brain

We are asked to design a system of roads on an $n times n$ grid of buildings where each road has a positive integer length and the sum of all road lengths does not exceed 48,000.

We are given a sequence of integers $B1, B2, dots, Bn$. Each value $Bi$ represents a power of two, so we can think of the underlying numbers as $Ai = 2^{Bi}$.

We are given two arithmetic progressions, $B$ and $C$. The progression $C$ contains all numbers that are common to some unknown progression $A$ and $B$. Our goal is to count how many finite arithmetic progressions $A$ could exist that satisfy this property.

The problem defines a very strong constraint on a string: every substring must have almost equal counts of all characters that appear in the full string.

We are given several independent rounds. In each round, Alice and Bob repeatedly delete contiguous parts of a string until nothing remains. The deleted characters contribute to their personal scores, where each letter has a fixed value from 1 for a up to 26 for z.

Thank you for providing the detailed sample input and output. I carefully traced the prefix-suffix logic from the previous solution, and the bug is clear. The previous approach misaligns indices when determining which friends could be thieves. Specifically: 1.

We are given a directed acyclic graph where each vertex has some number of incoming and outgoing edges. We are allowed to delete edges, but only under a very specific local rule: a vertex can lose edges only if it actually had at least one edge of that type before, unless it…

We are given a 2D grid representing a desktop with icons marked as '' and empty cells as '.'. The desktop is considered "good" if the icons occupy a compact rectangle starting from the top-left corner: all icons fill some number of complete columns, and possibly a prefix of…

We are given an array a of integers and need to simulate a two-step procedure to form a new array c. The first step repeatedly moves elements from the end of a into the middle of a second array b.

We are given a row of wall segments, each with some durability. A segment is considered destroyed once its durability drops to zero or below.

We are given a tiny language where every valid word consists of exactly two different lowercase English letters. Because the alphabet is fixed and small, every possible word can be generated by picking an ordered pair of distinct letters, such as ab, ac, ba, and so on.

The problem gives you two strings for each test case: a string s composed entirely of the letter 'a' and another string t composed of arbitrary lowercase letters. You are allowed to repeatedly replace any single 'a' in s with the string t.

We observe a sequence of visitors entering a room one after another. Exactly one of them stole a painting at some moment, but we do not know when the theft happened.

The array represents stacks of pancakes placed on a row of dishes. You are allowed to repeatedly pick a single pancake and move it one step to an adjacent dish, paying a cost of one per such move.

We are given a city structured as a tree with n houses connected by n-1 roads, which guarantees there is exactly one simple path between any two houses.

We are given a rooted tree, but the root is not explicitly provided. Instead, every node tells us its parent, and exactly one node is its own parent. That node is the root. From this structure, we must partition all nodes into several directed paths.

We are given a string made of lowercase English letters, and we are allowed to repeatedly perform a global transformation on it.

We are given several independent arrays. For each array, we are allowed to repeatedly shrink elements, where one operation picks a position and replaces its value with half of it rounded down. No swapping is allowed, so the order of elements is fixed.

We are given a situation in a pet store where there are three types of food: dog food, cat food, and universal food that can feed either dogs or cats. Polycarp owns a certain number of dogs and cats. Our task is to decide if the store has enough food to satisfy all of his pets.

We are given two parallel rows of segments, top and bottom, both of length $n$. Each segment on the top row has a wire connecting it to a specific segment on the bottom row, defined by an array $a$, where $ai$ is the bottom segment connected to top segment $i$.

We are given a collection of wires, one per index, where wire i starts at position i on the upper rail and ends at position a[i] on the lower rail.

We are given a set of boxes, each containing some number of candies. There is exactly one box for each friend, and the goal is to make all boxes contain the same number of candies so that no friend feels left out. We can only eat candies from a box; we cannot add any.

We are given a rooted tree where vertex 1 is the root. Every vertex is colored either black or white. For any vertex, its subtree consists of the vertex itself and all nodes that have it on their path up to the root, meaning all descendants in the rooted structure.

We are given an array of integers and a threshold number $k$. Our goal is to find a contiguous range of integers $[l, r]$ such that every integer in this range appears at least $k$ times in the array.

We are given a two-dimensional grid representing a chessboard where each cell contains a non-negative integer. A bishop can be placed on any cell, and it attacks all cells along the four diagonals that intersect at its position. The cell the bishop occupies counts as attacked.

We are given multiple independent scenarios, each describing a collection of candies where each candy has a fixed amount of sugar.

We are given a collection of words where each word has the same length. Our task is to measure how “far apart” any two words are in terms of the minimum number of single-letter changes needed to make them identical.

We are given a sequence of short strings, each representing a six-digit ticket number. Each ticket should be split into two halves of equal length. The task is to decide whether the sum of digits in the left half matches the sum of digits in the right half.

We are asked to simulate a dynamic seating scenario. There is a hall with n rows and m columns, and students arrive one by one, each either serious or naughty.

We are given a line of positions, and each position contributes a stack of identical gems of a specific type. Position i holds ai gems of type i.

We are given a permutation that undergoes a very specific transformation: one “operation” is a full left-to-right bubble pass where adjacent inversions are fixed once. After applying this operation exactly $k$ times, we obtain a permutation $a$.

We are given a permutation of integers from 1 to $n$, and we are asked multiple queries about contiguous subsegments of this permutation. A subsegment is called beautiful if its maximum value can be expressed as the product of two elements inside that segment.

Each color appears exactly once on two parallel tapes, so every color can be seen as a pair of positions: one index on the first tape and one index on the second tape.

We are given a permutation of integers from 1 to n and asked to count how many quadruples of indices [a, b, c, d] satisfy two inequalities: the first element is smaller than the third (pa < pc) and the second element is larger than the fourth (pb pd).

We are given a binary string of even length and asked to transform it into a "good" string. A good string is defined by two rules.

The problem asks us to turn a sequence of numbers into all zeros using a particular operation. In each operation, we pick two distinct elements. If they are equal, we replace one of them with zero. If they are different, we replace both with the smaller of the two.

We are given a binary string and allowed to flip any individual character at unit cost. After all changes, the string is interpreted by compressing it into maximal runs of identical characters.

We are given a length $n$ string of decimal digits, and a set of constraints between digits in the form of allowed adjacent swaps.

We are given a string of length n composed of the first 17 lowercase letters (a through q) and question marks. Each question mark acts as a wildcard that can be replaced by a letter from a specific set provided in a query.

The problem gives a directed graph where each vertex has a positive integer value. Masha can place a coin on any vertex, and then move it along the graph edges exactly $k-1$ times. Every time the coin visits a vertex, the vertex’s number is recorded in a notebook.

We are asked to determine how many buses a fleet could have given the total number of wheels. Each bus comes in one of two types: two-axle buses with 4 wheels and three-axle buses with 6 wheels.

We are given an array of integers and a sequence of queries. Each query either replaces a single element in the array with a new value or replaces every element in the array with the same value. After each query, we need to report the sum of the array.

We are asked to simulate a sequence of operations on an $n times n$ chessboard involving rooks. Each rook attacks its entire row and column.

We are asked to construct arrays called beautiful arrays. A beautiful array is one in which the number of elements equal to the minimum value falls between two given bounds, $l1$ and $r1$, and the number of elements equal to the maximum value falls between two bounds, $l2$ and…

We are asked to find a lenient vertex cover for a connected undirected graph. A normal vertex cover is a set of vertices such that every edge touches at least one vertex in the set.

We are given a sequence of movements of a dog along an infinite line. Each minute, the dog moves a certain distance. Positive numbers move it to the right, negative to the left, and zero indicates unknown movement where we can pick any integer in the range $[-k, k]$.

We are given a $2 times n$ board where some cells contain chips, represented by '', and others are empty, represented by '.'. Our task is to move chips so that exactly one chip remains on the board.

We are given a small rectangular grid, with each cell either empty or containing a robot. Robots can move simultaneously in one of four cardinal directions, and if any robot tries to leave the grid, it explodes.

We are given a binary string and we are allowed to cut it in a very specific way: we choose some prefix and suffix to remove, leaving a single contiguous substring in the middle. The remaining substring is what we “keep”, while everything outside it is considered removed.

Two players each hold a multiset of integers. They play a turn-based game where the only rule is that every newly played number must be strictly larger than the previous one.

We are given a tree with $n$ vertices. Each edge has an integer label. For any pair of vertices $v$ and $u$, we define $f(v, u)$ as the number of edge labels that appear exactly once along the unique path connecting $v$ and $u$.

We are given a labyrinth structured as concentric layers in an $n times n$ grid. Each layer is a contiguous set of cells surrounding the previous layer. The first layer is just the bottom-left corner.

We are given an integer x and a target length n. We can repeatedly choose any digit y from the current number x and multiply x by y. The goal is to make x have exactly n digits using the minimum number of such operations.

We are given two arrays of the same length, a and b. We can perform a special swap operation: choose any two positions i and j, and simultaneously swap a[i] with a[j] and b[i] with b[j].

We are given a deck of n cards with distinct integer values, ordered from top to bottom. We then perform m shuffle operations, each defined by a number bj.

We are asked to construct a tree on n nodes arranged in a circle. Each node has a requirement: its degree must be even or odd, depending on the corresponding character in a binary string s.

We are given a sorted array a and an integer array b. Each query gives a segment [l, r], and we only care about it if the sum of b over this segment is zero. This condition guarantees that total “supply” and “demand” inside the segment balance perfectly.

We are given a permutation of integers from $1$ to $n$ and a set of $m$ swaps, which are guaranteed to be the minimum swaps required to sort the permutation. These swaps have been shuffled arbitrarily.

We are given a permutation of the integers from 0 to n−1, and we are allowed to rearrange it using a very unusual swap rule. A swap between two positions is only allowed when the bitwise AND of the two values currently stored at those positions equals a chosen value X.

We are given a string that is already a palindrome, meaning it reads the same from left to right and right to left. For each position in this string, we imagine removing exactly one character and then ask whether the remaining string is still a palindrome.

We are given several test cases. Each test case provides a multiset of numbers, and we are allowed to rearrange them in any order we want. After choosing an ordering, we look at two sequences: the chosen array itself, and its reverse.

We are given a binary string, which is a sequence of '0's and '1's. Our task is to partition this string into contiguous substrings in such a way that, if we interpret each substring as a binary number and sum them all, the result is a power of two.

We are given a multiset of positive integers, and we are told that this multiset was produced by repeatedly running Euclid’s algorithm on several unknown integer pairs.

We are given a sequence of traps, each with a base damage value. We traverse the traps in order, and for each trap we either take its damage or skip it by jumping over it.

We are given an array and several queries, where each query describes a segment of indices. The requirement is that inside every given query segment, all values must be pairwise distinct. If the array already satisfies this condition for all segments, we do nothing.

We are given an array of non-negative integers and we are allowed to perform up to k replacements, where each replacement changes any element to any non-negative value. After these edits, we evaluate the array using two quantities.

We are asked to construct three positive integers $x, y, z$ given three constraints on remainders. Each constraint relates one number to another through a modulo operation: the remainder when dividing $x$ by $y$ must equal $a$, the remainder when dividing $y$ by $z$ must equal…

We are given a matrix where each row represents a sequence of numbers arranged across columns. The goal is to make every row individually non-decreasing from left to right, but we are only allowed a single global operation: swap two entire columns.

We are given an integer with no zero digits, and two players, Alice and Bob, play a turn-based game on its digits. Alice always moves first and can swap any two digits at different positions. Bob always removes the last digit of the number.

We are asked to transform one permutation into another in a way that minimizes a specific “weight” function. The weight is the sum of absolute differences between each element of the new permutation and the element of the original permutation located at the next position in…

We are given a permutation of size $2n+1$, and this array keeps changing through swaps of two positions. After every swap, we are not asked to analyze the array itself directly, but to consider all its cyclic rotations and determine whether at least one rotation has a…

We are given a permutation p of the numbers from 1 to n, and our task is to construct another permutation q of the same numbers that minimizes a specific weight function.

We are asked to arrange a set of integers on a circle so that each number is either strictly larger or strictly smaller than both of its neighbors. The input consists of multiple test cases. Each test case provides the number of integers, followed by the integers themselves.

The solution addresses the exercise directly. It identifies the event $f(n)-f(n-1) = k$ as the occurrence of exactly $k-1$ consecutive terms outside the interval $[alpha,beta)$ followed by one term inside, which correctly models the "gap" between hits in the interval.

We are given a sequence of parentheses of length $2n$, containing exactly $n$ opening and $n$ closing brackets. The task is to transform this sequence into a correct balanced parentheses string using the minimum number of operations, where each operation consists of reversing…

We are given a permutation of the numbers from 1 to n. We may split this permutation into any number of consecutive pieces. Each piece is a subarray. For every subarray, we look at its inversion count. A subarray is called odd if its inversion count is odd.

We are given several independent test cases. Each test case starts with an array of integers. The only operation allowed transforms the array in a very specific way: we pick exactly one element to leave untouched and replace every other element by the average of the chosen group.

The solution addresses the exercise directly. It identifies the event $f(n)-f(n-1) = k$ as the occurrence of exactly $k-1$ consecutive terms outside the interval $[alpha,beta)$ followed by one term inside, which correctly models the "gap" between hits in the interval.

For every integer, look at the two consecutive perfect squares surrounding it. If $$m^2 le x < (m+1)^2,$$ then $g(x)=m^2$ and $f(x)=(m+1)^2$. The number is called cute when it is strictly closer to the lower square than to the upper square.

We are given a sequence of integers a and a number v initialized to 1. The task is to perform a series of operations on v so that it becomes the greatest common divisor of all pairwise products of elements from a.

The solution addresses the exercise directly. It identifies the event $f(n)-f(n-1) = k$ as the occurrence of exactly $k-1$ consecutive terms outside the interval $[alpha,beta)$ followed by one term inside, which correctly models the "gap" between hits in the interval.

We are asked to compute the maximum number of mushrooms Marisa can collect in a one-dimensional forest over a limited number of minutes.

We are given a railway network consisting of n stations and m bidirectional tracks, each with a positive length. The network may not be fully connected and can have multiple tracks between the same pair of stations.

We are asked to reconstruct the initial string of length 1 from which a sequence of operations produced a given final string. Each operation consists of selecting a substring of the current string and replacing it with another string, possibly of different length.

We are given a positive integer $x$, and the task is to find the smallest positive integer $y$ such that two bitwise conditions hold simultaneously: the bitwise AND of $x$ and $y$ is greater than zero, and the bitwise XOR of $x$ and $y$ is also greater than zero.

We are asked to transform a collection of magical tokens so that every token has an odd magical power. Each token starts with some positive integer power.

We are given a permutation $p$ of the numbers $1$ through $n$. We want to construct another permutation $q$ using the same numbers such that every position changes. For every index $i$, the value placed in $qi$ must be different from $pi$.

We are given an array of non-negative integers. Each array element corresponds to a vertex in a graph. Two vertices are connected by an edge whenever the bitwise AND of their values is positive. The graph is not guaranteed to be connected.

We are given a grid of size $n times m$, where each cell is either black or white. Our task is to pick a cell such that the maximum Manhattan distance from this cell to any black cell is minimized.

We are given a binary tree rooted at vertex 1. Each vertex has at most three neighbors, except the root which has at most two. Initially, only the root is infected.

We are given two strings, a and b, which do not share any letters. We can build a new string c by repeatedly taking the smallest available letter from either a or b. However, there is a restriction: we cannot take more than k characters from the same string consecutively.

Each carriage has its own maximum speed. When all carriages start moving, a carriage cannot move faster than any carriage in front of it, so its actual speed becomes the minimum value seen so far from the left.

We are given two increasing arrays. The array s contains the arrival time of each task. Task i becomes available at time s[i]. The array f contains the completion time of each task. Task i finishes exactly at time f[i]. Polycarp processes tasks in FIFO order.

We are given a string s of length n and a permutation p of the integers from 1 to n. Each permutation p defines a reordering operation: after one application, the character at position i in the new string moves to position p[i].

We are given an even number of goods, each with a weight, and we need to pack them into pairs. The cost of a pair is calculated by taking the sum of its weights, dividing by a fixed number $k$, and rounding down to the nearest integer.

We are given a stripe of tiles, each either white or black, and the goal is to ensure that at least one segment of exactly k consecutive tiles is entirely black.

We are given two arrays of non-negative integers, a and b, both of length n. The allowed operation is to simultaneously decrement all positive elements of a by one.

We are given a number of identical building blocks and must split them into three stacks representing a podium. Each stack corresponds to a rank: third place, second place, and first place.

We are given a binary string of length $n$ and we want to minimize a sum computed from all consecutive pairs of digits. Each pair of digits $si s{i+1}$ is treated as a decimal number, so "10" counts as ten, "01" as one, and so on.

We are given an undirected tree with $n$ vertices. For every choice of a root $r$, the tree becomes rooted. For every $k$-element vertex set $S$, we look at the smallest rooted subtree that contains all vertices of $S$. A rooted subtree is not just any connected subgraph.