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We are asked to construct a permutation of integers from 1 to $n$ such that, no matter which of two typewriters Misuki uses, the number of required carriage returns to complete the permutation is identical.

We are asked to count sequences of integers with a very particular merging property. For a given sequence of length n, each element between 1 and m, we are allowed to repeatedly take three consecutive elements ai, a{i+1}, a{i+2} and merge them into a single number defined as…

We are constructing a sequence of rounds where exactly one of two players gains a point in each round. After each round we maintain two counters, one for each player, and the process is only valid if at every prefix the greatest common divisor of the two counters never exceeds 1.

We are asked to plan a match between two performers, Lelle and Flamm, where each round produces a winner. The goal is to maximize the total score, which is computed as l WL + f WF, where WL and WF are the number of wins for Lelle and Flamm, respectively.

The problem asks only for existence or nonexistence in three classes. For part (1), it suffices to exhibit a tetrahedron that tiles space by congruent copies. For part (2), we must decide whether a congruent tiling by equifacial tetrahedra exists.

This is a Type C problem: a precise numerical count of admissible fillings is requested. The solution produces a combinatorial formula $an = 2^{n-1}$ and rigorously justifies it via bijection arguments and induction.

For the first values,

We are given a rooted tree where vertex 1 is the root. Alongside the tree, we maintain a permutation stored in an array indexed by positions, and we repeatedly swap two positions in this array.

We have two players, Alice and Bob, each holding a permutation of the numbers from 1 to n. They will alternately remove elements from either end of their respective arrays. After n-1 turns, only one element remains in each array.

We are given a perfect binary tree whose vertices are numbered in heap order. Vertex 1 is the root, and for every vertex i 1, its parent is i // 2. A permutation p of all vertices is maintained. After every query, two positions in the permutation are swapped.

We are given a rectangular grid with n rows and m columns. Each cell must be assigned a color, but colors are restricted by a distance rule: if two cells share the same color, then they must be sufficiently far apart in the Chebyshev sense, meaning the maximum of their row…

This is a counting problem, so it falls under Type C. The goal is to determine the exact number of admissible fillings of the strip. The solution must provide a precise count and justify the enumeration rigorously.

This is a counting problem, so it is closest to Type C in the supplied classification. The requirement is to determine the exact number of admissible fillings and justify the count rigorously.

We are asked to find the maximum number of inversions in permutations of length $n$ that satisfy a specific "interesting" property defined by a set of intervals.

The problem gives us a permutation of integers from 1 to $n$ and asks us to consider certain "interesting" permutations based on a set of intervals.

The problem involves a turtle playing with several integer sequences and a non-negative integer $x$. For each sequence, the turtle can perform an operation exactly once: he sets $x$ to the $text{mex}$ of its current value together with the numbers in the chosen sequence.

The point $M$ is defined from the circle through $P,Q,K$.

We are given several sequences of non-negative integers, and an initial integer $x$. Turtle can repeatedly choose any sequence and replace $x$ with the mex of $x$ together with all numbers in that sequence.

Let the stone masses be $x_1,\dots,x_n$ with $0<x_i\le 2$ and $\sum_{i=1}^n x_i=50$.

We are given a string consisting of lowercase letters, and our task is to reorder it to maximize the number of "good pairs.

We are asked to determine whether a given string can be split into two or more contiguous substrings such that no substring starts with a character that appears at the end of a later substring.

We are given a sequence of numbers and two players who repeatedly shrink it until only one value remains at the front position. Each move removes one element, but before removing it, the left element of the chosen adjacent pair is updated.

We are given a sequence of cities arranged on a line, each equipped with exactly two portals of different colors from a fixed set of four: blue, green, red, and yellow.

We have a sequence $a$ and an $n times m$ matrix $b$. The game follows the sequence from left to right. On move $k$, the current player must choose a cell whose value equals $ak$.

This is a Type B (prove) problem. The solution correctly interprets the statement: it aims to prove that any triangle admitting a circle tangent to two sides and two medians must be isosceles.

We are given two arrays of equal length. At every index we can either keep the pair as it is, or swap the two values at that position. The operation we are allowed to perform is more structured: we must choose one contiguous segment and swap all pairs inside it simultaneously.

We are asked to simulate a pursuit on a one-dimensional line with cells numbered from 1 to $n$. David, a student, starts in a given cell and can move left, right, or stay in place.

This is a Type B (prove) problem. The solution correctly interprets the statement: it aims to prove that any triangle admitting a circle tangent to two sides and two medians must be isosceles.

We are asked to construct a string consisting solely of the English vowels a, e, i, o, u of a given length n such that the number of palindrome subsequences in the string is minimized.

This is a Type B (prove) problem. The solution correctly interprets the statement: it aims to prove that any triangle admitting a circle tangent to two sides and two medians must be isosceles.

For $n=1$ the inequality becomes

We are building a tree incrementally, one node at a time. After each insertion, we are asked to evaluate how “close” the current tree is to a perfect full binary tree shape.

This is a Type B (prove) problem. The solution correctly interprets the statement: it aims to prove that any triangle admitting a circle tangent to two sides and two medians must be isosceles.

For every subarray of the given array, we throw all of its values into a set, removing duplicates. Starting from that set, we may repeatedly pick two existing numbers whose average is an integer and insert that average.

I can't reliably provide a complete editorial with a correct accepted Python solution for Codeforces 2006D from the problem statement alone.

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