brain

We are given two distinct lattice points in the plane, and we must choose a third lattice point such that the triangle formed by all three points has strictly positive area while that area is as small as possible.

We start with an array and are allowed to repeatedly transform it exactly $m$ times. Each transformation replaces the current array with a new one formed in one of two ways: either we duplicate the array and append it to itself, or we take a reversed copy and place it in front…

We are given a one-dimensional strip of cells. Each cell is either empty or contains a lily, represented by a character string where L means a lily is already planted and . means the cell is empty soil.

We are given a strictly convex polygon, and we imagine choosing a point inside it. For any fixed direction, we draw the maximal chord of the polygon that passes through this point and is parallel to that direction.

The problem statement for Codeforces 104009L - Transafagarasan is missing from your prompt, so I don’t have the actual rules, input format, or task definition needed to write a correct editorial.

Let $\mu_n(x_1,\ldots,x_{2n})$ be the Boolean function whose truth table encodes a monotone Boolean function in the sense of Section 7.

I can’t write a correct editorial for this yet because the actual problem statement is missing. “Codeforces 104009K - Restricted Tree” is just a title here, but the key details are exactly what determine the solution: what “restricted” means, what operations or queries…

We are given a long row of seats indexed from 0 to N + 1, where the two boundary seats are permanently occupied. The interior seats start empty, but this configuration changes over time through events. There are two kinds of events.

I don’t have the actual statement for Codeforces 104009G - Genetics, so I can’t reliably reconstruct the problem, constraints, or intended solution path without risking inventing details.

I don’t have the statement for Codeforces 104009H - Lottery, so I can’t reliably reconstruct the problem, constraints, or solution without risking inventing details.

I can’t reliably write a correct Codeforces editorial for 104009F - Engine without the actual problem statement. Right now the prompt has empty sections for the statement/input/output, which usually means the content wasn’t included or was stripped.

We are given a sequence of edges of an undirected graph, presented in a fixed order. We are not allowed to reorder edges, but we can cut this sequence into consecutive blocks. Each block forms its own independent graph using exactly the edges inside it.

I’m missing the actual problem statement for Codeforces 104009E - Coins, so I can’t reliably reconstruct the intended solution, constraints, or edge cases.

The problem describes a very small “accounting system” where each record in the input represents a set of monetary transactions, and the task is to determine the final net balance after processing all of them in order.

I don’t have the actual problem statement for Codeforces 104009B - AGaMe in your message, and I can’t reliably reconstruct it from the title alone without risking inventing details.

I can’t produce a correct full editorial for “Codeforces 104009C - Bar hopping” yet because the problem statement is missing from your prompt. Right now the input and output sections are empty, so there is no way to know what the algorithm is supposed to compute.

We are given a collection of straight road segments in the plane. Each road is just a line segment with real geometry: two endpoints in 2D, and the segment is the asphalt between them. From these segments we must choose exactly three.

We are given an even number of words, all of identical length, and we want to pair them up. A pair is considered valid for a chosen value $k$ if the two words share a common prefix of length at least $k$.

We are dealing with a rectangular arena that contains a smaller rectangular grass field in its center. Around this grass field, there are multiple concentric rectangular “lanes”.

We are given a line of positions that will contain alternating objects: a pine, then a lamp, then a pine, then a lamp, and so on. Since there are n lamps, there are n + 1 pines placed at the pine positions.

We are given a collection of acrobats, each described by two attributes: a height-like value $ai$ and a weight-like value $bi$. We need to arrange all acrobats in a line, producing a permutation of indices.

We are given a target value $S$, and we must construct a contiguous interval of integers $[l, r]$, where $0 le l le r le 10^{18}$. If we write all integers from $l$ to $r$ in decimal form and concatenate them without separators, we obtain a single long string.

We are given a sequence of problem scores laid out in a fixed order. Each position has a positive value, and solving a problem yields that many points.

We are standing at position zero on an infinite number line and want to reach some target coordinate $x$, which can be positive, negative, or zero. In one move, we choose a non-negative integer $k$, and then jump either left or right by exactly $2^k$.

We are given a strictly convex polygon, and we imagine choosing a point inside it. For any fixed direction, we draw the maximal chord of the polygon that passes through this point and is parallel to that direction.

We are given an $n times m$ grid. One unknown cell contains a magnet. Every other cell contains a compass that points toward the magnet using one of 8 discrete directions: four axis-aligned directions and four diagonals.

We are working with an infinite complete binary tree where every node has a left child and a right child. Initially, every node is uncolored. We receive two kinds of operations.

Let $U={0,1,\dots,n-1}$ and let each subset $S\subseteq U$ be identified with its characteristic integer $s=\sum_{e\in S}2^e$.

We are given a group of $n$ people. Each person $i$ has an associated number $di$, which represents how many friends that person has. The friendship rule is unusually rigid: two distinct people $i$ and $j$ are friends if and only if they have the same value $di = dj$.

We are given a strictly convex polygon, and we imagine choosing a point inside it. For any fixed direction, we draw the maximal chord of the polygon that passes through this point and is parallel to that direction.

We are given several threads, and each thread contains a fixed sequence of assignment operations. Each operation writes a value to a named variable, and these writes happen in a strict order inside each thread.

We are given a collection of independent trees. For each tree, every vertex comes with two numbers that describe how many incident edges of two different colors it should have in a valid reconstruction.

We are given a final picture of an $n times m$ grid consisting of white and black cells. This picture was produced by repeatedly stamping two fixed, non-rotated, non-mirrored patterns, called the letters Q and F.

We are given an undirected graph with weighted edges, where the weight is called colorfulness. The task is to travel from city 1 to city n, but not just by any path.

We are working on a one-dimensional street of fixed length $L$. One endpoint is Julia’s home at position $0$, and the other endpoint is Jane’s home at position $L$.

We are given a hidden permutation of size $n$, where the values are a rearrangement of the integers from 1 to $n$. We cannot see the array directly. Instead, we can interact with it using two operations.

Each contestant has a personal list of problems they are capable of solving. For contestant $i$, problem $j$ has a nonzero time $a{i,j}$ if it is solvable, otherwise it is unusable.

We are given a number $n$, and we consider all integers from $1$ up to $n-1$. From this range, we keep only those numbers that share no common divisor with $n$ except 1. In other words, we filter the range by a coprimality condition relative to $n$.

We are working on a small fixed integer grid of size 21 by 21, where both coordinates range from minus 10 to 10. The task is not to compute anything dynamically but to construct a function over this grid and output it in reverse Polish notation using a restricted set of…

Exercise 225 constructs a ZDD whose paths encode all simple paths between two fixed vertices $s$ and $t$. The construction proceeds by a controlled search over partial edge sets: each ZDD node represents a state of the partial path, and each decision corresponds to including…

We are given a sequence of timestamps when problems were solved, already sorted in increasing order. Each timestamp represents a moment in continuous time, but the platform converts these moments into discrete “days” using a floor division after shifting time by a chosen…

We are given two points on a simplified timeline where every year is labeled either in the AD system or in the BC system. Each input line describes a year like “AD 2022” or “5508 BC”.

We are given a collection of file names, all consisting of lowercase letters. Charlie wants to delete all of them using a restricted version of the rm command.

Exercise 225 constructs a ZDD whose paths encode all simple paths between two fixed vertices $s$ and $t$. The construction proceeds by a controlled search over partial edge sets: each ZDD node represents a state of the partial path, and each decision corresponds to including…

We are given two moments in a day written on a 24-hour digital clock. The first is the current displayed time, and the second is the target correct time.

We are given a multiset of cards where each card has a value between 0 and n − 1. At any moment, we know how many copies exist of each value. We are allowed to partition all cards into any number of non-empty groups.

We are given a permutation of numbers from 1 to n, initially arranged on a line of cubes. We are also given a target permutation of the same numbers.

We are asked to construct a set of dice for up to five players. Each player gets one die, and all dice have the same number of sides, denoted by k, with k at most 120.

Let $x_1\ldots x_{2^n}$ be a truth table of length $2^n$.

We have a circular arrangement of $n$ sectors, each sector initially holding at most one envelope. After several rounds, only $k$ envelopes remain, and their exact positions on the circle are known in clockwise order.

We are given an upper bound $n$, and we consider all ordered pairs $(x, y)$ where both values lie between 1 and $n$. For each such pair, we run a modified version of the Euclidean algorithm.

We are given a sequence of digits of length $n$. For each position $i$, we impose a condition on the number $x + i$: when written in decimal, it must contain the digit $di$ somewhere in its representation.

We start with a calculator that only stores a single real number and repeatedly applies one of six unary functions to it: sine, cosine, tangent and their inverses. The initial value is fixed at zero.

We are given an $n times n$ grid where each cell is either already painted black, already white, or empty. Empty cells are candidates where Eva may optionally draw a special black triangle.

We are given an $n times n$ grid where each cell has a fixed color value. A cube starts at the top-left cell and must be moved to the bottom-right cell.

We are given a rectangular grid representing a wall, where each cell is either blocked or free. On the free cells we are allowed to place up to two additional rectangular bricks.

We are given a set of computers, each equipped with a single outgoing wire. If a computer uses its wire directly, sending one bit takes a fixed amount of time equal to its own delay value. In addition to these wires, there is a hub with a limited number of ports.

We are given four numbers representing the current lengths of the legs of a table. The table is stable only when all four legs end up having exactly the same length.

We are working with a set of binary “sticks”, each stick being a sequence of length n where every position is either quartz or onyx. A finished product, a nunchuck, is formed by choosing two sticks and joining them end to end.

We are given an unknown permutation π on n elements, but we are not allowed to see it directly. Instead, we can feed the system any permutation f, and we receive back a transformed permutation g defined by conjugation through π, meaning that every value is relabeled by π…

We are given a set of distinct integers representing cards held by different players. There is also an initial pile that starts with a single card of value 0.

We are given a single year in the range from 1995 to 2019. Each year has a fixed contest winner (or a pair of winners in one special case), and the task is to output exactly the winner string corresponding to the given year.

We are given several passwords, each a short string of length at most 50 made of digits and English letters. For every password, we must construct a collection of strings, all of the same length as the password, such that if we take the bitwise XOR of ASCII codes column by…

We are given a boolean formula built from variables a through h and the operators not, and, and or with standard precedence rules. The formula defines which subsets of variables are considered valid. Each variable corresponds to a band member who may or may not be present.

We are dealing with a complete knockout tournament with $2^k$ participants, where every match is a fair coin flip between two players.

We are given two separate structures that must interact: a context-free grammar in Chomsky Normal Form and a directed graph whose edges are labeled by lowercase letters.

We are given a single year in the range from 1995 to 2019. Each year has a fixed contest winner (or a pair of winners in one special case), and the task is to output exactly the winner string corresponding to the given year.

We are given a collection of operations on a single integer variable that starts with value c. Each operation has the form “if the current value equals a, then replace it with b”, otherwise it does nothing.

We are given a sequence of daily oil prices and we want to inspect every contiguous subarray of length at least three. For each such segment, we look at the sequence of day-to-day differences inside it.

We are given a hidden array of length $n$, containing a permutation of numbers from $1$ to $n$. We cannot see the array directly. Our only way to interact with it is to pick two positions $i < j$ and ask the judge to compare the values at those positions.

We are given a library of pixel fonts where each printable character is represented as a fixed bitmap of size $w times h$. Each bitmap is a grid of and ., where means the pixel is lit and . means it is dark.

A cyclist is choosing between two monthly bike rental plans, and we need to compute the total cost of each plan over a fixed month. Each day, the cyclist uses the bike for a total of T minutes.

We are given a mine placement on a strip of length $N$, where each position in the top row either contains a mine or is empty.

Let $f(x_1,\dots,x_n)$ be a monotone Boolean function.

We are given a directed acyclic structure with $N$ glades (nodes) and $M$ directed paths (edges). Each path has two fixed weights: the number of mushrooms and the number of berries collected when traversing it. Importantly, these values do not change across days.

I don’t have the actual problem statement for Codeforces 104014K - Old Barabarian Rap in your message (it’s empty in the prompt), so I can’t reliably reconstruct the intended task or produce a correct editorial.

I can’t reliably write an editorial for “Codeforces 104014H - Match of the Millennium” because the actual problem statement (and even the contest archive context) isn’t included in your prompt.

I can write the full editorial in the required format, but I’m missing the actual problem statement for Codeforces 104014I - Piecewise Linear Functions.

There is a row of cities, each city containing a known number of attractions. In every city, a telescope is installed. Each telescope has a non-negative integer power.

We are given a text consisting of multiple words, and each word is supposed to follow a simple phonetic rule of a fictional language. A word is valid if it is either a “noun” or a “verb” under this language definition.

Let $F$ denote the family of 5757 SGB words represented on variables $a1,dots,z5$ as in (131), and let the associated ZDD be constructed in the standard ordered way with variables processed in lexicographic order.

We are given a final version number of a project, written as a positive integer $N$. The project evolves month by month according to a fixed rule: if the current version has $k$ digits, then after one month it increases by the number consisting of $k$ ones.

We are given a vending machine that contains a collection of snack packs, each with a name and a price expressed in rubles and kopeks. The buyer also has a fixed amount of money, also expressed in the same format.

We are given a fixed sequence of digits 1 2 3 4 0 in that order, and we are allowed to insert arithmetic operators between them, optionally group parts using parentheses, and optionally concatenate adjacent digits to form multi-digit numbers.

We are given a long digit string s that was formed by taking an array of positive integers and concatenating them without separators. Each original array element is a positive integer that does not contain the digit zero.

Let $F$ denote the family of 5757 SGB words represented on variables $a1,dots,z5$ as in (131), and let the associated ZDD be constructed in the standard ordered way with variables processed in lexicographic order.

We are given several strings, each consisting only of opening and closing brackets. We are allowed to reorder these strings arbitrarily and then concatenate them into one long sequence. While scanning this final sequence from left to right, every position defines a prefix.

We are working on an $n times m$ grid where each cell is either usable or blocked, and the grid changes over time as we toggle individual cells. After each toggle, we must report how many distinct “staircase paths” exist in the current grid.

We are given a virtual address representation p = \pi(p)2^e + \sigma(p), \quad \pi(p)=p \gg e,\quad \sigma(p)=p \bmod 2^e, and we must show that a BDD node stored at address $p$ does not need to store...

We are given three piles of balls, each pile having a different color. In one move, we pick two balls of different colors, remove both, and replace them with a single ball of the third color.

We are given a string of lowercase English letters. The goal is to transform it into a string where characters never decrease when read from left to right, meaning each character is at least as large in alphabetical order as the previous one.

We are given a sequence of piles of coconuts, where each pile has some integer size. We are allowed to choose a single positive integer base value $x$, and then we only collect coconuts from those piles whose size is an exact positive power of $x$, meaning values of the form…

We are given a collection of problems, where each problem is described by two attributes: a topic label and a difficulty value.

Let $F$ denote the family of 5757 SGB words represented on variables $a1,dots,z5$ as in (131), and let the associated ZDD be constructed in the standard ordered way with variables processed in lexicographic order.

We are given an array of integers and we first compute its average value, which is the total sum divided by the number of elements. This average is not necessarily an integer, but it is a fixed rational value determined by the full array.

We are dealing with a rectangle with unknown side lengths, say $a$ and $b$, both strictly positive real numbers. We are not given the rectangle directly. Instead, we are told two aggregated pieces of information about its sides.

We are given a very small game board with exactly two rows and $n$ columns. A player starts at the top-left cell and wants to reach the bottom-right cell. Some cells are blocked by traps, and stepping onto a trap immediately makes the path invalid.

We are given a fixed number of candies and a school split into two groups of students: boys and girls. The principal must choose a positive integer amount of candies for each boy, and a different positive integer amount for each girl.

I don’t have the actual statement for Codeforces 104017K - Gastronomic Event in your prompt (the input/output sections are empty), so I can’t reliably reconstruct the problem or produce a correct editorial without guessing.

Let $F$ denote the family of 5757 SGB words represented on variables $a1,dots,z5$ as in (131), and let the associated ZDD be constructed in the standard ordered way with variables processed in lexicographic order.