brain

We are given two points moving in the plane over time, but their motion is only specified at discrete timestamps. Between consecutive timestamps, each friend moves in a straight line at constant speed, so the position at any intermediate time is obtained by linear interpolation.

We are given a leaderboard of astronauts sorted by their current total points in non-increasing order, meaning the first astronaut currently has the highest score and the last has the lowest. One specific astronaut, identified by their position $D$, is the one we care about.

I can’t reliably write a correct 3200-rated editorial from just the title and a broken/empty statement link. The structure you provided points to Codeforces 921 Problem 1, but the actual problem content (grid/graph rules, input format, and what needs to be computed) is…

The task describes a large grid maze where each cell is a room and adjacent rooms may or may not be traversable directly. Between neighboring cells there can be empty space, a wall, or a locked door. The robot starts at a fixed cell and must reach any one of several exit cells.

I can’t write a correct 3200-rated editorial from just the problem link because the statement content isn’t included here and I don’t have access to fetch it.

I can’t reliably access the full statement from that Codeforces link in this environment, and for a 3200-rated problem I’d need the exact rules to avoid hallucinating details.

I can write the full editorial in the exact format you want, but I need the actual problem statement content. The link alone (921/problem/01) is not accessible from here, and without the statement I’d have to guess the rules, which would make the editorial unreliable at a…

I can’t safely write a correct full editorial for “Labyrinth-6” from Codeforces 921 without the actual problem statement content. Right now, the only thing available is the title and rating, and the link itself does not expose the statement text in a way I can rely on.

I can’t reliably access the full statement from that Codeforces link in this environment, and for a 3200-rated problem I can’t safely reconstruct it from the title alone without risking inventing details.

I cannot write a correct 3200-level editorial for this problem yet because the actual problem statement content for Labyrinth-3 (92103) is not available in your prompt.

I cannot responsibly write a correct 3200-rated editorial for “Codeforces 92102 - Labyrinth-2” from the link you provided because the actual problem statement is not accessible in the material included in your prompt.

We can view the labyrinth as a large grid graph where each cell is a vertex and adjacency exists between cells sharing a side. Movement between adjacent cells is sometimes blocked by a permanent wall, sometimes by a door, and otherwise is free.

We are simulating a single-server queue that evolves over time. Each student appears at a known second and joins the end of a line. The server is a teapot that can serve exactly one student per second, and serving always goes to the student currently at the front of the queue.

We are working with an array of integers where two operations are repeatedly applied over ranges. One operation replaces every value in a segment with its number of positive divisors, and the other asks for the sum of values in a segment after all previous updates.

The task is to decide how to buy a fixed amount of apples while minimizing total cost, given multiple supermarkets with different pricing schemes. Each supermarket does not directly give a per-kilogram price.

We are given a sequence of integers, each of which can be thought of as a 17-bit mask. From this array we must form ordered selections of five positions.

We are given a tree where each node carries a lowercase character from a limited alphabet of size 20. The task is to examine every simple path in the tree and determine whether the multiset of characters along that path can be rearranged into a palindrome.

We are asked to enumerate a very specific set of integers and then pick a particular element from that ordered list. The allowed numbers are those whose prime factors all come from a fixed small set of given primes.

We are given an integer $N$. From the points $0, 1, 2, dots, N$ on a number line, consider every possible segment whose endpoints are chosen among these points. That means every pair $(l, r)$ with $0 le l < r le N$ defines a segment.

The program consists of tasks that must be executed in an order consistent with a dependency graph. Each task may depend on earlier tasks, forming a directed acyclic graph where execution is only allowed once all prerequisites are completed.

The board is a 9 by 9 grid, but it is conceptually split into nine 3 by 3 sub-boards arranged in a 3 by 3 macro layout. Each cell can contain either a mark from the first player, a mark from the second player, or be empty. The game is not ordinary tic-tac-toe.

We are given four small integers representing sizes of three bears and Masha. The bears require three car sizes that are strictly decreasing from father’s car to son’s car.

We are given a set of teleporters placed along a one-dimensional number line. Pig starts at position 0 and wants to reach position m.

We are given two integer sequences of length $n$. One sequence $b$ acts as a fixed “key”, and the other sequence $c$ is the observed encrypted output of some unknown sequence $a$ under that key.

We are given a connected undirected graph where every edge must eventually receive an integer weight. For each vertex, we are also given a target value, and the requirement is that if you look at all edges incident to a vertex and sum their weights, that sum must exactly match…

We are given a rooted tree described indirectly through a “level size profile.” Instead of edges, we only know how many nodes exist at each distance from the root.

We are given a graph whose vertices are arranged in a fixed line from 1 to n. The edges are arbitrary, but the structure is restricted so that the graph does not contain any cycle of even length when edges are treated as simple undirected edges.

We are asked to construct two integer polynomials, each of degree at most $n$, with very small coefficients (each coefficient is either -1, 0, or 1), and both leading coefficients equal to 1.

We are given several Bitcoin wallets, each holding some amount of satoshis. The goal is to redistribute these coins into possibly more wallets so that no wallet ends up holding more than a fixed limit $x$.

We are given a set of points on a plane. We must place a very specific triangle shape so that every point lies either inside it or on its boundary.

We are given a single integer $n$, and we need to represent it as a sum of three positive integers $a, b, c$. The extra restriction is that none of these three numbers is allowed to be divisible by 3.

We are given a growing sequence of points on the integer line. Each point has a position value and a weight. As we reveal more points one by one, we want to maintain a configuration where all positions are distinct, but we are allowed to move points left or right using very…

We are working with a very small geometric object: a grid with exactly two rows and $n$ columns. Each cell can independently be painted in one of two colors, which we can think of as black or white.

We are given a connected undirected weighted graph with up to one hundred thousand vertices and edges, but with a crucial structural restriction: the number of edges exceeds the number of vertices by at most twenty.

We are given a sequence of numbers, but it behaves like a multiset, meaning duplicates matter and order only matters for output assignment. Each occurrence of a value must be assigned to one of two groups, A or B.

We are given one very long decimal string a, and we want to split it into a sequence of contiguous pieces. Each piece is a substring of a, taken in order, and the concatenation of all pieces must reproduce a exactly.

We are given a contiguous segment of integers from $l$ to $r$, and the task is to partition all these numbers into pairs. Every number must appear in exactly one pair, so the segment is fully covered.

We are given a password string consisting of digits and Latin letters in mixed case. The goal is to end up with a string that contains at least one lowercase letter, at least one uppercase letter, and at least one digit.

We are given a sequence of length $2n - 1$, which is supposed to represent the order in which a squirrel visits vertices of some tree during a full traversal.

We are given two sequences, one indexed by $i$ and one indexed by $j$, and we are asked to combine every pair $(i, j)$ into a single weighted contribution. The weight is not linear or even separable in the usual sense: each pair contributes $$ai cdot bj cdot c^{i^2 j^3}.

We are asked to construct a tree on $n$ labeled vertices, representing citizens, using exactly $n-1$ edges so that the graph is connected and acyclic.

We are given a set of points in the plane, each point being an “event location” where a horizontal wire and a vertical wire must cross. Every wire is axis-aligned: horizontal wires lie on a fixed y-coordinate, vertical wires lie on a fixed x-coordinate.

We are given a line of children, and we need to reconstruct any valid assignment of candy counts to them. Each child has already reported two numbers: how many children to their left have strictly more candies than them, and how many children to their right have strictly more…

We are given a grid where each cell contains a short string made of characters 0 and 1. Think of each cell as a small stack of chips written in a row, where we can see the order of chips from left to right.

We are given an array of integers where each value is represented using exactly $k$ bits. Alongside the array, we are allowed a very specific transformation: for any position, we can flip all bits of the number, turning it into its bitwise complement within the $k$-bit space.

We are given a sequence that is claimed to be constructed step by step starting from an empty array. At each step, the builder is allowed to look at any subset of elements that already exist in the array, compute the mex of that subset, and append that mex as the next element.

Masha starts on a floor $x$ and wants to reach floor $y$. She has two ways to travel vertically inside the building: stairs or an elevator. The stairs always work in a simple linear way, every move between neighboring floors costs a fixed amount of time $t1$.

The tree gives you a system where every vertex can be assigned a value: the XOR of edge weights on the path from an arbitrary root (say vertex 1) to that vertex.

We are given an array of values laid out on a line, and a collection of candidate intervals on that line. From these intervals, we must pick exactly $m$ of them.

We are maintaining an array of hair lengths on a line of positions, and the system evolves over time as some positions grow.

The problem describes a scheduling process where each task depends heavily on a continuously changing “skill level”. You are given several problems, each with a difficulty value and a reward.

We are given a circular metro line with stations numbered from 1 to n. Moving along the circle is always possible in both directions, so from any station we can go clockwise or counter-clockwise with wrap-around.

The city is a graph where crossroads are vertices and roads are undirected edges. Each ride is a simple path in this graph, meaning the driver never revisits a vertex within that ride. Across all rides, we observe many such simple paths.

Each cell in an $n times n$ grid defines a number of candies equal to $i^2 + j^2$, where $i$ and $j$ are the row and column indices. For every cell, we imagine taking that many identical candies and trying to split them evenly among $m$ friends.

We are given a rooted tree, with node 1 fixed as the root. Only leaf nodes initially receive colors, and all other nodes are uncolored. A leaf, in this problem, is defined as a node whose subtree consists of only itself, which in a rooted tree means a node with no children.

We are given a binary template string s, where each character is either 0 or 1, and a target string t consisting of lowercase letters.

We are given a sequence of moments during a tram ride, where each moment corresponds to a stop where Arkady briefly observed which tram lines serve that stop. Each stop lists a set of line identifiers, and Arkady remembers the full set at every observed stop.

We are given a growing network of routers that forms a tree rooted at router 1. Every router except the first was added one by one, and each new router was directly connected to exactly one earlier router.

We are given a rooted tree where every vertex contains a positive weight. The task is to partition all vertices into a minimum number of vertical paths.

We are given a set of points in the plane, each representing an animal’s location. We need to place a circle that covers all these points. At the same time, there is a fixed horizontal river, which after transformation becomes the x-axis, so the line is $y = 0$.

We start with a fixed sequence containing the integers from 1 up to n. We repeatedly perform an operation where we look at all remaining elements, compute their greatest common divisor, record that value, and then delete exactly one element of our choice.

We are given a binary grid representing a target drawing made of ink cells. Each cell is either already filled or empty, and we want to determine whether this final pattern could have been produced from an initially empty grid using a specific stamping tool.

We are given a workday of fixed length and a sequence of service intervals where a cashier must be continuously present. Between these service intervals, there are gaps of free time.

We are given an infinite line of positions starting from zero. Initially, each position i holds a ball labeled i, so the configuration is perfectly aligned: position equals ball number. Some positions are marked as pockets.

We are given a tree with up to 50 vertices, and we repeatedly compress it until only one vertex remains. Each operation picks an edge uniformly at random. The two endpoints of that edge disappear, and they are replaced by a single merged vertex.

We are working in a very unusual computational model: instead of directly manipulating variables, we interact with an array of hidden memory cells. Only two of them initially contain unknown values, while all others are initialized to one.

We start with a tree of subway stations. Every station is a node, and every tunnel is an edge, so there is exactly one simple path between any two stations.

We are given a single large integer $n$, and we want to split it into two non-negative integers $a$ and $b$ such that their sum stays exactly $n$.

We are given several guests, and each guest has a personal requirement about how they sit in a circular arrangement.

The matrix in this problem is not given explicitly. Instead, every cell is formed by multiplying an element from array a with an element from array b. This creates a grid where each row is a scaled version of b, and each column is a scaled version of a.

We are given a multiset of digit cards, each card containing a single character from 0 to 9. From these cards, we want to assemble as many valid phone numbers as possible.

We are given a tree that is known to be a perfect $k$-ary tree, meaning it has a root, every internal node has exactly $k$ children, and all leaves are at the same depth.

We are given two different spanning trees over the same set of cities. Each tree represents how one candidate organizes the country, with a chosen root city acting as their capital.

We are given a collection of time intervals, each representing a TV show that occupies a continuous range of minutes. The key constraint is that a single TV cannot be used to watch two shows whose intervals overlap in time.

We are given several vertical stacks of blocks placed side by side. Each position $i$ has a stack of height $ai$. From above, the camera only cares whether a position is occupied or empty, so every column contributes exactly one visible cell as long as it has at least one block.

We are asked to count how many subsequences of a given array are “valid” under a position-based divisibility rule.

We are given a collection of coin denominations that includes every integer value from 1 up to n, and we are allowed to use any number of coins of any of these values.

We are given a directed acyclic graph of cities connected by one-way roads. From the constraints, the graph has no directed cycles, so it behaves like a partial order: some cities are comparable through reachability, and others are incomparable.

We are given a single integer as a starting point, and we are allowed to transform it using two operations. One operation multiplies the current number by any positive integer we choose, effectively letting us inflate the number arbitrarily.

The company hierarchy forms a rooted tree where employee 1 is the root, and every other employee has exactly one direct boss. This defines a parent relationship and also induces depths from the root, where depth is the number of edges from employee 1.

We are given a set of integers from the range of absolute values 2 up to n, and each number also has its negative counterpart available. Think of every integer i in this range as a node in a graph, including both i and -i.

We are given a binary array where each position has a value of either zero or one. We repeatedly remove elements from a chosen segment, and every time we remove an element, two things happen: we gain its value immediately, and all remaining elements in that segment get…

We are given a strictly increasing array of integers, all between 1 and 1000. Someone is allowed to erase exactly one contiguous block of elements, leaving a gap in the sequence. The remaining elements stay in their original positions.

We are given a single string and we want to decompose it into a sequence of progressively shorter substrings, where each substring in the sequence must appear inside the previous one.

We are simulating a deterministic passing process on a circular arrangement of $n$ children. A box starts at position $l$ and is passed clockwise.

The maze is an $n times n$ grid where each cell can either be empty or contain a fixed type of mirror that behaves like a 45-degree reflector. A laser starts from each column on the southern boundary and travels northward into the grid.

We are interacting with an unknown assignment of colors to points, where the opponent can choose each point’s color after seeing where we place it. We must output a sequence of distinct integer-coordinate points.

We are given a multiset of lowercase letters. We are allowed to rearrange these letters into any order we want, forming a new string of the same length. For any resulting string, we look at every contiguous substring and check whether it is a palindrome.

The task places us on a grid maze where each cell is either open or blocked. We start from a given cell and can move in the four cardinal directions as long as we stay inside the grid and avoid obstacles.

We are given three positive integers representing the current lengths of three sticks. In one move, we are allowed to pick exactly one stick and increase its length by one unit.

The equation links a number a with a variable x using two operations: subtraction and bitwise XOR. For each given value of a, we are asked to count how many non-negative integers x satisfy the identity a - (a XOR x) - x = 0. The input consists of several independent values of a.

We are given a rooted tree with vertex 1 as the root, and every vertex except the root has exactly one parent defined by the input. Some vertices are leaves in the usual sense of the tree structure.

We are working with a family of binary strings built in the same recursive way as Fibonacci numbers, except instead of addition we concatenate strings.

We are given strings of length n over an alphabet of size Instead of tracking how a single string evolves, the real question is to understand which positions in the string are fundamentally indistinguishable under repeated application of these operations.

We are given an $N times N$ grid where every cell contains a unique number from $1$ to $N^2$. These numbers define a forced visiting order: we must start at the cell containing 1, then eventually reach the cell containing 2, then 3, and continue in increasing order until $N^2$.

We are given a simple undirected graph with a fixed number of vertices and edges. The graph has no self-loops and no duplicate edges.

Vasya has a fixed amount of money and wants to maximize how many chocolate bars he ends up with under a repeating promotion. Every bar has a fixed price, so his initial purchasing power is simply how many bars he can buy directly.

We are given a collection of vertical towers, each represented by a positive integer height. In one operation we choose a target height $H$, and then every tower that is taller than $H$ is cut down to exactly $H$, while shorter towers remain unchanged.

We are given a set of points in the first quadrant of the grid, and Maksim starts at the origin. In one move he can step to any of the four neighboring lattice points, so every move costs one unit of Manhattan distance. The key restriction is not geometric but structural.

We are given a one-dimensional house represented as a binary array. Some positions contain heaters, marked with 1, while empty positions are 0. Every heater, if activated, warms a continuous interval around its position.