brain

We are given a rooted tree with root fixed at vertex 1. Each vertex carries a label that is either 0, 1, or unknown. The unknown entries are the only things we are allowed to choose during the game, and the two players alternate assigning them values until every vertex is fixed.

This is a Type B (prove) problem. The requirement is to show that any triangle admitting a circle tangent to two of its sides and to two of its medians must necessarily be isosceles. The statement is exact and requires a complete geometric or analytic proof.

We start with all integers in the interval $[l,r]$. In one operation, we must choose three distinct numbers whose pairwise greatest common divisors are all equal to $1$. After choosing them, those three numbers are removed permanently.

For the planar analogue, take a triangle $ABC$ and a point $P$ inside it.

We are given an array, but the operations do not directly modify positions. Instead, each operation targets values: whenever we see a range $[l, r]$, every element whose current value lies in that numeric interval is incremented or decremented by exactly one.

We are given an array and a parameter $x$. With this $x$, we are allowed to repeatedly pick any element that is at least $x$ and reduce it by exactly $x$.

This is a Type B (prove) problem. The requirement is to prove that for every point $P$ in the plane and for every choice of the distinguished vertex of the isosceles trapezoid, the sum of the distances from $P$ to the other three vertices is strictly greater than the distance…

We are given a string of lowercase letters. An alternating string has a very specific form. All characters at even indices must be identical, all characters at odd indices must be identical, and the final length must be even.

A convex $n$-gon $A_1A_2\dots A_n$ is inscribed in a circle of radius $R$ with center $O$.

We are given a binary string that was originally formed by flattening a rectangular grid row by row. The grid is known to be “beautiful”, meaning its border cells are all 1 and all interior cells are 0.

This is a Type B (prove) problem. The requirement is to prove that for every point $P$ in the plane and for every choice of the distinguished vertex of the isosceles trapezoid, the sum of the distances from $P$ to the other three vertices is strictly greater than the distance…

We are given a permutation of integers from 1 to n, and each number is colored black or white. From any starting integer i, we can repeatedly apply the permutation to jump from i to pi, then to p{pi}, and so on.

We are asked to decide whether it is possible to assign a positive or negative sign to each element in an array made up of a certain number of ones and twos such that the total sum becomes zero.

We are asked to construct an array of integers between two given bounds, l and r, such that the array is strictly increasing and the differences between consecutive elements are also strictly increasing. Our goal is to find the maximum possible length of such an array.

We are given an array and many range queries. For each query range $[l, r]$, we look at every subarray inside it whose length is at least $k$.

The answer is **yes**.

Let

Represent each chosen vector by an oriented edge of a directed graph whose vertices are the given points.

We are given an array a of size n. For each subarray of a, Yunli can perform an operation on any element to set it to any integer she wants. The goal is to form a consecutive increasing sequence of length at least k within that subarray.

We are given an array a of length n, and we are asked to handle queries on a much larger array b. This array b is constructed by taking all n cyclic shifts of a and concatenating them in order.

We are given an array of integers and need to answer multiple queries. Each query asks for the minimum number of changes required to create a consecutive increasing subarray of length exactly k within a specified slice of the array.

We are given a conceptual array that contains a strictly increasing sequence of integers starting at $k$ and ending at $k+n-1$.

Let $ABCD$ be an isosceles trapezoid with $ABparallel CD,$ and let $AB$ be the longer base. Then $ABCD.$ For a point $P$ in the plane, consider the quantity $S=PA+PB+PC+PD.

Each test case describes a small osu!mania beatmap with 4 columns. Every row contains exactly one note, marked by , and the remaining cells contain ..

We are given multiple independent queries. Each query provides two integers a and b, and we are allowed to choose any integer c in the inclusive range between them. For each choice of c, we compute a cost defined as the distance from a to c plus the distance from c to b.

Codeforces 2009C: The Legend of Freya the Frog

We are asked to determine whether a given string could have resulted from a very specific transmission error in a network. The error occurs when a message s is sent twice consecutively, and the second copy partially overlaps the first at some non-zero length.

The task is to compute an alternating sum over a sequence of integers. Specifically, for each sequence, you start with the first number, add it, subtract the second number, add the third, subtract the fourth, and continue in this alternating pattern.

This is a Type B (proof) problem. To succeed, the solution must rigorously establish the stated vector identities for: 1. An equilateral triangle. 2. An arbitrary regular $n$-gon. 3. A regular tetrahedron.

We receive a single string t that represents a message captured by a server. The server originally sends some string s twice in a row.

We maintain two evolving ordered structures over patients: one behaves like a FIFO queue (first in, first out) and the other like a stack (last in, first out). Each patient has a fixed service duration.

We are asked to count all contiguous subarrays of a given integer array such that each subarray can be generated by a process that starts with a single integer and repeatedly appends either the same number, one less, or one more than an element already in the array.

We are given a 2-row by $n$-column grid representing a field, where each cell can be empty, contain a sheep, or contain a wolf. There is exactly one sheep, and some cells contain wolves.

This is a Type B (proof) problem. The required task is to prove the stated vector identities for: 1. An equilateral triangle. 2. A regular $n$-gon. 3. A regular tetrahedron. A valid solution must establish the identities themselves.

Connection interrupted. Waiting for the complete answer

We are given a string consisting of blocks of digits separated by plus signs, where each block has between 2 and 13 digits, and every digit is non-zero.

Working

The condition says that corresponding sides are parallel, but the directions are reversed.

Jane has a list of problems with known difficulties. The last problem in the list is strictly harder than every earlier problem. Her skill level is some integer $x$.

We need to construct a permutation of numbers from 1 to n such that every adjacent pair follows a rule that depends on divisibility. If two neighboring numbers have a divisibility relationship, the sequence must increase at that point.

Let the regular tetrahedron be $ABCD$, with edge length $a$.

We are given a tree. One player, Alice, always starts at vertex 1. The other player, Bob, starts at a vertex that lies on a fixed simple path between two vertices u and v.

Zhan has a number of fruits, and he wants to blend all of them as quickly as possible using a blender with two constraints. Each second, he can put at most a fixed number of fruits into the blender, and the blender can process a maximum number of fruits per second.

For each query we look only at the subarray $al, a{l+1}, dots, ar$. Robin moves first. On every turn a player chooses any remaining target, gains its value, and removes it.

We are tracking Little John's milk consumption over time. He receives milk in discrete batches, each associated with a day and a quantity. Milk has a freshness limit k days, meaning that milk acquired on day di will spoil after day di + k - 1.

We are given a tree where each node represents a camp holding some amount of gold. We may choose certain camps to “strengthen”. Strengthening a camp does not change its own gold directly, but it reduces the gold of all its neighbors by a fixed amount $c$.

Each job occupies an interval of days [l, r]. A visitor stays for exactly d consecutive days, so choosing a start day s means the visit covers the interval [s, s + d - 1]. A job is counted if it overlaps the visit at least once. The amount of overlap does not matter.

We are working on a weighted undirected graph where two people start from opposite ends and want to meet as quickly as possible. One starts at vertex 1 and the other at vertex n.

We are asked to determine how much extra gold must be added to the richest person in a town so that strictly more than half of the population becomes unhappy. Each person's happiness depends on whether their wealth is less than half of the average wealth of the town.

This is a construction and impossibility problem (type D/B hybrid). For each case, the task is either to construct an explicit set of numbers such that all sums of three distinct elements are prime, or to prove that no such set exists. Each of the four subcases is independent.

The tree produces new leaves every year. In year $i$, it grows $i^i$ leaves. Leaves do not stay forever. A leaf created in year $i$ remains on the tree for exactly $k$ years, meaning it is present during years $i, i+1, dots, i+k-1$.

Assume that an equilateral triangle has all three vertices at lattice points.

We are asked to count arrays of integers representing "time limits" for conquering cities in a row, and then categorize those arrays by how many starting cities allow a valid conquest.

The problem asks us to count arrays of length $n$ where each element $ai$ is a positive integer between 1 and $n$, and to categorize these arrays based on how many starting cities allow a sequential conquest without violating the upper limits $ai$.

I can't provide a complete, correct editorial and implementation for this problem from the information available here. I was able to retrieve the official editorial hints and the high-level solution outline.

We are given a collection of closed segments on a number line, each defined by a left and right endpoint. The goal is to identify the largest possible subset of these segments that forms a "complex" set.

Producing a fully correct editorial for Codeforces 2018E2 requires reconstructing a fairly intricate DSU-based optimization and the divide-and-conquer evaluation of all values of $f(m)$.

We are given a one-dimensional array of positive integers and the ability to color some elements red. The restriction is that no two consecutive elements can be red.

We are given a tree rooted at node 1, and we can remove leaves one by one along with their connecting edges. The goal is to make all remaining leaves equidistant from the root. A leaf is any node with degree 1 that is not the root.

We are given multiple independent scenarios. In each one, we start with a multiset of cards where each integer value from $1$ to $n$ appears a certain number of times. We are allowed to add up to $k$ extra cards of any values we choose.

We are given a line of cities, each city indexed from left to right, and each city comes with a deadline that tells us how late it is allowed to be conquered.

We are given a set of points on the number line, each at a distinct positive integer coordinate, and we want to understand coverage by segments formed between every pair of points.

We are given an array of positive integers and asked to select a subset of elements to color red, subject to a simple restriction: no two red elements can be adjacent.

We are given an array of integers, each with an independent probability of being included in a random multiset. The task is to compute the expected value of the square of the XOR of all selected elements.

A six-digit number can be written as the product of two three-digit numbers if and only if it belongs to the set

Codeforces 2020F: Count Leaves

We are given a set of points on a line, initially completely disconnected. Each operation takes a starting position, a fixed step size, and a length, and then connects all points that lie in that arithmetic progression segment.

We are given three non-negative integers b, c, and d. We must construct a non-negative integer a such that $$(a mid b) - (a & c) = d$$ where The task is not to optimize some value of a. Any valid a is acceptable. If no such value exists, we print -1.

We are given a process on a line of bulbs indexed from $1$ to $n$. Initially every bulb is on. We then repeatedly flip bulbs in a structured way: for each $i$, every multiple of $i$ has its state toggled once.

We are asked to reduce a given integer n to zero by repeatedly subtracting powers of a second integer k. Each subtraction can use any power of k (including k^0 = 1), and the goal is to minimize the number of subtractions.

We are given a connected village represented as a graph. Each node is a house, and edges are internet cables with a latency weight. Some subset of houses specifically need internet.

Let the weights in the first group be

In this problem, we are given a village represented as a connected graph with houses as nodes and internet cables as edges. Each edge has a latency, representing the delay of transmitting data along that cable.

We are given a connected undirected graph representing a village, where nodes are houses and edges are internet cables with latencies. A subset of houses requires internet, and we are allowed to place servers at up to k houses.

We are asked to plan drink sales over several days in a canteen. Each day, there are multiple drink types, each with a projected profit that can be positive or negative. On any day, we must select a contiguous segment of drink types to sell.

We have a fixed permutation a describing the initial order of team members in a line. During the presentation, the person currently at the front presents the next slide.

We are given an initial lineup of people, where each person appears exactly once. A sequence of presentations must be delivered in a fixed order, and each presentation is assigned to a specific person.

We are given an array of positive integers and a process that repeatedly combines two elements into a single new element: we pick two distinct elements, compute the floor of their average, remove the original two, and append the new number.

We start with an array of non-negative integers. The only operation allowed is to pick an element and add x to it. We may repeat this operation any number of times on any element. Adding x repeatedly has a very specific effect.

We are asked to count the number of ways to fill an integer grid with certain cells already preassigned so that the grid is "beautiful." A grid is beautiful if, for any rectangle defined by four corners, the XOR of the corner values equals zero.

Let the route produced by the minimum-greedy algorithm be

We are given a partially specified n × m grid. Some cells already contain integers in the range [0, 2³⁰), while the remaining cells are unknown. A grid is called beautiful if every axis-aligned rectangle satisfies the XOR condition on its four corners.

The task is to find the single Impostor among $n$ players in an interactive game where each player has one of three roles: Knight, Knave, or Impostor. Knights always tell the truth, Knaves always lie, and the Impostor behaves like a Knave from the perspective of all others.

In this problem, we are asked to identify a single Impostor among n players, where every player is either a Knight, a Knave, or the Impostor.

Each bus row contains exactly two seats. We have several families, and family i contains ai people. A person is happy in one of two situations. They sit next to a member of the same family, or they occupy a row alone with the other seat empty.

Each test case describes a dealership with several car models, where model i has ai identical cars that must all be sold.

We are given a voting grid with exactly two rows and n columns, where each cell is labeled either “A” or “J”, representing how that house will vote.

We are given an array of nonzero integers representing a one-dimensional terrain. Positive values mean surplus sand that must be removed, and negative values mean deficits that must be filled.

We are given a tree of n nodes, which are connected by n-1 edges. Each node can be thought of as a source of magical energy, and each edge as a channel through which this energy flows.

The problem presents a casino with multiple games, each offering a chance to win a certain amount. Each game has two parameters: a probability of winning, given as a percentage, and a payout if you win.

Let each square correspond to its center.

We are given a sequence of problems in an olympiad, each with a score and a skip parameter. The competition begins with the first problem.

We are given n arrays, and every array contains exactly two numbers. We may reorder these arrays in any way we want, but inside each array the order of the two elements must remain unchanged. After choosing an order, we concatenate all arrays and obtain one sequence of length 2n.

We are given two directed graphs, each with the same number of vertices. Each graph is already strongly connected, and every directed cycle inside each graph has length divisible by a fixed integer $k$.

We are given two types of bank deposits and a number of coins Alice has. The "Profitable" deposit requires at least a certain amount, b, to open, while the "Unprofitable" deposit has no minimum.

Let $D_{16}$ denote the closed disk of radius $16$ centered at the origin, and suppose $650$ points $P_1, \dots, P_{650}$ are placed in $D_{16}$.

Let the convex pentagon be $A_1A_2A_3A_4A_5$, with indices taken modulo $5$.

The core issue is not a boundary bug or a missing case. The previous approach was fundamentally overconstrained: it tried to force both segments to be aligned with axes and also tied them to a single corner or a simplistic placement rule.

The problem models a turn-based battle between a dragon and an army of heroes protected by artifacts. Each hero has a health value and each artifact has a durability value. A hero can hold at most one artifact, and artifacts can only protect heroes while they are active.

We are given a sequence of game records where each element represents either acquiring an attribute point or encountering an attribute check. The character starts with Strength and Intelligence at zero.