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We are given a tree with $n$ vertices and a multiset of $n-1$ integers. One vertex is special: it is not assigned any value from the list, while every other vertex must be assigned exactly one number from the list.

We are given a multiset of $2n$ integers, which may include both primes and positive integers greater than one. The goal is to count how many positive integers $m$ have a prime factorization that produces exactly this multiset when each prime and its exponent are included as…

We are given an array of positive integers. The goal is to modify the array so that no element divides the next element, using only increments of 1. Each element can be incremented multiple times, but the total number of operations must not exceed twice the array length.

We are given a hidden string $s$ of length $n$. Instead of seeing $s$ directly, we are shown a multiset of strings consisting of every non-empty prefix of $s$ and every non-empty suffix of $s$, except for $s$ itself.

We are given a non-decreasing array. For every prefix a[1...k], we must determine a special value called the cost. For any subsequence of length d, its score is the product of its elements divided by d!.

We are given a tree, which is a connected acyclic graph with n vertices. Some of these vertices initially contain chips, each located on a distinct vertex. Every vertex that contains a chip is colored black, and the rest are white.

We are given a graph with $n$ vertices and $m$ edges. Each vertex $i$ has an associated number $ai$. A vertex can only be removed if its current degree equals $ai$, and when a vertex is removed, its incident edges disappear, reducing the degrees of its neighbors.

Consider the equation

We are given a row of monsters, each with a certain health, and two types of attacks: a basic spell that reduces any monster's health by 1 for 1 MP, and a single-use "Explosion" that can be targeted on one monster with an arbitrary power.

The graph is not arbitrary. It is composed of independent groups of three vertices, and inside each group all three pairs are connected, forming a triangle.

We are given two vertical stacks of colored blocks. Each stack is described from bottom to top as a string consisting of only two symbols: B for blue and R for red.

Connection interrupted. Waiting for the complete answer

Consider the problem of determining whether both sums

We are working with a set of closed intervals on a number line, and we are allowed to discard any subset of them. For any integer coordinate $x$, we define its coverage value $f(x)$ as the number of remaining segments that include $x$.

We are asked to count triples of positive integers (a, b, n) that satisfy a specific strange property. Here, a can range from 1 up to A-1, b from 1 up to B-1, and n from 1 up to N-1.

We are given a tree with $n$ vertices, and we are allowed to choose any vertex $r$ as a root. Once the root is fixed, every vertex gets a distance label $dv$, which is simply how far it is from $r$. After this, we assign colors to vertices under two constraints.

Producing a complete editorial of the quality and length you requested for Codeforces 1796D would require first deriving and verifying the full intended solution in detail.

We are asked to reason about an infinite string that is not built directly character by character, but generated from the positive integers in order.

The configuration contains two coupled constructions on the base line $BC$ and two circumcenter loci attached to triangles $NEC$ and $MFB$.

We are asked to construct a “template” string that can describe two given strings simultaneously. A template consists of lowercase letters and asterisks. Each asterisk can be replaced with any string, including the empty string, to match the original strings.

We are asked to work with sets of integers that are “beautiful,” meaning that any two numbers in the set are comparable under divisibility.

We are given an array of integers and two types of operations. The first operation transforms elements in a range by applying Euler's totient function, which reduces a number to the count of integers coprime to it.

We are working with a rooted but otherwise unrooted tree where every vertex carries a unique label from 1 to n, and these labels matter structurally rather than just as identifiers. For any two vertices u and v, consider the unique simple path connecting them.

We are given a rooted tree with n vertices, where each vertex has a numerical importance. The root is vertex 1. Two types of operations can be performed on this tree. The first operation asks for the sum of importance values in the subtree rooted at a given node.

We are dealing with a hidden position on an extremely large grid, up to one billion in both dimensions. A king is placed on one cell, and our task is to determine its exact coordinates. We cannot directly access the position.

This is a Type B (prove) problem: the task is to prove that the intersection point of the diagonals of quadrilateral $ABCD$ coincides with the intersection of the common internal tangents of the two circles. The solution correctly addresses a proof,

We are given an $n times n$ grid where each cell is either 0 or 1, representing two colors. We are also given a fixed number $k$ of moves. A single move flips the color of exactly one chosen cell.

We are given an array of integers whose sum is zero. The task is to rearrange the elements so that the maximum absolute sum over any contiguous subarray is strictly less than the difference between the largest and smallest elements in the array.

We are given a collection of gift boxes, each box containing some positive number of gifts. There are $n$ existing boxes, and we must add exactly one additional box with an integer number of gifts between 1 and $10^6$. After that, we will have $n+1$ boxes in total.

Each candy type has two numbers attached to it. There are $ai$ candies of that type in stock, and each candy costs $bi$ coins. For every type, we must choose a pack size $di$.

We are given an array, and we look at every suffix starting from position i. For each suffix, we are allowed to change some elements arbitrarily to non-negative integers.

We are given a sequence of days. On each day, a group of people participates in a lottery, and exactly one of them is chosen as the winner for that day. The key restriction is that once someone wins on day i, they are forbidden from appearing in any later day i+1 through m.

We are given two arrays of equal length, which we can think of as parallel sequences of tiles. Each tile has two numbers: one on the "a" side and one on the "b" side. We are allowed to swap the numbers on a single tile as many times as we want.

We have a group of $n$ people, each belonging to a specific team, and each person wants to receive a certain number of votes. The rules restrict voting so that no one can vote for themselves and no one can vote for someone on their own team.

We are given an array of positive integers and two types of optional operations that can be applied to elements, with global limits and per-element limits.

I can't reliably write a correct, complete editorial for Codeforces 1799H from the problem statement alone. This is a 3200-rated problem with a highly nontrivial tree DP over subsets, and the solution involves delicate state design and transition conditions.

We are given a rectangular grid of size $n times m$, where each cell is either filled or empty. A "city" is defined as a connected component of filled cells, where connectivity is through sides (no diagonals). Initially, there are exactly two cities on the grid.

We are given a fixed execution order of programs, and we must execute them sequentially on a machine with two independent CPUs. Each program has two possible execution costs: a normal cost and a reduced cost.

We are given a sequence of programs that must be executed in order, using two CPUs. Each program has two run times: a cold time, which applies if the CPU last ran a different program, and a hot time, which applies if the CPU last ran the same program.

We are given a string consisting of lowercase letters, and the task is to reorder its characters into a new string t such that when we consider t and its reverse, the lexicographically larger of the two-denoted tmax-is as small as possible in lexicographic order.

Consider a tree with $n$ vertices, each initially containing a single chip.

We are given an array of positive integers. We can repeatedly pick two distinct indices $i$ and $j$, and set the element at index $i$ to the ceiling of its division by the element at index $j$.

We are asked to track the "Recent Actions" field of posts on a site like Codeforces. Initially, the field contains the first $n$ posts, numbered $1$ through $n$, from top to bottom. New posts with numbers $n+1$ and higher can appear through actions.

We are given a rooted tree with vertices numbered from 1 to $n$, where vertex 1 is the root. Each vertex can have zero or more children, and the input specifies the edges connecting vertices. The task is to determine whether the tree is symmetrical.

We have two strings of equal length, s and t. Starting from s, we may repeatedly swap characters whose positions differ by exactly k or exactly k + 1. The question is whether these allowed swaps are powerful enough to transform s into t.

We are given a list of words and asked to count pairs of words whose concatenation satisfies several strict properties.

Assume that a function $f:\mathbb R\to\mathbb R$ satisfies

We are given a string consisting of lowercase Latin letters, and the task is to count how many distinct strings can be formed by removing any two consecutive characters. Each test case consists of a string of length at least three.

We are given two strings of equal length, and we are allowed to transform the first string into the second by swapping characters, but only under a restricted rule: we can swap positions whose indices differ by either 3 or 4.

We process a deck of cards from top to bottom. Positive values represent bonus cards, while 0 represents a hero card. Whenever we encounter a bonus card, we may keep it for later or throw it away.

We are given a sequence of cards arranged in a fixed order, and we process them from top to bottom. Each card is either a hero or a bonus. A hero card has value zero and represents an opportunity to finalize one hero in our army.

Kristina has a string of letters that can be lowercase or uppercase. She earns one burl for every valid pair consisting of a lowercase letter and its corresponding uppercase version. Each letter can only participate in a single pair.

For a query polynomial $Q$, the answer is the set of real roots of

We are asked to determine if a given string represents the sound of a cat meowing. The sound must strictly follow the pattern “m-e-o-w” in order, where each letter can appear multiple times consecutively, and both uppercase and lowercase letters are allowed.

Consider an isosceles triangle $ABC$ with $CA=CB$ and axis of symmetry $CM$, where $M$ is the midpoint of $AB$.

We are given a very long text string t and a set of distinct dictionary words s₁, s₂, ..., sₙ. For each query [l, r], we look only at the substring t[l..r]. Inside that interval we must count how many substrings are equal to one of the dictionary words.

We are given a chocolate bar in n dimensions. Along each dimension, the bar is already divided into discrete units: for dimension i, there are ai units. Vasya wants to cut the chocolate further along these pre-existing divisions to produce at least k pieces.

We are given a directed network of cities where each city has a per-day income, and each flight has a fixed cost. We start in city 1 with some initial amount of money, and the goal is to reach city n.

The reason your solution produces no output is that it never actually writes to stdout. In the previous drafts, you were appending results to a list ans but never called print or sys.stdout.write. This is why the program runs without error but shows an empty output.

We are given a country, Berland, structured as a tree of cities rooted at city 1. Each city has a gas station with a price range $[li, ri]$. Every year, the king’s two sons inspect gasoline prices along two paths of equal length.

We are asked to construct an $n times m$ grid filled with integers such that a specific XOR condition holds for every $4 times 4$ subgrid.

We are given several independent test cases. In each test case, there are $n$ departments, and each department offers two possible prices $ai$ and $bi$.

We are simulating a timeline where new guinea pigs arrive over time, and occasionally a “reveal” event happens where all currently owned animals suddenly have their genders revealed.

This is a **C: optimization problem**.

The problem describes a sequence of user interactions on a post, where each interaction is either a like or an unlike. The input sequence is shuffled, so the chronological order is lost.

Label the marked points by the elements of $\mathbb Z_{2n}$ in cyclic order.

We are tasked with simulating a network line shared by multiple users, where each user transmits data at a rate that can grow or shrink depending on network congestion. Each user has a start time, end time, and initial data rate.

A king starts at the origin of a grid and wants to reach a target cell (a, b). Unlike a normal king, it can only move one step vertically or horizontally, or stay in place. The unusual restriction is that the same action cannot be used twice in a row.

We are given a grid of size $n times m$ where each cell represents a window in a high-rise building. Each cell is either lit or dark. The building is structured in floors, and each floor is an independent row of the grid.

A cube in three-dimensional space with vertices in $\mathbb{Z}^3$ can be described by a vertex and three vectors $\mathbf{u}, \mathbf{v}, \mathbf{w} \in \mathbb{Z}^3$ that are pairwise orthogonal and…

We are given a sorted list of arrival times of patients. Each patient arrives at a specific moment and is willing to wait for a limited number of time units, meaning there is a window during which they can be vaccinated.

We have a roulette wheel with n sectors numbered from 0 to n-1. The wheel starts with an arrow pointing at sector x, and we can spin it by pulling a handle with an integer force f between 1 and p.

I’m sorry, but I can’t reliably write a correct editorial and solution for Codeforces 1805F2 from memory alone. This is a 3100-rated problem with a fairly intricate mathematical reduction, and I don’t want to invent a proof, algorithm, or code that may be incorrect.

We are given a tree with $n$ vertices and asked to generate a sequence of graphs $Gk$ for $k = 1$ to $n$. In each graph $Gk$, an edge exists between vertices $u$ and $v$ if the distance between $u$ and $v$ in the original tree is at least $k$.

We are given an array of non-negative integers and asked to repeatedly transform it by taking all pairwise sums, sorting them, and keeping only the smallest $n-1$ sums. This operation is applied $n-1$ times until a single number remains, which is the output modulo $10^9+7$.

We place square $ABCD$ in the coordinate plane with $A=(0,0)$, $B=(1,0)$, $C=(1,1)$, and $D=(0,1)$.

We are given a tree, which is a connected acyclic graph, with n vertices. Each vertex has an integer label. The task revolves around a parameter called MAD, or “maximum double,” which is defined as the largest integer that occurs at least twice among the vertices of a tree.

We are asked to choose, for each parabola, a straight line that passes through the origin and does not intersect the parabola at all. Each line is given by its slope $k$, so the line is $y=kx$. Each parabola is given by $y = ax^2 + bx + c$ with $a0$, so it opens upwards.

We are given a string and allowed to perform exactly one operation: pick a character at some position and move it to the start of the string. The goal is to produce the lexicographically smallest string possible after this operation.

Working

We are given an array of integers where each element is at most $m$, and we are allowed to perform exactly $k$ operations. In each operation, we select two elements, remove them, and append their greatest common divisor (gcd) to the array.

We are given a binary array that encodes how edges are directed between values, and we are asked to evaluate a rather indirect quantity over all permutations of each prefix length. For a fixed length $k$, we take every permutation of ${1,2,dots,k}$.

The tree is rooted at vertex 1. Every vertex has a value a[v], and every vertex except the root has a parent. For a query (x, y), both vertices are guaranteed to lie at the same depth. Starting from these two vertices, we repeatedly move both upward one edge at a time.

We are given an array p of length 2n. We want to modify its values and obtain another array q of the same length. The target array is not arbitrary.

We are asked to compute the minimum number of moves for a character, YunQian, to reach a target point on an infinite Cartesian plane.

We are asked whether a given array c can be generated starting from an initial array [1] by repeatedly adding a new element equal to the sum of any subsequence of the current array.

We have a ball moving inside an n × m rectangular grid. The ball always travels diagonally, so each move changes both coordinates by ±1. A direction consists of a vertical component and a horizontal component.

We start with an array containing a single element [1]. We are allowed to repeatedly pick any subsequence of the current array, sum its elements, and append this sum as a new element to the array.

I’ve carefully analyzed this problem. The root cause of the previous wrong output is twofold: 1. Incorrect input handling: Each test case is a string of digits.

We are given several independent test cases. In each one there is a collection of piles of stones. Every pile contains some number of stones, and all stones have weight 1 except for a single hidden special stone located in exactly one pile, which contributes an extra unit of…

We are given an array of integers and a sequence of queries. Each query asks whether, if we were to overwrite all elements in a certain subarray with a fixed value, the total sum of the array would become odd. The queries do not modify the array permanently.

We are given a string made of lowercase letters, and we are allowed to repeatedly “collapse” entire character classes into binary digits. One operation picks a letter, say x, and globally rewrites every occurrence of x in the string into either 0 or 1.

Introduce Cartesian coordinates adapted to the symmetry of the equilateral triangle.

Let $a, a+1, \dots, a+n-1$ be $n>1$ consecutive natural numbers.

We have several bags of candies. A bag containing an even number of candies always goes to Mihai, while a bag containing an odd number of candies always goes to Bianca. The order of the bags is not fixed. We may rearrange them however we want before the game starts.

A ticket is a sequence of n digits in base k, so every position contains a value from 0 to k - 1. Let the total digit sum modulo k be $$S = a1 + a2 + dots + an pmod k.$$ A ticket is lucky if there exists some position whose digit equals the sum of all remaining digits modulo k.

Place the square in coordinates with $A(0,0)$, $B(1,0)$, $C(1,1)$, $D(0,1)$.

For every subarray of odd length k, we want to know how many element replacements are needed to turn that subarray into a palindrome. The answer is the sum of those values over all length-k subarrays.

Each player receives a card containing m integers. For every pair of players, the amount won in their game is the sum of absolute differences between the corresponding positions on their cards. If player i has card values c[i][1...m] and player j has card values c[j][1...

We are given a sorted list of participant ratings, and we are allowed to permute these participants into a line. After fixing a permutation, a sequential tournament is played: the first two players fight, then the winner immediately fights the third player in line, then the…