brain

We are given a tree with nodes connected by edges, each labeled with a lowercase letter. In addition, we have a collection of strings. For any two nodes in the tree, the path connecting them defines a string by concatenating the letters along that path in traversal order.

We are building an array step by step from left to right, and after each position we look at the prefix we have constructed so far. From that prefix we compute its MEX, which is the smallest nonnegative integer that does not appear in the prefix.

A mass $m$ is attached to a rope wound around the shaft of a dynamo armature.

We are given a collection of dominoes, each represented by two cells: left and right. Some cells are already colored black or white, while others are undecided.

We are asked to determine, for each player in a tournament, whether they can possibly win if the tournament is conducted optimally. There are two maps, and each player has a strength on both maps.

We are asked to construct a permutation of the numbers from 1 to n that has exactly a local maxima and b local minima in the interior positions. A local maximum is a number larger than its immediate neighbors, and a local minimum is a number smaller than its immediate neighbors.

We are asked to construct an increasing sequence of integers for each test case, with a specific restriction on consecutive elements. The sequence must be strictly increasing, and additionally every next element must not be divisible by the previous one.

**Solution to patch**

The mismatch you are seeing (“correct count but invalid orientation”, sometimes even extra vertices counted) is a strong signal that the previous construction is not just buggy, but conceptually wrong.

We are asked to simulate the movement of robots on a number line. Each robot occupies a unit segment and follows a cyclic program of instructions that can push it left, right, or leave it stationary.

We are working with two integer arrays, one of size $n$ and one of size $m$. Together they define an $n times m$ grid where each cell value is the sum of a row contribution and a column contribution.

We are given a long array of non-negative integers. The task is to count how many contiguous subarrays have a specific property that depends on two extreme values inside the subarray: its minimum and its maximum.

We are working with a mutable string consisting only of the characters a, b, and c. After each update, we must answer a structural question about the string: how many positions must be changed so that the string no longer contains abc as a subsequence.

We are given a group of people who initially have no relationships at all. Over time, we are shown a sequence of pairs, and each pair represents a constraint that must eventually be satisfied: the two people in the pair must end up connected, meaning there exists a chain of…

We are given an array of integers and a step size e. For each starting index i, we can form a subsequence by taking every e-th element: a[i], a[i+e], a[i+2e], ... up to the point where the index does not exceed the array bounds.

We are given several independent test cases. Each test case consists of a small array of integers, and we are allowed to repeatedly redistribute factors of two between elements.

We are given an undirected weighted graph where each edge has a weight of either 1 or 2. Our task is to assign a direction to each edge to maximize the number of vertices whose outgoing and incoming edge sums differ by exactly 1.

A flexible rope passes over an ideal pulley.

The physical system consists of two dolphins moving through water along the same straight line toward one another.

A planet is modeled as a sphere of radius $R$ filled with an incompressible liquid of constant density $\rho$.

We are given a tree of $n$ vertices. The game is played on the tree by two players who take turns deleting edges. After an edge is removed, any connected component that contains no “pinned” vertex is completely discarded. The first player who cannot make a move loses.

We are given a tree with n vertices and m players. Each player selects two vertices on the tree, which we will call xi and yi. A player can only be eliminated if Mashtali chooses a vertex v and the closest vertex w to v on the path from xi to yi is strictly between xi and yi.

We are given a string of parentheses S that has been modified by repeatedly performing a wrapping operation. Each operation allows you to take any contiguous substring B and surround it with a pair of parentheses, inserting it back into its original place in the string.

We are given a nondecreasing array and asked to remove as few elements as possible so that the remaining array avoids a very specific kind of “badness” condition.

We are given an array and we look at every non-empty subsequence of it. Each subsequence is interpreted as a list of lengths, and for every length $bi$ we are allowed to build a contiguous block of exactly $bi$ consecutive integers.

A neon-filled discharge tube is placed between two large parallel plate electrodes separated by a distance

We are given an array and allowed to perform a very specific type of transformation: we may pick one value x and remove any occurrences of that value from the array, not necessarily all of them.

We are given a collection of people where each person has a fixed wealth equal to their index, so person 1 has 1 dollar, person 2 has 2 dollars, and so on up to n. We want to choose a subset of these people to invite to a party. Each person comes with two constraints.

We are given a rectangular grid with $n$ rows and $m$ columns. Somewhere inside this grid, an adversary hides a single cell $(x, y)$. We are allowed to “probe” a fixed set of $k$ cells of our choice. For each chosen cell, we receive its Manhattan distance to the hidden cell.

Consider a planet of radius $R$ and mass $M$ surrounded by an atmosphere of uniform density $\rho$.

We are given a rectangular grid of size $n times m$, where each cell either contains a candy ('1') or is empty ('0'). A robot can start on any cell in the top row and move diagonally down-left or down-right until it leaves the grid.

The problem involves simulating an ATM that starts with a fixed amount of money and must serve a queue of students. Each student either deposits a positive amount or withdraws a negative amount from the ATM.

The maze is an undirected tree rooted at room 1. Vlad starts at room 1 and wants to reach any leaf node that is not room 1. A leaf is any node with exactly one adjacent corridor. At the same time, several friends are already placed on distinct nodes.

We are given a tree where each vertex has a pointer to its parent, except for the root which points to itself. This parent array b encodes the tree unambiguously.

Two vertical cylinders of cross-sectional areas $S_1$ and $S_2$ are open to the atmosphere at their upper ends.

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.