brain

We are given a grid where some cells are blocked and the rest are free. A game starts when the first player chooses any free cell and places a stone there.

We are simulating a repeated attempt to clear a sequence of at most four game levels, where each full attempt either succeeds or fails depending on independent probabilistic choices made inside each level.

Each meteor starts at a known point in the plane and moves in a fixed direction with constant velocity. After time $t ge 0$, meteor $i$ is at $$(xi(t), yi(t)) = (xi + v{x,i} t,; yi + v{y,i} t).

The situation is a simple road trip with a mistake in direction. Two distances are relevant: the first is how far the travelers went after taking the wrong direction, and the second is the remaining distance from their starting point to the destination along the correct route.

Let $K$ be the number of cycles in the graph $G = M_1 \cup M_2$, where $M_1$ is a fixed perfect matching on $\{0,1,\dots,2n-1\}$ and $M_2$ is a uniformly random perfect matching.

Each uppercase letter must be translated into its position inside the fixed 5 by 5 Knock Code matrix. Every letter is represented by a pair consisting of its row and column. The numbers themselves are encoded as consecutive asterisks, so row 3 becomes and column 5 becomes .

We are given an integer parameter $N$ and a rational function $$f(x) = frac{5xN^2}{x^2 + 3xN - 5N^2}.$$ The task is to find all real values $a$ such that plugging $a$ into the function returns the same value, meaning $f(a) = a$.

We are given a string of length $N$ consisting of three kinds of characters: fixed opening parentheses, fixed closing parentheses, and wildcard positions written as ?. Each wildcard can independently be replaced by either ( or ).

We are given a string made only of the characters a and b. This string is not arbitrary: it is supposed to be an encoding of a binary password where each 0 was replaced by the block ab and each 1 was replaced by the block aba.

Two athletes alternate doing fixed-size blocks of repetitions at a gym. Marcel always performs the first block, then Joãozão, then Marcel again, and so on. Each block contains exactly M repetitions.

We restart from the correct structural fact that the previous argument failed to justify: in an Omega network, _switch settings are not independent generators_, but for permutations with a prescribed...

The problem gives an undirected, unweighted graph of cities connected by roads. Lex starts at city 1 and must reach city N. Some cities are marked as “debt cities”, meaning passing through them is undesirable. A path is evaluated in two stages.

Let $N=2^d$.

We are given a function that depends on two layers of summation. First, for a fixed interval $[l, r]$, we look only at numbers divisible by 3 inside that interval. For each such number $x$, we count how many digit ‘3’ appear in its decimal representation and sum this count.

We are repeatedly simulating a farming process that produces two types of items. Each run of the stage independently yields either a sniper chip with probability $p = frac{a}{100}$ or a caster chip with probability $1 - p$.

We are given a tree with nodes labeled from 1 to n. Each query selects a subset of these nodes, and we are allowed to delete any subset of the selected nodes. After deletions, the remaining nodes form a forest.

We are given a line of fighters, each carrying a power value. A tournament repeatedly takes the first two remaining fighters in the current sequence and makes them fight. The loser is removed, while in the case of a tie both are removed.

We are given a fixed string and then many independent queries, each query selecting a contiguous segment of that string.

We are given a tree where every edge is equipped with two directed weights. If we stand at node u, each neighbor v has a positive weight Cu(v).

Each item in the input represents a topping. The topping has a value, and it also carries a single restriction pointing to another topping index. If you decide to include topping i in your final selection, then the topping bi is no longer allowed to appear together with it.

We restart from the actual combinatorial structure of $P(2^d)$ as used in TAOCP: a recursive permutation network built from $2 \times 2$ switches arranged in $2d-1$ stages, where each stage consists o...

We are given a string and a number $k$. From the string, we look at all subsequences that have exactly length $k$. The question is whether any two different ways of choosing positions in the string can produce the same resulting length-$k$ sequence of characters.

We are given an array of length n and a fixed sequence of swaps. Each swap exchanges two positions in the array, and this sequence is always applied in the same order.

We are given an array of numbers, each number representing a panda’s label. For any ordered pair of distinct indices $(i, j)$, we form a new number by writing $xi$ directly followed by $xj$.

Number the bits of the 64-bit register from $0$ (least significant) to $63$.

The previous solution fails because it never instantiates a valid TAOCP register program: it introduces informal “tensor elimination”, undefined data layouts, and unsupported cost sharing.

Let $d=\lceil \lg m\rceil$.

Let $\upsilon$ be a permutation of ${0,1,\ldots,d-1}$, and let $j = (j_{d-1}\ldots j_1 j_0)_2$.

Let scheme (71) be interpreted in its actual routing form: for each stage $k\in\{0,\dots,5\}$, the exchange operates on the $k$-th bit position, and the masks must be chosen so that a datum located at...

Let the butterfly network (71) act on bit strings $x \in \{0,1\}^d$.

The proof must avoid any assumption that arbitrary integers can be freely “normalized” into powers of 3.

The previous solution failed because it never used the actual structural property of δ-swaps and introduced unrelated complexity measures.

We are given an array and a variable integer $x$. For any choice of $x$, we compute a value by XORing $x$ with every array element and summing the results. This produces a single integer $S(x)$.

The previous solution fails because it replaces δ-swaps by XOR translations on indices, which is unrelated to Knuth’s construction.

Let δ be the mask selecting the positions to be swapped in the general δ-swap (69).

The task reduces a contest joke into a binary decision. We are given a single integer $T$, but the important part is not the value itself, it is the meta-information: we are asked whether it is possible to obtain a correct solution submission simply by printing the sample output.

We are given an undirected graph where each vertex carries an integer value. The task is not to compute anything over all paths in the usual shortest-path sense, but instead to consider all simple paths in the graph, pick any one of them, take the XOR of the vertex values…

The problem gives us a single line of input that represents a user's feedback or selection of their favorite problem. The actual content of this line does not influence any computation. The task is to produce a fixed response regardless of what the input string contains.

We are given a fixed reference list of company names ordered by prestige, where position 1 corresponds to the most prestigious company. Each query consists of a company name, and we must determine whether that name appears in the reference list.

I don’t have the actual statement of Codeforces 105071G (“:wink:”) available from your prompt, so I can’t safely reconstruct the problem or write a correct editorial without risking inventing details.

The task gives a single string of length up to one hundred thousand characters and asks for another string as output. There is no transformation rule described in a structured way such as parsing, filtering, or reordering.

The problem presents two quantities defined through limits, infinite sums, and definite integrals, and asks for a single integer derived from them. The final output is the floor of the hypotenuse of a right triangle whose legs are these two quantities.

We are given an array of integers and we need to output a positive integer with a very specific property: when you reduce every array element modulo this integer, at least half of the elements land in the same residue class.

We are given a rectangular grid where each cell already has a color assigned to it. The game involves repeatedly applying painting operations, where each operation paints an entire row or an entire column with a single color, overwriting whatever was there before.

The task describes a grid coloring process where operations repaint entire rows or entire columns, overwriting previous colors.

We are given a tree where each node carries a single character, either an opening bracket or a closing bracket. For multiple queries, we are asked to look at the characters along the unique path between two nodes and decide whether that resulting sequence satisfies a special…

We are given a set of planar points, and the task is to reason about “beautiful angle pairs” formed by these points.

We are given a line of positions, and on this line there are constraints that forbid certain patterns from being formed inside chosen segments.

We are given two rooted trees built on the same set of labeled leaves. The first tree defines a notion of distance between any two leaves through their lowest common ancestor, so any pair of leaves has a fixed distance determined entirely by the structure of that tree.

We are looking at a planning problem where a person repeatedly performs an action that has a trade-off: each unit of work produces some amount of “food value”, but also consumes some amount of stamina or effort.

We are given an array of numbers and multiple independent queries. Each query focuses on a subarray defined by a left and right boundary, and a number $k$.

I can’t write a correct editorial for this yet because the actual problem statement content is missing. Right now we only have the title “除法与取模 (Division and Modulo)” and placeholder Input/Output sections, but none of the key details: what is given, what…

We are given an array of integers and the task is to transform it into a non-decreasing sequence by removing elements.

We are given a set of rainfall events, each event covering a continuous segment of cities on a line. Each event has a value representing how much “rain contribution” we gain if we choose it.

I can’t reliably write a correct editorial for this problem yet because the actual statement for “rain (easy version)” (Codeforces Gym 105069A) is not present in the prompt you provided, and the accessible attachment content is incomplete and mixed with unrelated…

The problem revolves around a sequence of numbers that is repeatedly updated using bitwise OR operations, together with online queries that ask for the current value of a particular element after all updates that affect it have been applied.

We are given a sequence of books arranged in a line, where each book has an identifier representing its type. The goal is to perform a minimal number of moves so that the final configuration matches a very restricted structure: the books end up split into two consecutive…

The task describes a robot moving on a 2D grid made of different types of cells, where some cells are passable and others are blocked. The robot has a facing direction at all times, and its movement rules distinguish between turning and moving forward.

We are working on a grid where each cell describes a different kind of terrain in a factory. Sally starts at the top-left corner and wants to reach the bottom-right corner.

We are simulating a dynamic process on an $N times M$ grid that starts empty and gradually becomes filled cell by cell. Each incoming operation places a cupcake of one of three flavors into a specific empty cell.

We are given a row of cupcakes, each starting with some number of sprinkles. Over time, Alice performs a sequence of operations. In each operation she increases every cupcake’s sprinkles by a fixed amount, taken from an array in a fixed order.

We are given a circular arrangement of positions indexed from 1 to n, where each position may contain a cupcake with a given “deliciousness” value. Suzie starts at position 1 and repeatedly moves forward by one index at a time, wrapping from n back to 1.

We are working with a circular “base” region centered at the origin, and several additional circular regions placed on top of it.

We are given a party with a fixed number of guests, and each guest names exactly one cupcake flavor they would be satisfied with. Each flavor is identified by an integer from 1 to M.

We are given a sequence of cupcake types, each type having a fixed spiciness value. The judge eats all cupcakes in some order, exactly one of each type.

The task is deceptively simple. You are given a single line of input describing a 10-pull action in a gacha game, but the input carries no meaningful constraints or parameters that influence the result.

The task is about counting how many longest strictly increasing subsequences exist in a given array. A subsequence is formed by deleting elements without changing the order of the remaining ones.

The statement provides no meaningful input format and no explicit output requirement beyond the problem title. Interpreting this in the way Codeforces sometimes frames puzzle or joke problems, the only consistent reading is that the program is not expected to process any data…

The task does not involve computation over an input in the usual sense. There is no dataset to transform and no structure to analyze. Instead, the judge is waiting for a single fixed string: Alice’s forgotten five-digit passcode. The interaction rules are simple but strict.

You are given several independent test cases. Each test case describes a short string consisting of characters that represent a state or response sequence produced by a system that might or might not be a robot.

Let $x$ be a register containing bits $x_0, x_1, \ldots$, and fix distinct positions $i \neq j$.

The error in the previous solution is the implicit claim that one must rely on structural invariance of the relation under permutations.

We are given a square grid of size $N times N$, where each cell is either blocked or usable. A blocked cell is marked with $-1$, and cannot be entered. Every other cell contains a non-negative number representing cupcakes available in that cell.

We are given a minimum threshold $n$, and we want to choose a number $x$ representing the number of polkadots on a cupcake. This number must satisfy two constraints at the same time. First, $x ge n$. Second, $x$ must not be a prime number.

Let ordinary generating functions be taken in the sense $A(z)=\sum_{n\ge 0} a_n z^n,$ and extend the functions by $a_0=0$ for $\rho,\lambda,\nu$.

We control a small robot moving on an $N times N$ grid, where each cell is either free or blocked. The robot starts in the bottom-left cell, which we can treat as coordinate $(1,1)$, and its goal is to reach the top-right cell $(N,N)$. It always begins facing upward.

Let $\lambda x$ denote the index of the most significant $1$ in $x$, with the convention $\lambda 0 = 0$, so that $2^{\lambda x - 1} \le x < 2^{\lambda x}$ for $x > 0$.

Let $x = (x_{w-1}\ldots x_1 x_0)_2$ be a word of fixed width $w$.

The error in the previous solution is that it reconstructs a generic “parallel prefix” algorithm and then assigns instruction counts without grounding them in the actual MMIX operations used in proced...

Let (55) and (56) define the function $\lambda x$ recursively in terms of shifts and bit tests, with the standard convention that the recursion terminates at $x = 0$ by assigning a base value $\lambda...

Let $x = (x_{63}\ldots x_0)_2$.

We address each error by restarting from a correct signed-digit construction and then showing how it is obtained by constant-time bitwise operations.

We work with 2-adic integers and interpret $\rho(x)$ as the 2-adic valuation $v_2(x)$, with $\rho(0)=\infty$.

The proposed procedure maintains $\rho$ as the number of trailing zero bits of $x$ by repeatedly replacing $x \leftarrow x \gg 1$ while $x \mathbin{&} 1 = 0$.

We are given a network of cities connected by bidirectional roads, where each road has a travel time. The key difference from a standard shortest path problem is that only some cities contain Nexters, and we only care about distances between those Nexter cities.

The previous solution fails because it treats the case $\rho = 64$ as requiring a structural change to the algorithm, when in fact the MMIX conventions already make $\rho = 64$ perfectly well-defined...

We are asked to work with a specific infinite sequence derived from primes. We consider odd integers greater than 1 that are not prime. Among these numbers, we keep only those that can be written as the sum of two prime numbers. This filtered increasing list is called $G$.

We maintain a dynamic collection of balloon volumes. Each event either inserts a new value into this collection or asks a query about the current state.

We are building a structure in a 2D grid where blocks fall from above and form a growing “tower”. Each block is a domino of size 2×1, and it can be placed either horizontally or vertically.

We are working with a grid that has exactly two rows and a large number of columns. Every cell starts at zero, and we are allowed to perform a very specific local operation.

We are given a square park whose sides are aligned with the axes and span from $(0,0)$ to $(100,100)$. Inside this square, there are three fixed points representing fountains.

Let $\mu_k$ denote Pratt’s magic mask from (47).

We are given several independent scenarios. In each one, there is a list of customer service times, and the task is to split these customers into two checkout queues.

We are given a tree of galaxies. Each edge of the tree connects two galaxies and comes with two parameters: an initial distance and a yearly growth rate. If an edge connects nodes $u$ and $v$, then after $t$ years the distance on that edge becomes $a + b cdot t$.

We are given a directed graph of cities and one-way roads. City 0 is the starting point and city $N-1$ is the destination. Each road can be “blocked” by assigning one farmer to it, and blocking removes that directed edge from the graph.

We are given a chess endgame on a rectangular board of size $T times T$. Only two pieces matter: a white pawn and a black king. The pawn always moves upward (towards increasing row index), and the king moves one square in any direction, including diagonals.

We are asked to output a large number of triples of positive integers $(a, b, c)$, each bounded by $10^{18}$, with the additional requirement that every triple must satisfy a fixed cubic polynomial identity in three variables.

We are given an initial array and a sequence of swap operations that act on it. Each operation exchanges the values at two positions, and applying the full sequence produces a final arrangement of the array. The twist is that we are not executing the swaps in the given order.

We are given a sequence of bosses fought in a fixed order. For each boss, there are two ways to handle the fight.

Let $y = (x + 1) ,&, \bar{x}$.