brain

We are given a fixed array of integers. We are allowed to choose a single integer x and apply it to every element using XOR. After this transformation, each original value ai becomes bi = ai XOR x.

We are given a starting integer and a target integer. From the starting value, we are allowed to repeatedly apply one of two transformations: if the current value is even, we may replace it with three halves of itself, and if the value is greater than one, we may reduce it by…

We are given several arrays, and for each one we need to find the shortest contiguous segment that has a strict majority element.

We are given a sequence of monsters that must be defeated strictly from left to right. Each monster has a required strength threshold, and none can be skipped or reordered. We also have a pool of heroes.

We are given a multiset of white boards, each with a fixed integer length, and a very small set of red boards. From these boards we want to form a “mountain shaped” fence. The fence uses exactly one red board, which must be the unique maximum element in the sequence.

Each voter comes with two independent ways to make them support you. You may directly pay a fixed cost to activate any voter.

We are given several independent scenarios. In each scenario, there are an odd number of employees, and each employee has a salary interval from which their final salary must be chosen.

We are given a set of voters, and each voter can be activated in one of two ways. Either we directly pay a fixed cost to convince them, or we can exploit a dependency: if enough other voters are already convinced, they will join for free.

We are given a final string that appeared on a screen after someone pressed keyboard keys one by one. Each key corresponds to a lowercase Latin letter, and each key is either always healthy or always broken during the entire typing process.

We are given a very long decimal string, but the digits are not free to move arbitrarily. The only allowed move is swapping two neighboring digits, and even that swap is restricted: the two digits must have different parity, meaning one is even and the other is odd.

We are given several binary strings and allowed to repeatedly swap characters between any two positions in any strings.

We are given a collection of digit candles, where each digit from 0 to 9 appears a certain number of times. Each candle can be reused infinitely, so the counts do not deplete when we form numbers.

We are given a target area $n$, and we want to build a rectangle whose sides are integers and whose area is exactly $n$. Every valid rectangle corresponds to choosing two integers $a$ and $b$ such that $a cdot b = n$.

We are simulating a social media feed where posts continuously swap positions based on incoming likes. Initially, posts are arranged in a fixed vertical order from top to bottom, with post 1 at the top and post n at the bottom.

We are given a collection of integer segments on a number line. Each segment covers every integer point between its endpoints. A point becomes problematic if it is covered by more than $k$ segments at the same time.

We are given several independent scenarios. In each scenario there is a list of distinct integers representing student skill levels.

We are given two families of straight lines on the plane. The first family consists of lines of the form $y = x + pi$, and the second family consists of lines of the form $y = -x + qj$. Every $pi$ is distinct within its own group, and every $qj$ is distinct within its own group.

We are given a range of integers from l to r, and we need to count ordered pairs (a, b) inside this range such that adding them behaves exactly like XOR. In other words, the usual addition of a and b produces the same result as their bitwise XOR.

We are given two step sizes, a and b. Starting from zero, we paint every nonnegative integer in increasing order, but whether a number becomes white depends on whether it can be reached from already white numbers by repeatedly adding either a or b. Formally, zero starts as white.

We are given a sequence of rock-paper-scissors moves played by Bob. Alongside this, Alice has a fixed inventory of moves: she must play exactly a specified number of Rocks, Papers, and Scissors across all rounds.

We are given a tree where each node must be assigned one of three colors, and each assignment has a cost depending on the chosen color.

Two vertical cylinders of cross-sectional areas $S_1$ and $S_2$ are filled with water between two weightless, frictionless pistons.

A prismatic wooden block of constant square cross-section of side $a$ and length $L$ floats on the surface of water of density $\rho_w$.

We are given several test cases, each consisting of a list of plank heights. Every plank has width 1 and some integer height.

Two large parallel plates of area $S$ are separated by distance $L$, with $L$ much smaller than the lateral dimensions so edge effects are neglected.

We are given a path of length n, where each position is a tile arranged in a straight line. We assign a color to every tile. The constraint is not local adjacency, but global structure tied to divisors of n.

A rigid hemispherical bell of radius $R$ rests on a horizontal table with its rim in tight contact with the table, preventing fluid flow under the rim until lift-off.

A small block of mass $m$ moves without friction on a rigid surface consisting of two horizontal half-planes connected by a smooth spatial transition.

A body of mass $M$ is attached to an ideal spring of stiffness $k$, whose upper end is fixed.

Two identical thin-walled cylindrical tubes of mass $m$ and radius $R$ move on a horizontal rough plane.

Two identical direct current motors are rigidly connected by their shafts, so they share the same angular velocity $\omega$ and produce torques that add algebraically.

A long cylinder of radius $R$ and uniform material density contains a cylindrical hole parallel to its axis.

Two one-dimensional periodic structures represent the combs.

A parallel-plate capacitor with large identical plates of area $A$ is short-circuited, so both plates are connected by an external conducting wire and always remain at the same electric potential.

A refrigerator maintains its internal air at temperature $T_1 = 5^\circ\text{C}$ while it is placed in a room at temperature $T_2 = 20^\circ\text{C}$.

Two pistons of masses $m_1$ and $m_2$ move inside two rigid tubes of cross-sections $S_1$ and $S_2$.

Two identical steel balls of mass $m$ move on rigid, massless rods that constrain motion to circular trajectories of fixed radii $l$ and $2l$.

Three large open barrels contain water and have free surfaces located at fixed heights $H_1$, $H_2$, $H_3$ above a common reference level, with $H_1 > H_2 > H_3$, measured in meters.

A small mass $m$ is attached to a fixed point on a horizontal table by a spring of stiffness $k$.

A grounded conducting sphere of radius $r$ is fixed in vacuum.

A fixed mass $m = 1,\mathrm{kg}$ of an unknown gas is considered under two thermodynamic processes: heating at constant pressure and heating at constant volume.

Two helical springs are made from identical steel wire segments of equal total wire length $L_w$ and identical wire diameter.

Three identical communicating vessels contain water of density $\rho$ in a uniform gravitational field $g$.

We are counting a very specific family of binary search trees built on the keys from 1 to n. The tree structure must satisfy the usual BST ordering, but that is not the main constraint that drives the solution. The real restriction comes from two additional rules.

We are given a timeline of cars entering a tunnel and a separate timeline of the same cars exiting it. Every car appears exactly once in each list, so both sequences are permutations of the same set of identifiers. Inside the tunnel, overtaking is only detectable indirectly.

We are given several kinds of items, where each kind is unlimited in supply. We also have several distinct boxes, each belonging to a different friend, so boxes are labeled and cannot be swapped. For each kind of item, Alice chooses a subset of boxes to place that kind into.

We are given several independent scenarios. In each one, a shop has a list of item prices. The goal is to replace all of these different prices with a single uniform price so that selling all items at this single price does not reduce the total revenue compared to the original…

We are given an undirected graph with up to 100,000 vertices and up to 300,000 edges. The task is to split all vertices into exactly three non-empty groups so that the structure between every pair of groups is perfectly regular.

We are given a closed interval of integers from l to r, and we need to find any number inside this interval whose decimal representation does not repeat any digit. In other words, when writing the number as a string of digits, every character must be unique.

We are given a length-n sequence of questions, and for each position i there is a correct answer h[i]. We construct another sequence a of length n, where each a[i] is chosen independently from 1 to k.

We are given several closed intervals on a number line. Each interval represents a set of integer or real points between its endpoints, and the endpoints themselves are included.

We are asked to construct two positive integers $a$ and $b$ such that they differ by exactly one unit in the sense that $a + 1 = b$. We are not given the numbers themselves. Instead, we are only given the first digit of $a$ and the first digit of $b$.

We are walking in a grid from the top-left cell to the bottom-right cell, moving only right or down. The grid is not empty in a passive sense: some cells contain rocks, and those rocks behave dynamically. When we step into a rock cell, the rock does not block us.

We are given a schedule of TV shows over a sequence of days. Each day broadcasts exactly one show, and if we buy a subscription to a show, we gain access to all of its episodes for the entire timeline.

We are given multiple independent sequences, and for each one we want to transform it into a non-decreasing array using a very specific operation.

Each query describes a pool of students split into three groups. Some students are coders, some are mathematicians, and some have no specialization at all. A valid team must contain exactly three students and must include at least one coder and at least one mathematician.

We are given an $n times n$ table that was originally generated from some hidden array $a1, a2, dots, an$. Every entry of the table was formed by multiplying two elements of this array, so the cell in row $i$, column $j$ equals $ai cdot aj$.

Alan follows a fixed schedule of gym visits over $N$ days. On day $i$, he must lift exactly $X[i]$ grams, and he starts with a base strength $K$. He can permanently increase his strength by buying a pre-workout drink on any day he visits a gym.

We are given a directed functional graph defined by an array A. From every node i, there is exactly one outgoing edge to A[i]. Repeated application of this mapping defines a process where starting from a node x, we move along the graph m times.

We are given a creature with an initial number of heads, and a collection of attack types. Each attack type behaves in a very specific way: when used, it removes some number of heads, but after the strike, if the creature is still alive, it immediately regrows a fixed number…

We are given a conceptual infinite string formed by concatenating blocks of digits. The first block is just the number “1”. The second block is “12”.

We are given a ticket represented by a string of even length. Each position corresponds to a digit from 0 to 9, except that some positions are unknown and marked with a question mark. The unknown positions must eventually be filled with digits.

We are given a sequence of nonzero integers and asked to count how many subarrays produce a positive product and how many produce a negative product. A subarray is defined by choosing two indices $l le r$ and multiplying everything from $al$ to $ar$.

We are given three integers describing a small combinatorial setup. There are a fixed number of boys and girls in total, and exactly $n$ participants will attend an event, but we do not know how many of them are boys or girls.

We are given a weighted tree where every edge has a cost, and we are asked multiple questions about how “expensive” paths between pairs of vertices can be. For any two vertices $u$ and $v$, there is exactly one simple path between them because the graph is a tree.

We are given a list of integers, and we are allowed to repeatedly reduce any chosen number by replacing it with its half rounded down. Each such replacement costs one operation.

We are given an undirected graph with up to 500 vertices. The graph is known to come from a very specific hidden labeling process: each vertex originally had a number between 0 and 15, and an edge existed between two vertices if and only if their labels differ in exactly one…

We are given a multiset of player ratings. Each player can be used at most once, and our goal is to form as many disjoint teams as possible under a strict structure constraint. Every valid team consists of two homogeneous groups of players.

We are given a binary string s consisting only of '0' and '1'. A string is called coherent if it does not contain either of the following patterns as a subsequence: - "011" - "110" Recall that a subsequence does not need to occupy consecutive positions.

We are given an undirected weighted graph with n nodes and a list of weighted edges. We must travel from source to target. The key concept in this problem is the definition of a threshold.

We are given an array sides containing exactly three positive integers. Each integer represents the length of one side of a potential triangle. The task is to determine whether these three side lengths can form a triangle with positive area.

Each light in this problem behaves like a binary switch that flips its state over time. You are given an initial configuration where each light is either on or off.

We are given a small restaurant model where each query describes available ingredients and selling prices for two kinds of burgers. One burger requires buns plus beef, the other requires buns plus chicken.

We are given two collections of positive integers. One collection represents the possible values we can pick as the first number, and the second collection represents the possible values we can pick as the second number.

We are asked to build permutations of numbers from 1 to n such that a specific structural property holds on subarrays, and to control how many subarrays satisfy it.

We are asked to arrange the numbers from 1 to 2n around a circle. Once the circle is fixed, we look at every contiguous block of exactly n elements. Each such block produces a sum, and since the circle has 2n positions, there are 2n such blocks (wrapping around at the end).

We are given an unknown binary grid of size $n times n$, where $n$ is odd. Each cell contains either 0 or 1. Two facts are guaranteed: the top-left cell is 1 and the bottom-right cell is 0. We cannot directly read the grid.

We are given a tree with $n$ nodes. Each edge must be assigned a non-negative integer. Once the edges are labeled, every pair of nodes defines a path, and each path has a sum obtained by adding the values on its edges.

We are looking at all possible sequences of length $n+m$ made of exactly $n$ ones and $m$ minus ones. Every arrangement is considered once, so this is a multiset permutation problem.

We are given a binary string and we are allowed to construct another binary string of the same length. The requirement is not local, it is global over all substrings: for every interval of positions, the value of the longest non-decreasing subsequence inside that interval must…

We are given a binary string and asked to construct another binary string of the same length. The key requirement is not about matching the strings themselves, but about preserving a structural value computed on every substring: the length of the longest non-decreasing…

We are given a length n array of positive integers, but the array is not arbitrary. Every value in it must satisfy a strict structural rule: each number is either 1, or it is an even number whose half is also present somewhere in the array.

Each project can be thought of as an action that becomes available only when Polycarp’s current rating is high enough. Once he starts a project, his rating may go up or down, but it is never allowed to become negative at any point.

We are given two strings, s and t. The second string is guaranteed to already appear inside the first one as a subsequence, meaning we can pick characters from s in order and obtain t.

We are given several independent scenarios. In each scenario, a group of students is arranged in a fixed circular order, and we are shown that order as a linear list representing clockwise traversal around the circle.

We are given a source string s and a target string t, where t is already guaranteed to be a subsequence of s. In other words, we can pick characters from s in order and obtain t without reordering anything.

We are given two binary strings, which represent two integers. One of them, call it $x$, is fixed. The other, $y$, is effectively shifted left by some number of positions $k$, then added to $x$. After computing this sum, we take its binary representation and reverse it.

We are given a list of numbers and allowed to repeatedly increase any single element by one. Each such increment has a cost of one operation, and we have a total budget of at most $k$ operations.

We are given a grid with rows increasing upward and columns increasing left to right. We start at the bottom-left cell and want to collect all treasures placed on various cells.