brain

We are given a complete bipartite graph, meaning every vertex on the left side is connected to every vertex on the right side, and there are no edges inside a side. We are also given the numbers from 1 to n + m, and we must place each number exactly once on one of the vertices.

We are given a single lowercase string of length $n$. From this string, we consider every suffix, meaning for each position $i$, we look at the substring starting at $i$ and continuing to the end.

We are given a recursively defined ordering of all binary strings of length $n$. This ordering is the classic Gray code construction: for $n=1$ it is simply $0,1$, and for larger $n$ it is built by taking the previous sequence, prefixing all elements with $0$, then taking the…

A string is written over a totally ordered infinite alphabet.

We are working with an array of values indexed from 1 to n, where each index carries a weight. For each query, we are given a segment of indices from l to r, and we are allowed to pick some indices from this segment and arrange them into pairs.

We are given several types of items, each type having a value and a weight, and we can pick items repeatedly, including picking the same type multiple times. A shopping plan is an ordered sequence of picks, and each pick chooses one item type independently.

We are given two arrays of length $n$. Alice has $n$ cards with fixed values, and Bob also has $n$ cards but plays them in a fixed order. Over $n$ rounds, Alice is allowed to choose the order in which she plays her cards. In each round, the two revealed values are compared.

We are given a one-dimensional game world with positions on the integer line. There are pillars at coordinates from 0 to n, where pillar i has a treasure value ai for i ≥ 1, while pillar 0 is the starting point and has no treasure.

We are asked to take all integers from 1 to n and partition them into several cycles. A cycle is an ordered sequence, and every number must appear in exactly one cycle.

We are given a chronological log of kill events in a match. Each event states that one player eliminates another player. Although the victim immediately respawns and can be killed again, we only care about the sequence of who killed whom over time.

We are given a network of cities connected by roads, where the structure forms a tree. Every city may contain several ACMers, and each person wants to attend a gathering held in exactly one city.

We are given a collection of straight line segments drawn on an infinite plane. Each segment is defined by two endpoints with integer coordinates. As more segments are added, they intersect each other at most once per pair, and possibly only at endpoints or not at all.

We are working on a grid of integer lattice points forming an $(N+1)times(N+1)$ square. Some of these grid points are marked as supply depots. Each depot is equally likely to be chosen. Independently, Tanya’s starting position is also a uniformly random grid point.

We are asked whether we can construct two permutations of the same length, but drawn from two different value ranges, such that each index forms a pair with a very rigid arithmetic constraint. One array is a permutation of the integers from 1 to N.

We are given several independent scenarios of an orange orchard. In each scenario, there are $n$ trees, and the $i$-th tree contains $ai$ oranges. From each tree, we are allowed to pick any integer number of oranges between $0$ and $ai$, independently of other trees.

We are given a system of constraints over an array of values, one value per city. Each city $i$ has a non-negative integer $xi$, representing the number of friends in that city.

We are given a scenario where there are $n$ identical bananas and $m$ monkeys. Every monkey must receive at least one banana, so we are really distributing $n$ into $m$ positive integers $a1, a2, dots, am$ with total sum fixed.

We are given the total number of points scored by two volleyball teams across an entire match, but we are not told how those points are distributed across individual sets or who won each set.

We are given a game played on a single string of lowercase letters. Two players alternate turns, starting with the first player.

We are given a set of students, each student being distinguishable but otherwise identical in role. We are allowed to form multiple groups, where each group is just a chosen subset of these students. Each group must satisfy two structural constraints.

We are given a sequence of integers representing elements placed in a row. The allowed operation depends on position: if the element currently sitting at position $k$ (1-indexed) has value exactly $k$, then that element is eligible to be removed. The process is not arbitrary.

I can’t reliably write an editorial for this problem as-is because the actual problem statement (inputs, operations, and required output) is missing. Right now I only have the title, constraints placeholders, and formatting instructions.

Let $S_n$ be the set of permutations of ${1,2,\dots,n}$.

We are simulating a propagation process where a fixed number of “patch cables” acts as a limited resource to spread updates across a growing set of computers.

We are given several closed integer intervals, each defined by a left endpoint and a right endpoint. The task is to determine how many integers lie inside every single one of these intervals at the same time. In other words, imagine each interval as a segment on the number line.

We are given a line of N hostel rooms, each room having a certain cleaning cost or weight associated with it. The management wants to assign sweepers in a structured way: instead of choosing arbitrary rooms, they must select a periodic pattern.

We are given an array of integers, and the only allowed operation is to pick one occurrence of the current maximum value in the array and decrease it by one. There is a restriction that prevents choosing the same position in two consecutive operations.

We are given a number of identical bricks and we want to build a staircase using them. Each stair has a positive integer height, and the staircase must strictly increase from one step to the next.

We are given a collection of identical objects and a container capacity. Each container can hold at most $K$ objects, and we want to place all $N$ objects into containers. The task is to determine the minimum number of containers required when we pack optimally.

We are given a grid of contestants by problem, where each pair $(c, p)$ represents one contestant solving one problem. Over time, we receive a chronological stream of submissions, each submission belonging to one such cell and carrying a verdict.

We are given a long sequence of days encoded as a string. Each character represents whether Cosenza has an exam on that day or not. A day marked E is an exam day, while F is a free day.

Let $f(x1,dots,xn)$ be a Boolean function, and let $G(z)$ be its generating function in the sense of Exercise 25, so that $$G(z)=sum{xin{0,1}^n} f(x), z^{w(x)},$$ where $w(x)=x1+cdots+xn$ is the Hamming weight of $x$.

Something went wrong while generating the response. If this issue persists please contact us through our help center at [help.openai.com](https://help.openai.com/).

We are given a fixed set of student registration numbers, each consisting of exactly six digits (leading zeros are allowed, so numbers like 000123 are valid and distinct from 123000).

The task is to simulate how a group of students occupy seats in a bus and accumulate the total time spent until everyone is seated. The bus can be viewed as a grid with N rows, each row containing four fixed seats: two window seats and two aisle seats.

We are given an array of length $N$, initially filled with zeros, representing how many bullets each student currently has. Two types of operations are performed online.

Let $p$ be a prime.

Let $$G(z)=sum{xin{0,1}^n} f(x),z^{x1+cdots+xn}.$$ Then $$G(-1)=sum{x} f(x),(-1)^{ where $ Write $f$ in its unique multilinear expansion over $mathbb{R}$, $$f(x)=sum{Ssubseteq [n]} aS prod{iin S} xi,$$ so that $a{[n]}$ is the coefficient of the full monomial $x1x2cdots xn$.

We are given a collection of precedence rules between named people, where each rule says that one person must appear before another in a valid ordering. Each name is just a string, and every rule is directed from a prerequisite to a dependent.

We are given a sentence split into a sequence of words. Each word consists only of uppercase English letters. The task is to determine whether the sentence satisfies a specific structural property involving its first word.

Let the alphabet have size $m$, totally ordered.

We are given a polynomial built as a product of independent factors. Each factor contributes a small set of possible powers of $x$, and when we multiply all factors together we obtain a final expanded polynomial.

Let $f(x1,dots,xn)$ be represented by an ordered reduced BDD with root node $r$. For each node $k$ in the BDD, write $V(k)$ for its variable index, and write $mathrm{LO}(k)$ and $mathrm{HI}(k)$ for its two successors. The sinks are $bot$ and $top$.

We are given a two-pan balance where each side can hold at most n identical items at once. There are m items in total, and exactly one of them is “bad”, meaning it differs in weight from all others, though we do not know whether it is heavier or lighter.

We are given a collection of items, where each item has two values: a label and a cost. The label is treated as a string or number that can be concatenated with others. We are allowed to pick any multiset of these items, meaning we may reuse the same item multiple times.

We are given an $n times n$ grid where each cell contains a distinct integer. Two agents move on this grid simultaneously. Ginger starts at the top-left cell and initially faces downward. His clone starts at the bottom-right cell and initially faces upward.

We are asked to reconstruct a full password of length $k$ over lowercase English letters. The password is not arbitrary: it must be non-decreasing in lexicographic order, meaning each character is at least as large as the previous one in the alphabet order.

The input is essentially a long encoded list of integers, where each integer is wrapped by vertical bars and appears in order. If we strip the formatting, we obtain an array of weapon attack values, each associated with its position in the original string.

We are given a permutation of the numbers from 1 to n. You can think of it as a directed graph where every node points to exactly one other node, and because it is a permutation, every node also has exactly one incoming edge. This structure breaks into disjoint directed cycles.

We are given a collection of straight lines in the plane, each described by an equation of the form $y = ai x + bi$. The task is to count how many unordered pairs of distinct lines intersect at exactly one point. Geometrically, two lines intersect if they are not parallel.

We are given an array of monster health values. Before anything starts, we are allowed to reduce each value independently, but we can never increase it beyond its original value. After that preparation, we choose a starting position $i$.

We are given an array of integers and a modulus value k. From the array, we can choose any non-empty subsequence, meaning we pick some indices while preserving order, but order itself does not affect the sum so effectively we only care about which elements are selected.

We are given two positive integers for each test case, call them $x$ and $y$. In one move, we are allowed to pick two divisors $a$ of $x$ and $b$ of $y$, with an extra constraint that $a$ and $b$ share no common prime factors.

We are given a line of houses indexed from 1 to n. Each house starts with an initial bank balance. Then a sequence of events happens over time. Some events modify balances over a whole interval of houses by adding a value that can be positive or negative.

We are given an array of integers and a fixed modulus value $m$. One player, Berta, gets to choose two elements that end up in the same group, and she wins if the sum of those two elements is divisible by $m$.

We are interacting with a judge that has fixed but hidden integers $x$ and $y$, each up to $10^{18}$. Our only way to learn about them is by asking queries of the form $(a,b)$, and receiving back the value of $gcd( The goal is to determine both coordinates exactly, using at…

We are given a process that evolves on an undirected connected graph with $n$ vertices. The process depends on a sequence of integers $a1, a2, dots, am$, which we can think of as “distance jumps over time”. A starting vertex is chosen and marked as painted at day 0.

We are given several independent scenarios, each describing a group of friends attending a theater. Each friend has already been assigned a seat, specified by a pair of integers representing a row and a column. The seating layout is unusual in how columns are numbered.

We are given a rewriting system over strings. Each rule describes how a single character can expand into a two-character string. If a rule says c → ab, then every time we choose a position in the string containing c, we are allowed to replace that character by the pair ab.

Let $f(x1,dots,xn)$ be represented by an ordered reduced BDD with root node $r$. For each node $k$ in the BDD, write $V(k)$ for its variable index, and write $mathrm{LO}(k)$ and $mathrm{HI}(k)$ for its two successors. The sinks are $bot$ and $top$.

We are given a row of $n$ connected cups. Each adjacent pair of cups is linked by a straw placed at a certain height $Ai$. Water is poured only into the first cup, and then it can propagate through these connections depending on how much water has accumulated.

We are given several independent test cases. In each test case, there are n types of cookies, and the i-th type has a pile size ai. The task is to completely consume all cookies, but the consumption is constrained by a very specific daily rule.

We are building a sequence that depends on a starting value and a rule that keeps “rounding up” to multiples of increasing indices. We choose a positive integer $m$, which becomes the first element $a1$.

We are given a set of points in the plane. Each point behaves like a source that emits a ray. Initially every ray points straight down. Over time, all rays rotate counterclockwise at the same constant speed.

Let $f(x1,dots,xn)$ be represented by an ordered reduced BDD with root node $r$. For each node $k$ in the BDD, write $V(k)$ for its variable index, and write $mathrm{LO}(k)$ and $mathrm{HI}(k)$ for its two successors. The sinks are $bot$ and $top$.

We are given a grid where each cell either contains one diagonal segment or is blocked. A diagonal connects two opposite corners of a cell, so every non-blocked cell contributes a single edge between two grid vertices.

We are given a collection of sets, each set containing some elements from a universe whose size is also bounded by the number of sets. Each set also has an associated cost, which can be negative or positive. We are allowed to pick any subset of these sets, possibly empty.

We are given an integer array and we are allowed to modify it using a very specific operation: in one move we pick a non-negative integer m and a positive increment k, and then we add k either to a prefix of length m+1 or to a suffix of length m+1.

Each test case describes a football squad split into four fixed groups: goalkeepers, defenders, midfielders, and forwards. Every player has a fixed skill value.

Let $S_n$ be the set of permutations of ${1,2,\dots,n}$.

Let $f(x1,dots,xn)$ be represented by an ordered reduced BDD with root node $r$. For each node $k$ in the BDD, write $V(k)$ for its variable index, and write $mathrm{LO}(k)$ and $mathrm{HI}(k)$ for its two successors. The sinks are $bot$ and $top$.

The problem statement for “Codeforces 103810E - Экспедиция” is missing from your prompt, so there’s no way to reliably reconstruct the intended task, constraints, or solution strategy.

I don’t have the actual statement for Codeforces 103810D - Высадка in your prompt, and without it I’d be guessing the entire solution structure.

I’m missing the actual problem statement for Codeforces 103810A - right now only the title “Упаковка (Packing)” is provided, with no input/output description or constraints.

We are missing the actual statement of Codeforces 103810C, and this is not publicly inferable from contest metadata alone. Without the problem text, any “editorial” would be fabrication rather than explanation, which would defeat the purpose of a Codeforces-style writeup.

Let the contribution of a minterm corresponding to an assignment $x1 ldots xn$ be $$C(x1,ldots,xn)=prod{i=1}^n (1-pi)^{1-xi}pi^{xi}.

We are given a probabilistic system that simulates what happens when Justin clicks on a video. Each time he enters or refreshes the page, exactly one of several outcomes occurs according to fixed probabilities.

The problem statement is missing in your prompt (the Input/Output sections are empty), so there isn’t enough information to reconstruct what “Kario Mart” is asking.

I can’t write a correct, meaningful editorial for this problem yet because the actual statement of Codeforces 103811I - Inno Per Gli Sconfitti is not reliably available from the information provided, and the version that appears in search results is inconsistent and clearly…

I can’t reliably reconstruct Codeforces 103811G - Gold Medal Bout from the ID alone, and I don’t want to invent a problem statement and accidentally write a convincing but wrong editorial. Please paste the full problem statement (or even a screenshot / text dump).

I can’t write a correct editorial for this yet because the actual problem content is missing. Right now the prompt only includes the title and empty “Problem Statement / Input / Output” sections, but nothing describing the rules, constraints, or what “getting rice”…

I’m missing the actual problem statement for Codeforces 103811E - Escape the Cube, and without it I can’t reliably reconstruct the intended solution or write a correct editorial.

I can’t reliably write a correct editorial for Codeforces 103811F - Furthest Travel because the actual problem statement (graph/array rules, constraints, and what “travel” means here) is missing from your prompt.

I can’t reliably write an editorial for Codeforces 103811D - Double Queue without the actual problem statement.

I don’t have the actual problem statement for Codeforces 103811A - Allowance Exhaustion in your message, so I can’t safely reconstruct the solution or write a correct editorial without risking inventing details.

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

The problem statement for Codeforces 103811B - Boat Assignment is not included in your prompt, so I don’t have the actual rules, input format, or required output to base an editorial on.

Let $w(x1,ldots,xn)$ denote the contribution of a minterm $$(1-p1)^{1-x1}p1^{x1}cdots (1-pn)^{1-xn}pn^{xn}.$$ Maximizing this quantity over all assignments satisfying $f(x1,ldots,xn)=1$ is equivalent to maximizing a product of independent local factors along a path in the BDD…

Let $f_n(k)$ be the binary de Bruijn cycle of order $n$ constructed in Exercise 97, so that the infinite periodic sequence f_n(0), f_n(1), \ldots, f_n(2^n-1) contains every $n$-bit string exactly once...

We are given a permutation of the numbers from 1 to N, and we define a score by scanning the permutation from left to right. Every time we place a value P[i], we look at all earlier values P[j] and add 1 if P[j] divides P[i].

We are given a timeline from moment 1 to moment M and a collection of movies, each represented by a closed interval [L, R], meaning the movie starts at time L and finishes at time R.

We are given a collection of balls where each ball has a color, so the input is essentially a multiset over colors. Alongside this, we are asked to distribute all balls into exactly K boxes.

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 a set of points on a plane. Each point represents a tree placed at a fixed coordinate, and each tree independently “survives” with a given probability. If a tree survives, it becomes part of the final active set.

We are given a grid of digits where equal digits that touch by edges form connected components, exactly like standard 4-direction flood-fill regions.

Each test case describes one week of planning. For every week, we are given seven small integers, each representing how many jokes are told on a specific day from Saturday through Friday.

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 $(n,n)$.