brain

We are given a fishing session defined by a single time interval within one day. Alongside this, we have a large set of fish “activity intervals”, each labeled with a species name. During an activity interval of a species, that fish is actively biting.

We are given a collection of tower heights. There are $n$ towers, where $n$ is guaranteed to be odd, and each tower has some initial height. We are also given $k$ extra unit cubes. Each cube can be added to exactly one tower, increasing its height by one.

We are given two configurations of the same number of points on an integer grid. Think of them as two drawings of indistinguishable particles placed on lattice points. The particles can move, but only through a very specific collective operation.

There are 30 independent positions, each associated with a weight equal to twice its index. Over a sequence of seconds, each position can experience at most one event per second: the occupant either enters its hole, leaves it, or stays unchanged.

We are given a fixed collection of existing family names, each written as a string of lowercase letters and hyphens. Then we are given several candidate names for a newborn. For each candidate, we must decide whether it is acceptable. A candidate is rejected in two situations.

We are given a line of houses, each with a fixed height. A resident who lives in house i wants to reach their own roof starting from the ground, but movement is constrained by a single ladder of fixed length. A ladder of length L allows two kinds of actions.

The problem describes a circular cake that behaves like a clock. The cake is first cut at noon, and then a sequence of people arrive at fixed integer hours between 12 and 24. Each arrival creates a cut at that hour, and the cake is divided into segments between consecutive cuts.

We are given a list of flower beds, each associated with a number of shells. For the i-th bed, there are ai shells that must all be used to form a decorative border.

We are given multiple independent scenarios. In each scenario, a reader has a sequence of reading amounts over days and a list of book lengths. Each day contributes a certain number of pages that can be used to progress through books in order.

We are counting ordered pairs of integers $(a, b)$ where $0 le a le b$, but not all pairs are valid. The restriction comes from a bitwise condition: when we take the bitwise OR of $a$ and $b$, the result must not exceed $n$.

We are asked to evaluate a large sum where each term combines Fibonacci numbers and factorial exponents, but we only care about the last digit of the result. For each test case, an integer $n$ is given. We conceptually build the value $$S = f0^{0!} + f1^{1!} + f2^{2!

We are given multiple independent test cases. In each test case, there is an array of integers and a threshold value $k$. We call a set of values “good” if the largest and smallest elements in that set differ by at most $k$.

We restart the argument from the actual structure of Knuth’s swap-in-place algorithm (Exercise 147) and then isolate exactly what changes in the ZDD setting.

We are given many independent queries. Each query provides a non-negative integer $n$, and we must count how many integers $x$ in the range from $0$ to $n$ can be written as the sum of two integer squares, meaning $x = a^2 + b^2$ for some integers $a$ and $b$.

We are given a very large integer written in decimal form, and for each such number we need to count how many positive integers not exceeding it consist only of the digits 4 and 7.

We are given a pool of students, each student knows a subset of up to 60 topics. A valid team is any subset of students such that two conditions are simultaneously satisfied: every topic from 1 to p is covered by at least one team member, and for each topic, at most one team…

We are given a binary string, where each position is either 0 or 1. We are allowed to change at most k zeros into ones.

Let $x in [0,1)$ have ternary expansion $x = 0.x1 x2 x3 cdots quad (xj in {0,1,2}),$ where nonterminating representations are used. Define $omega = e^{2pi i/3}$, so $omega^3 = 1$ and $1 + omega + omega^2 = 0$.

We are given a string made only of the characters X, T, and U. For each test case, we need to count how many substrings have the property that the number of X, T, and U characters inside that substring are all equal.

We are given a two-versus-two game where each round reduces to a comparison between two independent “targets” produced by the two main players, Mo and Larro. In each round, Mo and Larro each pick one number from their personal hand. These numbers become target sums.

We are given a directed network of people where each person knows the addresses of some other people. When someone receives a message, they immediately forward it to everyone they know. The process starts from a specific person and repeats indefinitely.

For each query, we are given a prime number $p$. We look at all integers from $1$ to $p-1$, and for each such integer $a$, we compute its multiplicative inverse modulo $p$. That means we find a number $b$ in the range $[1, p-1]$ such that $a cdot b equiv 1 pmod p$.

We are given a ticket-printing system with two identical machines that can be used to generate reimbursement slips. Each machine can produce at most one ticket per operation, and after producing a ticket it becomes unavailable for a cooling period of a minutes.

We are given three groups of employees with sizes A, B, and C. Every employee must be placed into pairs, meaning each employee is matched with exactly one other employee, and no one is left unmatched.

A projection function $x_j$ corresponds to the Boolean function that is $1$ exactly on those assignments where the $j$-th variable is $1$.

We are given multiple independent scenarios. In each scenario, a student starts with a fixed number of items that must be carried, and there are several checkpoints along a path.

We are given a directed graph with possibly multiple edges between the same pair of vertices and also self loops. Each directed edge represents a single-step move between cities.

We are given a binary sequence arranged on a circle. Each position contains either 0 or 1, and indices wrap around so that position n−1 is adjacent to position 0.

We are given a rectangular grid of lowercase letters. From this grid, we can choose any sub-rectangle by selecting a contiguous block of rows and a contiguous block of columns.

Let $F = \mathrm{MUX}(f,g,h)$ denote the Boolean function defined by selecting $g$ when $f=1$ and selecting $h$ when $f=0$, so that F = (f \wedge g)\ \vee\ (\neg f \wedge h).

We are given a string composed only of the characters q and a. We are allowed to insert exactly x additional characters, each of which can independently be either q or a, at arbitrary positions in the string.

We are given a collection of static segments on a number line, each segment having integer endpoints within a bounded universe up to $L$.

We are given a sequence of $n$ independent “draws”. In the $i$-th draw, we choose an integer score $xi$ uniformly from the range $[0, ai]$, where $ai < k$. After each draw, we maintain a running prefix sum of all chosen values.

We are asked to construct up to 20 distinct integer vectors in at most three dimensions. Each vector has non-negative coordinates up to 10^9. After constructing them, we look at the sum of all vectors.

We are given a lineup of enemy units, each with an integer attack value. A special effect card removes every unit whose attack is odd after all modifications are applied. Before using this card, we are allowed to cast a collection of single-use spells.

We are given two intervals of positive integers, one for a and one for b. We need to count how many pairs (a, b) can be formed such that a is chosen from the first interval and b from the second interval, and the pair satisfies a bitwise and arithmetic constraint: the XOR of a…

Let $t(m)$ denote the parity of the binary digit sum of $m$, so that $t(m)=0$ when $m$ has an even number of 1s in binary representation and $t(m)=1$ otherwise.

We are given several “ability strings”, where each ability is made of distinct characters and no character appears in more than one ability.

The earth map can be modeled as a directed graph where each city is a node and each one-way road is a directed edge.

We are given a vertical stack of items, where each item has a color. The top of the stack is position 1, and positions increase as we go downward. A sequence of queries is performed on this stack.

A ZDD represents a family of finite sets over an ordered universe of items $x_1 < x_2 < \cdots$.

We are tracking how a quantity evolves over time when a deterministic growth rule starts applying only after a delay. Saimon begins with some number of identical units, specifically pairs of Emm coins.

We are given multiple test cases. Each test case contains a single lowercase string representing text Roshid wants to type. However, his keyboard has a hardware failure: a specific group of letters no longer works, corresponding to the bottom row of a standard keyboard layout.

We are asked to construct an array of length n where each value is a 30-bit non-negative integer. The construction must satisfy a set of constraints that relate elements either by inequality to a fixed value or by XOR relationships between pairs.

We are given an integer $m$ and an array $a$. The task is to look at every divisor $d$ of $m$, and decide whether $d$ is “safe” or “bad”. A divisor $d$ is considered bad if there exists at least one array element $ai$ such that $d$ divides $ai$. Otherwise, $d$ is safe.

We are given a sequence of integers and then a sequence of queries. For each query value $b$, we are interested in all positive divisors of $b$. Among those divisors, some may appear inside the given array, and others may not.

The input describes several independent “landscapes” made of vertical stacks of unit-width bricks. Each landscape is an array where the value at position i represents how tall the wall is at that point. When rain falls, water can accumulate in the gaps between taller walls.

We are given several test cases. In each test case, there is an even-length array. We must partition the array into disjoint pairs so that every element belongs to exactly one pair. For each pair, its contribution to the answer is the larger of the two values inside that pair.

Each test case describes a process of completing identical forms, where each form requires collecting signatures from several offices. For every office i, a single form requires ai signatures from that office.

Let $u$ and $v$ be ZDD nodes representing families of sets for Boolean variables ordered as $x_1 < x_2 < \cdots < x_n$.

We are given a multiset of integers where value i appears exactly mi times. From this multiset we consider every possible permutation of the full expanded array.

We are given two binary grids of the same size, call them the starting grid and the target grid. The only allowed move is to choose a contiguous segment of length l either horizontally within a row or vertically within a column, and flip all bits in that segment.

We are given an array of length n where the value at position i is i-1, so the array is fixed as [0, 1, 2, ..., n-1]. The task is to consider every contiguous subarray, compute the bitwise XOR of its elements, and sum all those XOR results.

Let $f$ be a Boolean function on variables $x_1,\dots,x_n$, and let its BDD be given in the ordered and reduced form described in Section 7.

We start with an array that initially contains the numbers from 1 to n in order. Then we repeatedly apply a fixed sequence of operations until only one element remains.

We are given two binary arrays of equal length. From each array we are allowed to pick one contiguous segment, and the only restriction on each segment is that its length must fall inside a given range.

We are given several test cases. In each test case we receive an array of integers, and we are asked to analyze a faulty piece of code that tries to compute the maximum value of the array.

Let variables $x_1,\dots,x_n$ be interpreted as characteristic bits of a subset $S \subseteq {1,\dots,n}$, where $x_i=1$ means $i \in S$.

We are given three piles of chips. On each turn, a player can either take chips from exactly one pile, choosing any positive number up to the size of that pile, or perform a global move where they take the same positive number of chips from all three piles simultaneously, but…

We are given a base string s of length N, where each position has an associated weight ai. From this string, we are allowed to “erase” characters by choosing a subset of letters from the alphabet.

We are given a line of monsters, each with a power value, positive or negative. Saitama can choose any consecutive segment of these monsters and defeat exactly that group.

We are given a fixed $N times N$ grid of elevations. Every query gives an interval $[a, b]$, and we must find the largest axis-aligned square subgrid such that every cell inside it has elevation within that interval. The answer is the area of that square, not its side length.

Let $M_2(x_1,x_2,x_3,x_4)$ denote the 4-way multiplexer.

We are given a multiset of cards, where each card has an integer label called a quirk number. From this initial collection, we want to end up with a very strict final collection: it must contain exactly one card of each quirk number from 1 up to some chosen value K, and…

We are given a rectangular grid of integers representing pixel intensities. The grid behaves like a torus, meaning moving off any edge wraps around to the opposite side.

We are maintaining a dynamic collection of “habitats”, where each habitat stores multiple named dragons, and every dragon has a unique size value. The system supports two operations over time. One operation inserts a new dragon into a chosen habitat.

Let $f(x_1,\ldots,x_n)$ have truth table $\tau$, and let $f^Z$ have truth table $\tau^Z$.

We are given three independent walls, each described as an array of section heights. Each array contains $n$ integers, where each integer represents the height of a segment in that wall.

We are given a collection of short text messages, each independent from the others. For every message, we need to decide whether it represents a “battle” or just casual conversation.

We are given a tree of houses. Each house initially has a certain number of friends living there. Over time, two kinds of events happen.

We are given a list of students, where each student comes with a name and four numerical attributes: kicking skill, magic skill, speed, and demon slaying skill. The task is to produce a ranking of all students based on a strict multi-level priority system.

The task is a pure transformation problem on text. We are given an ASCII picture of Bessie the cow, represented as multiple lines of characters. We must output what the picture looks like after being rotated 180 degrees.

We are given an undirected connected graph representing cities and bidirectional roads. Some roads have a special property: if removing such a road would disconnect the graph, then that road is considered expensive.

We are given a hidden position on a one-dimensional strip of cells numbered from 1 to n, where n can be as large as 10^9. Exactly one cell contains a buried treasure. We can interact with the judge by choosing a cell i and effectively placing a detector there.

Let $f(x_1,\ldots,x_n)$ have truth table $\tau$, and let $f^Z$ have truth table $\tau^Z$.

We are given a final sequence of cards that appeared on a table during a game. In the game, the players start with a hidden initial deck, and repeatedly remove either the top or the bottom card of the deck.

We are given a fixed 4×4 grid made of two types of cells: road cells represented by dots and fence cells represented by hashes. The grid encodes a small road junction.

We are given an array of integers, and we are allowed to modify it using a very specific operation. Each position in the array can be used at most once, and when we use position i, we multiply the value at that position by i.

We are given a permutation of size $n$, meaning it contains every integer from $0$ to $n-1$ exactly once. For every contiguous subarray, we compute two values. The first value is the mex of the subarray, which is the smallest non-negative integer missing from that subarray.

We are given a collection of straight lines on the plane, each defined by an equation of the form $y = kx + b$. We are not asked to analyze intersections between arbitrary pairs of lines or to find a geometric intersection point.

We are given an array of integers that is modified over time, and we must answer queries about its subarray GCD. Two operations happen online. The first operation adds a fixed value to every element in a prefix or a range.

We imagine a huge infinite tape formed by writing the integers from 1 up to 10¹⁰ in order, without any separators. So the string begins as 1234567891011121314... and continues by appending each next integer in decimal form.

We are given several independent test cases. In each test case there is an array of integers, and then a sequence of operations.

We are given a group of $n$ students who must be split into exactly two teams. Both teams must be non-empty. There is also a lower bound $k$, meaning each team must contain at least $k$ students.

We are given two disjoint integer intervals: one interval for x and one interval for y. Specifically, x must be chosen strictly greater than a and at most c, and y must be strictly greater than b and at most d.

We are given a group of n children who each collected some number of chestnuts and then placed them into a single pile. The first child in order, Sasha, puts his chestnuts into the pile first. Every next child contributes twice as many chestnuts as the previous child.

We are given a single month where the weekday of one specific day is known. That known anchor consists of a day number between 1 and 31 and a weekday name such as Monday or Sunday. Using this anchor, we must determine the weekday of another day in the same month.

The Z-transform is defined recursively on binary strings with special behavior depending on whether the second argument is a block of zeros, identical to the first half, or a general concatenation cas...

The problem statement is not included in your message, so there isn’t enough information to reconstruct what needs to be solved.

We are given two non-negative integers, a and n. A program starts with a value b = 0 and then applies the same update step exactly n times: b := (b - a) & a Here subtraction and bitwise AND are done on 64-bit integers using two’s complement arithmetic.

We are given a chain of $N$ numbered rings arranged in a line. The traveler wants to be able to pay exactly one ring per day for $N$ consecutive days, but he is allowed to cut the chain beforehand into separate usable pieces.

We are given a small set of cards, each card showing a pair of integers. One of these cards is secretly the “prize” card. The first player sees only the left number of that card, the second player sees only the right number.

We are repeatedly building a growing sum of very specific numbers. The k-th summand is a number made of a single digit 2 at both ends, with zeros filling the middle as the number grows, starting from 2, then 22, then 202, then 2002, and so on.

We are given a state consisting of two integers, and we are allowed to transform this pair using exactly three reversible operations. One operation subtracts the second value from the first, another adds the second value to the first, and the third swaps the two coordinates.

We are looking at a continuous analog clock where the hour and minute hands move smoothly rather than jumping once per minute. The minute hand completes a full circle in 60 minutes, while the hour hand completes a full circle in 12 hours.

We are given a grid that represents an archipelago. Each cell is either land, marked as 1, or water, marked as 0. Any two land cells that touch up, down, left, or right belong to the same island, so the grid naturally splits into multiple connected components of 1s.

I can write the full editorial, but I need the actual problem statement in a clean, uncorrupted form first.

We are given a row that alternates between empty positions and fixed comparison symbols. There are $N+1$ positions that must be filled with distinct numbers from 1 to $N+1$, and between every two neighboring positions there is exactly one constraint, either “<” or “”…

We are told that Natasha has cats, and each cat behaves in a very rigid way during the night. Every time a cat “acts”, it produces exactly the same effect: a fixed number of items fall, from level A down to level B, and Natasha hears a total of N falling events in total.

We are asked to construct a single regular expression over decimal digits that accepts exactly those integers whose digits can be rearranged to form a number divisible by 6. A number is divisible by 6 if and only if it is divisible by 2 and by 3.