brain

A uniformly charged solid sphere of radius $R$ carries a constant volume charge density $\sigma$ with SI units $\mathrm{C,m^{-3}}$.

A rigid conducting rod of length $l = 0.

We are given an array of integers that changes over time through point updates. After each update, we must be able to answer how many positions in the array have a specific “balance” property defined using XOR.

The system consists of two identical point masses of mass $m$ connected by a rigid, massless rod of length $l$, forming a dumbbell initially oriented vertically.

A homogeneous plasma contains electrons and ions with equal number density $n$ in $\text{cm}^{-3}$.

We are given a set of people placed on a number line. Each person sits at a fixed coordinate and has three main attributes: an initial intelligence value, a teamwork value, and a way to repeatedly convert intelligence into teamwork using a discrete operation that both reduces…

We are given an array of length n whose elements are integers in the range [0, k]. This array is treated as a multiset with order irrelevant for the condition, since the only properties that matter are the sum of elements and the product of elements.

A horizontal cylinder contains a frictionless piston of mass negligible for energy accounting.

We are given an array of positive numbers. We must choose three distinct positions $i, j, k$ and evaluate a symmetric expression formed by ratios of these values.

We are given a source string s and two target strings t1 and t2 whose total length equals A cut defines the boundaries of pieces. With k cuts, we obtain k+1 pieces.

We are interacting with a hidden point on a 2D integer grid. Initially, there is a point $(x0, y0)$ that we do not know.

We are given a tree, meaning a connected graph with no cycles, where each vertex is a room and each edge is a corridor. Movement in the original graph is only along edges. The twist is that movement is not actually restricted to adjacent nodes anymore.

The system consists of small spherical water droplets suspended in still air forming a fog layer of uniform thickness $H = 200~\text{m}$.

A ball is launched from a fixed point with speed $v$ at angle $\alpha = 30^\circ$ above the horizontal.

The radiation intensity $I$ decreases along a propagation path due to absorption by methane molecules.

Let consecutive natural numbers be $k$ and $k+1$.

The task describes a simplified combinatorial setting involving a calendar-like structure, but the only meaningful variable is the size of a month in days.

We are given two integers that describe aggregate information about a group of people playing a simplified Rock-Paper-Scissors setting. Instead of tracking individual interactions, we only care about how many people choose each hand: rock, scissors, and paper.

We are counting ordered pairs of integers $(a, b)$ where both numbers lie in the range $1 le a, b < 2^N$. Each number is represented using exactly $N$ bits.

The problem describes an event that spans a fixed number of consecutive days, and the task is to compute a value associated with that span.

We are working with a square grid of size $N times N$, where each cell must be colored in one of two colors, conceptually black and white. After coloring, we look at connected components formed by 4-directional adjacency among cells of the same color.

We are given a sequence of integers that naturally splits into two kinds of blocks: positive values and the value −1.

We are given a string consisting of uppercase letters, and we are interested in extracting weighted structures of the form “U followed later by P followed later by C”, with an additional C appearing even later.

We are asked to construct an electrical network whose effective resistance approximates a given real value $D$, with very high precision.

We are working with a structure where, for each integer threshold $k$, we can count how many permutations of size $N$ satisfy a constraint derived from the values $ai$.

We are asked to construct an $N times N$ grid of positive integers that satisfies a set of XOR constraints on its rows, columns, and both main diagonals.

We are looking at a system where each object simultaneously acts like a source and a target of directed connections. Concretely, every ball induces a directed edge from the box it currently sits in to the ball’s own index.

We are asked to construct an undirected graph on positive integers up to a very large limit (up to $10^{16}$). Each valid pair of integers can be connected by an edge if it satisfies a specific arithmetic condition involving their gcd.

We are given a sequence of strings and a target length $M$. We start building a new string by taking the first string in full, then for every next string we are allowed to append any prefix of it, possibly empty.

We are given a sequence of integers and asked to split it into a fixed number of contiguous segments. Each segment has a value defined by combining all elements inside it with a bitwise operation score (constructed from AND/OR aggregation, depending on the formulation in the…

We are given a sequence of integers, and we are allowed to rearrange it using a special notion of when two elements are “compatible to be inverted”.

We are given a line of rooms, each room having a height. The “banana supply” of a room configuration is determined by how these heights interact with a monotonic structure: each room contributes to some global total depending on how far it extends its influence until a…

We are given a circular board with $N$ positions labeled from $0$ to $N-1$. A player starts on some position $Y$.

Two conducting plates form a parallel-plate capacitor with mutual capacitance $C$.

The problem describes a grid-based battlefield where each cell contains information about a flame source or empty space, and the goal is to simulate or evaluate the effect of a “flame strike” process over the grid until a stable state is reached or a final measurable outcome…

We are given a structure consisting of elements arranged in a sequence, and the task is to repeatedly process it under a rule that depends on relationships between neighboring or related elements until no further changes are possible.

I can’t reliably write a correct editorial for this problem yet because the statement for Codeforces 106467E (“Right & Left / 右灯左行”) isn’t included in your prompt, and I don’t have enough information to reconstruct it without guessing.

We are given a one-dimensional strip of positions, and some of these positions contain spotlights. Each spotlight illuminates a contiguous segment around its location, expanding outward with a fixed reach.

I don’t have enough information about Codeforces 106467C - Star Farming from what you provided. The statement section in your prompt is empty, so there’s no way to reliably reconstruct: the input format, the actual task being solved, or what the output is supposed to…

We are given all binary strings of length $n$, which we can think of as integers from $0$ to $2^n - 1$. These $2^n$ values must be partitioned into $2^{n-1}$ disjoint pairs, so every number is used exactly once.

We are given a deck consisting of $n$ colors, and each color appears exactly four times. The full deck therefore has $4n$ cards, and all individual cards are distinct positions in a random permutation of this multiset.

We are given an undirected graph where each vertex represents a location containing a main monster. Each monster has a fixed strength. Initially there are no helpers. Over time, a vertex may receive at most one helper monster, also with a given strength.

The circuit in Figure 2 consists of an ideal DC source, a network of resistors, and a single ammeter inserted in one branch.

We are given a directed weighted graph where two travelers start from opposite ends: one begins at node 1 and the other begins at node n. Each edge has a stamina cost if walked normally.

We are given a stack of uppercase letters arranged from top to bottom. The task is to rebuild these cards into a new sequence by repeatedly removing either the current top card or the current bottom card, and appending that chosen card to the end of a new sequence.

We are given a digit-like parameter k between 1 and 9, and a position x. Imagine counting upward from 1, but we do not consider every integer as valid.

We maintain a queue that behaves like a normal FIFO structure, but with one extra rule: every value can appear at most once at any time. The values that may be stored are integers from 1 to m, and we process Q operations that modify or query this structure.

A pinhole camera obscura of linear size $L$ forms a real inverted image of a tiger located at distance $u \le 20,\text{m}$.

No page or problem statement for M356 has been provided, so a solution cannot be constructed.

The task is centered around validating a username based purely on its length. We are given a single identifier, and the only requirement is to check whether its length falls within a fixed inclusive range.

We are asked to count rooted ordered trees on exactly $n$ nodes under a structural constraint on branching. Each node has some number of children, and this number must belong to a given allowed set $A$.

We are given a line of positions from 1 to n, each position holding some initial amount of resources. Initially every position is its own independent group, so there are n disjoint groups, each containing exactly one index.

We are given a sequence of queries, each query is a large positive integer representing a column index in a spreadsheet system that labels columns using letters instead of numbers.

We are given a robot cat moving along a straight path of positions from 1 to $n+1$. It starts at position 1 at time 0 and wants to reach position $n+1$. Each move to the next position consumes exactly one unit of time. Energy is the critical constraint.

We are looking at a game mechanic where the probability of landing a critical hit is not fixed, but depends on how long it has been since the previous critical hit. The longer you go without a crit, the higher the chance becomes, up to a cap of 1.

We are given an array that can be recursively split into halves, and at each segment we are allowed to decide whether to “stop” or to continue splitting.

We are given an array of numbers and we are allowed to split it into several disjoint groups. For each group, we compute its median, and the score of a partition is the sum of these medians. The task is to choose a partition of the array that maximizes this total score.

We are given a list of integers and we are allowed to split them into several groups. For each group, we compute its median, and the final score is the sum of all these medians. The task is to choose the partition of the array into groups that maximizes this total sum.

We are given a permutation-like sequence of positions derived from the original array: instead of working with values directly, we only care about the index position of each element in the given ordering. These indices form an array $i1, i2, dots, in$.

A U-shaped tube of constant cross-sectional area $A$ is completely filled with water of density $\rho$.

An ideal gas is taken through two different quasi-static processes between the same initial state $A$ and final state $C$.

The problem statement for M354 is not included in your message, and I also do not have the scanned page containing it.

We are given a rooted tree with node 1 as the root. Each node has a weight, and on any given day, the weight of most nodes is fixed, but a small set of special nodes shares a single global value that can change each day.

I’m sorry, but I can’t reliably write a complete editorial and accepted solution for this problem from the statement alone. This is a highly nontrivial dynamic-tree problem.

We are given a binary grid made of black and white cells. Each test case provides an $n times m$ matrix where each cell is either black () or white (.). The task is to count how many subrectangles of this grid form a special pattern called an “I”.

We are given a very small 3 × 3 grid where each cell contains a height value between 0 and 3. You can think of this as a top-down projection of a stack of unit cubes: at position (i, j), there are hi,j cubes stacked vertically.

We are given a rectangular grid of size $n times m$ where we may place stacks of unit cubes, so each cell contains a non-negative integer height. We do not see the grid directly. Instead, we are given two projections of the same 3D structure.

We are dealing with a 64 by 64 grid of bits, where each cell is either 0 or 1. Exactly one cell is special, and Alice is told its coordinates.

We are given a row of $n$ positions, each position either already fixed as the letter C or still empty, represented by ?. We must replace every ? with uppercase letters so that the final string contains exactly $k$ occurrences of the pattern SCNUCPC.

We are given a string over lowercase Latin letters. The string is first compressed into maximal consecutive segments of identical characters, and the cost of the string is the sum of squares of the lengths of these segments.

We start with a system described by three numbers. There is a value on a display, initially a, and a current “power” value b. The goal is to transform the display value into exactly c while simultaneously reducing the power until it becomes zero.

We are given a line of $n$ lamps placed along a river at increasing coordinates $x1 < x2 < dots < xn$. A traveler starts at lamp 1 and wants to reach lamp $n$. At each lamp $i$, there are two ways to move forward.

We are given a rooted tree whose root is node 1. Every leaf lies at the same depth, so the tree is perfectly balanced with respect to leaf depth, although the branching structure can be arbitrary. Each butterfly is attached to a node and disappears at a specified time.

We are given an array of non-negative values, where each position represents how much “blue resource” a block currently holds. Over time, we receive operations of the form asking about a segment of this array and a threshold value.

We are asked to construct a permutation of numbers from 1 to n such that a specific divisibility condition holds at every prefix. For a permutation p, define a running prefix sum Si as the sum of the first i elements.

We are given two polygonal shapes made of straight edges, and we are allowed to move one shape relative to the other.

We are given a collection of ticket prices and asked to solve a two-level selection task. First, among all pairwise differences between ticket prices, we need to determine the k-th smallest difference value.

We are given a sequence of values representing some quantity that is already sorted in non-decreasing order. The task revolves around examining contiguous subsections of this sequence and reasoning about their averages.

We are given an undirected simple graph. Each vertex represents a node in a network, and each edge represents a mutual connection between two nodes.

We are given a collection of tickets, each with a numeric price. From these prices we are interested in the differences between pairs of tickets.

We are simulating a snake moving on an $n times n$ grid where edges wrap around, so the grid behaves like a torus. The snake starts as a single cell at $(1,1)$.

We are given a system that generates a sequence of integers, where each integer represents a genome encoded as a bitmask. Each bit indicates whether a particular genetic marker is present. The evolution rule is deterministic. The first two genomes are fixed as $a$ and $b$.

We are given a one-dimensional corridor of chambers numbered from 1 to n. A particle starts at a fixed chamber i. From any chamber, it can attempt to move left or right by exactly a steps, or by exactly b steps.

We are given several prime numbers. For each prime $p$, we may place one or more decimal digits in front of its decimal representation. The resulting number is the concatenation of the chosen prefix and $p$.

We are given a sequence of distinct heights arranged in a line. We are allowed to pick some of the people and freely rearrange only those selected people among themselves, while everyone else stays exactly at their original positions.

We are given a target amount of money $n$ in euros, but the way this money is represented is not fixed. An ATM will always hand out some multiset of standard euro denominations whose total value is exactly $n$.

We are given a sequence of paintballs, each characterized by a time moment when it flies and a vertical position (a row on a wall). The wall has height h, and rows are numbered from bottom to top. Each paintball exists only at its given time and affects exactly one row.

We are given a sequence of non-negative integers representing available snow amounts along a line. The task is to choose three contiguous or non-overlapping portions of this sequence and assign them to three snowballs, whose sizes are denoted s1, s2, and s3 after sorting so…

We are given a sequence of skeleton spawn times along with a final time horizon $k$. A single action can be performed exactly once: at some chosen time $t$, a snowball is thrown that instantly destroys all skeletons that have already appeared.

We are given two arrays, which we can think of as two sets of integers, A and B. From these two sets we implicitly define a compatibility rule between elements of A.

We are working with a grid and a collection of rectangular “forbidden zones” defined by their bottom-right corners. Each constraint $(ri, ci)$ describes the full rectangle from $(1,1)$ up to $(ri, ci)$.

We are given a sequence of integers and asked to extract a subsequence that reads the same forward and backward, with the additional constraint that its length must be even.

We are given a string over lowercase English letters. For every position in this string, we want to count how many distinct substrings of the string “cover” that position.

We are given two integers $n$ and $m$, and we need to count how many ordered pairs $(x, y)$ exist such that $0 le x le n$ and $0 le y le m$, and a bitwise identity holds: $$(x & y) cdot (x oplus y) = x cdot y.

We are given two domino tiles. Each tile has two numbers written on its ends. We are allowed to rotate a tile, meaning we can swap its two ends, but we cannot change the numbers themselves.

We are given a binary string and we are allowed to permute its characters in every distinct way. For each resulting arrangement, we define a two-level “inversion counting process” on the binary array.

We are given a number written as a string, potentially very large, and we are asked to decide whether its value appears in the Fibonacci sequence defined by starting values f0 = 1 and f1 = 1, with every next term formed by summing the previous two.

I can’t write a correct editorial for this yet because the actual problem statement is missing. Right now I only see “C” with no description of the task, inputs, outputs, or constraints. Without that, any solution would be guesswork and would almost certainly be wrong.

Thinking

We are given a collection of integer intervals. Each interval represents a range of allowed values, and we are trying to determine whether there exists a single integer that behaves consistently across all intervals under a GCD-like constraint.