wants you to help her. | [] |
The Government of Mars is not only interested in optimizing space flights, but also wants to improve the road system of the planet.One of the most important highways of Mars connects Olymp City and Kstolop, the capital of Cydonia. In this problem, we only consider the way from Kstolop to Olymp City, but not the reverse path (i. e. the path from Olymp City to Kstolop).The road from Kstolop to Olymp City is $$$\ell$$$ kilometers long. Each point of the road has a coordinate $$$x$$$ ($$$0 \le x \le \ell$$$), which is equal to the distance from Kstolop in kilometers. So, Kstolop is located in the point with coordinate $$$0$$$, and Olymp City is located in the point with coordinate $$$\ell$$$.There are $$$n$$$ signs along the road, $$$i$$$-th of which sets a speed limit $$$a_i$$$. This limit means that the next kilometer must be passed in $$$a_i$$$ minutes and is active until you encounter the next along the road. There is a road sign at the start of the road (i. e. in the point with coordinate $$$0$$$), which sets the initial speed limit.If you know the location of all the signs, it's not hard to calculate how much time it takes to drive from Kstolop to Olymp City. Consider an example: Here, you need to drive the first three kilometers in five minutes each, then one kilometer in eight minutes, then four kilometers in three minutes each, and finally the last two kilometers must be passed in six minutes each. Total time is $$$3\cdot 5 + 1\cdot 8 + 4\cdot 3 + 2\cdot 6 = 47$$$ minutes.To optimize the road traffic, the Government of Mars decided to remove no more than $$$k$$$ road signs. It cannot remove the sign at the start of the road, otherwise, there will be no limit at the start. By removing these signs, the Government also wants to make the time needed to drive from Kstolop to Olymp City as small as possible.The largest industrial enterprises are located in Cydonia, so it's the priority task to optimize the road traffic from Olymp City. So, the Government of Mars wants you to remove the signs in the way described above. | [] |
When registering in a social network, users are allowed to create their own convenient login to make it easier to share contacts, print it on business cards, etc.Login is an arbitrary sequence of lower and uppercase latin letters, digits and underline symbols («_»). However, in order to decrease the number of frauds and user-inattention related issues, it is prohibited to register a login if it is similar with an already existing login. More precisely, two logins s and t are considered similar if we can transform s to t via a sequence of operations of the following types: transform lowercase letters to uppercase and vice versa; change letter «O» (uppercase latin letter) to digit «0» and vice versa; change digit «1» (one) to any letter among «l» (lowercase latin «L»), «I» (uppercase latin «i») and vice versa, or change one of these letters to other. For example, logins «Codeforces» and «codef0rces» as well as «OO0OOO00O0OOO0O00OOO0OO_lol» and «OO0OOO0O00OOO0O00OO0OOO_1oI» are considered similar whereas «Codeforces» and «Code_forces» are not.You're given a list of existing logins with no two similar amonst and a newly created user login. Check whether this new login is similar with any of the existing ones. | ['strings'] |
Vasya's birthday is approaching and Lena decided to sew a patterned handkerchief to him as a present. Lena chose digits from 0 to n as the pattern. The digits will form a rhombus. The largest digit n should be located in the centre. The digits should decrease as they approach the edges. For example, for n = 5 the handkerchief pattern should look like that: 0 0 1 0 0 1 2 1 0 0 1 2 3 2 1 0 0 1 2 3 4 3 2 1 00 1 2 3 4 5 4 3 2 1 0 0 1 2 3 4 3 2 1 0 0 1 2 3 2 1 0 0 1 2 1 0 0 1 0 0Your task is to determine the way the handkerchief will look like by the given n. | [] |
A substring of some string is called the most frequent, if the number of its occurrences is not less than number of occurrences of any other substring.You are given a set of strings. A string (not necessarily from this set) is called good if all elements of the set are the most frequent substrings of this string. Restore the non-empty good string with minimum length. If several such strings exist, restore lexicographically minimum string. If there are no good strings, print "NO" (without quotes).A substring of a string is a contiguous subsequence of letters in the string. For example, "ab", "c", "abc" are substrings of string "abc", while "ac" is not a substring of that string.The number of occurrences of a substring in a string is the number of starting positions in the string where the substring occurs. These occurrences could overlap.String a is lexicographically smaller than string b, if a is a prefix of b, or a has a smaller letter at the first position where a and b differ. | ['graphs', 'strings'] |
Everything got unclear to us in a far away constellation Tau Ceti. Specifically, the Taucetians choose names to their children in a very peculiar manner.Two young parents abac and bbad think what name to give to their first-born child. They decided that the name will be the permutation of letters of string s. To keep up with the neighbours, they decided to call the baby so that the name was lexicographically strictly larger than the neighbour's son's name t.On the other hand, they suspect that a name tax will be introduced shortly. According to it, the Taucetians with lexicographically larger names will pay larger taxes. That's the reason abac and bbad want to call the newborn so that the name was lexicographically strictly larger than name t and lexicographically minimum at that.The lexicographical order of strings is the order we are all used to, the "dictionary" order. Such comparison is used in all modern programming languages to compare strings. Formally, a string p of length n is lexicographically less than string q of length m, if one of the two statements is correct: n < m, and p is the beginning (prefix) of string q (for example, "aba" is less than string "abaa"), p1 = q1, p2 = q2, ..., pk - 1 = qk - 1, pk < qk for some k (1 ≤ k ≤ min(n, m)), here characters in strings are numbered starting from 1. Write a program that, given string s and the heighbours' child's name t determines the string that is the result of permutation of letters in s. The string should be lexicographically strictly more than t and also, lexicographically minimum. | ['strings'] |
You helped Dima to have a great weekend, but it's time to work. Naturally, Dima, as all other men who have girlfriends, does everything wrong.Inna and Dima are now in one room. Inna tells Dima off for everything he does in her presence. After Inna tells him off for something, she goes to another room, walks there in circles muttering about how useless her sweetheart is. During that time Dima has time to peacefully complete k - 1 tasks. Then Inna returns and tells Dima off for the next task he does in her presence and goes to another room again. It continues until Dima is through with his tasks.Overall, Dima has n tasks to do, each task has a unique number from 1 to n. Dima loves order, so he does tasks consecutively, starting from some task. For example, if Dima has 6 tasks to do in total, then, if he starts from the 5-th task, the order is like that: first Dima does the 5-th task, then the 6-th one, then the 1-st one, then the 2-nd one, then the 3-rd one, then the 4-th one.Inna tells Dima off (only lovingly and appropriately!) so often and systematically that he's very well learned the power with which she tells him off for each task. Help Dima choose the first task so that in total he gets told off with as little power as possible. | [] |
Currently Tiny is learning Computational Geometry. When trying to solve a problem called "The Closest Pair Of Points In The Plane", he found that a code which gave a wrong time complexity got Accepted instead of Time Limit Exceeded.The problem is the follows. Given n points in the plane, find a pair of points between which the distance is minimized. Distance between (x1, y1) and (x2, y2) is .The pseudo code of the unexpected code is as follows:input nfor i from 1 to n input the i-th point's coordinates into p[i]sort array p[] by increasing of x coordinate first and increasing of y coordinate secondd=INF //here INF is a number big enoughtot=0for i from 1 to n for j from (i+1) to n ++tot if (p[j].x-p[i].x>=d) then break //notice that "break" is only to be //out of the loop "for j" d=min(d,distance(p[i],p[j]))output dHere, tot can be regarded as the running time of the code. Due to the fact that a computer can only run a limited number of operations per second, tot should not be more than k in order not to get Time Limit Exceeded.You are a great hacker. Would you please help Tiny generate a test data and let the code get Time Limit Exceeded? | [] |
You are given an array $$$a$$$ of length $$$n$$$. Let $$$cnt_x$$$ be the number of elements from the array which are equal to $$$x$$$. Let's also define $$$f(x, y)$$$ as $$$(cnt_x + cnt_y) \cdot (x + y)$$$.Also you are given $$$m$$$ bad pairs $$$(x_i, y_i)$$$. Note that if $$$(x, y)$$$ is a bad pair, then $$$(y, x)$$$ is also bad.Your task is to find the maximum value of $$$f(u, v)$$$ over all pairs $$$(u, v)$$$, such that $$$u \neq v$$$, that this pair is not bad, and also that $$$u$$$ and $$$v$$$ each occur in the array $$$a$$$. It is guaranteed that such a pair exists. | [] |
Burenka came to kindergarden. This kindergarten is quite strange, so each kid there receives two fractions ($$$\frac{a}{b}$$$ and $$$\frac{c}{d}$$$) with integer numerators and denominators. Then children are commanded to play with their fractions.Burenka is a clever kid, so she noticed that when she claps once, she can multiply numerator or denominator of one of her two fractions by any integer of her choice (but she can't multiply denominators by $$$0$$$). Now she wants know the minimal number of claps to make her fractions equal (by value). Please help her and find the required number of claps! | ['math', 'number theory'] |
Kuro is currently playing an educational game about numbers. The game focuses on the greatest common divisor (GCD), the XOR value, and the sum of two numbers. Kuro loves the game so much that he solves levels by levels day by day.Sadly, he's going on a vacation for a day, and he isn't able to continue his solving streak on his own. As Katie is a reliable person, Kuro kindly asked her to come to his house on this day to play the game for him.Initally, there is an empty array $$$a$$$. The game consists of $$$q$$$ tasks of two types. The first type asks Katie to add a number $$$u_i$$$ to $$$a$$$. The second type asks Katie to find a number $$$v$$$ existing in $$$a$$$ such that $$$k_i \mid GCD(x_i, v)$$$, $$$x_i + v \leq s_i$$$, and $$$x_i \oplus v$$$ is maximized, where $$$\oplus$$$ denotes the bitwise XOR operation, $$$GCD(c, d)$$$ denotes the greatest common divisor of integers $$$c$$$ and $$$d$$$, and $$$y \mid x$$$ means $$$x$$$ is divisible by $$$y$$$, or report -1 if no such numbers are found.Since you are a programmer, Katie needs you to automatically and accurately perform the tasks in the game to satisfy her dear friend Kuro. Let's help her! | ['math', 'strings', 'number theory', 'trees'] |
Vasya is developing his own programming language VPL (Vasya Programming Language). Right now he is busy making the system of exceptions. He thinks that the system of exceptions must function like that.The exceptions are processed by try-catch-blocks. There are two operators that work with the blocks: The try operator. It opens a new try-catch-block. The catch(<exception_type>, <message>) operator. It closes the try-catch-block that was started last and haven't yet been closed. This block can be activated only via exception of type <exception_type>. When we activate this block, the screen displays the <message>. If at the given moment there is no open try-catch-block, then we can't use the catch operator.The exceptions can occur in the program in only one case: when we use the throw operator. The throw(<exception_type>) operator creates the exception of the given type.Let's suggest that as a result of using some throw operator the program created an exception of type a. In this case a try-catch-block is activated, such that this block's try operator was described in the program earlier than the used throw operator. Also, this block's catch operator was given an exception type a as a parameter and this block's catch operator is described later that the used throw operator. If there are several such try-catch-blocks, then the system activates the block whose catch operator occurs earlier than others. If no try-catch-block was activated, then the screen displays message "Unhandled Exception".To test the system, Vasya wrote a program that contains only try, catch and throw operators, one line contains no more than one operator, the whole program contains exactly one throw operator.Your task is: given a program in VPL, determine, what message will be displayed on the screen. | [] |
You are given an array $$$a_1, a_2, \ldots, a_n$$$ and an integer $$$x$$$.Find the number of non-empty subsets of indices of this array $$$1 \leq b_1 < b_2 < \ldots < b_k \leq n$$$, such that for all pairs $$$(i, j)$$$ where $$$1 \leq i < j \leq k$$$, the inequality $$$a_{b_i} \oplus a_{b_j} \leq x$$$ is held. Here, $$$\oplus$$$ denotes the bitwise XOR operation. As the answer may be very large, output it modulo $$$998\,244\,353$$$. | ['math'] |
You are given $$$n$$$ integer numbers $$$a_1, a_2, \dots, a_n$$$. Consider graph on $$$n$$$ nodes, in which nodes $$$i$$$, $$$j$$$ ($$$i\neq j$$$) are connected if and only if, $$$a_i$$$ AND $$$a_j\neq 0$$$, where AND denotes the bitwise AND operation.Find the length of the shortest cycle in this graph or determine that it doesn't have cycles at all. | ['graphs'] |
Alice has a cake, and she is going to cut it. She will perform the following operation $$$n-1$$$ times: choose a piece of the cake (initially, the cake is all one piece) with weight $$$w\ge 2$$$ and cut it into two smaller pieces of weight $$$\lfloor\frac{w}{2}\rfloor$$$ and $$$\lceil\frac{w}{2}\rceil$$$ ($$$\lfloor x \rfloor$$$ and $$$\lceil x \rceil$$$ denote floor and ceiling functions, respectively).After cutting the cake in $$$n$$$ pieces, she will line up these $$$n$$$ pieces on a table in an arbitrary order. Let $$$a_i$$$ be the weight of the $$$i$$$-th piece in the line.You are given the array $$$a$$$. Determine whether there exists an initial weight and sequence of operations which results in $$$a$$$. | [] |
You are given $$$n$$$ segments $$$[l_1, r_1], [l_2, r_2], \dots, [l_n, r_n]$$$. Each segment has one of two colors: the $$$i$$$-th segment's color is $$$t_i$$$.Let's call a pair of segments $$$i$$$ and $$$j$$$ bad if the following two conditions are met: $$$t_i \ne t_j$$$; the segments $$$[l_i, r_i]$$$ and $$$[l_j, r_j]$$$ intersect, embed or touch, i. e. there exists an integer $$$x$$$ such that $$$x \in [l_i, r_i]$$$ and $$$x \in [l_j, r_j]$$$. Calculate the maximum number of segments that can be selected from the given ones, so that there is no bad pair among the selected ones. | [] |
In one little known, but very beautiful country called Waterland, lives a lovely shark Valerie. Like all the sharks, she has several rows of teeth, and feeds on crucians. One of Valerie's distinguishing features is that while eating one crucian she uses only one row of her teeth, the rest of the teeth are "relaxing".For a long time our heroine had been searching the sea for crucians, but a great misfortune happened. Her teeth started to ache, and she had to see the local dentist, lobster Ashot. As a professional, Ashot quickly relieved Valerie from her toothache. Moreover, he managed to determine the cause of Valerie's developing caries (for what he was later nicknamed Cap).It turned that Valerie eats too many crucians. To help Valerie avoid further reoccurrence of toothache, Ashot found for each Valerie's tooth its residual viability. Residual viability of a tooth is a value equal to the amount of crucians that Valerie can eat with this tooth. Every time Valerie eats a crucian, viability of all the teeth used for it will decrease by one. When the viability of at least one tooth becomes negative, the shark will have to see the dentist again. Unhappy, Valerie came back home, where a portion of crucians was waiting for her. For sure, the shark couldn't say no to her favourite meal, but she had no desire to go back to the dentist. That's why she decided to eat the maximum amount of crucians from the portion but so that the viability of no tooth becomes negative. As Valerie is not good at mathematics, she asked you to help her to find out the total amount of crucians that she can consume for dinner.We should remind you that while eating one crucian Valerie uses exactly one row of teeth and the viability of each tooth from this row decreases by one. | [] |
You are given an array of integers $$$a$$$ of length $$$n$$$. The elements of the array can be either different or the same. Each element of the array is colored either blue or red. There are no unpainted elements in the array. One of the two operations described below can be applied to an array in a single step: either you can select any blue element and decrease its value by $$$1$$$; or you can select any red element and increase its value by $$$1$$$. Situations in which there are no elements of some color at all are also possible. For example, if the whole array is colored blue or red, one of the operations becomes unavailable.Determine whether it is possible to make $$$0$$$ or more steps such that the resulting array is a permutation of numbers from $$$1$$$ to $$$n$$$?In other words, check whether there exists a sequence of steps (possibly empty) such that after applying it, the array $$$a$$$ contains in some order all numbers from $$$1$$$ to $$$n$$$ (inclusive), each exactly once. | ['math'] |
Vanya wants to pass n exams and get the academic scholarship. He will get the scholarship if the average grade mark for all the exams is at least avg. The exam grade cannot exceed r. Vanya has passed the exams and got grade ai for the i-th exam. To increase the grade for the i-th exam by 1 point, Vanya must write bi essays. He can raise the exam grade multiple times.What is the minimum number of essays that Vanya needs to write to get scholarship? | [] |
You are given an array $$$a$$$ of $$$n$$$ elements. You can apply the following operation to it any number of times: Select some subarray from $$$a$$$ of even size $$$2k$$$ that begins at position $$$l$$$ ($$$1\le l \le l+2\cdot{k}-1\le n$$$, $$$k \ge 1$$$) and for each $$$i$$$ between $$$0$$$ and $$$k-1$$$ (inclusive), assign the value $$$a_{l+k+i}$$$ to $$$a_{l+i}$$$. For example, if $$$a = [2, 1, 3, 4, 5, 3]$$$, then choose $$$l = 1$$$ and $$$k = 2$$$, applying this operation the array will become $$$a = [3, 4, 3, 4, 5, 3]$$$.Find the minimum number of operations (possibly zero) needed to make all the elements of the array equal. | [] |
In Chelyabinsk lives a much respected businessman Nikita with a strange nickname "Boss". Once Nikita decided to go with his friend Alex to the Summer Biathlon World Cup. Nikita, as a very important person, received a token which allows to place bets on each section no more than on one competitor.To begin with friends learned the rules: in the race there are n sections of equal length and m participants. The participants numbered from 1 to m. About each participant the following is known: li — the number of the starting section, ri — the number of the finishing section (li ≤ ri), ti — the time a biathlete needs to complete an section of the path, ci — the profit in roubles. If the i-th sportsman wins on one of the sections, the profit will be given to the man who had placed a bet on that sportsman. The i-th biathlete passes the sections from li to ri inclusive. The competitor runs the whole way in (ri - li + 1)·ti time units. It takes him exactly ti time units to pass each section. In case of the athlete's victory on k sections the man who has betted on him receives k·ci roubles.In each section the winner is determined independently as follows: if there is at least one biathlete running this in this section, then among all of them the winner is the one who has ran this section in minimum time (spent minimum time passing this section). In case of equality of times the athlete with the smaller index number wins. If there are no participants in this section, then the winner in this section in not determined. We have to say that in the summer biathlon all the participants are moving at a constant speed.We should also add that Nikita can bet on each section and on any contestant running in this section.Help the friends find the maximum possible profit. | [] |
A row of $$$n$$$ cells is given, all initially white. Using a stamp, you can stamp any two neighboring cells such that one becomes red and the other becomes blue. A stamp can be rotated, i.e. it can be used in both ways: as $$$\color{blue}{\texttt{B}}\color{red}{\texttt{R}}$$$ and as $$$\color{red}{\texttt{R}}\color{blue}{\texttt{B}}$$$.During use, the stamp must completely fit on the given $$$n$$$ cells (it cannot be partially outside the cells). The stamp can be applied multiple times to the same cell. Each usage of the stamp recolors both cells that are under the stamp.For example, one possible sequence of stamps to make the picture $$$\color{blue}{\texttt{B}}\color{red}{\texttt{R}}\color{blue}{\texttt{B}}\color{blue}{\texttt{B}}\texttt{W}$$$ could be $$$\texttt{WWWWW} \to \texttt{WW}\color{brown}{\underline{\color{red}{\texttt{R}}\color{blue}{\texttt{B}}}}\texttt{W} \to \color{brown}{\underline{\color{blue}{\texttt{B}}\color{red}{\texttt{R}}}}\color{red}{\texttt{R}}\color{blue}{\texttt{B}}\texttt{W} \to \color{blue}{\texttt{B}}\color{brown}{\underline{\color{red}{\texttt{R}}\color{blue}{\texttt{B}}}}\color{blue}{\texttt{B}}\texttt{W}$$$. Here $$$\texttt{W}$$$, $$$\color{red}{\texttt{R}}$$$, and $$$\color{blue}{\texttt{B}}$$$ represent a white, red, or blue cell, respectively, and the cells that the stamp is used on are marked with an underline.Given a final picture, is it possible to make it using the stamp zero or more times? | [] |
William has two arrays of numbers $$$a_1, a_2, \dots, a_n$$$ and $$$b_1, b_2, \dots, b_m$$$. The arrays satisfy the conditions of being convex. Formally an array $$$c$$$ of length $$$k$$$ is considered convex if $$$c_i - c_{i - 1} < c_{i + 1} - c_i$$$ for all $$$i$$$ from $$$2$$$ to $$$k - 1$$$ and $$$c_1 < c_2$$$.Throughout William's life he observed $$$q$$$ changes of two types happening to the arrays: Add the arithmetic progression $$$d, d \cdot 2, d \cdot 3, \dots, d \cdot k$$$ to the suffix of the array $$$a$$$ of length $$$k$$$. The array after the change looks like this: $$$[a_1, a_2, \dots, a_{n - k}, a_{n - k + 1} + d, a_{n - k + 2} + d \cdot 2, \dots, a_n + d \cdot k]$$$. The same operation, but for array $$$b$$$. After each change a matrix $$$d$$$ is created from arrays $$$a$$$ and $$$b$$$, of size $$$n \times m$$$, where $$$d_{i, j}=a_i + b_j$$$. William wants to get from cell ($$$1, 1$$$) to cell ($$$n, m$$$) of this matrix. From cell ($$$x, y$$$) he can only move to cells ($$$x + 1, y$$$) and ($$$x, y + 1$$$). The length of a path is calculated as the sum of numbers in cells visited by William, including the first and the last cells.After each change William wants you to help find out the minimal length of the path he could take. | ['math'] |
There are $$$n$$$ Christmas trees on an infinite number line. The $$$i$$$-th tree grows at the position $$$x_i$$$. All $$$x_i$$$ are guaranteed to be distinct.Each integer point can be either occupied by the Christmas tree, by the human or not occupied at all. Non-integer points cannot be occupied by anything.There are $$$m$$$ people who want to celebrate Christmas. Let $$$y_1, y_2, \dots, y_m$$$ be the positions of people (note that all values $$$x_1, x_2, \dots, x_n, y_1, y_2, \dots, y_m$$$ should be distinct and all $$$y_j$$$ should be integer). You want to find such an arrangement of people that the value $$$\sum\limits_{j=1}^{m}\min\limits_{i=1}^{n}|x_i - y_j|$$$ is the minimum possible (in other words, the sum of distances to the nearest Christmas tree for all people should be minimized).In other words, let $$$d_j$$$ be the distance from the $$$j$$$-th human to the nearest Christmas tree ($$$d_j = \min\limits_{i=1}^{n} |y_j - x_i|$$$). Then you need to choose such positions $$$y_1, y_2, \dots, y_m$$$ that $$$\sum\limits_{j=1}^{m} d_j$$$ is the minimum possible. | ['graphs'] |
There are one cat, $$$k$$$ mice, and one hole on a coordinate line. The cat is located at the point $$$0$$$, the hole is located at the point $$$n$$$. All mice are located between the cat and the hole: the $$$i$$$-th mouse is located at the point $$$x_i$$$ ($$$0 < x_i < n$$$). At each point, many mice can be located.In one second, the following happens. First, exactly one mouse moves to the right by $$$1$$$. If the mouse reaches the hole, it hides (i.e. the mouse will not any more move to any point and will not be eaten by the cat). Then (after that the mouse has finished its move) the cat moves to the right by $$$1$$$. If at the new cat's position, some mice are located, the cat eats them (they will not be able to move after that). The actions are performed until any mouse hasn't been hidden or isn't eaten.In other words, the first move is made by a mouse. If the mouse has reached the hole, it's saved. Then the cat makes a move. The cat eats the mice located at the pointed the cat has reached (if the cat has reached the hole, it eats nobody).Each second, you can select a mouse that will make a move. What is the maximum number of mice that can reach the hole without being eaten? | [] |
The problem describes the properties of a command line. The description somehow resembles the one you usually see in real operating systems. However, there are differences in the behavior. Please make sure you've read the statement attentively and use it as a formal document.In the Pindows operating system a strings are the lexemes of the command line — the first of them is understood as the name of the program to run and the following lexemes are its arguments. For example, as we execute the command " run.exe one, two . ", we give four lexemes to the Pindows command line: "run.exe", "one,", "two", ".". More formally, if we run a command that can be represented as string s (that has no quotes), then the command line lexemes are maximal by inclusion substrings of string s that contain no spaces.To send a string with spaces or an empty string as a command line lexeme, we can use double quotes. The block of characters that should be considered as one lexeme goes inside the quotes. Embedded quotes are prohibited — that is, for each occurrence of character """ we should be able to say clearly that the quotes are opening or closing. For example, as we run the command ""run.exe o" "" " ne, " two . " " ", we give six lexemes to the Pindows command line: "run.exe o", "" (an empty string), " ne, ", "two", ".", " " (a single space).It is guaranteed that each lexeme of the command line is either surrounded by spaces on both sides or touches the corresponding command border. One of its consequences is: the opening brackets are either the first character of the string or there is a space to the left of them.You have a string that consists of uppercase and lowercase English letters, digits, characters ".,?!"" and spaces. It is guaranteed that this string is a correct OS Pindows command line string. Print all lexemes of this command line string. Consider the character """ to be used only in order to denote a single block of characters into one command line lexeme. In particular, the consequence is that the given string has got an even number of such characters. | ['strings'] |
An identity permutation of length $$$n$$$ is an array $$$[1, 2, 3, \dots, n]$$$.We performed the following operations to an identity permutation of length $$$n$$$: firstly, we cyclically shifted it to the right by $$$k$$$ positions, where $$$k$$$ is unknown to you (the only thing you know is that $$$0 \le k \le n - 1$$$). When an array is cyclically shifted to the right by $$$k$$$ positions, the resulting array is formed by taking $$$k$$$ last elements of the original array (without changing their relative order), and then appending $$$n - k$$$ first elements to the right of them (without changing relative order of the first $$$n - k$$$ elements as well). For example, if we cyclically shift the identity permutation of length $$$6$$$ by $$$2$$$ positions, we get the array $$$[5, 6, 1, 2, 3, 4]$$$; secondly, we performed the following operation at most $$$m$$$ times: pick any two elements of the array and swap them. You are given the values of $$$n$$$ and $$$m$$$, and the resulting array. Your task is to find all possible values of $$$k$$$ in the cyclic shift operation. | ['math', 'graphs'] |
Petya got interested in grammar on his third year in school. He invented his own language called Petya's. Petya wanted to create a maximally simple language that would be enough to chat with friends, that's why all the language's grammar can be described with the following set of rules: There are three parts of speech: the adjective, the noun, the verb. Each word in his language is an adjective, noun or verb. There are two genders: masculine and feminine. Each word in his language has gender either masculine or feminine. Masculine adjectives end with -lios, and feminine adjectives end with -liala. Masculine nouns end with -etr, and feminime nouns end with -etra. Masculine verbs end with -initis, and feminime verbs end with -inites. Thus, each word in the Petya's language has one of the six endings, given above. There are no other endings in Petya's language. It is accepted that the whole word consists of an ending. That is, words "lios", "liala", "etr" and so on belong to the Petya's language. There aren't any punctuation marks, grammatical tenses, singular/plural forms or other language complications. A sentence is either exactly one valid language word or exactly one statement. Statement is any sequence of the Petya's language, that satisfy both conditions: Words in statement follow in the following order (from the left to the right): zero or more adjectives followed by exactly one noun followed by zero or more verbs. All words in the statement should have the same gender.After Petya's friend Vasya wrote instant messenger (an instant messaging program) that supported the Petya's language, Petya wanted to add spelling and grammar checking to the program. As Vasya was in the country and Petya didn't feel like waiting, he asked you to help him with this problem. Your task is to define by a given sequence of words, whether it is true that the given text represents exactly one sentence in Petya's language. | [] |
One important contest will take place on the most famous programming platform (Topforces) very soon!The authors have a pool of $$$n$$$ problems and should choose at most three of them into this contest. The prettiness of the $$$i$$$-th problem is $$$a_i$$$. The authors have to compose the most pretty contest (in other words, the cumulative prettinesses of chosen problems should be maximum possible).But there is one important thing in the contest preparation: because of some superstitions of authors, the prettinesses of problems cannot divide each other. In other words, if the prettinesses of chosen problems are $$$x, y, z$$$, then $$$x$$$ should be divisible by neither $$$y$$$, nor $$$z$$$, $$$y$$$ should be divisible by neither $$$x$$$, nor $$$z$$$ and $$$z$$$ should be divisible by neither $$$x$$$, nor $$$y$$$. If the prettinesses of chosen problems are $$$x$$$ and $$$y$$$ then neither $$$x$$$ should be divisible by $$$y$$$ nor $$$y$$$ should be divisible by $$$x$$$. Any contest composed from one problem is considered good.Your task is to find out the maximum possible total prettiness of the contest composed of at most three problems from the given pool.You have to answer $$$q$$$ independent queries.If you are Python programmer, consider using PyPy instead of Python when you submit your code. | ['math'] |
There're $$$n$$$ robots placed on a number line. Initially, $$$i$$$-th of them occupies unit segment $$$[x_i, x_i + 1]$$$. Each robot has a program, consisting of $$$k$$$ instructions numbered from $$$1$$$ to $$$k$$$. The robot performs instructions in a cycle. Each instruction is described by an integer number. Let's denote the number corresponding to the $$$j$$$-th instruction of the $$$i$$$-th robot as $$$f_{i, j}$$$.Initial placement of robots corresponds to the moment of time $$$0$$$. During one second from moment of time $$$t$$$ ($$$0 \le t$$$) until $$$t + 1$$$ the following process occurs: Each robot performs $$$(t \bmod k + 1)$$$-th instruction from its list of instructions. Robot number $$$i$$$ takes number $$$F = f_{i, (t \bmod k + 1)}$$$. If this number is negative (less than zero), the robot is trying to move to the left with force $$$|F|$$$. If the number is positive (more than zero), the robot is trying to move to the right with force $$$F$$$. Otherwise, the robot does nothing. Let's imaginary divide robots into groups of consecutive, using the following algorithm: Initially, each robot belongs to its own group. Let's sum up numbers corresponding to the instructions of the robots from one group. Note that we are summing numbers without taking them by absolute value. Denote this sum as $$$S$$$. We say that the whole group moves together, and does it with force $$$S$$$ by the same rules as a single robot. That is if $$$S$$$ is negative, the group is trying to move to the left with force $$$|S|$$$. If $$$S$$$ is positive, the group is trying to move to the right with force $$$S$$$. Otherwise, the group does nothing. If one group is trying to move, and in the direction of movement touches another group, let's unite them. One group is touching another if their outermost robots occupy adjacent unit segments. Continue this process until groups stop uniting. Each robot moves by $$$1$$$ in the direction of movement of its group or stays in place if its group isn't moving. But there's one exception. The exception is if there're two groups of robots, divided by exactly one unit segment, such that the left group is trying to move to the right and the right group is trying to move to the left. Let's denote sum in the left group as $$$S_l$$$ and sum in the right group as $$$S_r$$$. If $$$|S_l| \le |S_r|$$$ only the right group will move. Otherwise, only the left group will move. Note that robots from one group don't glue together. They may separate in the future. The division into groups is imaginary and is needed only to understand how robots will move during one second $$$[t, t + 1]$$$. An illustration of the process happening during one second: Rectangles represent robots. Numbers inside rectangles correspond to instructions of robots. The final division into groups is marked with arcs. Below are the positions of the robots after moving. Only the left of the two rightmost groups moved. That's because these two groups tried to move towards each other, and were separated by exactly one unit segment.Look at the examples for a better understanding of the process.You need to answer several questions. What is the position of $$$a_i$$$-th robot at the moment of time $$$t_i$$$? | [] |
This is an interactive problem.To prevent the mischievous rabbits from freely roaming around the zoo, Zookeeper has set up a special lock for the rabbit enclosure. This lock is called the Rotary Laser Lock. The lock consists of $$$n$$$ concentric rings numbered from $$$0$$$ to $$$n-1$$$. The innermost ring is ring $$$0$$$ and the outermost ring is ring $$$n-1$$$. All rings are split equally into $$$nm$$$ sections each. Each of those rings contains a single metal arc that covers exactly $$$m$$$ contiguous sections. At the center of the ring is a core and surrounding the entire lock are $$$nm$$$ receivers aligned to the $$$nm$$$ sections. The core has $$$nm$$$ lasers that shine outward from the center, one for each section. The lasers can be blocked by any of the arcs. A display on the outside of the lock shows how many lasers hit the outer receivers. In the example above, there are $$$n=3$$$ rings, each covering $$$m=4$$$ sections. The arcs are colored in green (ring $$$0$$$), purple (ring $$$1$$$), and blue (ring $$$2$$$) while the lasers beams are shown in red. There are $$$nm=12$$$ sections and $$$3$$$ of the lasers are not blocked by any arc, thus the display will show $$$3$$$ in this case. Wabbit is trying to open the lock to free the rabbits, but the lock is completely opaque, and he cannot see where any of the arcs are. Given the relative positions of the arcs, Wabbit can open the lock on his own. To be precise, Wabbit needs $$$n-1$$$ integers $$$p_1,p_2,\ldots,p_{n-1}$$$ satisfying $$$0 \leq p_i < nm$$$ such that for each $$$i$$$ $$$(1 \leq i < n)$$$, Wabbit can rotate ring $$$0$$$ clockwise exactly $$$p_i$$$ times such that the sections that ring $$$0$$$ covers perfectly aligns with the sections that ring $$$i$$$ covers. In the example above, the relative positions are $$$p_1 = 1$$$ and $$$p_2 = 7$$$. To operate the lock, he can pick any of the $$$n$$$ rings and rotate them by $$$1$$$ section either clockwise or anti-clockwise. You will see the number on the display after every rotation.Because his paws are small, Wabbit has asked you to help him to find the relative positions of the arcs after all of your rotations are completed. You may perform up to $$$15000$$$ rotations before Wabbit gets impatient. | [] |
And now the numerous qualifying tournaments for one of the most prestigious Russian contests Russian Codec Cup are over. All n participants who have made it to the finals found themselves in a huge m-floored 108-star hotel. Of course the first thought to come in a place like this is "How about checking out the elevator?".The hotel's elevator moves between floors according to one never changing scheme. Initially (at the moment of time 0) the elevator is located on the 1-st floor, then it moves to the 2-nd floor, then — to the 3-rd floor and so on until it reaches the m-th floor. After that the elevator moves to floor m - 1, then to floor m - 2, and so on until it reaches the first floor. This process is repeated infinitely. We know that the elevator has infinite capacity; we also know that on every floor people get on the elevator immediately. Moving between the floors takes a unit of time.For each of the n participant you are given si, which represents the floor where the i-th participant starts, fi, which represents the floor the i-th participant wants to reach, and ti, which represents the time when the i-th participant starts on the floor si.For each participant print the minimum time of his/her arrival to the floor fi. If the elevator stops on the floor si at the time ti, then the i-th participant can enter the elevator immediately. If the participant starts on the floor si and that's the floor he wanted to reach initially (si = fi), then the time of arrival to the floor fi for this participant is considered equal to ti. | ['math'] |
Pasha and Akim were making a forest map — the lawns were the graph's vertexes and the roads joining the lawns were its edges. They decided to encode the number of laughy mushrooms on every lawn in the following way: on every edge between two lawns they wrote two numbers, the greatest common divisor (GCD) and the least common multiple (LCM) of the number of mushrooms on these lawns. But one day Pasha and Akim had an argument about the laughy mushrooms and tore the map. Pasha was left with just some part of it, containing only m roads. Your task is to help Pasha — use the map he has to restore the number of mushrooms on every lawn. As the result is not necessarily unique, help Pasha to restore any one or report that such arrangement of mushrooms does not exist. It is guaranteed that the numbers on the roads on the initial map were no less that 1 and did not exceed 106. | [] |
There is unrest in the Galactic Senate. Several thousand solar systems have declared their intentions to leave the Republic. Master Heidi needs to select the Jedi Knights who will go on peacekeeping missions throughout the galaxy. It is well-known that the success of any peacekeeping mission depends on the colors of the lightsabers of the Jedi who will go on that mission. Heidi has n Jedi Knights standing in front of her, each one with a lightsaber of one of m possible colors. She knows that for the mission to be the most effective, she needs to select some contiguous interval of knights such that there are exactly k1 knights with lightsabers of the first color, k2 knights with lightsabers of the second color, ..., km knights with lightsabers of the m-th color.However, since the last time, she has learned that it is not always possible to select such an interval. Therefore, she decided to ask some Jedi Knights to go on an indefinite unpaid vacation leave near certain pits on Tatooine, if you know what I mean. Help Heidi decide what is the minimum number of Jedi Knights that need to be let go before she is able to select the desired interval from the subsequence of remaining knights. | [] |
Further research on zombie thought processes yielded interesting results. As we know from the previous problem, the nervous system of a zombie consists of n brains and m brain connectors joining some pairs of brains together. It was observed that the intellectual abilities of a zombie depend mainly on the topology of its nervous system. More precisely, we define the distance between two brains u and v (1 ≤ u, v ≤ n) as the minimum number of brain connectors used when transmitting a thought between these two brains. The brain latency of a zombie is defined to be the maximum distance between any two of its brains. Researchers conjecture that the brain latency is the crucial parameter which determines how smart a given zombie is. Help them test this conjecture by writing a program to compute brain latencies of nervous systems.In this problem you may assume that any nervous system given in the input is valid, i.e., it satisfies conditions (1) and (2) from the easy version. | ['graphs', 'trees'] |
Ilya is very fond of graphs, especially trees. During his last trip to the forest Ilya found a very interesting tree rooted at vertex 1. There is an integer number written on each vertex of the tree; the number written on vertex i is equal to ai.Ilya believes that the beauty of the vertex x is the greatest common divisor of all numbers written on the vertices on the path from the root to x, including this vertex itself. In addition, Ilya can change the number in one arbitrary vertex to 0 or leave all vertices unchanged. Now for each vertex Ilya wants to know the maximum possible beauty it can have.For each vertex the answer must be considered independently.The beauty of the root equals to number written on it. | ['math', 'graphs', 'number theory', 'trees'] |
For a connected undirected weighted graph G, MST (minimum spanning tree) is a subgraph of G that contains all of G's vertices, is a tree, and sum of its edges is minimum possible.You are given a graph G. If you run a MST algorithm on graph it would give you only one MST and it causes other edges to become jealous. You are given some queries, each query contains a set of edges of graph G, and you should determine whether there is a MST containing all these edges or not. | ['graphs'] |
A and B are preparing themselves for programming contests.B loves to debug his code. But before he runs the solution and starts debugging, he has to first compile the code.Initially, the compiler displayed n compilation errors, each of them is represented as a positive integer. After some effort, B managed to fix some mistake and then another one mistake.However, despite the fact that B is sure that he corrected the two errors, he can not understand exactly what compilation errors disappeared — the compiler of the language which B uses shows errors in the new order every time! B is sure that unlike many other programming languages, compilation errors for his programming language do not depend on each other, that is, if you correct one error, the set of other error does not change.Can you help B find out exactly what two errors he corrected? | [] |
You are given an array consisting of $$$n$$$ integers $$$a_1, a_2, \dots, a_n$$$, and a positive integer $$$m$$$. It is guaranteed that $$$m$$$ is a divisor of $$$n$$$.In a single move, you can choose any position $$$i$$$ between $$$1$$$ and $$$n$$$ and increase $$$a_i$$$ by $$$1$$$.Let's calculate $$$c_r$$$ ($$$0 \le r \le m-1)$$$ — the number of elements having remainder $$$r$$$ when divided by $$$m$$$. In other words, for each remainder, let's find the number of corresponding elements in $$$a$$$ with that remainder.Your task is to change the array in such a way that $$$c_0 = c_1 = \dots = c_{m-1} = \frac{n}{m}$$$.Find the minimum number of moves to satisfy the above requirement. | [] |
Little girl Tanya climbs the stairs inside a multi-storey building. Every time Tanya climbs a stairway, she starts counting steps from $$$1$$$ to the number of steps in this stairway. She speaks every number aloud. For example, if she climbs two stairways, the first of which contains $$$3$$$ steps, and the second contains $$$4$$$ steps, she will pronounce the numbers $$$1, 2, 3, 1, 2, 3, 4$$$.You are given all the numbers pronounced by Tanya. How many stairways did she climb? Also, output the number of steps in each stairway.The given sequence will be a valid sequence that Tanya could have pronounced when climbing one or more stairways. | [] |
It is nighttime and Joe the Elusive got into the country's main bank's safe. The safe has n cells positioned in a row, each of them contains some amount of diamonds. Let's make the problem more comfortable to work with and mark the cells with positive numbers from 1 to n from the left to the right.Unfortunately, Joe didn't switch the last security system off. On the plus side, he knows the way it works.Every minute the security system calculates the total amount of diamonds for each two adjacent cells (for the cells between whose numbers difference equals 1). As a result of this check we get an n - 1 sums. If at least one of the sums differs from the corresponding sum received during the previous check, then the security system is triggered.Joe can move the diamonds from one cell to another between the security system's checks. He manages to move them no more than m times between two checks. One of the three following operations is regarded as moving a diamond: moving a diamond from any cell to any other one, moving a diamond from any cell to Joe's pocket, moving a diamond from Joe's pocket to any cell. Initially Joe's pocket is empty, and it can carry an unlimited amount of diamonds. It is considered that before all Joe's actions the system performs at least one check.In the morning the bank employees will come, which is why Joe has to leave the bank before that moment. Joe has only k minutes left before morning, and on each of these k minutes he can perform no more than m operations. All that remains in Joe's pocket, is considered his loot.Calculate the largest amount of diamonds Joe can carry with him. Don't forget that the security system shouldn't be triggered (even after Joe leaves the bank) and Joe should leave before morning. | ['math'] |
There is a trampoline park with $$$n$$$ trampolines in a line. The $$$i$$$-th of which has strength $$$S_i$$$.Pekora can jump on trampolines in multiple passes. She starts the pass by jumping on any trampoline of her choice. If at the moment Pekora jumps on trampoline $$$i$$$, the trampoline will launch her to position $$$i + S_i$$$, and $$$S_i$$$ will become equal to $$$\max(S_i-1,1)$$$. In other words, $$$S_i$$$ will decrease by $$$1$$$, except of the case $$$S_i=1$$$, when $$$S_i$$$ will remain equal to $$$1$$$. If there is no trampoline in position $$$i + S_i$$$, then this pass is over. Otherwise, Pekora will continue the pass by jumping from the trampoline at position $$$i + S_i$$$ by the same rule as above.Pekora can't stop jumping during the pass until she lands at the position larger than $$$n$$$ (in which there is no trampoline). Poor Pekora!Pekora is a naughty rabbit and wants to ruin the trampoline park by reducing all $$$S_i$$$ to $$$1$$$. What is the minimum number of passes she needs to reduce all $$$S_i$$$ to $$$1$$$? | [] |
Ghosts live in harmony and peace, they travel the space without any purpose other than scare whoever stands in their way.There are $$$n$$$ ghosts in the universe, they move in the $$$OXY$$$ plane, each one of them has its own velocity that does not change in time: $$$\overrightarrow{V} = V_{x}\overrightarrow{i} + V_{y}\overrightarrow{j}$$$ where $$$V_{x}$$$ is its speed on the $$$x$$$-axis and $$$V_{y}$$$ is on the $$$y$$$-axis.A ghost $$$i$$$ has experience value $$$EX_i$$$, which represent how many ghosts tried to scare him in his past. Two ghosts scare each other if they were in the same cartesian point at a moment of time.As the ghosts move with constant speed, after some moment of time there will be no further scaring (what a relief!) and the experience of ghost kind $$$GX = \sum_{i=1}^{n} EX_i$$$ will never increase.Tameem is a red giant, he took a picture of the cartesian plane at a certain moment of time $$$T$$$, and magically all the ghosts were aligned on a line of the form $$$y = a \cdot x + b$$$. You have to compute what will be the experience index of the ghost kind $$$GX$$$ in the indefinite future, this is your task for today.Note that when Tameem took the picture, $$$GX$$$ may already be greater than $$$0$$$, because many ghosts may have scared one another at any moment between $$$[-\infty, T]$$$. | ['math', 'geometry'] |
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