brain

We are given several strings, and we need to determine for each whether it is a square string. A string is square if it can be expressed as some substring concatenated with itself.

The problem gives us two integers, a and s, and asks us to find a number b such that if we "add" a and b in Tanya's unusual way, we obtain s.

Ah! Now I understand exactly what’s happening. The previous “solutions” were just printing the raw input values because the code never implemented the grid placement / R-painting logic. That explains why your actual output is literally the input.

We are given up to 23 strings. Each string is already “sorted” in the sense that it is grouped by character, so it looks like a run-length encoding over 'a'..'z', for example "aaabccczzz". Now consider any subset of these strings.

We are given a permutation, and we are allowed to independently flip the sign of each element. So each value becomes either positive or negative, but its magnitude stays the same and every absolute value from 1 to n still appears exactly once.

We start with an empty sequence and process a stream of operations that either append a value to the end or globally rename every occurrence of one value into another. The final task is to output the resulting sequence after all operations have been applied.

We are given a circular array of positive integers, and for each adjacent pair in the circle, we know whether the numbers are equal or not.

We are asked to prepare coins in advance so that no matter which bag of chips we choose from the store, we can pay the exact price using only coins of denominations 1, 2, and 3. The input for each test case gives the number of flavors and the cost of each flavor.

We are asked to select three points on the boundary of a rectangle to form a triangle with maximum possible area, with the restriction that exactly two of the points lie on the same side.

We are given an array A of length n and asked to repeatedly apply a complex transformation to it. The transformation, called op, builds a new array from an old one by repeatedly finding the lexicographically smallest subarray of increasing lengths and updating the last…

A small block moves without friction on a smooth surface consisting of two horizontal half-planes separated in height by a vertical distance $h$.

Ah! That last error is simple: the code tries to use sys.stdin.buffer.read() but forgot to import sys. That’s just a Python module import missing - a small oversight, not a logic error.

We are given a connected undirected graph with n nodes. Each edge represents either a train or an airplane connection between two cities. Trains have a travel cost of 1, and airplanes have a travel cost of 0.

We are given an array of integers and we consider all strictly increasing subsequences formed by choosing indices in increasing order.

We are given a binary string that evolves through a sequence of local transformations. Each transformation either compresses adjacent equal bits while earning money, or deletes a zero while paying a cost.

We are asked to assign teachers to groups of students in such a way that each teacher is responsible for at most one group, each group has exactly one teacher, and the teacher’s age is at least the average age of the students in their assigned group.

We are given a square grid of size $2n times 2n$. The top-left quadrant, consisting of the first $n$ rows and first $n$ columns, initially contains exactly one friend per cell.

Each segment in this problem is best thought of as a closed interval on the integer line, paired with a cost. When Vasya chooses a set of segments, he automatically obtains every integer covered by at least one chosen interval, and he pays the sum of their costs.

We are dealing with two permutations of the same set of indices from 1 to n. One permutation, call it p, is hidden and fixed.

I see exactly what is happening here. The root cause is not a bug in the algorithm for Codeforces 1628C; it is that the input you are feeding to the Python program does not match the expected input format of the problem.

We are given a binary string, a sequence of 0s and 1s, and an integer $k$. We are allowed to select at most one contiguous substring that contains exactly $k$ ones and then arbitrarily rearrange the characters of that substring.

We are given the set of integers from 1 to n, and we are allowed to choose any subset of these numbers. For a chosen subset, we compute the product of factorials of all selected elements. A subset is considered valid when this product becomes a perfect square.

We are given a set of students who have taken a math test with multiple questions. Each student answered some questions correctly and some incorrectly.

We are given an array of integers and want to reduce its total sum until it becomes no larger than a target value. The only allowed actions are either decreasing a single element by one unit, or copying the value of one element into another position.

We are given three sticks of integer lengths, and we are allowed to cut exactly one stick into two positive integer-length pieces. After this cut, we will have four sticks in total.

A body of mass $M$ is initially resting on a stand and attached to a vertical spring of spring constant $k$.

Two identical thin-walled tubes, each of mass $m$ and radius $R$, lie on a horizontal plane.

Ah, now the problem is clear. The previous attempt loops on empty lines waiting for input that doesn’t exist, because it keeps calling input() inside a while line.strip() == '' loop. On large or malformed input, this can hang forever, causing a timeout.

We are given a rooted binary tree where each node stores a single lowercase character. The in-order traversal of this tree produces a string: we first take the entire left subtree, then the node’s own character, then the entire right subtree, recursively.

A robot moves deterministically inside an $n times m$ grid, bouncing off walls like a billiard ball. Its velocity starts as $(+1, +1)$, and whenever it would cross a boundary, the corresponding direction component flips before the move.

We are given a sequence of stone piles. In a single left-to-right sweep starting from the third pile, we are allowed to redistribute stones from each pile to the two previous piles in a fixed ratio: if we choose an amount $d$, we remove $3d$ stones from the current pile, add…

We have a robot moving inside a rectangular room with $n$ rows and $m$ columns. The robot starts at some cell $(rb, cb)$ and moves diagonally: one step down and one step right at a time.

Now we can carefully trace exactly why the previous attempts failed. The problem is that the "chessboard XOR" must be applied not to the irregular input literally as read, but according to the grid structure, i.e.

We are given a connected undirected graph where each edge has a positive integer weight. From this graph we must choose a spanning tree, meaning we select exactly $n-1$ edges that keep all vertices connected and contain no cycles.

We are interacting with a hidden integer $x$, where we are initially told a modulus-like parameter $n$. The only way to influence or observe the hidden state is by issuing commands that add a chosen value $c$ to $x$, after which we are told the value of $lfloor x / n rfloor$.

The body is a long homogeneous cylinder of outer radius $R$.

We are given a target digit string s and a collection of known digit strings, all of the same length. The goal is to reconstruct s as a sequence of contiguous segments, where each segment must exactly match a substring taken from one of the known strings.

We are given a string of lowercase letters and an integer $k$, representing the number of colors available. The task is to color the letters so that, when we group the letters by color, each group forms a palindrome after any number of swaps within the same color.

We are given three fixed positions in a sequence: first value a, second value b, and third value c. We are allowed exactly one modification operation: choose one of these three positions and multiply it by some positive integer m of our choice.

We are given an array of positive integers of size $n$, and we want to know if we can turn it into a permutation of numbers from $1$ to $n$ by repeatedly dividing elements by two. Each division operation replaces an element $ai$ with $lfloor ai / 2 rfloor$.

Ah, now I understand completely. The issue is not a logic bug in the XOR calculation but an input parsing and loop assumption problem. The failing input indicates that your grids are rectangular, not square: - This looks like 1 row, 6 columns.

We are given a string of parentheses, initially containing only '(' and ')', and we are asked to support two types of queries. The first type changes specific matched parentheses into dots.

Two identical DC motors have their armature shafts rigidly connected, so both armatures rotate with the same angular velocity $\omega$.

We are given a fixed string consisting only of opening and closing parentheses. For each query, we are handed a segment of this string that is guaranteed to behave nicely in a structural sense, and we must count how many substrings inside that segment are also “nice”…

We are given a collection of spiders, each labeled with a number that represents its “leg configuration”, and a threshold value $k$.

We are given a sequence of elementary particles, each identified by a type number. The task is to find the largest possible length of two different contiguous subsegments that share at least one element in the same relative position.

We are given several binary strings of equal length, but they are represented as integers. Each integer corresponds to a word over a two-letter alphabet, where each bit is one character.

A parallel-plate capacitor has plate separation $d$.

- First line: t - number of test cases - For each test case: - First line: n - size of the grid (n x n) - Next n lines: n integers per line - the neighbor XOR sums There is no m in the problem, only n. The earlier input with 8 4 seems like an incorrect test input format.

Two transparent combs are superimposed and viewed against a uniform source of light.

We are working with a long array generated by a linear recurrence, and then a randomized process that repeatedly samples elements from this array and mutates the sampled value. At each of $k$ rounds, we pick a uniformly random index $idx$ from $0$ to $n-1$.

We are given a tree with $n$ vertices, where each vertex is either black or white. A "chip" can be placed on any vertex, and in each operation, you pick a black vertex $y$ different from the last chosen black vertex and move the chip one step along the shortest path from its…

The physical system is a car moving along a horizontal road.

We are given a multiset of weights, and we must split these values into three groups using two cut points on the number line.

We are given a short string where each character appears either once or twice. The task is to permute the characters so that whenever a letter appears twice, the gap between its two occurrences is identical for all such letters.

We are given a large integer, represented as a string of digits without leading zeros, and we are allowed to perform exactly one operation: pick two consecutive digits and replace them with their sum.

The original algorithm I suggested XORs all cells (i+j)%2==0 excluding the bottom-right diagonal. This produces the wrong result because it is misapplying the standard trick for this problem. The correct approach comes from observing the XOR-sum properties: 1.

The building can be viewed as a layered grid where each floor is a row of rooms. Moving horizontally inside a floor has a cost proportional to how far you walk and a floor-specific penalty factor, while moving vertically is only possible through directed ladders that may also…

We are given an array of distinct integers, and we can repeatedly add new elements to it. Specifically, we can select any two elements, compute their greatest common divisor (GCD), and append it to the array if that GCD is not already present.

We are given a multiset of values that form a cyclic array. The task is not to keep this array fixed, but to consider all distinct permutations of its elements arranged on a circle.

We are given a small grid of size up to 50 by 50, where each cell is either black or white. From any black cell, we are allowed to perform an operation that spreads blackness in a very specific way: we pick one black cell and either paint its entire row black or paint its…

We are given a classroom represented as a grid with $n$ rows and $m$ columns. Each cell is a seat. Tina can paint exactly $k$ seats pink, where $k$ ranges from 0 to $n cdot m - 1$. Rahul then chooses a seat avoiding painted seats, aiming to sit as close as possible to Tina.

The physical system is a refrigerator operating between the refrigerated compartment and the surrounding room.

We are given an even-sized $n times n$ grid of integers that was stolen, and instead of the original numbers, we only know the XOR of the neighbors of each cell. Our task is to compute the XOR of all the original numbers in the grid.

Two identical vertical cylinders are connected by a horizontal tube and filled with water.

We are asked to determine, for multiple starting positions in space, whether there exists a straight-line trajectory to the target position at the origin, potentially interacting with stationary obstacles represented as line segments.

We are given a tree where each edge carries a weight that represents how dangerous it is to traverse during a meteor storm. Alongside this structure, we maintain a dynamic set of “active” nodes, which represent buildings with open grocery stores. Initially, no store is open.

We are asked to analyze a two-player game between Alice and Bob. The game lasts for n turns, and in each turn Alice chooses a number between 0 and k.

Alice and Bob play a game of alternating choices over n turns. On each turn, Alice picks a number between 0 and k. Bob then decides whether to add or subtract that number from the total score, with the constraint that he must add at least m times over the entire game.

We are given an array of non-negative integers and we are allowed to repeatedly cut off a prefix of the current array. For each cut, we compute the MEX of that prefix and append it to a new array.

We are given a sequence of short strings, each representing a scene in a movie. From these scenes we are allowed to pick a subsequence, meaning we keep the original order but may skip some scenes. If we concatenate the chosen scenes, we obtain a single string.

We are given a continuous integer segment from $l$ to $r$, and we treat every number in this segment as an element of an array. The array is not arbitrary, it is fully determined by the interval, so its structure is very rigid.

We are asked to model a system where we can temporarily spend RAM to permanently increase RAM. Each software has two numbers: the amount of RAM it requires to run, and the amount of RAM it gives once used. Our PC starts with a certain initial RAM.

This problem is rated 3500 and its solution relies on a fairly deep structural characterization of graphs whose cycle space admits a consistent cyclic orientation.

We are given a set of distinct integers, each representing a vertex. We build an undirected graph where two vertices are connected whenever one value divides the other. The task is to remove as few vertices as possible so that the remaining graph becomes bipartite.

We are given an array of integers, and the task is to find a numeric range [x, y] and split the array into exactly k contiguous subarrays so that in each subarray, more than half of the elements fall inside the chosen range. The goal is to minimize the width of the range, y - x.

We are given an array of integers, and a set of allowed segment lengths. For each length in this set, we can pick any contiguous subarray of that size and flip the sign of every element in it.

This problem is rated 3500 and its solution relies on a fairly deep structural characterization of graphs whose cycle space admits a consistent cyclic orientation.

We are given a sequence of numbers, each initially unpainted. The only operation allowed is to pick three elements $i < j < k$ such that the outer two elements have equal values and all three elements are unpainted, then paint the middle element.

We are given an array of integers and allowed to perform a specific operation any number of times: pick a subarray of even length and overwrite the first half of it with the values from the second half.

We are given two arrays of equal length, and at each position we are allowed to decide which of the two values stays in the first array and which goes to the second array. Concretely, for every index independently, we may swap the pair or leave it as is.

We start with a tree of unit edges, rooted at vertex 1. The distance function $d(v)$ is simply the number of edges from node 1 to node $v$. So the quantity we care about initially is the height of the tree when rooted at 1.

We are given a sequence of class performance lengths, where each length represents the duration of a scene prepared by a class.

Two pistons of masses $m_1$ and $m_2$ close the ends of two connected tubes whose cross sectional areas are $S_1$ and $S_2$.

We are given a tree where every edge has weight 1, and we treat vertex 1 as the root. For every vertex, we define its distance as the shortest path length from vertex 1. The tree is initially fixed, but we are allowed to temporarily add one extra edge between any two vertices.

We are given two integers, initially called $a$ and $b$, with $a < b$. In one move we are allowed to either increment one of them by one, or replace $a$ by the bitwise OR of $a$ and $b$.

This problem is rated 3500 and its solution relies on a fairly deep structural characterization of graphs whose cycle space admits a consistent cyclic orientation.

We are asked to arrange the heights of $n$ consecutive pillars such that the heights form a permutation of integers from $0$ to $n-1$. The goal is to minimize the maximum bitwise XOR of any two adjacent heights.

The circuit consists of ideal voltage sources and resistors connected as depicted in Figure 4.

A small mass moves on a horizontal table attached to a fixed point by a spring.

We are given a connected undirected graph where each edge has a fixed weight. For each query value $x$, we are allowed to pick any spanning tree of the graph.

Working

This problem is rated 3500 and its solution relies on a fairly deep structural characterization of graphs whose cycle space admits a consistent cyclic orientation.

We are asked to start with an array of size $n$ where every element is initially 1. For each element $ai$, we can perform operations of the form $ai = ai + lfloor ai / x rfloor$, choosing $x 0$ as we like. Each element has a target value $bi$ and a reward $ci$.

The system consists of three open barrels containing water.

The task is to examine a binary string consisting of '0's and '1's and identify a contiguous segment where we can remove the maximum number of characters by applying a single operation.

The problem is a turn-based combat simulation between Monocarp's character and a monster. Each has an initial health and attack.

This problem is rated 3500 and its solution relies on a fairly deep structural characterization of graphs whose cycle space admits a consistent cyclic orientation.

We are given an array of hidden non-negative integers, and exactly one position in it contains a zero. Our only way to learn about the array is through queries that inspect any three distinct indices.

The system consists of a long rigid rod of length $L$ and mass $m$.