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We are given a small pattern grid, called a motif, and a larger grid, called a mosaic. Each cell contains a color value, except that in the motif some cells are empty and behave like wildcards.

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Each island is a simple polygon lying on the ground plane, and each flight path is a 3D line segment with a positive altitude. A plane flies along that segment, and a downward-facing camera observes a strip of ground directly under the aircraft.

We are given a deck split into several marking categories. Each category contains a known number of distinct cards, and the total deck size can be extremely large. A random group of $k$ cards is selected, and one of these $k$ cards is hidden face down.

We are given a simple polygon representing the floor plan of a gallery. Inside this polygon there are two points: one is the guard’s starting position and the other is the center of a small circular sculpture.

Let $f^{D}(x1,dots,xn)=overline{f(overline{x1},dots,overline{xn})}$ and $f^{R}(x1,dots,xn)=f(xn,dots,x1)$. Composition yields $$f^{DR}(x)=overline{f(overline{xn},dots,overline{x1})},qquad f^{RD}(x)=overline{f(overline{xn},dots,overline{x1})},$$ so $f^{DR}=f^{RD}$ follows from…

We are given a weighted tree. Every query describes a scenario where a player starts at one room, must collect a special key located at another room, and must avoid permanently failing by entering a trap room before the key has been collected.

The solution answers all parts, but part (b) is incorrect and breaks subsequent reasoning.

We are given a rectangular grid of size $dx times dy$. Each cell $(x, y)$ can either contain a molecule or be empty. The true arrangement is unknown, but we are given several “wind experiments” that partially reveal it.

Let $f^{D}(x1,dots,xn)=overline{f(overline{x1},dots,overline{xn})}$ and $f^{R}(x1,dots,xn)=f(xn,dots,x1)$. Composition yields $$f^{DR}(x)=overline{f(overline{xn},dots,overline{x1})},qquad f^{RD}(x)=overline{f(overline{xn},dots,overline{x1})},$$ so $f^{DR}=f^{RD}$ follows from…

We are given an initial sequence and a target sequence, both permutations of the same multiset of values. We are allowed to repeatedly perform a very specific operation: pick two positions i and j with i < j where the value at i is larger than the value at j, and then take the…

We are given a positive integer $n$, and we need to find the smallest integer $k$ such that $k ge n$ and every digit of $k$ is identical. Such numbers look like $1, 2, 3, dots, 9, 11, 22, 33, dots, 9999$, where a single digit is repeated some number of times.

We are given a sequence of integers, and we are allowed to cut it into contiguous non-empty pieces. For each piece, we compute a value called its score, defined as the bitwise XOR of all elements inside that piece.

Let $f^{D}(x1,dots,xn)=overline{f(overline{x1},dots,overline{xn})}$ and $f^{R}(x1,dots,xn)=f(xn,dots,x1)$. Composition yields $$f^{DR}(x)=overline{f(overline{xn},dots,overline{x1})},qquad f^{RD}(x)=overline{f(overline{xn},dots,overline{x1})},$$ so $f^{DR}=f^{RD}$ follows from…

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I’m missing the actual problem statement for Codeforces 1042904 - “Ультра mex”. Right now only the title is provided, but the editorial you’re asking for depends completely on the formal rules of the task (what the input is, what operations are allowed, what…

I can’t write a correct editorial yet because the actual problem statement for Codeforces 1042902 - “Тайное послание” is not included in your prompt, and it also does not appear in the retrieved references.

The problem statement section is empty, so there isn’t enough information to derive the actual task for “Codeforces 1042903 - Рекорды и антирекорды”.

I can’t write a correct editorial yet because the actual problem statement for Codeforces 1042901 - “Видеонаблюдение” is missing from your prompt.

I can write the full editorial in your required format, but I’m missing the actual problem statement for Codeforces 1042918 - “Выполнить план, но не перевыполнить”.

I can’t produce a correct Codeforces-style editorial for “1042916 - Конференция” without the actual problem statement.

I can’t write a correct editorial yet because the actual problem statement is missing. Right now I only see the title “Улитка на склоне”, but no description of the grid/array rules, input format, or what is being computed.

I can absolutely write this editorial in the requested style, but I’m missing the one thing that makes it possible to be correct: the actual problem statement for Codeforces 1042917 - “Яблоки по корзинам”.

We are given a directed graph where cities are nodes and magical portals are directed edges. Each portal represents a one-way travel route. Misaka wants to “investigate” as many portals as possible using multiple independent agents called clones.

We are given a rectangular grid of integers representing an image. Each cell is a pixel, and its value is an intensity. The grid is not just a flat array, but a torus: moving off the right edge brings you back to the left, moving off the top brings you back to the bottom.

We are given a collection of entities, each initially isolated, and a sequence of historical statements describing pairwise interactions between them.

We are given a multiset of anime trading cards, where each card has an integer label called a quirk number. Midoriya wants to end up with a very specific final collection: it must contain exactly one copy of each integer from 1 up to some value K, with no gaps and no extras.

We are given three piles of chips. On each move, a player can either take chips from exactly one pile, choosing any positive number up to what remains in that pile, or take chips from all three piles simultaneously, choosing a positive number up to the smallest current pile…

We are given a tree of houses, where each house initially contains a certain number of friends. The roads form a connected acyclic structure, so between any two houses there is exactly one simple path. There are two types of events.

We are simulating a point moving in the plane under mirror reflections, and we care about two things: how close the moving point ever gets to the origin, and how many times it reflects off two fixed lines passing through the origin. The geometry is completely deterministic.

We are given a string s of length N, where each position also carries a weight ai. We are allowed to choose any subset of alphabet characters (both lowercase and uppercase).

Each leaf behaves like an object that falls straight down while being pushed horizontally by a time dependent wind. The wind at second t is a linear function of a global parameter k, so every second contributes a term of the form at + k dt.

We are given a tree with $N$ rooms connected by $N-1$ hallways. Each hallway is either usable or blocked, and blocking hallways partitions the tree into several connected components.

We are given a line of monsters, each with an integer power value. We want to choose a contiguous block of these monsters and compute the sum of their powers. Among all possible contiguous blocks, we need the one whose sum is as large as possible, and we output that sum.

We are given several independent groups of angels. Each group contains a multiset of combat powers, and whenever a defense is formed, exactly one angel must be chosen from each group. The defense power is the sum of the chosen angels’ powers.

We are given an $N times N$ grid representing terrain heights. Each cell has an integer elevation, and the grid has a structural monotonic property: every cell is no higher than its right, bottom, and bottom-right neighbors.

We are given three separate sequences, each representing the section heights of a wall. Every wall has the same number of sections, but the ordering is irrelevant for the task. What matters is only which heights appear in each wall at least once.

We are given an undirected simple graph, and we need to decide whether its structure can be interpreted as a very specific “frog shape”. This shape consists of a simple cycle that represents the body. From this cycle, exactly four attachment points are chosen.

We are given a tree with n rooms. Each room initially contains a distinct number written on it. The rooms are connected by corridors, so the structure is a single connected acyclic graph. We must choose exactly x rooms that will remain in use.

We are given a runner moving through a long 3-lane track, where each row is a step in time and each of the three columns represents a lane. Each cell can either be empty, contain a coin, contain an obstacle, or contain a trampoline. The runner starts at row 1 in the middle lane.

We are given a rectangular wall represented as a grid of lowercase Latin letters with $n$ rows and $m$ columns. Inside this grid we want to place a square frame of fixed size $k times k$.

Given f=\{\emptyset,\{1,2\},\{1,3\}\}, \quad g=\{\{1,2\},\{3\}\}.

We are given a fishing session defined by a single time interval within one day. Alongside this, we have a large set of fish “activity intervals”, each labeled with a species name. During an activity interval of a species, that fish is actively biting.

We are given a collection of tower heights. There are $n$ towers, where $n$ is guaranteed to be odd, and each tower has some initial height. We are also given $k$ extra unit cubes. Each cube can be added to exactly one tower, increasing its height by one.

We are given two configurations of the same number of points on an integer grid. Think of them as two drawings of indistinguishable particles placed on lattice points. The particles can move, but only through a very specific collective operation.

There are 30 independent positions, each associated with a weight equal to twice its index. Over a sequence of seconds, each position can experience at most one event per second: the occupant either enters its hole, leaves it, or stays unchanged.

We are given a fixed collection of existing family names, each written as a string of lowercase letters and hyphens. Then we are given several candidate names for a newborn. For each candidate, we must decide whether it is acceptable. A candidate is rejected in two situations.

We are given a line of houses, each with a fixed height. A resident who lives in house i wants to reach their own roof starting from the ground, but movement is constrained by a single ladder of fixed length. A ladder of length L allows two kinds of actions.

The problem describes a circular cake that behaves like a clock. The cake is first cut at noon, and then a sequence of people arrive at fixed integer hours between 12 and 24. Each arrival creates a cut at that hour, and the cake is divided into segments between consecutive cuts.

We are given a list of flower beds, each associated with a number of shells. For the i-th bed, there are ai shells that must all be used to form a decorative border.

We are given multiple independent scenarios. In each scenario, a reader has a sequence of reading amounts over days and a list of book lengths. Each day contributes a certain number of pages that can be used to progress through books in order.

We are counting ordered pairs of integers $(a, b)$ where $0 le a le b$, but not all pairs are valid. The restriction comes from a bitwise condition: when we take the bitwise OR of $a$ and $b$, the result must not exceed $n$.

We are asked to evaluate a large sum where each term combines Fibonacci numbers and factorial exponents, but we only care about the last digit of the result. For each test case, an integer $n$ is given. We conceptually build the value $$S = f0^{0!} + f1^{1!} + f2^{2!

We are given multiple independent test cases. In each test case, there is an array of integers and a threshold value $k$. We call a set of values “good” if the largest and smallest elements in that set differ by at most $k$.

We restart the argument from the actual structure of Knuth’s swap-in-place algorithm (Exercise 147) and then isolate exactly what changes in the ZDD setting.

We are given many independent queries. Each query provides a non-negative integer $n$, and we must count how many integers $x$ in the range from $0$ to $n$ can be written as the sum of two integer squares, meaning $x = a^2 + b^2$ for some integers $a$ and $b$.

We are given a very large integer written in decimal form, and for each such number we need to count how many positive integers not exceeding it consist only of the digits 4 and 7.

We are given a pool of students, each student knows a subset of up to 60 topics. A valid team is any subset of students such that two conditions are simultaneously satisfied: every topic from 1 to p is covered by at least one team member, and for each topic, at most one team…

We are given a binary string, where each position is either 0 or 1. We are allowed to change at most k zeros into ones.

Let $x in [0,1)$ have ternary expansion $x = 0.x1 x2 x3 cdots quad (xj in {0,1,2}),$ where nonterminating representations are used. Define $omega = e^{2pi i/3}$, so $omega^3 = 1$ and $1 + omega + omega^2 = 0$.

We are given a string made only of the characters X, T, and U. For each test case, we need to count how many substrings have the property that the number of X, T, and U characters inside that substring are all equal.

We are given a two-versus-two game where each round reduces to a comparison between two independent “targets” produced by the two main players, Mo and Larro. In each round, Mo and Larro each pick one number from their personal hand. These numbers become target sums.

We are given a directed network of people where each person knows the addresses of some other people. When someone receives a message, they immediately forward it to everyone they know. The process starts from a specific person and repeats indefinitely.

For each query, we are given a prime number $p$. We look at all integers from $1$ to $p-1$, and for each such integer $a$, we compute its multiplicative inverse modulo $p$. That means we find a number $b$ in the range $[1, p-1]$ such that $a cdot b equiv 1 pmod p$.

We are given a ticket-printing system with two identical machines that can be used to generate reimbursement slips. Each machine can produce at most one ticket per operation, and after producing a ticket it becomes unavailable for a cooling period of a minutes.

We are given three groups of employees with sizes A, B, and C. Every employee must be placed into pairs, meaning each employee is matched with exactly one other employee, and no one is left unmatched.

A projection function $x_j$ corresponds to the Boolean function that is $1$ exactly on those assignments where the $j$-th variable is $1$.

We are given multiple independent scenarios. In each scenario, a student starts with a fixed number of items that must be carried, and there are several checkpoints along a path.

We are given a directed graph with possibly multiple edges between the same pair of vertices and also self loops. Each directed edge represents a single-step move between cities.

We are given a binary sequence arranged on a circle. Each position contains either 0 or 1, and indices wrap around so that position n−1 is adjacent to position 0.

We are given a rectangular grid of lowercase letters. From this grid, we can choose any sub-rectangle by selecting a contiguous block of rows and a contiguous block of columns.

Let $F = \mathrm{MUX}(f,g,h)$ denote the Boolean function defined by selecting $g$ when $f=1$ and selecting $h$ when $f=0$, so that F = (f \wedge g)\ \vee\ (\neg f \wedge h).

We are given a string composed only of the characters q and a. We are allowed to insert exactly x additional characters, each of which can independently be either q or a, at arbitrary positions in the string.

We are given a collection of static segments on a number line, each segment having integer endpoints within a bounded universe up to $L$.

We are given a sequence of $n$ independent “draws”. In the $i$-th draw, we choose an integer score $xi$ uniformly from the range $[0, ai]$, where $ai < k$. After each draw, we maintain a running prefix sum of all chosen values.

We are asked to construct up to 20 distinct integer vectors in at most three dimensions. Each vector has non-negative coordinates up to 10^9. After constructing them, we look at the sum of all vectors.

We are given a lineup of enemy units, each with an integer attack value. A special effect card removes every unit whose attack is odd after all modifications are applied. Before using this card, we are allowed to cast a collection of single-use spells.

We are given two intervals of positive integers, one for a and one for b. We need to count how many pairs (a, b) can be formed such that a is chosen from the first interval and b from the second interval, and the pair satisfies a bitwise and arithmetic constraint: the XOR of a…

Let $t(m)$ denote the parity of the binary digit sum of $m$, so that $t(m)=0$ when $m$ has an even number of 1s in binary representation and $t(m)=1$ otherwise.

We are given several “ability strings”, where each ability is made of distinct characters and no character appears in more than one ability.

The earth map can be modeled as a directed graph where each city is a node and each one-way road is a directed edge.

We are given a vertical stack of items, where each item has a color. The top of the stack is position 1, and positions increase as we go downward. A sequence of queries is performed on this stack.

A ZDD represents a family of finite sets over an ordered universe of items $x_1 < x_2 < \cdots$.

We are tracking how a quantity evolves over time when a deterministic growth rule starts applying only after a delay. Saimon begins with some number of identical units, specifically pairs of Emm coins.

We are given multiple test cases. Each test case contains a single lowercase string representing text Roshid wants to type. However, his keyboard has a hardware failure: a specific group of letters no longer works, corresponding to the bottom row of a standard keyboard layout.

We are asked to construct an array of length n where each value is a 30-bit non-negative integer. The construction must satisfy a set of constraints that relate elements either by inequality to a fixed value or by XOR relationships between pairs.

We are given an integer $m$ and an array $a$. The task is to look at every divisor $d$ of $m$, and decide whether $d$ is “safe” or “bad”. A divisor $d$ is considered bad if there exists at least one array element $ai$ such that $d$ divides $ai$. Otherwise, $d$ is safe.

We are given a sequence of integers and then a sequence of queries. For each query value $b$, we are interested in all positive divisors of $b$. Among those divisors, some may appear inside the given array, and others may not.

The input describes several independent “landscapes” made of vertical stacks of unit-width bricks. Each landscape is an array where the value at position i represents how tall the wall is at that point. When rain falls, water can accumulate in the gaps between taller walls.

We are given several test cases. In each test case, there is an even-length array. We must partition the array into disjoint pairs so that every element belongs to exactly one pair. For each pair, its contribution to the answer is the larger of the two values inside that pair.

Each test case describes a process of completing identical forms, where each form requires collecting signatures from several offices. For every office i, a single form requires ai signatures from that office.

Let $u$ and $v$ be ZDD nodes representing families of sets for Boolean variables ordered as $x_1 < x_2 < \cdots < x_n$.

We are given a multiset of integers where value i appears exactly mi times. From this multiset we consider every possible permutation of the full expanded array.

We are given two binary grids of the same size, call them the starting grid and the target grid. The only allowed move is to choose a contiguous segment of length l either horizontally within a row or vertically within a column, and flip all bits in that segment.

We are given an array of length n where the value at position i is i-1, so the array is fixed as [0, 1, 2, ..., n-1]. The task is to consider every contiguous subarray, compute the bitwise XOR of its elements, and sum all those XOR results.

Let $f$ be a Boolean function on variables $x_1,\dots,x_n$, and let its BDD be given in the ordered and reduced form described in Section 7.

We start with an array that initially contains the numbers from 1 to n in order. Then we repeatedly apply a fixed sequence of operations until only one element remains.

We are given two binary arrays of equal length. From each array we are allowed to pick one contiguous segment, and the only restriction on each segment is that its length must fall inside a given range.

We are given several test cases. In each test case we receive an array of integers, and we are asked to analyze a faulty piece of code that tries to compute the maximum value of the array.

Let variables $x_1,\dots,x_n$ be interpreted as characteristic bits of a subset $S \subseteq {1,\dots,n}$, where $x_i=1$ means $i \in S$.