brain

We are given two strings, s and t, of equal length composed only of the letters "a" and "b". The task is to make s and t identical using a sequence of allowed swap operations. Each operation lets us pick one character from s and one from t and swap them.

In this problem, we have two football teams playing a match. Each team starts with a certain number of players, denoted by a1 and a2. Throughout the match, the referee shows n yellow cards to players.

We have a line of rooms numbered from 1 to n. Every room must end up connected to the Internet. There are two ways to achieve that. A room can be connected directly, paying a cost equal to its index. Connecting room i directly costs i.

The sequence is built by concatenating “blocks”. The first block is just the string “1”. The second block is “12”.

We are given three rectangles on a plane aligned with the axes. The first rectangle is a white sheet, and two subsequent rectangles are black sheets. Each rectangle is defined by its bottom-left and top-right coordinates.

We are given a line of cans, each with a durability value. Vasya has to shoot all cans exactly once, but the number of shots required to knock down a can depends on how many cans have already been knocked down.

We are given a binary string and asked to count its "good" substrings. A substring is good if its length equals the decimal value of its binary representation, allowing leading zeros.

We are given a string s and we need to predict the outcome of a two-player game for each starting position k in the string. The game begins with a substring consisting of a single character at position k.

We have an undirected graph. Every vertex must be assigned either 0 or 1. For each edge, we write the sum of the values on its two endpoints.

We are given several independent game states. Each state consists of tiles whose values are powers of two. In one move, we may pick two tiles with the same value and merge them into a single tile whose value is their sum.

We have a collection of ticket prices, and we are free to choose the order in which tickets are sold. Two donation programs exist. Every a-th sold ticket contributes x% of its price, and every b-th sold ticket contributes y% of its price.

We have a set of $n$ positive integers on a blackboard and a number $k ge 2$. None of the integers is divisible by $k$. We are allowed to repeatedly pick any two numbers $x$ and $y$, erase them, and write $f(x+y)$ instead.

We are given the TV schedule for the next n days. Each day broadcasts an episode from exactly one show, represented by an integer. A subscription is purchased per show, not per episode. If we subscribe to show x, we can watch every occurrence of x in the schedule.

We are asked to compute a product over a sequence of numbers derived from a pair of integers $x$ and $n$. The function $f(x, y)$ looks at all prime factors of $x$ and for each such prime $p$ determines the largest power of $p$ that divides $y$.

We are given a sequence x of length m. For every value i from 1 to n, we construct a special permutation: pi = [i, 1, 2, ..., i-1, i+1, ..., n] This permutation differs from the identity permutation only because the value i has been moved to the front.

We are simulating the behavior of a smartphone chat interface. The smartphone screen can show at most $k$ conversations. Each incoming message comes from a friend identified by a unique ID. If a conversation with that friend is already on the screen, the screen does not change.

We are given a tree and need to find the largest connected subgraph that can be represented as an intersection graph of line segments on a line.

We are given two integers, x and y, with x y. The task is to determine whether it is possible to choose a single prime number p and repeatedly subtract it from x until we reach y. Importantly, once a prime is chosen, it must be used exclusively; we cannot switch primes mid-way.

We are asked to count the number of "random pictures" of size $n times m$, where each cell is either black or white. A picture is considered random under Ivan's definition if every cell has at most one adjacent cell of the same color. Here, adjacency is horizontal or vertical.

We are asked to arrange soldiers of various heights into a parade formation with exactly $k$ rows. Each row must have the same number of soldiers, and within a row, no two soldiers can differ in height by more than one.

We are tracking whether a single employee, Randall, remains employed after a sequence of yearly “pruning” operations in a company where employees are ranked by a fixed performance value. The company always keeps exactly $N$ employees.

We are given a tree with $N$ junctions and $N-1$ roads connecting them. A tree means that every pair of junctions is connected by exactly one path.

We are given a permutation of integers from 1 to N, which represents Danang's completed homework. Darto wants to submit his own permutation, different enough from Danang's, but still using numbers 1 through N exactly once.

There is not enough information to diagnose the bug or provide a corrected solution. The sample with expected output appears to belong to a string construction problem, but you have not provided: - the problem statement, - the original Python solution, - or even the name of…

We are given a list of sweets, each with a positive integer value representing its sweetness level. Yui wants to eat exactly $k$ sweets for each $k$ from 1 to $n$. However, she cannot eat more than $m$ sweets in a single day. Time is divided into days starting from day 1.

We are asked to rearrange chocolates among boxes so that Alice will be happy. Each of the $n$ boxes has either zero or one chocolate, and at least one box contains a chocolate.

There is not enough information to diagnose the bug or provide a corrected solution. The sample with expected output appears to belong to a string construction problem, but you have not provided: - the problem statement, - the original Python solution, - or even the name of…

We are asked to determine whether a target sum S can be constructed using two types of coins. One type has value n and there are a coins of it, the other type has value 1 and there are b coins of it.

The input does not give the number directly. Instead, it gives its prime factorization as a list of prime factors, where equal primes may appear many times. Suppose a prime $q$ appears $c$ times. Then every divisor chooses an exponent between $0$ and $c$ for that prime.

We have three participants and a set of problems numbered from 1 to n. Each participant currently holds some subset of problems, but the subsets may not respect the desired division: the first participant should hold a prefix of the problems, the third participant should hold…

We have a line of students numbered from 1 to n. Among them, two students are rivals, located at positions a and b. The gym teacher wants to maximize the distance between these two students, where distance is simply the absolute difference of their positions.

We are given a tree where each vertex does not have a fixed color. Instead, every vertex has a range of possible colors, and we imagine choosing one integer color independently for each vertex within its allowed interval.

There is not enough information to diagnose the bug or provide a corrected solution. The sample with expected output appears to belong to a string construction problem, but you have not provided: - the problem statement, - the original Python solution, - or even the name of…

We are given a house with n rooms. For each room, we know two numbers: ci, the maximum number of radiators we can install, and sumi, the total number of radiator sections required to adequately heat that room.

We have two independent electrical grids in a palace, each forming a rooted tree with node 1 as the head. The trees supply electricity to devices through their leaves.

We are given three piles of candies: red, green, and blue. Each pile has a certain number of candies, represented by integers $r$, $g$, and $b$. Every day, Tanya eats exactly two candies, but they must be of different colors.

We are given a small arithmetic sequence of positive integers: $a, a+d, a+2d, dots, a+(n-1)d$. The goal is to find another arithmetic sequence $b, b+e, b+2e, dots, b+(n-1)e$ such that the last 18 digits of the corresponding Fibonacci numbers contain the original numbers as…

There is not enough information to diagnose the bug or provide a corrected solution. The sample with expected output appears to belong to a string construction problem, but you have not provided: - the problem statement, - the original Python solution, - or even the name of…

There is not enough information to diagnose the bug or provide a corrected solution. The sample with expected output appears to belong to a string construction problem, but you have not provided: - the problem statement, - the original Python solution, - or even the name of…

We are given several independent strings, each consisting of the characters a, b, c, and ?. The task is to replace every ? with one of the letters a, b, or c so that the final string has no two equal adjacent characters.

The problem asks us to figure out whether it is possible to build a vertical tower of standard six-sided dice such that the sum of all visible pips equals a given integer. We have an unlimited number of dice, and for each die, we can choose its orientation.

Before I dive into writing the full editorial, I want to clarify one thing: Codeforces Problem 1267F, Foolprüf Security, involves a graph with weighted edges and some combinatorial security constraint.

There is not enough information to diagnose the bug or provide a corrected solution. The sample with expected output appears to belong to a string construction problem, but you have not provided: - the problem statement, - the original Python solution, - or even the name of…

We are given a row of colored balls represented as a string of uppercase letters. The task is to insert a single ball of any color at any position, including at the ends, so that after the insertion, a chain reaction of eliminations occurs until no segment of length three or…

The Young diagram can be viewed as a histogram whose column heights are given by a non-increasing array $a1,a2,dots,an$.

We have two arrays of equal length. We may choose a single non-negative value x, add it to every element of the first array, and take all results modulo m. After this transformation, the order of elements does not matter.

There is not enough information to diagnose the bug or provide a corrected solution. The sample with expected output appears to belong to a string construction problem, but you have not provided: - the problem statement, - the original Python solution, - or even the name of…

Each test case describes a single student group that must be split into two subgroups. Every subgroup has three independent “resource limits”: an auditorium limit for maths attendees, a computer lab limit for programming attendees, and a gym limit for PE attendees.

There is not enough information to diagnose the bug or provide a corrected solution. The sample with expected output appears to belong to a string construction problem, but you have not provided: - the problem statement, - the original Python solution, - or even the name of…

We are working on a grid where movement is restricted to four directions, so distance between two points is measured using Manhattan distance. A school is fixed at one coordinate, and each student lives at another coordinate on the same infinite grid.

We are given two sequences of brackets, s and t, which may individually be invalid as bracket sequences. The task is to produce the shortest string of brackets that is valid (balanced and properly nested) and contains both s and t as subsequences.

There is not enough information to diagnose the bug or provide a corrected solution. The sample with expected output appears to belong to a string construction problem, but you have not provided: - the problem statement, - the original Python solution, - or even the name of…

There is not enough information to diagnose the bug or provide a corrected solution. The sample with expected output appears to belong to a string construction problem, but you have not provided: - the problem statement, - the original Python solution, - or even the name of…

There is not enough information to diagnose the bug or provide a corrected solution. The sample with expected output appears to belong to a string construction problem, but you have not provided: - the problem statement, - the original Python solution, - or even the name of…

We are asked to simulate a massive array defined recursively using C-style macros. The macros expand in a very structured way: each macro level quadruples the number of elements generated by the previous level, with simple offsets added.

We are given a string s consisting of lowercase English letters, and we want to count all distinct substrings that appear in s or in any version of s where exactly one character has been replaced by an asterisk.

We are given a connected undirected graph of cities. Among all cities, two special nodes are distinguished, call them a and b, representing two fairs.

There is not enough information to diagnose the bug or provide a corrected solution. The sample with expected output appears to belong to a string construction problem, but you have not provided: - the problem statement, - the original Python solution, - or even the name of…

We are given a collection of binary strings, and we are allowed to optionally reverse some of them. After doing so, we want to arrange all strings in a single sequence such that every adjacent pair is compatible: the last character of a word must match the first character of…

There is not enough information to diagnose the bug or provide a corrected solution. The sample with expected output appears to belong to a string construction problem, but you have not provided: - the problem statement, - the original Python solution, - or even the name of…

We are given two integers, and we are allowed to repeatedly perform operations where the cost of the k-th operation is exactly k. Each operation lets us choose one of the two numbers and add the current operation index to it.

We are given a sequence of durations, where each value represents how long Vasya needs to recite one part of a verse. He must recite the parts in order, from left to right, and he earns a reward equal to how many parts he manages to fully complete before time runs out.

We are asked to rearrange a scrambled 2-row grid of alien organs into a prescribed order. The grid has dimensions 2 × (2k + 1) with 4k + 1 organs numbered 1 through 4k + 1 and exactly one empty cell.

The process starts with a short string made only of digits 1, 2, and 3. Alongside this string, we maintain a cursor that moves from left to right over positions between characters, and a clipboard that can store a copied suffix of the string.

We are given a person moving along a straight line segment from position a to position b at constant speed 1 unit per minute. Independently, there is a network base station at position c that covers everything within distance r.

We are given a directed assignment problem on $n$ people where each person must end up choosing exactly one other person to give a gift to, and at the same time must receive exactly one gift from someone else.

We need the total number of framed segments across all permutations of length n. For a fixed permutation, a segment [l, r] is framed when the values inside it form a set of consecutive integers.

There is not enough information to diagnose the bug or provide a corrected solution. The sample with expected output appears to belong to a string construction problem, but you have not provided: - the problem statement, - the original Python solution, - or even the name of…

We have an array representing the tastiness of cupcakes, and two players. Yasser always takes all the cupcakes, summing up the full array. Adel can pick any contiguous subarray that does not cover the entire array.

We are asked to maintain a string S and an integer array W dynamically as queries arrive. Each query adds a new character to S and a weight to W.

We are given a row of n bulbs, each numbered from 1 to n. Some bulbs are missing, indicated by zeros in the input array.

There is not enough information to diagnose the bug or provide a corrected solution. The sample with expected output appears to belong to a string construction problem, but you have not provided: - the problem statement, - the original Python solution, - or even the name of…

We need to count pairs of arrays $(a,b)$, both of length $m$, whose values lie between $1$ and $n$. Array $a$ must be non-decreasing, array $b$ must be non-increasing, and at every position we require $ai le bi$. The answer can be very large, so we output it modulo $10^9+7$.

For each test case, we have a deadline of n days and a program that normally needs d days to finish. Before running the program, we may spend x days optimizing it.

We have a line of n lamps, each either on or off. The goal is to compute, for every prefix of the lamps, the minimum number of subset toggle operations required to turn all lamps in that prefix on.

We are given a collection of n distinct toy boxes, each labeled with a positive integer ai. ROBO can perform a very specific action multiple times: pick three distinct boxes where one box's label divides the other two, and then remove the third box to place it on his pile.

We are given a 2-row by $n$-column grid representing a maze. The player starts at the top-left cell $(1,1)$ and wants to reach the bottom-right cell $(2,n)$. Movement is allowed only between orthogonally adjacent cells. Initially, all cells are passable ground.

There is not enough information to diagnose the bug or provide a corrected solution. The sample with expected output appears to belong to a string construction problem, but you have not provided: - the problem statement, - the original Python solution, - or even the name of…

We are given an $n times m$ matrix. The target matrix is completely fixed: row-major order from $1$ to $n cdot m$. The value that should end up in position $(r,c)$ is $$(r-1)cdot m + c.$$ We may perform two kinds of operations.

For each test case, we are given a single integer n. We must determine whether n can be written as the product of three distinct integers a, b, and c, where each of them is at least 2. If such a decomposition exists, we print "YES" and one valid triple.

We are asked to decide whether a given number of coins can be distributed among three sisters so that they all end up with the same total number of coins. Each sister already has some coins: Alice has a, Barbara has b, and Cerene has c.

We are given a contest with n problems, where each problem i has a range [li, ri] representing the possible number of accepted solutions. Each integer in this range is equally likely, and selections for different problems are independent.

There is not enough information to diagnose the bug or provide a corrected solution. The sample with expected output appears to belong to a string construction problem, but you have not provided: - the problem statement, - the original Python solution, - or even the name of…

We are given two strings, s and t. Initially, we have an empty string z, and we want to transform z into t. The allowed operation is to append any subsequence of s to z. A subsequence is formed by selecting characters from s without changing their relative order.

We have a string of lowercase letters. Every position must be assigned one of two colors, 0 or 1. After choosing the colors, we are allowed to swap neighboring characters only when their colors are different. We may perform such swaps as many times as we want.

There is not enough information to diagnose the bug or provide a corrected solution. The sample with expected output appears to belong to a string construction problem, but you have not provided: - the problem statement, - the original Python solution, - or even the name of…

We have an array of integers. An operation allows us to pick two different positions and copy the value from one position into the other. Since we may repeat this operation any number of times, values can be duplicated throughout the array.

There is not enough information to diagnose the bug or provide a corrected solution. The sample with expected output appears to belong to a string construction problem, but you have not provided: - the problem statement, - the original Python solution, - or even the name of…

The task is to convert a raw number of likes into a compact, human-readable format for display. Numbers less than a thousand remain as-is, numbers in the thousands are rounded to the nearest thousand with a 'K' suffix, and numbers in the millions are rounded to the nearest…

We are given a strictly convex polygon $P$ in the plane as a list of its vertices in counterclockwise order. Each vertex has integer coordinates, and no three vertices are collinear.

We are given an array of integers and want to make two properties simultaneously true: the sum of the elements must not be zero, and the product of the elements must not be zero. The only operation allowed is incrementing a single element by one.

We are given a rectangular grid where every cell represents a city, and each city has one of a small number of colors.

There is not enough information to diagnose the bug or provide a corrected solution. The sample with expected output appears to belong to a string construction problem, but you have not provided: - the problem statement, - the original Python solution, - or even the name of…

We are given a sequence of length n. Although the statement talks about a string, the characters are represented by integers, and the alphabet is extremely large. The large alphabet size means we cannot rely on tricks that depend on a small number of distinct symbols.

There is not enough information to diagnose the bug or provide a corrected solution. The sample with expected output appears to belong to a string construction problem, but you have not provided: - the problem statement, - the original Python solution, - or even the name of…

We are asked to maintain an array of integers that starts with all zeros and answer a series of queries. Each query is either an assignment, setting a specific element to a value, or a range sum query, asking for the sum of a contiguous subarray.

We are given a binary string, a sequence of 0's and 1's, and the goal is to manipulate it so that all the 1's appear in one contiguous block. The only allowed operation is to remove some 0's.

We are given a tree with $n$ vertices, where each vertex is connected such that there is exactly one path between any two vertices. Then we are asked multiple queries.

Yes, the inequality $nu(n) le 2^{l(n) - lambda(n)}$ holds for all positive integers $n$. Consider an addition chain of minimal length $l(n)$ and let $lambda(n)$ be the length of a shortest chain consisting only of doubling steps.

The algorithm itself is correct. The bug is in the editorial's test harness code, not in the algorithm. In the posted solution, the pairs initialization was accidentally split across lines: That formatting artifact can cause issues when copied, and more importantly, the helper…