brain

Let $f$ be represented by a reduced ordered binary decision diagram, and let $F(p)$ denote the reliability polynomial under the specialization $p1=cdots=pn=p$.

We are given two teams of size $N$. Team A starts with strengths $1, 2, 3, dots, N$, while team B starts with strengths $2, 3, 4, dots, N+1$. So the two teams are identical sequences shifted by one.

We are given a one-dimensional corridor of $N$ cells arranged in a line. A robot starts in the middle cell, specifically at index $lceil N/2 rceil$, and initially faces to the right.

We are given two horizontal rows of points on a grid. One team stands on the bottom edge at positions $(1,0)$ through $(n,0)$, and the other team stands directly above at $(1,n)$ through $(n,n)$.

We are given a rectangular grid of integers. Before the game starts, we are allowed to flip signs of entire rows and entire columns any number of times, where flipping a row or column multiplies every value in it by -1. After all flips are chosen, the board is revealed.

Let $v$ be a node of the reduced ordered BDD for $f$, and let $Fv(p)$ denote the reliability polynomial of the subfunction represented at $v$ under the specialization $p1=cdots=pn=p$. Let $F'v(p)$ denote its derivative with respect to $p$.

We are given a sequence of distinct integers representing rating changes from upcoming contests. We are allowed to reorder these values arbitrarily and feed them to a process that simulates how many contests a player ends up actually playing.

We are given a string of lowercase letters and a target number of occurrences of the pattern “awa”. We are allowed to modify characters freely, but each modification changes exactly one position to any lowercase letter.

We are given a grid with dimensions $n times m$. Each operation allows Rabbit to shrink the grid by removing some positive number of rows or some positive number of columns.

We start with a permutation of size $n$, and we are allowed to perform swaps of any two positions freely. Each swap exchanges the values at two indices, so in effect we are working in the full symmetric group where any transposition is allowed.

Let $f(x1,ldots,xn)$ be a Boolean function and let $$G(z)=sum{x1=0}^1 cdots sum{xn=0}^1 z^{x1+cdots+xn} f(x1,ldots,xn)$$ be its generating function as defined in the preceding exercise.

The Chinese ring puzzle (Baguenaudier) has a standard representation as a binary state vector $(a_1,\dots,a_n)$ in which each $a_j \in {0,1}$ encodes whether ring $j$ is disengaged or engaged, and leg...

Let $f(x1,ldots,xn)$ be a Boolean function and let $$G(z)=sum{x1=0}^1 cdots sum{xn=0}^1 z^{x1+cdots+xn} f(x1,ldots,xn)$$ be its generating function as defined in the preceding exercise.

I don’t have the actual statement of Codeforces 103828N - Moving grid in the prompt, so I can’t safely reconstruct the solution or write a correct editorial without risking inventing the wrong problem.

The problem statement for “Codeforces 103828K - No Server? !Problem” is not included in your prompt, so there is no way to reconstruct the task, constraints, or required output.

I can’t write a correct editorial for this yet because the actual problem statement is missing. Right now I only have the title “Codeforces 103828M - Mobile Game”, but no description of the rules, input/output, or constraints.

I can write the full editorial in exactly that format, but I’m missing the actual problem content. The statement section is empty in your prompt (“ Problem Statement / Input / Output” have no details), so I don’t know what “Even Adjacent Product” is asking us to…

Let $G(z)=sum{x1=0}^{1}cdotssum{xn=0}^{1} z^{x1+cdots+xn} f(x1,ldots,xn)$ be the generating function defined in Exercise 25, and let $F(p)$ denote the reliability polynomial when $p1=cdots=pn=p$, so that $$F(p)=sum{x1=0}^{1}cdotssum{xn=0}^{1} (1-p)^{1-x1}p^{x1}cdots…

I can’t write a correct Codeforces editorial for “103828I - Bombing buildings” because the actual problem statement is missing from your prompt. Right now there is no description of: - what the buildings are (array, graph, coordinates, etc.

I cannot responsibly write a correct Codeforces editorial for “103828H - Kite” yet because the actual problem statement is not available in the prompt, and it is not retrievable from standard Codeforces archives.

I can’t reliably write a correct editorial for this yet because the actual problem statement is missing from your prompt.

I can’t write a correct editorial for Codeforces 103828F - Subgrid without the actual problem statement. Right now the prompt only includes formatting instructions, but the core information is missing: what the grid contains, what defines a valid subgrid, what needs to be…

We are simulating a very small “text editor” that starts empty and receives a sequence of key presses. Each key press is one of a few control actions that manipulate three conceptual pieces of state: the current text, the clipboard, and whether the entire text is currently…

I can’t reliably write a correct editorial yet because the actual problem statement for “Codeforces 103828A - 2 Arrays Problem” is missing from your prompt. Without the statement, anything I write would be guesswork, and that would break correctness for an editorial.

Let $H$ be an $mtimes n$ parity-check matrix over $mathbb{F}2$, and let $$f(x)= [Hx=0], qquad x=(x1,dots,xn)^T.$$ Fix a variable order $x1,dots,xn$.

I can write the full editorial in the exact format you want, but I’m missing the actual problem content for Codeforces 103828B - “Too simple for a hard problem?”.

The problem describes a fictional currency system with two types of money: wooden coins with several fixed denominations, and a single gold coin whose value in wooden coins is unknown.

There are up to 17 shops, and each pair of shops has a travel cost, possibly zero meaning no direct connection. You are allowed to move between shops in any order, paying those costs, and you start at shop 1 for free.

The Chinese ring puzzle (Baguenaudier) has a standard representation as a binary state vector $(a_1,\dots,a_n)$ in which each $a_j \in {0,1}$ encodes whether ring $j$ is disengaged or engaged, and leg...

Let $H$ be an $mtimes n$ parity-check matrix over $mathbb{F}2$, and let $$f(x)= [Hx=0], qquad x=(x1,dots,xn)^T.$$ Fix a variable order $x1,dots,xn$.

We are given a circular arrangement of beads, where each bead is painted with one of M colors. The structure is a necklace, so positions wrap around: after position N comes position 1 again.

I can’t responsibly write a full editorial yet because the problem statement I initially saw is incomplete and likely mismatched.

The problem can be viewed as navigating a maze laid out on a grid. Each cell of the grid represents either free space that can be walked on or a blocked cell that cannot be entered.

The problem statement section is empty, so there isn’t enough information to reconstruct what “Rhomboid tiles” is asking.

I can’t write a correct editorial yet because the actual problem statement for Codeforces 103831B - Skis sorting is not available in the provided prompt, and it is not accessible from standard problem archives.

I can’t write a correct editorial yet because the actual problem statement for Codeforces 103833H - Parallel computing is missing from your message.

I can write the full editorial in exactly that structure, but I’m missing the actual problem content. “Codeforces 103833G - Clustering evaluation” isn’t included in your message, and I don’t have the statement, input/output format, or constraints needed to reconstruct…

Let $H$ be an $mtimes n$ parity-check matrix over $mathbb{F}2$, and let $$f(x)= [Hx=0], qquad x=(x1,dots,xn)^T.$$ Fix a variable order $x1,dots,xn$.

We are given a rooted tree. Each vertex carries a value that is either +1 or −1. You start at a fixed vertex at time zero with an initial life value of 1. Time advances in discrete steps.

I can’t reliably proceed yet because the actual problem statement for Codeforces 103833F - Yet another unusual equation is not included in your prompt, and this specific gym/problem ID does not have a stable canonical statement in the standard archive I can reconstruct with…

We are modeling a penalty shot as a discretized grid over the goal. Each cell in the grid corresponds to a possible shot placement. For the goalkeeper, each cell contains a value describing how likely it is that the goalkeeper saves a shot directed there.

The problem statement for Codeforces 103833A - Liberdance is not included in your prompt, so there is no way to reconstruct the required logic or produce a correct editorial.

The problem statement section is empty, so there isn’t enough information to reconstruct the task or produce a correct editorial. Please paste the full statement of Codeforces 103833B (input, output, constraints, and description).

The problem starts with a sequence of numbers and asks us to compute a global expression over all pairs of elements. For each array that appears in the input, we are effectively aggregating a function that depends on pairwise differences between elements.

Let $H$ be an $mtimes n$ parity-check matrix over $mathbb{F}2$, and let $$f(x)= [Hx=0], qquad x=(x1,dots,xn)^T.$$ Fix a variable order $x1,dots,xn$.

We are working on a circular arrangement of $N$ positions, each position carrying a label, most naturally interpreted as a binary type such as 0 or 1, or more generally two kinds of “buttons”.

We are given a game played on a grid-like structure where each state can be thought of as a rectangular region with two dimensions. Two players alternate moves, and each move modifies the current active region by effectively trimming it along its boundary.

Let $f = \text{COLOR}(x_1,\dots,x_n)$ be the Boolean function encoding proper 4-colorings of the US map, where each vertex variable $x_i$ takes values in ${0,1,2,3}$, represented in binary as in (73).

We are working on a grid where every cell contains a value, and we care about paths that move from the top-left corner to the bottom-right corner.

We are given a system that builds and manipulates collections of marbles, where each marble can be thought of as an atomic unit that may later be grouped into larger sets.

We are working with a sequence of stones, each stone having a color and a value. The operation that generates contribution is not local to a single stone, but depends on a triple of indices $i < j < k$.

We are given a sequence of stones arranged in a line, each stone having a color and a weight. The first structural observation is that consecutive stones of the same color can be compressed: within any maximal block of identical colors, only the maximum weight in that block…

We are given a set of weighted points on a grid. Each point represents a monster located at some coordinate $(x, y)$ and contributing some value (or implicitly one unit of value if weights are not explicitly stated in the statement variant).

We are given a string formed from characters that can be compared in a fixed “Rock Paper Scissors” style cycle, but the important restriction is that the dynamics we simulate only meaningfully depend on how characters compare pairwise.

We are working with a system that has a fixed target configuration of colors over several regions, and a set of tools that can modify those colors. Each region can take one of several possible colors, and each tool changes colors in a structured way.

We are repeatedly taking two groups of points in the plane and transforming them into a new pair of groups using a deterministic geometric rule.

We are given a configuration of chords drawn inside a convex polygon. Each chord connects two boundary vertices, and different chords may intersect each other inside the polygon.

Let $H$ be an $mtimes n$ parity-check matrix over $mathbb{F}2$, and let $$f(x)= [Hx=0], qquad x=(x1,dots,xn)^T.$$ Fix a variable order $x1,dots,xn$.

We are given a timeline of $n$ days. Some days are fixed rest days (legal holidays). All other days are working days initially. We are allowed to convert additional working days into rest days, and these converted days are called paid leave.

We are given an $n times m$ grid. We may choose any subset of cells and place a camera in each chosen cell. Each camera placed at $(i, j)$ does not directly “cover” a fixed shape; instead, it can be configured by selecting another cell $(p, q)$, and this configuration turns…

Each test case describes a hidden “answer key” for a 10-question multiple choice quiz, where every question has exactly one correct option among A, B, C, and D.

We are given a binary string that is not written explicitly, but compressed as alternating runs of equal characters. Each run tells us how many consecutive zeros or ones appear, and runs always alternate between the two characters.

We are given several independent games, each consisting of multiple integer intervals. Each interval represents a set of currently “alive” integers, starting as all integers from $li$ to $ri$. There is also a fixed multiplier $p 1$.

Let $f = \text{COLOR}(x_1,\dots,x_n)$ be the Boolean function encoding proper 4-colorings of the US map, where each vertex variable $x_i$ takes values in ${0,1,2,3}$, represented in binary as in (73).

We are given a tree whose vertices are labeled from 1 to n. The task is to remove some edges so that the remaining connected components satisfy a strong ordering constraint: every component must correspond exactly to a contiguous segment in label order.

We are given an array of up to 21 positive integers and an initial value $x$. We are allowed to reorder the array arbitrarily. After fixing an order, we process the elements one by one, repeatedly updating $x$ by replacing it with $x bmod ai$.

Let $H$ be an $mtimes n$ parity-check matrix over $mathbb{F}2$, and let $$f(x)= [Hx=0], qquad x=(x1,dots,xn)^T.$$ Fix a variable order $x1,dots,xn$.

We are given a Tetris grid of width $w$ and a very small height, initially filled only in the bottom $n le 15$ rows. Above that, everything is empty. Some cells in these bottom rows are already occupied. No row is completely filled.

We are given a permutation of the numbers from 1 to n, and we run a modified selection sort on it. For each position i from left to right, we scan the suffix to the right of i and swap whenever we find a smaller element than the current value at position i.

We are given a short program consisting of two types of instructions stored in an array-like structure. The execution does not run this list directly in order once; instead, it builds a second structure, a queue, and executes instructions from it dynamically.

We are given a set of underground exits, each exit connects the cave to the outside world and has its own travel cost to a refrigerator.

Let $H$ be an $mtimes n$ parity-check matrix over $mathbb{F}2$, and let $$f(x)= [Hx=0], qquad x=(x1,dots,xn)^T.$$ Fix a variable order $x1,dots,xn$.

We are given a length-n array that starts completely zero. Instead of observing the array directly, we are told the final state only through a binary string: each position tells us whether the final value at that index is zero or not.

We are given an undirected graph where vertex 1 is the starting hub and vertex n is the final target we must eventually clear. Every vertex except 1 contains a boss with an initial strength. The player also has a strength value and evolves over time.

We are given several independent test cases. In each test case, there is an array of integers, and we need to count how many pairs of positions $(i, j)$ with $i < j$ satisfy a specific inequality involving multiplication and addition of the chosen elements.

Let $H$ be an $mtimes n$ parity-check matrix over $mathbb{F}2$, and let $$f(x)= [Hx=0], qquad x=(x1,dots,xn)^T.$$ Fix a variable order $x1,dots,xn$.

The prompt is missing essential information needed to write a correct editorial. The sample input and sample output in the statement are corrupted and do not line up, and there is no complete problem specification.

We are given a grid where each cell already has a fixed color, or is still undecided. The final goal is to assign colors to all undecided cells so that the resulting fully colored grid contains as many valid 2 × 2 “checker” blocks as possible.

We start from the binary representation of an integer $k$ with $n$ bits: k = (b_{n-1} b_{n-2} \dots b_0)_2,\quad b_j \in \{0,1\}, and we extend the notation by setting $b_n = 0$.

We are given a long sequence of days, each day having a numerical happiness value that can be positive, zero, or negative.

We are dealing with a three-player card game involving Alice, Bob, and Prof. Pang. Each player has a private hand of cards, and all cards are distinct. Cards are ranked only by their face value, with Ace being highest and 2 being lowest, while suits are irrelevant.

We are given a rooted tree with node 1 as the root. A depth-first search is run on this tree, and the only constraint is that each node can visit its children in any order.

We are given a digit string and asked to consider every substring of it. For each substring, we look at ways to split it into exactly six consecutive nonempty parts.

Let $H$ be an $mtimes n$ parity-check matrix over $mathbb{F}2$, and let $$f(x)= [Hx=0], qquad x=(x1,dots,xn)^T.$$ Fix a variable order $x1,dots,xn$.

The problem defines a sequence of values $f1, f2, dots, fn$ that must be computed in increasing order. Each $fi$ depends on a direct contribution term $ci$, a scaling term $bi$, and a history-dependent quantity $ai$.

We are given a sequence of values on cities, and we want to travel from city 1 to city N. Moving from a city i to a later city j has a cost that depends on the bitwise xor of the values between them, specifically the xor of the segment from i+1 to j, followed by a fixed shift…

Let $H$ be an $m times n$ binary matrix and let $$f(x) = [Hx = 0],$$ where arithmetic is over $mathbb{F}2$. The BDD for $f$ is constructed under a fixed ordering of variables $x1,dots,xn$ as in Section 7.1.4.

We are given a set of horizontal or slanted “roads” that can be thought of as intervals on the x axis, each equipped with a y coordinate representing its height.

An $n$-tuple $(a1,dots,an)$ is admissible when it satisfies the alternating constraint $$a1 le a2 ge a3 le a4 ge cdots .$$ Let $mathcal{A}n$ denote the set of all such binary $n$-tuples.

Let $n$ be fixed and let each array location be indexed by an $n$-bit integer.

We are working on a grid where a fixed “intended path” is already given implicitly, and the task is to place obstacles so that this path becomes the only viable way to traverse under the movement rules.

We are working with a weighted undirected graph where some vertices are marked as “cool”. The goal is not to compute ordinary shortest paths, but something stronger: for any pair of cool vertices, we consider all possible paths between them and look at the largest edge…

We are given a graph with labeled edges, and two entities moving on it simultaneously: Waymo and Thomas. Each state of the system is described by a pair of positions, one for Thomas and one for Waymo.

We are looking at a stochastic elimination process built around repeated rounds of a three-choice game where each participant independently picks one of three options.

The task revolves around the text that would be produced if we expand a piece of code that represents a matrix-style computation.

We construct a mixed-radix decrementing analogue of Algorithm M, operating in the same state space 0 \le a_j < m_j \quad (1 \le j \le n), but traversing it in reverse lexicographic order by repeatedly...

We are given a sequence of students, each associated with a trust value that changes as we process the system. At any moment, some students are considered to have positive trust, and we are interested in two things: how many students currently have positive trust, and the sum…

We are given a rectangular grid of integers with n rows and m columns. Each cell contributes a value, and we can compute the sum over any subrectangle using a function f(a, b, c, d), which means summing all cells in rows a through b and columns c through d.

I can’t produce a correct editorial for Codeforces 103870G - XOR Fun yet because the actual problem statement is not available from standard archives or mirrors, and it is not identifiable from known Codeforces contests or gym indexes with certainty.

The flaw in the previous solution is not in the identification of trailing maximal components, but in how step M4 is counted inside a single transition.