brain

We are asked to maintain two integer arrays, A and B, under a sequence of operations called Fibonacci additions. Each operation specifies a segment [l, r] and an array to update.

We are asked to arrange $n cdot k$ items, priced from $1$ to $n cdot k$, into a grid with $n$ shelves and $k$ items per shelf. The key restriction is that the mean price of any contiguous segment of items on a shelf must be an integer. Each number must appear exactly once.

We are given a string s of length n and a number k. We can perform exactly k operations on s, where each operation is either appending the reverse of the string to itself (s + rev(s)) or prepending the reverse (rev(s) + s).

We are given an array of integers, and a process that runs left to right over this array. A person starts with some initial value and, at each position, must choose one of two actions: add the current array value to their running number, or XOR the current array value with it.

We consider a horizontal rod of length $l$ that rotates about a vertical axis passing through the point $O_1$.

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 asked to reconstruct a hidden configuration of points on a line, one for each car, and assign each car a direction, left or right. After reconstruction, the configuration must explain two kinds of pairwise constraints.

We are given a sequence of points lying on a horizontal line, each with a coordinate and a positive weight. The task is to answer multiple queries about contiguous subarrays of these points.

Three communicating vessels contain water and are covered by pistons made of the same material and having the same thickness.

We have to modify an array so that no position is strictly greater than both of its neighbors. Such positions are called local maximums. The first and last elements are never local maximums because they only have one neighbor.

We are given an integer array and we are allowed to perform a very specific transformation: pick three indices in increasing order and overwrite the leftmost position with the difference of the two later values.

We are given several arrays, each containing small non-negative integers. For each array, we are allowed to repeatedly pick two different positions and replace both values with new numbers, as long as the bitwise OR of the chosen pair stays unchanged after the replacement.

We are given a permutation, meaning every number from 1 to n appears exactly once in some order. We are allowed to pick some subsequence of this array, remove those chosen elements, and then reinsert them as a block at the very front, preserving their relative order.

We are given the numbers from $1$ to $n$ for each test case, and we can repeatedly pick any two numbers $x$ and $y$, remove them, and add two numbers: $x+y$ and $ The input size allows $n$ up to $5 cdot 10^4$ per test case, with a total sum of $n$ over all test cases also…

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.

A conducting sphere of radius $r$ is connected to ground.

We are working on a tree where each vertex has a height value. The task is to place a set of “towers” on vertices, and assign each tower a positive integer efficiency.

We have two arrays of equal length. For every position, we may either keep the pair as it is or swap the two numbers inside that position. After deciding this independently for all indices, both arrays acquire new values.

We are given a line of piles, each pile containing some number of stones. The only allowed operation removes two stones from some middle pile and redistributes them as one stone to a pile on its left and one stone to a pile on its right.

We are given an array of integers and asked to compute a sum of “values” over all its subsegments. A subsegment is any contiguous slice of the array. The value of a subsegment is defined as the maximum cost of any partition of that subsegment.

We have an array and we are forced to perform exactly one operation. We choose a position that splits the array into two non-empty parts. Then we sort the left part and the right part independently.

Two balls of masses $m_1$ and $m_2$ are placed on a smooth horizontal plane and connected by a spring of stiffness $c$.

We are given a sequence of vertical panels, each panel having width one and a fixed height. Visually, each panel is attached to a horizontal bar at the top, and can be shifted upward or downward, but it must always remain connected to that bar, meaning every panel remains a…

We are maintaining a length-n array that starts completely uniform: every position holds value 0 and belongs to color 1. Over time, the array evolves through two kinds of updates. One type recolors a whole interval, replacing whatever colors were there with a new one.

We are given a painted canvas represented as a grid with n rows and m columns. Each cell in this grid contains a color, and the painting process was performed using only a 2 × 2 brush.

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 permutation of length $n$, which is an array containing all integers from $1$ to $n$ exactly once. We are allowed to choose exactly one contiguous subsegment of the permutation and reverse it.

We are given a permutation of size $n$, which is an array containing all integers from 1 to $n$ exactly once in some order. From this permutation, we construct an undirected graph on $n$ vertices, where each vertex corresponds to a position in the permutation.

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.

Two helical springs are made from identical pieces of steel wire.

A sample of an unknown purified gas is contained in a cylinder.

We are exploring an unknown connected graph, but we never see global labels of vertices or edges. Instead, we start at a known vertex and repeatedly move along incident edges chosen locally.

The problem presents an interactive treasure hunt on a hidden undirected graph. Each vertex corresponds to a junction with a treasure. You begin at a specified start vertex, and every time you visit a new vertex, you collect the treasure there.

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 exploring an unknown connected graph. We know the complete graph beforehand, including every edge, but once the interaction starts we lose the vertex labels.

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.