brain

We are asked to simulate a treasure hunt on an unknown undirected graph. Each vertex has a treasure and a flag. We start at a known vertex, and at each move, we are shown the degrees of the neighbors and whether they already have a flag.

We are placed inside an unknown connected undirected graph. We start at a known vertex, but after that we never see vertex IDs again in a useful way. Every time we arrive at a vertex, the interactor shows us only its degree and a list of its neighbors in random order.

We are asked to simulate a treasure hunt on a hidden, undirected graph. Each vertex represents a junction with a treasure, and each edge is a road connecting junctions.

This task is an interactive graph exploration problem. We start from a known vertex of a connected undirected graph. Every vertex contains a treasure, and the first time we visit a vertex we automatically mark it with a flag.

The problem presents a treasure hunt on an undirected graph where each vertex represents a junction and each edge represents a road connecting junctions. You start at a given vertex, and every time you visit a vertex for the first time, you dig up a treasure and place a flag.

We are given a treasure-hunting scenario mapped onto an undirected graph. Each vertex represents a junction containing a treasure, and edges represent roads connecting the junctions. The hunter starts at a specific vertex and can move along edges to adjacent vertices.

We are given a collection of small sets, each set contains exactly $m$ distinct integers and a weight attached to that set. The task is to choose two different sets and combine all numbers from both sets.

I can do this, but a complete editorial for Codeforces 1641F at rating 3500 would be extremely long, and the requested sections plus a fully correct implementation and detailed traces would exceed the response length limit. A practical way to proceed is to split it into parts.

This problem is rated 3500 and its solution relies on a fairly deep structural characterization of graphs whose cycle space admits a consistent cyclic orientation.

I can write the editorial, but a complete response matching every section you requested, including a full proof, worked examples, implementation explanation, edge-case analysis, and test suite, would be extremely long.

We have a line of n people. Every person is either sick or healthy. As information arrives, we receive two kinds of statements. A statement with x = 0 says that nobody in the interval [l, r] is sick. Every person in that range becomes definitely healthy.

We are given an integer array and allowed to perform an operation that inserts two identical values next to each other anywhere in the array.

Assign coordinates to the white squares by declaring that a white square has coordinates $(x,y)$ with $x+y$ even and $y\ge 0$.

This problem is rated 3500 and its solution relies on a fairly deep structural characterization of graphs whose cycle space admits a consistent cyclic orientation.

We are given a multiset of values, where each value represents a type of power-up. The task is not to assign individual power-ups arbitrarily, but to distribute all of them into exactly $k$ non-empty groups, where each group represents a child.

We are given an $n times n$ grid, with a robot starting at the top-left corner $(1,1)$. The robot can move right (R) or down (D) and is provided a sequence $s$ of such moves. Each move in $s$ is guaranteed not to take the robot outside the grid.

We are asked to cover all possible arrays of length n whose elements range from 1 to k using a sequence of arrays, where each array in the sequence can "generate" others through two operations.

We need to construct permutations of the numbers from 1 to n such that no element starting from the third position equals the sum of the previous two elements.

We are given a grid that starts completely uncolored, with $n$ rows and $m$ columns. We then apply a sequence of operations. Each operation selects a row and a column, and paints every cell in that row and that column with a chosen non-white color.

We are given an array of integers and a number $x$. The task is to compute, for each possible $k$ from 0 to $n$, the maximum sum of a contiguous subarray if we are allowed to add $x$ to exactly $k$ elements in the array.

The problem describes a narrow hallway with three doors and three keys, each uniquely colored red, green, or blue. The knight starts at the left end of the hallway, and the princess waits at the far right.

We are given a number and asked to express it as a sum of distinct “building blocks”. Each building block must be either a power of two or a factorial value.

We are given a sequence of length $n+1$. Each element is either a “small” number in the range $[0, n-1]$, or a special large value equal to $n^2$. We are told only two things: the parameter $n$ and the total sum of all elements in the sequence, denoted $s$.

We are given a sequence of non-negative integers, and we can paint each element red, blue, or leave it unpainted.

We are given a permutation-like array of distinct positive integers. The task is to split its elements into two subsequences so that each subsequence is “hill-shaped”: it strictly increases up to a single peak and then strictly decreases.

We are given a number x that is already divisible by d. A number is called good if it is divisible by d. A number is called beautiful if it is good, but cannot be split into a product of two good numbers.

We have a classroom with n desks, each initially occupied by a student with a number from 1 to n in some unknown order. After each lesson, every student moves to a desk according to a permutation p.

We are given an initially empty binary grid filled with zeros, and we want to transform it into a target 0-1 pattern.

We are given a binary grid. A black subrectangle is any axis-aligned rectangle consisting entirely of 1s. Among all black rectangles, a rectangle is called nice if it cannot be extended into a larger black rectangle that contains it.

We are asked to construct the largest number possible whose digits add up to a given sum n, under two restrictions: the number cannot contain zero, and no two consecutive digits can be equal. Each test case provides a different sum n, and we must produce a valid number for each.

We are given a multiset of positive integers, and we must decide whether it is closed under a very specific operation: taking integer division between any ordered pair of elements where the numerator is at least the denominator.

I can't write a correct editorial and solution for Codeforces 1648F from the problem statement alone. This is a 3500-rated problem whose accepted solution relies on several deep graph-theoretic facts about bridges, edge 3-connectivity structure, randomized XOR labels on DFS…

We start with a connected undirected graph. The graph represents Berlaflot flights, and every edge has a price. The cost of traveling between two cities is unusual. A route may contain many flights, but its cost is not the sum of edge weights.

This problem is rated 3500 and its solution relies on a fairly deep structural characterization of graphs whose cycle space admits a consistent cyclic orientation.

We are given a 2D grid of size $n times m$ where each cell contains a color represented by an integer. The task is to compute the sum of Manhattan distances between every pair of cells that share the same color.

We are given two sequences of integers, s and t, representing letters of two strings. Each integer corresponds to a distinct letter, and equal integers in s and t denote the same character.

We are asked to determine the minimum number of balls that could have been used in a football passing game given only the number of passes each player made.

We are given a linear sequence of locations, each either land or water, starting and ending with land. You begin at the first location and need to reach the last one.

We have a sequence of course assignments whose deadlines are already sorted. Assignment i must be fully completed by absolute time a[i]. Initially every assignment has 0% progress. There are m available training options. Option j belongs to exactly one assignment e[j].

We are asked to count paths in an undirected graph from a start vertex s to a target t whose lengths are either exactly the shortest distance or exceed it by at most one. The graph has n vertices and m edges, with no loops or multiple edges.

We are given a sequence of exam days for Dmitry, each day numbered from 1 to $d$. He can rest some days before each exam, and the key measure of schedule quality is $mu$, the minimum rest between consecutive exams (or between the start of the session and the first exam).

We are given a string of odd length and a target character. The string can be shortened repeatedly by removing any two adjacent characters at a time. The question is whether it is possible to perform a sequence of such deletions so that only the target character remains.

We are given a permutation of numbers from $1$ to $n$, initially in increasing order. Petya performed $n$ operations to transform it. In the $i$-th operation, he cyclically shifted the first $i$ elements any number of times to the right.

We are given points on a number line. Every point has a coordinate and a weight. We want to choose exactly $2n$ of these points and use them as endpoints of $n$ segments. The segments must form a perfectly nested structure.

We are asked to maximize a function defined as $fa(x) = lfloor x / a rfloor + (x bmod a)$ over a range of integers from $l$ to $r$, where $a$ is a fixed positive integer.

We are given a bipartite graph with n vertices on each side. The vertices on the left are numbered 1 to n and the right n+1 to 2n. Each vertex has degree exactly 2, so every vertex is connected to exactly two vertices on the other side.

We have a one-dimensional tower defense level with n towers lined up along the axis from position 1 to n. Each tower has two attributes: a mana capacity ci and a regeneration rate ri.

We are given a set of distinct integer lattice points on a 2D grid. For each of these points, we want to “escape” to the nearest grid point that is not part of the input set, where distance is measured in Manhattan metric.

We need to construct an array of length n such that applying the given operation to any pair of positions never decreases the total sum of the array. Suppose we choose two values x and y.

We have a network of computers arranged in two rows, each containing n computers. Each computer has an associated grade, represented as an integer. Computers in the same row are initially connected to their immediate neighbors, forming two independent chains.

We are simulating a knockout tournament where the number of players is a power of two. Players are labeled from 1 up to $2^n$, and they repeatedly face off in rounds until only one remains.

We want to change as few array elements as possible so that the final array becomes an arithmetic progression. An arithmetic progression is completely determined by two parameters: its first value and its common difference.

We are given the final weights of n cake pieces. These pieces were produced from a single initial cake by repeatedly choosing a piece of weight w and splitting it into two parts: - floor(w / 2) - ceil(w / 2) Exactly n - 1 such cuts were performed, resulting in n final pieces.

We are given a line of $n$ cakes, each with a weight $ai$. The tastiness of the cake is defined as the sum of two adjacent pieces. We are allowed to reverse one contiguous subsegment of cakes at most once and then measure the tastiness.

We are asked to find a "good pair" of indices in an array of positive integers. A good pair consists of two indices $i$ and $j$ such that, if you consider any element $ak$ in the array, the sum of distances from $ai$ to $ak$ and from $ak$ to $aj$ equals the distance from $ai$…

We are given an array of non-negative integers. In one operation, we choose an integer $x ge 2$ and simultaneously replace every element $ai$ with $ai bmod x$. The operation may be applied any number of times, including zero.

We start with an array of integers. In one operation, we pick an element, remove it from the array, and subtract its value from every remaining element. After exactly $n-1$ operations, only one number remains.

We are asked to label the vertices of a tree with lowercase letters so that a set of path constraints is satisfied.

The task is to repeatedly remove "good" prefixes from a bracket string. A prefix is good if it is either a regular bracket sequence or a palindrome of length at least two.

We have a complete graph on n labeled vertices. Every edge receives an integer weight between 1 and k. Vertex 1 plays a special role. Consider the star centered at vertex 1, consisting of all edges (1, v) for v 1.

We have a strategy game scenario where Monocarp recruits a squad to fight a sequence of monsters. Each battle starts with an empty squad, and Monocarp can spend up to C coins per battle. Each unit type has a cost, damage per second, and health.

We need to construct an array $a0, a1, dots, an$ starting from $a0 = 0$. For every next position, we have exactly two choices. We can either increase the current value by $x$, or decrease it by $y$. The only restriction is that every value in the sequence must stay at most $B$.

We start with a chip at the origin (0, 0) and want to reach a target point (x, y). A move is allowed whenever the Euclidean distance between the current position and the next position is an integer. The task is to find the minimum number of moves needed to reach the target.

We are given a binary string, and we want to select some non-overlapping segments of it whose total length is a specified number, $m$, such that the concatenation of these segments has the same proportion of ones as the original string.

We are asked to check if a lost permutation could exist given a sequence of "powers" for its cyclic shifts. The power of a permutation is defined by counting distinct elements in the prefix maximum array.

We are asked to count permutations of length $n$ where the greatest common divisor of the sequence $1 cdot p1, 2 cdot p2, dots, n cdot pn$ is strictly greater than 1. Here $p$ is a permutation of integers from 1 to $n$.

We are given a line of cosplayers represented as a binary string. Each 0 is a male and each 1 is a female. The goal is to ensure that the line is "beautiful," meaning that in every contiguous segment of at least two cosplayers, the number of males does not exceed the number of…

We start with an unknown binary array $A$. For every prefix length $k$, we sort the first $k$ elements of $A$, leaving the rest unchanged. This produces $n$ arrays $B1, B2, dots, Bn$. Instead of seeing those arrays directly, we are given their column-wise sum $C$.

We are given a tree where every vertex hosts a position in a permutation. Separately, there is a token sitting on one vertex. The game evolves in alternating moves between Alice and Bob. Alice’s move does not change the tree.

We start with a binary string. We must perform exactly (k) operations. In one operation we choose a position (i). The chosen bit stays unchanged, while every other bit in the string is flipped.

We are given an undirected, connected graph where each edge has a non-negative integer weight less than $2^{30}$. A query asks for the minimum possible "length" of a walk from vertex $u$ to vertex $v$.

We have a king starting at position zero on a number line, and there are several unconquered kingdoms at strictly increasing positions. The king wants to conquer all kingdoms at minimal cost.

We are asked to construct a sequence of match outcomes between two teams, Red and Blue, given the total number of matches, the number of wins for Red, and the number of wins for Blue.

Thank you for the clarification. Now we can carefully analyze the problem. The input/output mismatch shows that the previous solution produces RRRWW for the first test case, while the expected output is RWRRW.

We are given a string consisting only of plus and minus signs. A substring of this string is considered promising if, through repeated replacement of two consecutive minuses with a single plus, the substring can be transformed into a balanced string, meaning it has an equal…

We are given a string consisting of '+' and '-'. For every non-empty substring, we must determine whether it is promising.

We are given a binary square matrix, and we are allowed to rearrange it using cyclic shifts of rows and columns. These shifts do not change values, they only rotate positions, so what ultimately matters is how we align the matrix before we start paying to flip bits.

Vasya has two types of coins in his wallet: 1-burle coins and 2-burle coins. The problem asks for the smallest positive integer amount of money that he cannot pay exactly using the coins in his possession.

We are given an array of integers where each value is extremely small in magnitude, only from -2 to 2. From this array, we are allowed to delete a prefix and a suffix, leaving a contiguous middle segment. That remaining segment could even be empty.

We are given a string and we are allowed to delete characters anywhere, not necessarily in a continuous segment. After deletions, we want the remaining characters to form a very rigid structure: the string must be split into consecutive pairs, and each pair must consist of two…

We are given a sorted sequence of points on a number line: starting at position 0, followed by positions $a1, a2, dots, an$, where $an$ is the final destination. These positions act as teleportation stations.

We are given a set of trees, each with an initial height. Our goal is to make all trees reach the same final height using a daily watering process. On odd-numbered days, watering a tree increases its height by 1, while on even-numbered days, it increases by 2.

We are given a grid with three rows and n columns, where each cell is either free or blocked. Conceptually, this is a narrow vertical slice of a 2D space, only three cells high, but potentially very wide.

We are given an array b of length n, where each element represents a target value we need to reach or exceed in a corresponding array a. Initially, a consists of all zeros.

At every position i, we have a pair of values (a[i], b[i]). The only operation allowed is swapping the two values inside the same pair. We may do this independently for any positions. After choosing which pairs to swap, we obtain final arrays a and b.

We are given a list of integers, each less than 32768, and the goal is to reduce each number to zero using two operations: either increment the number by one modulo 32768 or double it modulo 32768. The input specifies the number of integers followed by the integers themselves.

We must build a row of n wine bottles. Each bottle is either red (R) or white (W). Every critic wants to find some contiguous segment of bottles whose contents match a requested pair (r, w), where r is the number of red bottles and w is the number of white bottles in that…

The maze is drawn in polar coordinates with the center as the starting point. Movement is allowed continuously in any direction as long as we do not cross or touch a wall.

We are asked to place a new ice cream shop along a beach where huts are positioned at regular intervals of 100 meters. Each hut contains a certain number of people who will buy ice cream only from the shop that is strictly closest to their hut.

We are asked to count the number of ways to select four contestants standing on the vertices of a rectangle in an $n times n$ grid, such that when forming a banner using the two youngest contestants as one pole and the two oldest as another, the poles do not cross.

We are given a sequence of moments in a football match when kicks happen, each kick occurring at a fixed time and a fixed position along the touch-line. At time zero, we start at position zero, and then we are allowed to move continuously along the line with a bounded speed.

We are tasked with finding a residence point in a 2D plane from which you can meet any pair of three friends such that the sum of distances from each attendee to the meeting point does not exceed a certain threshold $r$.

We are given an $n times n$ grid of kids. Each cell has two attributes: an age from $1$ to $n$, and a binary label saying whether the kid is good at programming.

Bethany wants to tile her rectangular bathroom with a specific pattern. The interior, excluding the boundary, must be covered with standard $1 times 1$ tiles. The boundary is a one-tile-thick frame around the interior.

We are given a tree with n rooms, connected by n-1 corridors, forming a connected acyclic graph. Each room must host a unique Italian dish rated from 1 to n. A pleasing tour is a path in the tree where the sequence of dishes encountered is strictly increasing.

We are asked to count special trips on a graph of cities. Each city is a node, and each road is an undirected edge connecting two cities. A trip of length k is a sequence of k+1 cities such that each consecutive pair is connected by a road.

We are given a line of antennas indexed from left to right. Each antenna has a power value that determines how far it can directly communicate.

We have n people sitting at a round table, numbered from 1 to n. The initial seating is the natural order [1, 2, 3, …, n] clockwise around the table. We are given a desired seating order in the form of a permutation p.

We are given two DNA strings over the alphabet {A, B, C}. The goal is to decide whether we can transform the first string into the second using a sequence of operations.

We are given three strings, each representing a name. Think of each string as a multiset of letters we need to be able to reconstruct. We are allowed to manufacture “sheets”, and each sheet has two letters written on its two sides.