brain

We are given a tree of rooms. Vlad starts from room 1 and tries to escape by reaching any leaf node that is not room 1. A leaf is a room with exactly one corridor. At the same time, several friends are placed on distinct nodes.

We are given a sequence a of length n that was generated by repeatedly removing the smallest element from either end of a hidden permutation p of the numbers from 1 to n. If the smallest element is at the left, it is prepended to a; if it is at the right, it is appended to a.

We are given a number written in decimal form with no zero digits. The only operation allowed is to take a prefix of the number and reverse the order of digits inside that prefix. The prefix length can be anything from 1 up to the full length of the number.

We are given two groups of students: programmers and mathematicians. From these students, we want to form as many teams as possible, where every team contains exactly 4 students.

In this problem, Monocarp starts with the weakest armor and weapon, labeled 1. He can spend one hour to acquire a new armor set or weapon, but there is a catch: to obtain the $k$-th armor or weapon, he must already possess some armor and weapon combination whose total power is…

We are given a multiset of labels, where label i appears exactly ci times. We must arrange all these occurrences into a single array. Once the array is fixed, we look at every pair of identical values and add the distance between their positions.

We have a group of students, each of whom Monocarp wants to read a specific message in a chat. Every student has a limit on how many messages they will read in a day.

We are given a pair of integers $a$ and $b$, and another integer $x$. We are allowed to repeatedly replace either $a$ or $b$ with the absolute difference $ The constraints allow $a, b, x$ to go up to $10^{18}$, and there can be up to $10^4$ test cases.

We are working on a grid where movement is measured using Manhattan distance, meaning the cost between two points is the sum of horizontal and vertical separations. One point is fixed at the origin, while the second point lies at coordinates $(x, y)$.

We are asked to construct a permutation of numbers from 1 to n, where n is even. The permutation is split into two equal halves: the first n/2 elements form the left half, and the remaining n/2 elements form the right half. Two conditions must be satisfied simultaneously.

We are asked to simulate a chat scenario where a user wants to send an "emote triangle" consisting of messages with increasing then decreasing numbers of emotes.

We are given a rooted tree where vertex 1 is fixed as the root. The task is to assign a distinct label from 1 to n to every vertex, so the labels form a permutation of 1 to n.

We are asked to count subsequences of an array where every prefix of the subsequence behaves in a very specific “MEX-stable” way.

We are given a rectangular grid of size n × m containing three types of cells: free cells, blocked cells, and a single lab. The robot can occupy any free cell, and at each step we can issue one of four commands: move up, down, left, or right.

We are given a sequence of distinct positive integers. Our task is to construct roughly half as many pairs of numbers as the length of the sequence, where for each pair $(x, y)$, both numbers come from the sequence, they are distinct, and the remainder when $x$ is divided by…

Each number in this problem is not given in its expanded form, but as a compact description. Instead of writing the full integer, we are given a base value x and a number of trailing zeros p, meaning the actual number is x × 10^p.

We are given a sequence of attack times and a dragon with h hit points. Each attack applies a poison that lasts k seconds and deals 1 damage per second. If the dragon is already poisoned when a new attack lands, the poison timer resets.

- a = [3,1,4,9,2,5,6] - The edges form a path: 1→2→3→4→5→6→7 - res = a[:] = [3,1,4,9,2,5,6] - Process topologically: 1. Node 1 → Node 2: res[1] = min(1,3) = 1 OK 2.

The physical system consists of a satellite in low Earth orbit photographing the surface of the Earth.

We are asked to simulate the evolution of the temperature inside Divan's cottage over a sequence of days, where the temperature adjusts by one unit per day towards the outside temperature.

A hollow insulated conducting sphere of radius $R$ is charged by water drops falling from a vertical tube.

We are given a multiset of positive integers and allowed to permute them in any order. After choosing an order, we repeatedly take prefixes of this order and compute the gcd of each prefix, then sum all those gcd values.

We are given a list of positive integers, and we are allowed to reorder them arbitrarily. After fixing an order, we look at all prefixes of this order.

We are asked to reconstruct the sum of XORs over all non-empty subsequences of an array, knowing only the bitwise OR of several contiguous segments of that array. Each array element appears in at least one segment, so every element contributes to the OR values we are given.

Divan wants to buy chocolate bars under three constraints: he only considers bars within a price range [l, r], and he cannot exceed his total budget k.

Your input: Structure: - t = 4 - Each test case: - one integer n - one line with n integers So format is consistent: Expected: Now observe carefully: [1, 7] → 1 [1, 5, 4] → -1 [12345678, 87654321, 20211218, 23571113] → 4 [1,2,3,4,18,19,5,6,7] → 2 So the output is: not…

Consider a sphere of radius $R$ illuminated by a parallel beam of sunlight.

We are given an array with some fixed integers and some positions marked as zero. Each zero must be replaced by any integer from 1 to n, and different zero positions can be filled independently, even with the same value.

We are given a tree with n nodes and n-1 edges, where each edge can store a non-negative integer representing lights, or -1 if the value is unknown.

We are given a one-dimensional Lego strip of length n, where some positions may or may not contain a Lego block. Our memory of the starting and ending configurations is partial: for each position we either know it has a block (1), we know it is empty (0), or we don't remember (?

We are given a rooted tree with $n$ nodes, rooted at node 1, where all nodes start white. Two players, Red and Blue, take turns coloring nodes. Red goes first and can color at most $k$ nodes red by choosing entire subtrees.

We have a row of candles on a Hanukkah menorah, and each candle is either lit or unlit. The current state is given as a binary string a, where 1 indicates a lit candle and 0 indicates unlit. We are asked to reach a target configuration b using a special operation.

Let the side length of the square cross section be $a$.

We are given many test cases, and each test case describes a contiguous array of integers from $l$ to $r$. The array is not arbitrary, it is always a full interval with no gaps.

We have a row of towers, each with a certain number of blocks stacked vertically. The goal is to make the heights of all towers as close as possible. On any day, we can take a block from one tower and place it on another tower.

The problem presents a directed acyclic graph (DAG) where every edge points from a smaller-numbered vertex to a larger-numbered one.

We are asked to count all non-empty subsets of an array where the XOR of every pair of elements does not exceed a given number x. Concretely, if we take any subset of indices from the array, and compute the XOR for every pair of values in that subset, each XOR must be ≤ x.

We are given an array of integers and a target value x. The task is to select as many elements as possible from this array, with a constraint on consecutive selected elements: any contiguous subarray of length at least two that is entirely selected must have an average at…

We are given a small undirected graph where every edge is supposed to end up painted with one of three colors. Some edges are already fixed with a color, while others start uncolored and must be assigned a value in {1, 2, 3}.

We are given two strings of equal length, and we are allowed to modify the first string only by swapping adjacent characters. Each swap costs one operation.

We are given a list of integers and we are allowed to negate any subset of them. The goal is to maximize the number of distinct integers in the array after possibly negating some elements.

A valid output is constructed by choosing a split point k in a string. After choosing k, the visible string becomes the prefix s[1..k] followed by its reverse. So every candidate answer is a palindrome that is “built from a prefix”.

We are given an array of integers, and we are allowed to freely replace elements with any real numbers. The goal is to transform the array so that a very strong structural property holds: every subarray must satisfy a linear averaging condition that ties its sum only to its…

Your “actual output” is: That means the program likely: - crashed early, or - never entered the solve loop, or - mis-parsed the first token and exited silently Now look at the input: Your code almost certainly does: So it reads: Correct.

We are given a box of chocolates, each with a distinct type represented by a non-negative integer. Icy wants to make at least one pair of chocolates have the same type, but before any exchanges, she chooses two chocolates, say at indices $x$ and $y$.

We are given a group of n players, each labeled from 1 to n, where n is always a multiple of 3. Some of these players are impostors, and the rest are crewmates. We do not know the number of impostors k in advance, but we are guaranteed that it is strictly between n/3 and 2n/3.

We are given a set of players, each of whom is either an impostor or a crewmate. The total number of players, $n$, is divisible by three.

Two beads of masses $m_1$ and $m_2$ move without friction on a smooth horizontal circular wire ring of circumference $L$.

We are asked to construct three distinct positive integers $a$, $b$, and $c$ such that their sum equals a given integer $n$ and the greatest common divisor of the first two numbers equals the third: $gcd(a, b) = c$.

We are given a string S of arbitrary lowercase letters and a string T that is always a permutation of "abc". The task is to rearrange the letters of S into a new string S' that is the lexicographically smallest possible while ensuring that T does not appear as a subsequence.

Thanks. Now we have more clarity. The previous solution was using brute-force divisor counting, which explains why: - Small numbers like 10 → 4 and 1 → 1 worked. - 25 → 3 (wrong) because brute-force counted divisors incorrectly.

A car moves along a straight road with its wheels rolling without slipping.

We are allowed to repeatedly transform a positive integer by operating directly on its binary representation. In one move, we take the current binary string of the number, append either a 0 or a 1 at the end, reverse the entire resulting string, strip leading zeros, and…

In this problem, Monocarp has a set of items, each with an integer price, and he can trade these items with another character’s items.

We have a circular arrangement of $n$ towns, each with a singer who has an initial repertoire of $ai$ minutes. Every singer tours all towns in clockwise order, performing in each town.

We are given a multiset of integers. We repeatedly remove two elements exactly $k$ times, and each removal produces a value that is added to a running score. After these $k$ operations, whatever elements remain are also added directly to the score.

The solution correctly addresses the exercise.

We are given an array of positive integers. The task is to pick a positive integer $d$ and color the elements of the array in two colors: red for elements divisible by $d$, and blue for elements not divisible by $d$.

We are given a sequence of overlapping two-letter strings, or bigrams, derived from a word consisting of only 'a' and 'b'.

Now we can clearly separate the real issue from all the noise in earlier attempts. Your “solution” is not consistently failing on a single algorithmic idea.

Each test case gives a set of mines placed on a 2D grid. Every mine has a position and a “natural lifetime”, meaning it will explode automatically after a given number of seconds if nothing triggers it earlier. The key twist is that explosions propagate.

We are given a permutation of integers from 1 to $n$. Conceptually, this is an array of length $n$ in which every integer from 1 to $n$ appears exactly once, so every index maps to a unique value. On top of this array, we have two types of queries.

We are asked to organize a fair schedule for a multi-table game. There are $n$ players, $m$ tables, and $k$ rounds. Each round, every table must host either $lfloor n/m rfloor$ or $lceil n/m rceil$ players.

We are given an array of non-negative integers and we can increment any element by 1 any number of times. The task is to determine for each integer from 0 up to the array length whether it is possible to transform the array so that its MEX (minimum excluded value) equals that…

We are given several independent scenarios. In each scenario, Vlad has a fixed number of friends and a fixed number of shops. Each shop offers a value for every friend, representing how much joy that friend would get if their gift is bought there.

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 $lfloor x / n rfloor$.