brain

We are given three piles of candies. Two of the piles are guaranteed to start with the same size, while the third may differ. Two players alternate turns.

Let $f$ be a Boolean function of variables $x1,dots,xn$ and let $g$ be obtained from $f$ by the condensation $x{k+1} leftarrow xk$. Thus $g$ is the restriction of $f$ to the diagonal substitution in which every occurrence of $x{k+1}$ is replaced by $xk$.

We are given three sorted sequences of positive integers. Each sequence represents the heights of saplings loaded in a separate truck, and within each truck the saplings are already sorted in non-decreasing order.

Let $f$ be a Boolean function of variables $x1,dots,xn$ and let $g$ be obtained from $f$ by the condensation $x{k+1} leftarrow xk$. Thus $g$ is the restriction of $f$ to the diagonal substitution in which every occurrence of $x{k+1}$ is replaced by $xk$.

The problem statement section is empty, so there isn’t enough information to reconstruct what Codeforces 103720B (“Мëд для Михаила”) is asking.

Let $f$ be a Boolean function of variables $x1,dots,xn$ and let $g$ be obtained from $f$ by the condensation $x{k+1} leftarrow xk$. Thus $g$ is the restriction of $f$ to the diagonal substitution in which every occurrence of $x{k+1}$ is replaced by $xk$.

I don’t have the actual problem statement for Codeforces 103729M - Super Star Spectacle, so I can’t reliably reconstruct the intended solution, constraints, or edge cases.

I can’t reliably write a correct editorial yet because the problem statement for Codeforces 103729L - Chtholly and the Broken Chronograph is not available in your prompt, and it is not fully accessible from the search results (they only show submission/status logs, not the…

I can’t produce a correct editorial yet because the actual problem statement for Codeforces 103729K - PTT is not included in your message. Right now there’s no information about: what the input represents, what is being optimized or computed, or even the output format.

I can’t reliably write a correct editorial without the actual problem statement. “Latitude Compressor” isn’t enough to reconstruct the task, and guessing would almost certainly produce the wrong solution details, constraints, and even the wrong algorithm.

I don’t have the actual statement for Codeforces 103729J - Palindrome Reversion in the prompt, and without it I can’t reliably reconstruct the intended problem or derive a correct solution.

We are given two binary strings of equal length. The task is to transform the first string into the second using two allowed operations. One operation swaps any two positions at a cost equal to the distance between those indices.

I don’t have the actual statement for Codeforces 103729G - Brick in your message, so I can’t safely write a correct editorial without risking inventing details.

Let $g$ be obtained from $f$ by setting $x{k+1} leftarrow xk$. Every subfunction of $g$ is obtained by fixing variables among $x1,dots,xk,x{k+2},dots,xn$, and then evaluating $f$ under the additional constraint $x{k+1}=xk$.

We are given a line of $n$ holes labeled from left to right. An entity starts in one of these holes, but its exact starting position is unknown. Every minute, before we act, we are allowed to inspect exactly one hole.

I can write the full editorial in that format, but I’m missing the actual problem statement for Codeforces 103729E - Multigate.

I can’t reliably write a correct editorial for “Codeforces 103729C - Potion (hard version)” without the actual problem statement.

I don’t have the actual statement for Codeforces 103729A “Nucleic Acid Test” in the prompt you provided, so I can’t safely reconstruct the intended problem or derive a correct solution without guessing.

A string is written over a totally ordered infinite alphabet.

We are given a set of points on a 2D plane. Each point is a node, and Klee can move directly between any pair of points. The cost of moving depends only on which quadrants the two endpoints lie in.

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 $x$.

We are maintaining a string that starts empty and is modified by a sequence of operations. Each operation either appends a lowercase character to the end, removes the last character if one exists, or performs a global substitution that replaces every occurrence of a given…

We are given an array of lower bounds on variables and a target sum constraint. Each variable $xi$ must be at least $ai$, and we are asked how many integer vectors $x1, x2, dots, xn$ satisfy $$x1 + x2 + dots + xn le s$$ After each update, one position of the array $a$ is…

We are given a straight road from position 0 to position p, and Alice starts at 0 and wants to reach p. Along this road there are several fixed bike stops where she is allowed to switch between walking and biking, but biking is only meaningful between consecutive stops she…

We are given a sequence of integers and asked to count how many contiguous subarrays have a sum divisible by a given integer $k$.

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 $x$.

We are given an $n times n$ grid that behaves like a small maze. Each cell is either free space, an obstacle, or contains one of two special starting positions labeled for two players.

We are working with a tree, meaning a connected acyclic graph where every pair of vertices is connected by exactly one simple path. For each query vertex x, we need to count how many distinct simple paths in the tree pass through x.

We are given a custom ordering of the English lowercase alphabet, where the 26 letters appear in a specific sequence that defines a strict total order. If a letter appears earlier in this ordering, it is considered smaller.

We are given a single lowercase string and asked to count how many times the fixed pattern "hznu" appears as a contiguous block inside it.

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 $x$.

We are given a 500 by 500 integer grid in the plane, so all relevant coordinates live in a small bounded box. Each input item describes a unit diagonal segment that connects a lattice point $(xi, yi)$ to $(xi-1, yi-1)$.

We are given a permutation of length n, and we conceptually process its prefixes one by one. For each prefix A[1..k], we imagine running a given sorting procedure called SORT, and we are asked to record how many times a specific variable m gets assigned during that run.

A string is written over a totally ordered infinite alphabet.

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 $x$.

We are maintaining a dynamic sequence of integers. The sequence starts empty and grows only by appending elements to the end. At any moment, we may be asked to compute a value derived from all pairs of elements where the first element is to the left of the second.

We are given several long strings, and we want to extract a single “good” substring that behaves consistently across all of them. The requirement is that this substring must appear inside every given string at least k times.

We are given a permutation of numbers from 1 to n, but we do not know its order. Instead of directly constructing it, we are given constraints between positions. Each constraint is of the form “the value placed at position x is smaller than the value placed at position y”.

We are given an interval of integers from l to r, and a parameter k. A number is considered “bad” if it is divisible by p^k for some prime number p. Equivalently, a bad number contains a prime factor whose exponent in its factorization is at least k.

We are given an array of length $n$. For every contiguous subarray $[l, r]$, we define a value equal to the difference between the maximum element and the minimum element inside that subarray.

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 $x$.

We are given a set of points indexed in a cycle. Each point is repeatedly updated by a geometric transformation that depends on the current centroid of all points and on its two cyclic neighbors.

We are given several independent jobs, where each job represents preparing a contest problem. Every problem consists of two sequential stages: first it must be created, and only after that it can be checked or validated. There are n problems in total and m available people.

We are given a connected undirected graph. Each query gives four vertices $u, v, x, y$. From these four vertices we must build two paths.

We are given a string made only of the characters A, B, and C. You should think of it as a row of diamonds arranged left to right. The process allows us to repeatedly pick a consecutive block of three diamonds forming the pattern A, B, C in the current configuration.

We are counting binary tree shapes under a strict structural rule. Each node either has two subtrees or none, and what matters is not keys or labels but only how many nodes are inside each subtree.

We are given multiple queries, each asking us to construct a string over lowercase English letters such that the number of special subsequences inside that string equals a given integer $n$.

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.