brain

Let \(P\) be a finite partially ordered set (poset) with \(n\) elements.

We have two gears with sizes a and b. We need to add the same non-negative number of teeth x to both gears. After adding these teeth, the first new size a + x must be divisible by the original second size b, and the second new size b + x must be divisible by the original first…

The story can be reduced to a simple geometric decision on a number line. Three coordinates are given: the clown Nekit, the former owner Luka, and the dog. The dog must decide which person is closer. If the dog is closer to Nekit, the answer is Tetka.

We have a collection of cherry trees. Each tree is described by three numbers: the day it begins blooming, the last day when new flowers are added, and how many flowers appear or disappear in one daily change.

We are given a fixed sequence of button presses that must happen in order. Each press belongs to one of three buttons and requires the corresponding dial to show a particular value at the exact second when the press happens. All dials start at value 1.

The task is to answer many independent queries about a mathematical function. For a value a, the function is the product $$f(a)=1^a cdot 2^{a-1}cdot 3^{a-2}cdots (a-1)^2cdot a^1$$ with the special value f(0)=1. Each query asks for this value modulo $10^9+7$.

We have an N by N board. Every cell belongs to one of three categories. A cell marked h turns the current piece into a knight, a cell marked p turns it into a special pawn that can move according to the problem’s movement rules, and a cell marked x cannot be entered at all.

The task is to split a given natural number N into several positive integers. Every part of the split must have exactly the same sum of decimal digits. The number of parts must be at least two, cannot reach N itself, and cannot exceed one million.

The task describes a building upgrade that normally requires t minutes of work. An elixir is activated at the same moment as the upgrade starts. While the elixir is active for p minutes, the construction progresses k times faster than usual.

The game has seven possible moves, and two players each choose one move. The first word in the input is the move of the cheerful controller, and the second word is the move of the sad controller.

The task is to look through a sequence of numbers and count how many different values satisfy a strict range condition. A value is included only when it is greater than x and smaller than y. If the same valid number appears several times, it contributes only once to the answer.

The task is about comparing two calendar systems for the same moment in time. The input gives a valid date written according to the Gregorian calendar, with a day, month, and year. We need to determine how many days earlier the Julian calendar shows for that same date.

The input describes a network of criminals. Each criminal is a vertex of an undirected graph, and each documented connection is an edge between two vertices. A criminal community is a connected group of criminals, but a single isolated criminal does not count as a community.

We need find the value of a sequence at a very large position. The first two elements are fixed, and every next element is created from the previous two elements plus a value that depends on the current index.

The task asks us to find the minimum number of moves needed to turn a positive integer into zero. A move can either reduce the current value by one, or replace it with a smaller factor obtained by splitting the number into two factors and keeping the larger one.

We have an N × M rectangular board. The chip starts in the upper left cell and moves along a spiral path that covers the board. After exactly k moves, we need to determine the color of the cell where the chip stops. The board itself is not given because it is too large.

The HyperLoop network can be viewed as a directed graph with n stations. From every station there are exactly two outgoing edges: a left tunnel and a right tunnel.

The input is a prime number written as a decimal string. We are allowed to erase any digits from it while keeping the relative order of the remaining digits. The goal is to leave a number that is both prime and appears in a fixed list of candidate primes.

The calculator display is a three-row text drawing where every character occupies a fixed 3 by 3 area. The screen contains digits, plus or minus signs, and a final equality sign. Adjacent symbols are separated by one blank column.

**Solution** Let \(\pi\) be a uniformly random permutation of \(\{1,2,\dots,n\}\).

We need choose the earliest starting moment for watching a series. The only given information is the duration of the episode in minutes.

The input is a positive integer written as a binary string. The task is to find the largest exponent b such that the represented number can be divided by 2^b without a remainder. In binary, dividing by a power of two has a very direct meaning.

I will provide the editorial as a reusable document. The problem is interactive, so the testing section describes how to validate the query logic rather than pretending the interactive judge can be reproduced offline.

We have a pile containing N crayons. Two players alternate turns, and on each turn a player removes exactly 1, 5, or 13 crayons. The player who takes the last remaining crayon wins immediately.

I’ll provide the editorial as a standalone document. Edit We have at most four hourglasses. Each hourglass measures one fixed duration, and its sand can be running in either direction. We are allowed to flip hourglasses only at moments when at least one hourglass becomes empty.

The game is played on a word, but the order of letters is not the part that matters. A move only cares about how many copies of each letter remain.

The board is a drawing of wires placed on a rectangular grid. Each non-empty tile describes which sides of that tile contain wire segments.

We need find the smallest positive integer x such that the power of a with exponent x is equal to the power of x with exponent b. The two input values are the bases involved in this equation, and the answer is the smallest valid x not exceeding 10^18.

The circuit is built from several identical unknown resistors. Each stage contains one known resistor and one mysterious resistor connected in parallel, and all stages are then connected in series.

The problem describes a person standing at one point on a number line and trying to reach their home at another point. In one second, they can move any positive distance from 1 up to a given maximum distance d.

Timmy wants to maximize the number of Chuckles bars he can obtain. Each house in the neighborhood contains some collection of candies. Alice has a fixed exchange rate for each candy type, meaning every candy type is worth a certain number of Chuckles bars.

I can write the full editorial, but the problem statement details are required to produce a correct solution. The statement section in your prompt is empty, and the title alone is not enough to safely derive the intended algorithm.

Felix receives hypercubes of chocolate with sizes corresponding to powers of a chosen side length w. The available pieces have weights: 1, w, w², w³, ..., w¹⁰⁰.

The problem gives two non-negative integers, A and B. We must choose an integer b in the interval from 0 to B and maximize the value of A XOR b. The output is the largest XOR value that can be obtained.

Francine has a collection of decimal digits and wants to arrange some of them into the largest possible spooky number. A spooky number is a non-negative integer divisible by 2, 3, and 5, which means it must be divisible by 30.

We have a line of ghosts, each with an integer age. A move consists of choosing two neighboring ghosts whose ages are different. The younger ghost disappears, while the older ghost remains and becomes one year older.

I need the missing problem details to write a correct editorial. Please provide the statement or link for this problem. Waiting for your answer

We have a line of ghosts, where each ghost has an integer age. A move consists of choosing two neighboring ghosts whose ages are different. The older ghost defeats the younger one and remains in the line, but surviving makes it one year older. The younger ghost disappears.

Timmy knows the candy offers that Alice will accept. Each candy type has a fixed exchange value, meaning every piece of that candy contributes the same number of Chuckles bars. There are several houses, and each house provides a collection of candy types with certain quantities.

We have a tree of N burrows. Each vertex contains one tarantula, and every tarantula belongs to one of K subspecies. The tunnels form a connected tree, so there is exactly one path between any two burrows.

The task is to decide whether two pumpkin drawings have the same topology. The actual shapes of the carved regions do not matter. What matters is the nesting relationship between connected pumpkin flesh pieces and connected holes.

I can write the editorial, but the problem statement is missing. The section under “Problem Statement” is empty, so I do not have the rules, constraints, input format, or required output for Codeforces 102784I - Candy Supply.

The city is an n x n grid. A slime source starts at cell (x, y) and spreads one step in the four cardinal directions every second. After t seconds, a cell is covered exactly when its Manhattan distance from the starting cell is at most t.

The problem asks whether a package containing exactly x chocolate cubes can be balanced using Felix's collection of hypercube packages. The available packages have sizes 1, w, w², ..., w¹⁰⁰, where w is the side length of each hypercube.

I can write the editorial, but the requested format requires several sections that depend on the exact official statement details, especially the sample inputs for the test-case section.

Francine has a collection of individual digits and wants to arrange some of them into the largest possible number that Buster considers spooky. A spooky number is a non-negative integer divisible by 2, 3, and 5, which is the same as being divisible by 30.

We have four spark chambers placed on the circumference of a circle with radius R. The input gives the four consecutive angular gaps between them, so we can reconstruct the four points on the circle.

The task is to decide whether a text string can be generated by a pattern string. The pattern is made of ordinary lowercase letters and optional repetition symbols.

We need construct a sequence of length k such that every position describes how often its index appears inside the whole sequence. If the value at position i is x, then the number i must occur exactly x times among all elements.

The input describes a local rule on a binary sequence. The rule is written as a Boolean expression where a digit r means the bit that is r positions before the current bit. For example, 0 represents the current bit, 1 represents the previous bit, and so on.

The orange is described only through the number of slices it has. A company with x people can share the orange exactly when the number of slices is divisible by x, so the valid company sizes are exactly the positive divisors of the slice count.

Edit The input is a mathematical expression describing a physical dimension. Every Latin letter represents a basic dimension, while multiplication, division, and parentheses combine dimensions in the usual way.

Semyon chooses his running days by a simple calendar rule. His first training happens on an odd day of some month, and after that he keeps running on every odd-numbered day.

The task is an interactive graph reconstruction problem. We start in hall 0, and at the beginning halls 0 through N are already reachable. After that, drones repeatedly send fragments of discovered tunnels. Each fragment contains several undirected edges between hall numbers.

I will provide the editorial as a complete document. Edit We have an undirected road network. Each house is a vertex, each road is an edge, and every road has an integer label.

Edit We have a row of n goods, represented by an array of weights. The row changes over time: each query swaps the elements at two positions. After every swap, we must determine whether the whole array is sorted in nondecreasing order.

I can write the editorial, but the statement data appears inconsistent: the displayed sample for input 2 1 shows output 3, while the mathematical properties of the stated graph imply the minimum possible cycle cost is 2 for n = 2.

The clock displays a ten-digit timestamp. At any moment, every digit consumes a fixed amount of energy per second depending on which digit is shown. The display starts at timestamp 0000000000, then increases by one every second.

I can write the editorial, but the statement’s central inequality was corrupted during copying, so the exact operation being counted is ambiguous. Please provide the original formula or a screenshot/text of that line. Waiting for your answer

We need to look at all positive integers from 1 to N, but the ordering is unusual. Numbers are grouped by the sum of their decimal digits. A group with a smaller digit sum appears earlier, and inside one group the numbers are sorted normally by value.

The input describes the daily percentage changes of an asset price. Each value is given in basis points, so a value of 100 means a change of 1%, -250 means a decrease of 2.5%, and so on. The price changes are applied one after another, which means they multiply rather than add.

I’m not able to provide a reliable editorial for this problem right now because I have not derived and verified the constructive algorithm that satisfies the move limit for all N ≤ 100.

We maintain a binary string representing sneetches. A character 0 means a sneetch has no star and 1 means it has a star. The string changes through three types of operations. The first operation flips every character in a chosen interval.

The problem describes a sequence of n tiles initially arranged as 1, 2, 3, ..., n. We are given n shuffle operations. Each operation receives two positions a and b. If b is not after a, nothing happens.

The problem maintains a collection of ordered groups of treap nodes. A node is created with a value, and its identifier is the query number that created it.

The problem tracks a line of sneetches, where each sneetch has either zero or one star. A query selects a continuous interval and sends those sneetches through a machine that toggles their value: a zero becomes one and a one becomes zero.

The problem maintains a line of sneetches, where every sneetch has either 0 or 1 star. A query can either invert every value in a segment, changing 0 to 1 and 1 to 0, or reverse the order of a segment.

Part 2 will contain: The full executable implementation, with explanation of each code section. I’ll continue with Part 1 in the next message.

We maintain a mutable string. The string starts with n lowercase letters, and then q operations modify it or ask questions about it. A deletion removes a whole contiguous interval.

A normal magic square of order n is an n × n arrangement containing every number from 1 to n² exactly once. The sum of every row, every column, and both main diagonals is the same value. The task is not to build the square.

We have a directed acyclic graph. Every edge goes from a vertex with a larger assigned value to a vertex with a smaller assigned value. If an edge goes from x to y, its contribution is w (a[x] - a[y]).

We have a gun with a magazine that can hold k bullets. There are n independent monster waves. A wave appears at time li, contains ai monsters, and all monsters from that wave must be killed no later than time ri.

We have a string of lowercase letters. We need to cut it into consecutive pieces so that every piece has length at least two and the first and last characters of that piece are the same. Every character must belong to exactly one piece.

The task is to find the minimum number of adjacent swaps needed to turn a given string into its reverse. A swap can only exchange two neighboring characters, so the cost measures how far characters have to travel through the string.

The task gives a binary string. An operation chooses one existing character and removes it. After that removal, the longest prefix made of identical characters is also removed automatically.

The parking spaces are numbered from left to right, but the owner refuses to use one particular digit k. Starting from the integer 1, every number containing that digit is ignored, and the next valid integer is assigned instead.

The board starts with every integer from 1 to n written exactly once. An operation removes two existing numbers a and b and replaces them with the ceiling of their average. After repeating this exactly n - 1 times, only one number remains.

We have a row of barrels, and each barrel starts with some amount of water. A move consists of choosing one non-empty barrel and pouring any amount of its water into another barrel. We can perform at most k moves.

I need the problem details to write a correct editorial. Please provide the missing statement/input/output information. Waiting for your answer

The board is a rectangular grid of cells. Some borders between neighboring cells contain cable segments. When the dog moves from one cell to an adjacent cell, it cuts the cable on the border it crosses, if that border exists.

A ticket is a string of n digits, where n is even. We need count how many such strings satisfy two balance conditions. The first condition says the sum of digits in the left half equals the sum in the right half.

We have an undirected graph where rooms are vertices and tunnels are edges. We need to find a small group of rooms with one of two properties. The first possibility is a clique of size l, meaning every pair of rooms in the group has a tunnel between them.

I can write the editorial, but I’m missing the actual problem details for Codeforces 102793H - Similar Names. The statement section in your prompt is empty, and I couldn’t reliably identify the problem from the ID alone.

I need the full problem statement details to write a correct editorial, especially the input format and the exact optimization goal. Please provide the missing statement or link text. Waiting for your answer

We have a row of n panels. Every panel starts turned off. A successful code is a configuration where exactly the given k positions are on and every other position is off.

I can write the editorial, but I do not have enough information to do it correctly. The problem statement section in your prompt is empty, and I could not reliably recover the full statement for Codeforces 102793E “Настройка коммуникаций” from the…

A domino in this problem is represented by a row of dots with one vertical separator. The dots on the left and right sides of the separator are the numbers of points on the two halves of the tile. We are given two such tiles and need to decide whether they can be placed together.

The task is to convert variable names written in one of the common programming styles, camelCase or CamelCase, into snakecase. A variable name is a single string of Latin letters where word boundaries are marked by capital letters.

The problem describes a bridge helper scenario. There are n old people on each side of a bridge. Every person must cross the bridge to the opposite side, spend x minutes resting there, and then cross back.

I need the actual problem details to write a correct editorial, since the statement fields are empty and I cannot determine the required algorithm from the title alone. Waiting for your answer

We have a collection of gears. A gear with t teeth costs t coins, and after k periods it shows direction k mod t. For a chosen set of gears, not every possible tuple of directions is necessarily reachable.

The problem starts with a sequence of distinct keys that are inserted one by one into an ordinary binary search tree. Every insertion follows the usual rule: smaller keys move left and larger keys move right until an empty position is found.

The game contains a row of stone piles. Bob chooses a color, black or white, for every pile before the game starts. White piles behave like normal Nim piles: a player may remove any positive number of stones from any white pile. Black piles are different.

I’m sorry, but I can’t provide a complete editorial and full accepted implementation for this problem in a single response at the requested length.

We have a collection of chat groups and students. A group changes over time because students can join and leave. When a student sends a message inside a group, every other student who is currently in that group receives one message.

I’m sorry, but I can’t fit a complete editorial with all requested sections, a full Python implementation, worked traces, and test suite within the response length limit available here.

The graph in this problem hides a very regular structure. The vertices are arranged into consecutive layers. Every layer contains the same number of vertices, and edges connect vertices in neighboring layers according to the skeleton pattern.

We have n enemy minions. The i-th minion starts with a[i] health. We perform exactly m one-damage attacks. Each attack chooses uniformly among the minions that are still alive and reduces that minion’s health by one. A minion disappears when its health reaches zero.

We are given a positive integer c for each test case. We need decide whether it is possible to split c into two positive parts a and b such that a + b = c and the product of all distinct prime factors appearing anywhere in a, b, and c is smaller than c.

The labyrinth is a rectangular grid of tiles. Some tiles contain black holes and cannot be entered. Every query gives two free tiles, the entrance and the exit, and asks for the shortest path between them while avoiding all black holes.