brain

We are standing on an infinite grid starting at the origin. Each test gives a target coordinate, and we must construct a walk that eventually reaches that point. The movement rule is unusual: time is split into phases.

We are given a set of rectangular boxes placed somewhere on a large 2D plane. Each box is described by four corner points in counterclockwise order, so every obstacle is a convex quadrilateral with arbitrary orientation, not necessarily axis-aligned.

We are given an array whose length is exactly a power of two, specifically $2^k$, where $k$ can be as large as 20. Each operation takes the current array and throws away exactly one half of it, either the left half or the right half, and keeps the other half unchanged.

We are given an integer array and a special value $k$. The process is dynamic: as long as the array still contains at least one element equal to $k$, we repeatedly apply an operation that decreases every positive element of the array by one.

We are interacting with a hidden string of length at most 100. The string uses exactly two distinct lowercase letters, but we are not told which ones. Our only way to learn about it is by asking whether a chosen pattern appears as a contiguous substring of the hidden string.

We are given a tree, meaning a connected acyclic graph on $n$ nodes with $n-1$ edges. The task is to count how many distinct subgraphs of a specific shape appear inside this tree.

We are given a binary string where some positions are already fixed as 0 or 1, while others are unknown and marked with ?. We are allowed to replace each ? with either 0 or 1, producing a fully binary string.

We are given a directed or undirected weighted graph of rooms connected by portals. The traveler starts in room 1 and wants to reach room n. Every time he enters a room, he collects exactly one item called a “longzhu”, and each longzhu has one of 7 types.

We are given a large integer range $[A, B]$, and for each query we must count how many integers inside this range satisfy a special divisibility condition. For a number $x$, we compute the product of its decimal digits. Call this value $P(x)$.

We are asked to evaluate a double summation over Fibonacci numbers indexed by pairs of integers up to a very large bound.

We are given a connected undirected graph with exactly one more edge than a tree would have. That means the graph contains exactly one cycle somewhere inside it, while the remaining edges form tree-like attachments to that cycle.

We are building an array of length $n$ using a process that grows it in chunks. Initially the array is empty. At any moment, if the current length is $m$, we choose a number $x$ between $1$ and $n-m$. After choosing $x$, we append the sequence $1, 2, 3, dots, x$ to the array.

Let $r_s,\dots,r_0$ satisfy t = r_s + \cdots + r_1 + r_0,\qquad 0 \le r_j \le m_j \quad (s \ge j \ge 0).

We are given a directed graph on up to one hundred thousand vertices, but the edges are not explicitly listed in the input. Instead, the graph is generated internally using a deterministic pseudo-random generator seeded by two integers.

A row of wine bottles is placed from left to right, each bottle having a fixed intrinsic value. Every day exactly one bottle is removed from either the left end or the right end of the current row.

We are given two line segments in the plane for each test case. Each segment is defined by two endpoints with integer coordinates. The task is to compute how much of these two segments overlaps along their geometry, but only when the overlap forms a segment of positive length.

We are modeling a deterministic transfer process that starts with a single pile of apples and moves through a line of m children. The first child receives all n apples.

We are given an initial text buffer whose only relevant property is its length, call it x. From this starting state, we want to transform the buffer so that its length becomes exactly n using a set of keyboard operations.

We are given a sequence of axis-aligned rectangles that all sit on the x-axis, meaning every rectangle has its bottom edge on $y = 0$. Each rectangle $i$ spans an interval $[Li, Ri]$ on the x-axis and has height $Hi$.

We are simulating a deterministic transfer process of apples across a line of $m$ children. The process starts with an initial number $n$, which is given to the first child.

We are given a rectangular grid with $n$ rows and $m$ columns. Each cell $(i, j)$ contains the value $i cdot j$. So row 1 is $1, 2, dots, m$, row 2 is $2, 4, dots, 2m$, and so on.

We are given two 64-bit non-negative integers, a target sum $S$ and a target xor value $X$. We are allowed to construct an array of non-negative integers, possibly empty, such that the sum of its elements equals $S$ and the bitwise xor of its elements equals $X$.

We are given a collection of words, and we must produce their corresponding translations in the same order. The only way to obtain translations is through an interactive machine. The machine behaves in two consistent ways. If we query a single word, it returns its translation.

We are given a contiguous range of integers from $a$ to $b$, and initially each number stands alone as its own pile. We are allowed to merge two piles if we can find a number in each pile that shares a prime factor $t$ with $t ge p$.

We are given a set of digits from 0 to n, and we are asked to consider all possible permutations of these digits. Each permutation is interpreted as a number by concatenating the digits in order.

We have a box initially containing two types of indistinguishable balls, n white and m black. We repeatedly remove one ball uniformly at random until the box becomes empty.

An additive alphametic in the sense of Section 7.2.1.2 assigns distinct decimal digits to distinct letters so that a formal arithmetic identity between words becomes a true equality of base-10 integers.

I don’t have the actual statement of Codeforces 103464D “A Task With Queries” in the prompt, so I can’t reliably reconstruct the operations, constraints, or intended solution path.

We are given a collection of stegosauruses, each associated with a single integer value representing the number of spikes it has.

I can’t reliably write a correct editorial for “Codeforces 103464C - Protected String” because the actual problem statement is missing.

Let a 4-note chord be a 4-combination $c_4c_3c_2c_1$ with $n > c_4 > c_3 > c_2 > c_1 \ge 0.$ A single “adjacent-key move” replaces exactly one $c_j$ by $c_j \pm 1$ while preserving strict inequalities...

Let $S{s,t}$ denote the set of all bitstrings $a{n-1}cdots a0$ with $n=s+t$ containing exactly $t$ ones, and let $C{s,t}$ denote the corresponding set of index lists $ctcdots c1$ with $nctcdotsc1ge 0$, ordered lexicographically as in (3).

We are given a fixed triangle in the plane, defined by three vertices with integer coordinates. Along with this triangle, we are given a point $P$, which is supposed to lie on the boundary of the triangle.

We are given a collection of space stations indexed from 0 to n. Each station has an integer energy value, and station 0 is our starting point.

We are given a closed world consisting of people distributed across rooms, with no empty rooms. Every person belongs to exactly one of three behavioral types: supporters of the marriage, opponents of the marriage, and neutral participants.

Let $S{s,t}$ denote the set of all bitstrings $a{n-1}cdots a0$ with $n=s+t$ containing exactly $t$ ones, and let $C{s,t}$ denote the corresponding set of index lists $ctcdots c1$ with $nctcdotsc1ge 0$, ordered lexicographically as in (3).

This is a simulated poker tournament where five fixed-strategy players repeatedly participate in a series of independent rounds.

We are given several test cases. In each test case, there is a range of integer distances from L to R. For each integer distance d in this range, the problem defines a value f(d), which counts how many integer-coordinate points in 3D space lie exactly at Euclidean distance d…

We maintain a dynamic list of strings, indexed from 1 to n, which can be reordered by swap operations. Alongside this evolving list, we receive queries asking about a segment of indices [l, r] and a query string q together with a threshold k.

We are given an $n times n$ grid where a subset of cells are marked as absorbing “holes”. A token starts from a fixed cell $(r, c)$ that is guaranteed not to be a hole.

We are given a set of integers from 1 up to n. The task is to determine the smallest number k such that no matter how we choose any subset of size k from this range, we are guaranteed to find two distinct numbers u and v inside that subset where one divides the other.

We are given an integer grid where each cell contains a value, and we need to count certain special paths formed by moving through adjacent cells in the grid.

We are given two strings, each of length $2n+1$, over the alphabet ${A,B,C}$ plus wildcard characters ?. Each ? can be replaced independently by any of the three letters.

Codeforces 103469M: Math

We are asked to construct a short integer array such that the number of its subsets whose sum is exactly zero equals a given value K.

We are given a fixed permutation $p$ of size $n$, and a partially specified permutation $q$ of the same size. Some positions in $q$ are known, others are zero and must be filled so that the final sequence becomes a valid permutation of $1$ to $n$.

We are given a collection of axis-aligned rectangles in the plane. Each rectangle is defined by a closed x-interval and a closed y-interval, so it represents a solid filled region.

We are given a complete graph where every pair of vertices is connected by an edge colored either blue or yellow. The input is essentially an $n times n$ symmetric matrix encoded as characters, where each off-diagonal entry tells us the color of an edge.

Let a 4-note chord be a 4-combination $c_4c_3c_2c_1$ with $n > c_4 > c_3 > c_2 > c_1 \ge 0.$ A single “adjacent-key move” replaces exactly one $c_j$ by $c_j \pm 1$ while preserving strict inequalities...

Codeforces 103469H: Hamiltonian

We are given a pair of values $(a, b)$ over a finite field modulo a prime $p$. Each query asks for the minimum number of operations needed to transform a starting pair $(ai, bi)$ into a target pair $(ci, di)$, where each operation applies one of two deterministic transformations.

We are given a hidden simple undirected connected graph on $n$ vertices. We cannot see its edges directly. Instead, we can query any subset of vertices and receive the number of edges whose both endpoints lie entirely inside that subset. The task is not to reconstruct the graph.

We start with a sequence of labels from 1 to n, arranged in increasing order. The only allowed operation is to pick two adjacent elements in the current sequence, remove them both, and pay a cost that depends on the original labels of the two removed elements.

We are given an array of integers that changes over time through point updates. After every update, we need to compute a special “weight” of the array. To compute this weight, we first sort the array in non-decreasing order.

We are given a target string length l and a set of distinct positions a1, a2, ..., an, each in the range [1, l]. These positions are known to be exactly the palindromic prefix lengths of some unknown string, except that some of them may have been deleted.

We are given a set of integers that is known to come from some hidden array. The process that produced this set is as follows: take every contiguous subarray of the hidden array, compute the bitwise AND of that subarray, and collect all distinct results.

I don’t have the actual problem statement for Codeforces 103470M - Windblume Festival, so I can’t reliably reconstruct the task or derive a correct solution.

I can’t reliably write a correct editorial for Codeforces 103470L - Secret of Tianqiu Valley because the actual problem statement is missing from your prompt.

I can’t write a correct editorial for this yet because the problem statement is missing. Right now, all we have is the title “Xingqiu’s Joke”, but no description of the input, output, constraints, or rules.

The problem statement is missing from your prompt (both the description and input/output sections are empty), so I can’t reconstruct the intended solution or write a correct editorial.

We are given a tree with $n+1$ vertices and an ordering of $n$ weights. The process starts by choosing any vertex as a root and marking it black.

The problem statement is missing from the prompt, so there isn’t enough information to derive the solution, constraints, or even the underlying task (graph, DP, combinatorics, etc.).

The problem statement is missing, so there isn’t enough information to write a correct editorial. “Cloud Retainer’s Game” could correspond to multiple unknown tasks, and without the actual rules, constraints, or input/output definition, any solution would be guesswork…

I can’t write a correct editorial for this yet because the actual problem content (operations, constraints, and required queries) is missing from your prompt.

The problem statement for Codeforces 103470F - Paimon Polygon is missing from your prompt, so I don’t have the actual rules, inputs, or required computation to build a correct editorial.

We are given an array of integers, and we repeatedly apply a very unusual sorting procedure. The procedure does not compare neighboring elements like bubble sort, instead it compares every pair of positions.

The problem statement is missing from your prompt (both the narrative and constraints are blank), so there isn’t enough information to reconstruct the task, edge cases, or solution strategy.

We are given an integer array representing values on a line, and we are allowed to optionally pick one contiguous segment and increase every value inside that segment by a fixed constant k.

The problem statement is missing from the prompt, so there isn’t enough information to reconstruct the task, constraints, or required algorithm.

I can’t reliably write a correct editorial for this yet because the problem statement is missing. “Codeforces 1034715 - Квантовые вычисления” isn’t enough to reconstruct the task, and anything I produce without the full statement would risk inventing…

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

Let $n > c_t > \cdots > c_1 \ge 0$ with the constraints from exercise 57 and the additional condition $c_{j+1} > c_j + 1 \qquad (t > j \ge 1).$ Define the shifted variables $d_j = c_j - (j-1), \qquad...

We are given a single integer T that represents time in seconds after a stone is dropped into an infinite grid of cells. The stone creates an evolving pattern starting from one central cell. At time zero, only the starting cell is active.

The problem statement section is empty, so there isn’t enough information to write a correct or meaningful editorial. A Codeforces editorial depends heavily on the exact operations, constraints, and output definition.

I don’t have the actual problem statement for “Codeforces 1034712 - Фишки на поле” in your prompt, so I can’t produce a correct editorial yet without guessing.

The problem defines a non-standard way of writing numbers using powers of two, where each power of two can be used 0, 1, or 2 times.

The problem statement section is empty, so there isn’t enough information to derive a correct solution or write a valid Codeforces-style editorial.

I can’t reliably write a correct, full editorial for “Codeforces 103476D - Planar Tree” as-is because I don’t have the actual problem statement or constraints.

The problem statement is empty in the prompt, so there isn’t enough information to reconstruct the task, constraints, or required output.

I don’t have the actual statement for Codeforces 103476C - Divisor Circle in the prompt, and I can’t reliably reconstruct the problem from the title alone without risking hallucinating key rules.

We are given several circles drawn on an infinite plane. These circles interact only in a very structured way: any two circles are either completely separate, one fully contains the other, or they just touch.

We are given a sequence of integers that is claimed to be generated by a linear recurrence with modular reduction. The sequence starts from a known value and evolves using fixed parameters $a$ and $b$, but the modulus $p$ is unknown.

We are given an array $A$ of length $n$. For every subarray of length at least three, we first compute a modified average: we remove the minimum and maximum element of that subarray and then take the average of what remains.

We are given a permutation of length $2n$, meaning it contains every integer from $1$ to $2n$ exactly once. We are allowed to perform swaps of adjacent positions, exchanging the values at positions $i$ and $i+1$. The target condition is not global sorting.

We are given a fixed word s and another string t. A student repeatedly tries to type s, but his typing process is not continuous in a clean way.

We are asked to construct a very special integer array of length $n$. The array $a$ uses indices from $0$ to $n-1$, and we also define another array $cnt$ of the same length where $cnt[i]$ is the number of times the value $i$ appears in $a$.

We are given a single uppercase string representing a “word” built from letters A to Z. The task is to determine whether this word can be segmented completely into a sequence of chemical element symbols, but only using a restricted set: the first 20 elements of the…

We are given a universe of integers from 0 up to $2^n - 1$. From this universe, a game is played starting at the number $x = 0$. Two players alternate moves, and each move consists of choosing a new value $y$ from the same universe and replacing $x$ with it.

We are given multiple Codeforces accounts, each starting with its own rating value. Over time, a sequence of contests happens, and each contest contributes a fixed rating change. For every contest, we must choose exactly one account to participate.

We are given a circular track with $n$ equally spaced positions labeled from $0$ to $n-1$. A frog starts at position $0$. Each move forces the frog to jump exactly $x$ steps forward along the circle, meaning from position $i$ it always lands at $(i + x) bmod n$.

We are given several independent lines of text. Each line represents a sentence that has been distorted by a specific reordering rule applied to its words.

We are given a single integer n, and we are asked to print a fixed message exactly n times, each occurrence on its own line. The message itself is always identical and does not depend on any input besides n.

We are given a very small hand of at most ten cards and a single boss monster with fixed combat statistics. Our goal is to determine whether we can remove this boss from the game in a way that prevents it from ever coming back. The boss interacts with our cards in two phases.

Let $n > c_t > \cdots > c_1 \ge 0$ with the constraints from exercise 57 and the additional condition $c_{j+1} > c_j + 1 \qquad (t > j \ge 1).$ Define the shifted variables $d_j = c_j - (j-1), \qquad...

We are given a collection of songs, where each song has a unique popularity value and a unique name. The popularity value acts like a strict ranking key: no two songs share the same score, and higher values mean a song is more desired.

We are given two time displays written in a digital, seven segment style where each digit is rendered as a fixed 3 by 3 character block. Each time consists of six digits: two for hours, two for minutes, and two for seconds.

We are simulating how an operating system manages a small, fixed-size memory cache using the Least Recently Used policy. Memory is divided into a tiny number of slots, and we receive a sequence of memory access requests.

We are given two square grids of the same size, each cell containing an integer color. The only operation allowed on the first grid is a rotation around its center by 90 degrees, either clockwise or counterclockwise, and we may apply this operation multiple times.

We are given several arrays, and from each array we must choose exactly one value. After that selection, we conceptually have a multiset of size n.

We are given an initial lineup of $n$ children, each child $i$ has a fixed happiness value $hi$. The toys are initially shuffled by a permutation: child $i$ receives a toy labeled $wi$, so the array $w$ is a permutation of $1 dots n$. The process then evolves in rounds.