brain

We are given a line of tiles, initially each tile has height 1. Over time, the heights only increase. Each operation selects a contiguous segment and adds the same value to every tile in that segment.

Let $H$ be an $m\times n$ parity-check matrix over $\mathbb{F}_2$, and let f(x)= [Hx=0], \qquad x=(x_1,\dots,x_n)^T.

We are given a quadratic curve that models a drunk coworker’s path across a rectangular room. At any horizontal position $x$, the coworker is located at height $f(x)$, where $f$ is a quadratic function.

We can think of the building as an infinite rooted structure starting from room 0. Each room generates new rooms in the level above it, but the branching factor depends on the parity of the room: even-indexed rooms expand into a rooms, and odd-indexed rooms expand into b rooms.

We are given pairs of large integers representing counts of two types of parts, bowls and lids. From each pair, the number of complete toilets Thomas can assemble is determined by the greatest common divisor of these two quantities.

We are given a queue of employees, each associated with a fixed typing duration. There are $N$ employees standing in order, and we can place $M$ computers in front of them. At time zero, the first $M$ employees each occupy one computer and start typing.

We are given a sequence of $N$ time slots, each slot containing three independent offers: one from Jim, one from Dwight, and one from Kevin. In slot $i$, choosing Jim yields $ai$ ideas, Dwight yields $bi$, and Kevin yields $ci$.

We are given a circular target on a 2D plane and a set of points representing where employees throw an object. The task is to count how many of these points land inside or exactly on the boundary of the circle. Each throw is just a coordinate on the plane.

We are given a target string consisting only of the characters T and C. The task is to count how many different ways this string can be formed using a fixed set of stamps. Each position in the string is produced by choosing one of four stamp types.

Let $U=\{1,\dots,n\}$ with variable order $1<2<\cdots<n$.

We are given a prefix of natural numbers from 1 up to some limit $n$, and we want to choose as many of them as possible under a single restriction.

Two trucks move along a straight road made of five consecutive segments. Each segment has a fixed length, and two of these segments are “bad road” segments where movement becomes slower for both vehicles.

We are given several short strings made of lowercase English letters. For each string, we need to decide whether it is “valid” under a rule that depends on how letters alternate between two classes: vowels and consonants. The rule is applied after a preprocessing step.

We are given a positive integer $N$. We need to construct the smallest positive integer that satisfies two conditions at the same time: it must be divisible by $N$, and its decimal representation must end in the digit zero. Ending in zero means the number is a multiple of 10.

We are given a tree of up to $n$ vertices, where each vertex represents a star. The tree is rooted implicitly by the input construction, but conceptually it is just an undirected tree defined by $n-1$ edges. For each star, Mu chooses it as a viewing center.

Two players build small combat teams, each consisting of at most seven units placed in a fixed left-to-right order. Every unit starts with a single attribute value, which simultaneously acts as its hit points and its attack power.

We are given a simple polygon described by its vertices in counterclockwise order. On each vertex sits an object, and we want to count how many subsets of these vertices a group of thieves could choose, under a strong geometric constraint.

We are given an $n times m$ grid where each cell contains a species label. The grid represents a rigid matrix formation of dancers. The only way the configuration can change is through operations triggered by showing cards. A white card labeled $k$ affects row $k$.

We are given $n$ quartz types, and each type has two prices: a first piece price and a second piece price. Every type has exactly two pieces, but the second piece only becomes available after the first one of that type has been bought.

We are given a collection of strings. From all substrings of all these strings, we are interested only in those substrings that are palindromes. Each such palindrome can be used as a building block.

We are given two independent weighted networks on the same set of cities. One network consists of roads and the other consists of railways.

We start with a connected simple undirected graph. We are allowed to insert any number of missing edges, as long as we never introduce self-loops or duplicate edges. Every different subset of edges that we choose to add counts as a different construction.

Let $F$ be a forest on $\{1,\dots,n\}$ whose vertices are labeled in preorder, and let a(F)=\{\operatorname{anc}(1),\dots,\operatorname{anc}(n)\}.

We are asked to fill an $n times m$ binary matrix, each cell being either 0 or 1, and then consider every subrectangle formed by choosing a contiguous block of rows and a contiguous block of columns.

We are given a fixed-length sequence of 5 characters describing the outcomes of a best-of-five series between DRX and T1.

We are given a sequence of numbers and asked to apply a single global “compression” operation defined by an interval $[l, r]$, where the interval length is limited by $r - l le d$.

We are given two players, Alice and Bob. Each of them does not pick from a discrete list, but from a continuous set of real numbers. Their allowed numbers are described as a union of several disjoint closed intervals.

We are given a length $n$, and we must construct a binary string of that length. The goal is not to satisfy any pattern constraint, but to maximize how many distinct nonempty substrings appear in the string.

I’m missing the actual problem statement for Codeforces 104162I - “Гладкие числа”, and without it I can’t produce a correct editorial.

We are given a string consisting of multiple types of brackets, specifically parentheses, square brackets, braces, and angle brackets. The interpretation of “correctness” here is not the standard single-pair matching rule used in classical bracket problems.

The problem statement for “Codeforces 104162H - Выращивание кроликов” is missing from your prompt, so there’s no way to reconstruct the solution logic, constraints, or intended algorithm correctly.

I can’t reliably write a correct editorial for this yet because the actual problem statement for Codeforces 104162G - “Очередная скобочная последовательность” is missing from your prompt.

We are given a sequence of mushrooms placed along a line, each with an initial weight. From this initial configuration, pairs of adjacent mushrooms can interact in a deterministic way: every unit of time, between any two neighboring mushrooms, a new mushroom appears whose…

I can’t write a correct Codeforces editorial for this yet because the actual problem statement (input, output, and rules) is missing. Right now I only know the title “104162D - Зима в городе К”, but that isn’t enough to reconstruct the task reliably.

We are given a city represented as a line of positions, where each position can be thought of as a point on a number line. Some of these positions contain restaurants that can prepare food, and we also have a starting point that represents the delivery hub.

I can’t write a correct editorial for 104162A without the actual problem statement. The title “Блочная сортировка” isn’t enough to reconstruct the task reliably, and guessing here would likely produce a completely wrong solution and analysis.

I can write the full editorial in the exact format you want, but I’m missing the actual problem statement for Codeforces 104162B (“Особенные числа”). Right now I only have the title, which is too ambiguous to reliably reconstruct the task.

I don’t have the actual problem statement for Codeforces 104164D - “Соберите станок”, and without it I can’t reliably reconstruct the intended model, constraints, or solution strategy.

We are given a collection of points in the plane, where each point has an associated direction. The task is to determine how these directed points relate to each other under the rules implied by their geometry, and compute a final quantity derived from these directional…

I can’t reliably write a correct editorial yet because the actual problem statement for Codeforces 104164B - “Объединение чисел” is missing from your prompt.

I can’t reliably reconstruct Codeforces 104164C (“Тотальный mex”) from the title alone, and I don’t want to invent a solution that might be wrong.

Let $f$ be a Boolean function of variables $x_1,\dots,x_n$ given by a reduced ordered BDD.

We are given a positive integer $n$, and we consider all factor pairs $(x, y)$ such that $x cdot y = n$ with both $x$ and $y$ positive integers.

We are given a rooted tree where each vertex carries a positive value and each edge carries a positive weight. The root is fixed at node 1, and every other node has exactly one parent.

Let $f(x_1,\dots,x_n)$ be a Boolean function represented by an ordered reduced binary decision diagram with variable order $x_1 \prec \cdots \prec x_n$.

We are given a permutation of length $n$, which means it is a rearrangement of numbers from $1$ to $n$. From this permutation, we want to end up with a smaller sequence of length $m$, consisting of distinct values, and we are told exactly which values must survive.

We are given an array of positive integers. We are allowed to repeatedly modify individual elements using an operation of the form “replace a value by its remainder when divided by some chosen positive integer”.

We are given a fixed set of points on a 2D plane, stored in an array order from 1 to n. Each query specifies a contiguous segment of this array, and asks for the closest pair of distinct points whose indices both lie inside that segment.

We are given a game built around a perfectly uniform n-sided dice whose faces contain all integers from 0 to n − 1 exactly once. The game has two stages. First, Putata rolls the dice and obtains a value x. After seeing x, Budada gets a single decision.

We are simulating a very simple but constrained decision process over time. A player encounters an event every fixed number of seconds, and at each encounter they may or may not be able to act depending on whether a cooldown has finished.

We are given a two-segment motion starting from a point. First a segment of fixed length $l1$ is drawn from the origin, producing a point $Y$. From $Y$, a second segment of fixed length $l2$ is drawn to a final point $Z$.

We are given a sequence of n geese arranged in a line, where each goose is associated with a task type ai. A “plan” is chosen by selecting a contiguous segment of geese, meaning an interval [l, r], and only those geese participate in completing their tasks.

Let $H$ be an $m\times n$ parity-check matrix over $\mathbb{F}_2$, and let f(x)= [Hx=0], \qquad x=(x_1,\dots,x_n)^T.

We are given a sequence of digits, each digit between 1 and 9, written as a single string. We are allowed to insert exactly k plus signs into this string, splitting it into k+1 contiguous groups.

We are given a directed acyclic graph where every edge goes from a smaller indexed node to a larger indexed node, with an additional guarantee that the gap between endpoints is small. Each edge is either black or white. From vertex 1, we can reach every other vertex.

We are given a grid that is slightly larger than the standard one, with $(n+1)$ rows and $(m+1)$ columns. Each cell of this grid is independently determined to be black or white, but the way black cells appear is not given directly per cell.

We are given a rooted tree where nodes are numbered from 1 to n and each node i (except the root) has a parent pi with pi < i. This means the tree is already given in a constructive order, where every node appears after its parent.

We are asked to construct a binary grid with $n$ rows and $m$ columns, where each cell is either white (0) or black (1). The grid must satisfy two structural constraints that enforce global uniqueness in both directions. First, every row must be distinct from all previous rows.

We are given a rooted tree with nodes numbered from 1 to n, where node 1 is the root and every other node has exactly one parent. This tree represents a hierarchy. Each node has a set of immediate children. We want to form a collection of disjoint groups of nodes.

We are given a string composed of lowercase Latin letters. From this string, we are allowed to construct new strings by repeatedly choosing a character, writing it down, and then splitting the remaining string into the part strictly to its left and the part strictly to its right.

We are given a graph of cities where each city currently belongs to one of two factions, labeled 1 or 2. Some cities are “modifiable”, meaning we are allowed to flip their faction, while others are fixed and cannot be changed.

We are given a sequence of rectangular building blocks that are added one by one to construct a larger rectangular base. After the first day, we start with a single rectangle.

The reviewer identifies the central defect: the assumption $B_0(f)=B(f)$.

We are given a sequence of points, each point having up to 10 coordinates. We must cut this sequence into several contiguous blocks. Every point belongs to exactly one block. For any block, its cost is defined as the largest L1 distance between any two points inside that block.

We are given a tree of cities. Two people start at two different nodes: one is the police, the other is you. Each second, both of you move simultaneously to an adjacent city or stay in place, and both have full knowledge of the tree.

We are given a set of chickens, each with a positive weight. We also have a fixed number of biscuits. The goal is to distribute biscuits so that every chicken receives a nonnegative integer amount, and all chickens receive biscuits in strict proportion to their weights.

We are given the final marks of all students in a class. Your own mark is hidden, but you know one extra fact: your mark is not the maximum among all students. The rank of a student is defined as one plus the number of students who scored strictly higher.

We are given a directed graph where each vertex represents a friend’s house and each directed edge represents a one-way road between two houses.

We are given a collection of independent encounters, each corresponding to a rose in a field. When Rose chooses a rose, she triggers a local “chase scenario” involving a monster that spawns relative to that rose.

We are given a very large rectangular canvas, conceptually a grid with coordinates up to $10^9 times 10^9$. On this canvas, Bob has already painted several axis-aligned rectangles.

We are simulating a deterministic falling process on integers. For every starting height from 1 up to a given limit $H$, we release a “raindrop”. Each raindrop moves downward in discrete one-second steps.

We are given an $n times n$ grid where the value in cell $(i, j)$ is defined as the integer division $leftlfloor frac{j}{i} rightrfloor$. Row index $i$ and column index $j$ both start from 1. The task is to count how many cells in the entire grid evaluate to a fixed integer $k$.

We are given a sequence of energy drinks that Alberto consumes in a fixed order. Each drink contributes some amount of energy, and once he starts a drink he fully consumes it before moving on.

Let $L \subseteq {0,1}^n$ be a language of fixed-length binary strings and let $f(x_1,\dots,x_n)$ be its characteristic Boolean function.

We are given two collections of integers. One represents required brownie sizes requested by friends, and the other represents available baking tins, where each tin produces exactly one brownie of its own fixed size.

We are given a starting amount of rainwater collected on day one, denoted by an integer $i$. This value determines everything about the rest of the week.

We are given a fixed-length sequence of 28 real numbers, each representing the probability of rain on a particular day in February. Each value lies between 0 and 1. A day is considered “rainy” only if its probability meets or exceeds 0.8.

Let $a_1 \dots a_n$ be a restricted growth string with a_1 = 0,\qquad a_{j+1} \le 1 + \max(a_1,\dots,a_j)\quad (1 \le j < n).

We are given a directed graph where every intersection has exactly one outgoing road. If we start at any node and keep following the outgoing edge, we deterministically move to another node in one minute per step.

We maintain a dynamic collection of integers, each representing a chemical. Over time, we insert new values into this set.

Algorithm C computes, for every node of the BDD, the number of satisfying assignments represented by the subgraph rooted at that node.

We are given a directed graph where each node represents a friend’s house and each directed edge represents a one-way road. You are allowed to choose any starting house, then repeatedly travel along directed roads, possibly revisiting houses and roads multiple times.

Each rose can be thought of as an independent “encounter” that offers a reward: if Rose successfully deals with that rose, she earns one point toward the total number of roses collected.

We are given a very large grid, conceptually of size $10^9 times 10^9$, but we never work with it explicitly. Instead, we are told about $N$ non-overlapping axis-aligned rectangles drawn on this grid. Each rectangle contributes a set of unit cells that are initially painted.

Solution to TAOCP 7.1.4 Exercise 259.

Each integer height from 1 up to H represents a raindrop that is released once, and each drop falls independently until it reaches height 1.

We are given a sequence of energy sources that Alberto consumes strictly in order. Each source contributes a fixed amount of energy, and once consumed it cannot be revisited or split. After every completed workout set, Alberto’s energy is fully reset to zero.

Let $f$ be a Boolean function on variables $x_1,\dots,x_k$ and let its BDD be ordered with $x_1 < x_2 < \cdots < x_k$.

We are given a set of friends, each of whom wants a brownie of at least a certain minimum size. We are also given a collection of baking tins, each tin producing exactly one brownie of a fixed size.

We are given an $n times n$ grid where each cell is determined by its row $i$ and column $j$. The value in that cell is the integer division result $leftlfloor frac{j}{i} rightrfloor$. In other words, each row $i$ is formed by dividing all column indices by $i$, rounding down.

The key mistake in the rejected solution is the attempt to encode coefficients as additional atoms.

We are working with permutations that can be transformed using a restricted swapping operation: only elements that are not adjacent in the array are allowed to be swapped, and swaps can happen through intermediate states.

We are asked to count how many ways we can split a sequence of integers into several consecutive segments such that each segment satisfies a bitwise OR condition that depends on a fixed target value.

We are given an array of integers, and we are allowed to perform an operation where we pick a contiguous subarray and sort it in non-decreasing order while keeping the rest of the array unchanged.

We are given three chips placed on integer coordinates on a line. A single move allows us to pick one chip and move it to another position under a fixed rule implied by the process: the relative structure of the three positions is what matters, not their absolute location.

The process in this problem evolves over time in discrete seconds. During a single full cycle of length n, the system behaves consistently: you perform some number of upgrades, you execute some number of clicks, and those clicks generate a certain number of paperclips.

We are given a set of participants, each described by two numbers: a strength value and a riding speed. We want to choose some of them and arrange them in a line so that strength never decreases from front to back, and speeds also never decrease, while also ensuring that…

We are given a circular string of length $n$, and from it we define $n$ “individuals” by taking every cyclic rotation of this string. So the $i$-th individual is simply the original string rotated so that position $i$ becomes the first character.

Let $x \in \mathbb{N}$ with binary expansion x = 2^{e_1} + \cdots + 2^{e_t}, \quad e_1 > \cdots > e_t \ge 0.

We are given a sequence of incoming attacks, each with a strength value. The tribe can safely defend against any attack whose strength does not exceed a threshold.