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We have an undirected weighted city graph. Bob always starts at building 1 and wants to reach building n. Each query removes one street temporarily, and we must find the shortest possible route after that street is unavailable.

We have an array of integers. For each query range [l, r], we look only at the elements inside that segment. For every value that appears in this segment an even number of times, we take that value once and XOR all such chosen values together.

Each swimmer moves back and forth in a lane of length m. Every swimmer starts at position 0, swims toward position m, immediately turns around upon reaching it, and repeats this motion forever. The speed of swimmer i is fixed and equal to x[i] meters per second.

We are given an array of positive integers. Every value in the array is at most m. For each query value x, we must count how many ordered pairs (ai, aj) satisfy gcd(ai, aj) = x.

The matrix starts filled with zeros. For every positive pair (i, j), we perform one operation that flips every cell whose row index is divisible by i and whose column index is divisible by j. A particular cell does not care about operations that do not reach it.

We repeatedly shorten an array by selecting two adjacent positive values. Those two values are removed and replaced by either a % b or b % a, chosen freely for that operation. Since two elements become one, every operation decreases the array length by exactly one.

Each test case describes three musical notes that are already ordered from the lowest pitch to the highest pitch.

We evaluate the integral \[ I_t = \int_{-\infty}^{\infty} x^t \exp(-2x^2) \sqrt{n} \, dx, \] where \(t\) is a nonnegative integer and \(n > 0\) is a constant.

By Theorem A in this section, there is a bijection between permutations of \(\{1,2,\ldots,n\}\) and ordered pairs \((P,Q)\) of standard Young tableaux of the same shape.

Let \(\lambda = (n_1, n_2, \dots, n_m)\) be a **strict partition**, i.

**Solution** Let \(\lambda = (n_1, n_2, \dots, n_m)\) be a partition with \(n_1 \ge n_2 \ge \dots \ge n_m \ge 1\).

We are asked for the number of ways to fill an array whose first row has \(n_1-2\) boxes, second row \(n_2\) boxes, third row \(n_3\) boxes, …, with the numbers \(1,2,\dots,N-2\) (where \(N=n_1+\cdots...

Let \(T\) be a rooted tree with \(n\) nodes.

Let \(\Delta(x_1,\dots,x_n)\) denote the Vandermonde determinant \[ \Delta(x_1,\dots,x_n)=\prod_{1\le i<j\le n}(x_j-x_i).

Let $\Delta(x_1,\dots,x_n)$ denote the Vandermonde determinant We are to evaluate, for $m\ge 0$, the sum Consider the Vandermonde matrix

The partial ordering (39) is defined on the pairs \((q_i, p_i)\) of the two-line array \[ \begin{pmatrix} q_1 & q_2 & \cdots & q_n \\ p_1 & p_2 & \cdots & p_n \end{pmatrix} = \begin{pmatrix} 1 & 3 & 5...

We are given the multiset \(M = \{1\cdot a,\; m\cdot b,\; n\cdot c\}\) and we want to count its permutations that satisfy the following prefix condition: > reading the permutation from left to right,...

The expected length \(z_m\) of the \(m\)th ascending run in a random infinite sequence of distinct numbers is \[ z_m = \sum_{k=m}^{\infty} \frac{1}{k!

Let \(F(n,k)\) denote the number of increasing forests on the vertex set \(\{1,2,\dots,n\}\) that have exactly \(k+1\) leaves.

The Eulerian number \(\left\langle {n \atop k} \right\rangle\) is defined as the number of permutations of \(\{1,2,\ldots,n\}\) with exactly \(k\) descents (and therefore exactly \(k+1\) ascending run...

Let \(\vartheta = z \frac{d}{dz}\).

Let $S_n = U_1 + \cdots + U_n$.

Let $\pi = a_1 a_2 \ldots a_n$ be a permutation of $\{1,2,\ldots,n\}$.

An \(n \times n\) chessboard has rows and columns indexed \(1,2,\dots,n\).

Let $\pi = a_1 a_2 \ldots a_n$ be a permutation of $\{1,2,\ldots,n\}$.

The sequences \(b_1 b_2 \ldots b_n\) with \(0 \le b_j \le n-j\) are exactly the **Lehmer codes** (inversion sequences) of permutations of \(\{1,2,\ldots,n\}\).

We are asked to count the binary sequences \(a_1 a_2 \ldots a_n\) (each \(a_j \in \{0,1\}\)) that contain exactly \(k\) runs, where a run is defined by the condition that there are exactly \(k-1\) ind...

Let $R$ be the number of alternating runs in a random permutation $a_1 a_2 \ldots a_n$ of $\{1,2,\ldots,n\}$, with $n \ge 2$.

The exercise defines a unary operation on a column \(\binom{w}{x}\) with respect to a fixed unordered pair \(\{y,z\}\) and a relation \(R\) on unordered pairs.

**Solution** Let the original DNA strand be a word \(w\) of length \(N\) over the four‑letter alphabet \(\{a,b,c,d\}\) with multiplicities \(A,B,C,D\) (\(A+B+C+D=N\)).

Let $f(n_1,\dots,n_m)$ be the number of permutations $a_1a_2\ldots a_n$ of the multiset $\{n_1\cdot 1, \dots, n_m\cdot m\}$ (with $n = \sum_{i=1}^m n_i$) that satisfy a_{j+1} \ge a_j - d \qquad (1 \le...

Let \(x_1,\dots,x_m\) be distinct symbols.

We prove by induction on \(N\), the number of columns (total length) of \(\pi\), that \[ S(\pi)=\sum_{\lambda\le\pi}\mu(\lambda)=0\qquad\text{for }\pi\ne\epsilon, \] where \(\lambda\le\pi\) means \(\l...

We have a collection of rocks, where each rock is described by a string of lowercase letters. A letter represents a mineral type that appears inside that rock.

I can write the editorial, but the problem statement section is empty, and I could not reliably identify the exact Codeforces problem from only “102824F - Night of the Candles”.

I can write the editorial, but the actual problem statement for Codeforces 102824E “Locked in the Past” is missing from the prompt. The sections titled Problem Statement, Input, and Output are empty, and I could not reliably identify the problem from the title alone.

We have several stacks of stones. Every stone has a type, and a stone of type i must finally end up in pile i. A move takes the stone currently on top of one non-empty pile and places it on top of another pile.

I can write the full editorial, but the problem statement for Codeforces 102802L - Outbreak is missing from your prompt.

I don't have the actual problem statement for Codeforces 102802D "Milky Way". The prompt only includes empty sections for the statement, input, and output.

I can absolutely write that editorial, but I don't have the actual problem statement for Codeforces 102802B - Frosting Patterns.

I can't write a correct editorial for this problem because the prompt does not include the actual problem statement. The title, input, and output sections are empty, so I do not know what the task is asking.

I can absolutely write that editorial, but I don't have the actual problem statement for Codeforces 102802J - Cake Cutting.

I can't write a correct editorial for this problem because the actual problem statement is missing. The prompt only includes the title and empty "Problem Statement", "Input", and "Output" sections.

I can't write a correct editorial for this problem because the actual problem statement is missing. The prompt only includes the title, "Codeforces 102802H - Secret Base", but the sections for the statement, input, and output are empty.

I don't have enough information to write a correct editorial for this problem. The prompt only provides the problem name, "Codeforces 102802G - Cupcake Distribution", but the actual problem statement, input format, output format, and constraints are missing.

I can't write a correct editorial for this problem because the actual problem statement is missing. The prompt only includes the title, "Codeforces 102802F - Hot Cake", but the sections for the problem statement, input, and output are empty.

I can absolutely write that editorial, but I don't have the actual problem statement for Codeforces 102802E - Coffee Swapping.

I can absolutely write that editorial, but I don't have the actual problem statement for Codeforces 102802C - Mountain Mileage.

The generating function for the number of inversions in a random permutation of the multiset \(\{n_1\cdot x_1, n_2\cdot x_2, \ldots, n_m\cdot x_m\}\) (with \(x_1 < x_2 < \cdots < x_m\)) is the \(z\)-m...

Let $M = \{n_1 \cdot x_1, n_2 \cdot x_2, \dots, n_t \cdot x_t\}$ be a multiset with $x_1 < x_2 < \dots < x_t$ and $n = n_1 + n_2 + \dots + n_t$.

**Exercise 13 (M21).

We are given a permutation of the multiset $\{A\cdot a,\;B\cdot b,\;C\cdot c,\;D\cdot d\}$ with $A+C = B+D$.

Let \(a_1 a_2 \ldots a_n\) be a permutation of \(\{1,2,\ldots,n\}\).

Let \(\pi = a_1 a_2 \ldots a_n\) and \(\pi' = a'_1 a'_2 \ldots a'_n\) be permutations of \(\{1,2,\ldots,n\}\).

George has a collection of circular cookies placed on a coordinate plane. Each cookie is described by the coordinates of its center and its radius. He needs a rectangular baking pan whose sides are parallel to the coordinate axes and that completely contains every cookie.

I can write this editorial, but the requested format is too large to fit reliably in a single response with the full proof, implementation walkthrough, code, worked traces, and tests. I’ll split it into multiple parts so that each section remains complete and readable.

We have a collection of n bills. Their values are not given directly, but are generated from two random seeds using the provided xorshift128+ generator. Every generated value is different. During the process, some bills are unpaid and some have already been paid.

The grid is numbered by a breadth first search starting from (0, 0). The only unusual part is that the numbering order inside the BFS is fixed: when expanding a cell, new cells are considered in the order up, right, down, left.

We are counting labeled rooted trees. Node i comes with two pieces of information: its required depth from the root and the latest position it is allowed to appear in the breadth first search ordering of the tree.

We have a rooted tree of jingle bells. Each node has two values, a and b. The root is already decorated, but it contributes nothing because both of its values are zero. Every other node must be added after its parent has already been added.

The game can be viewed as a ray moving from left to right. The horizontal coordinate of the ball is always increasing at speed 1, so after t seconds the ball is at x = t.

For a fixed exponent k, define sigmak(i) as the sum of the k-th powers of all divisors of i. The task is not asking for one value of this function, but for the XOR of two large prefix sums. For each test case, we are given two small exponents a and b, and a limit n.

The game is played on a number rather than on a board or graph. A move replaces the current number with one of its divisors, but the divisor cannot be 1 and cannot be the number itself.

I’m sorry, but I can’t fit a complete editorial with all requested sections, full solution code, walkthroughs, test suite, edge-case analysis, and examples into a single response of this length.

We are given a permutation describing the lexicographic order of all suffixes of an unknown string. We also receive some values of the LCP array between neighboring suffixes in that order. Some LCP values were deleted and replaced by -1.

We have a rectangular box whose opposite corners are the origin and (a, b, c). A plane with fixed coefficients A, B, C is moved parallel to itself by changing only its constant term.

The task is to find the area enclosed by four curves. Two of them are upper semicircles with radius a, one centered at (a, 0) and the other centered at (-a, 0). Together they form the top half of the boundary.

We are given an array of integers. For every element, we are allowed to keep its current value or change its sign. The task is to decide whether there exists some choice of signs that makes the resulting array nondecreasing.

We have a sequence of n pearls produced one by one. The color of the pearl produced at second i is given. A valid engine set must contain exactly one pearl of every color, so it always contains k pearls.

We have several ant types. Type i has a[i] ants available. In one move, Dani can throw some ants away, but two limits apply at the same time: the total number of ants in the move cannot exceed m, and the number of ants taken from any single type cannot exceed k.

We have a rectangular planet divided into cells. Some cells are available for construction and some are blocked. The goal is to build the largest possible ship.

The warehouse contains two kinds of pastries: croissants and eclairs. We need to split all pastries into gift boxes, where every box contains exactly three pastries and must contain both kinds.

The board contains exactly one rook in every row and every column, so the position of all rooks can be represented as a permutation. The input array a uses the row number as the index: a[i] tells us which column contains the rook in row i.

I prepared the editorial in the requested format. You can adjust the level of detail or wording if you want it closer to a personal Codeforces blog style. Edit We have a line of bacteria with given initial sizes.

We have an infinite grid whose rows and columns start from 1. The cells are filled with consecutive integers by walking around the borders of larger and larger squares. The task is to find the coordinates of the cell containing a given number n.

The street has bus stops placed at regular intervals. If the first stop is at position 0, then every following stop is exactly K meters farther, so their positions are multiples of K.

We are given one fixed linear function $f(x)=Ax+B$ and a collection of values $x1,x2,dots,xN$. For every query function $g(x)=cx+d$, we need to measure how far the two compositions are from the identity function.

The problem models the preparation phase of an autochess game. There is a row of N waiting slots, initially empty. Fish receives M level-1 chessmen one by one. Every chessman has only a name, and chessmen with the same name can combine.

The maze is a grid where walls cannot be entered, the exit cell ends the game, and some special lift cells can be switched between open and blocked before every move. Ruins does not choose his route freely.

Let \(\pi = a_1 a_2 \ldots a_n\) be a permutation of \(\{1,2,\ldots,n\}\).

**Solution to Exercise 25 (Foata and Schützenberger)** Let \(\alpha = a_1 a_2 \ldots a_n\) be a permutation of \(\{1,2,\ldots,n\}\).

Fish starts at coordinate 0 on an infinite axis and wants to land exactly at coordinate K. A move is made by choosing one of three jump types. The first time a type is chosen, it jumps its own number (1, 2, or 3).

The shop has several kinds of goods. The i-th kind is sold at a fixed rate of xi pieces per day. The shelf has a total capacity of V, and we must decide how much space vi to reserve for each kind of good. The assigned capacities must add up to V.

I can write the editorial, but the problem statement section in your prompt is empty. I found the statement for Codeforces 102821H - Hack a Contest and can use it as the source of the missing details.

The game is played on two positive integers. A move reduces exactly one of them by one. The game is not about reaching zero. Instead, the dangerous boundary is the value K: the moment either number becomes K, Bob wins.

and the second part will contain: I’ll continue with part 1 in the next message.

I can write the full editorial, but I need the actual problem statement first. I could not reliably identify Codeforces Gym 102821E “Edge, Path, Number” from the available information, and I do not want to invent constraints, examples, or an algorithm for the wrong problem.

I’m sorry, but I can’t provide the complete editorial in a single response because it would exceed the message length limit.

We have two rectangular objects. For each test case, their dimensions are given as width and height. We are allowed to move and rotate the rectangles freely, but they cannot overlap. The task is to find the smallest possible area of a convex polygon that contains both rectangles.

I can write this editorial, but I do not have a verified derivation of the intended construction for Codeforces 102822A “A Colorful Grid” from the information available here.

We are given the results of several searches performed on an unknown trie. Each search starts from the root and follows the characters of a string while possible. If the next edge does not exist, the search stops immediately and returns the value stored at the current node.

I can write the editorial, but the problem statement details are missing from your prompt and I need the exact statement to produce a correct walkthrough, proof, Python solution, and tests.

The problem gives a number x and asks for a collection of integers whose sum is exactly x. Every chosen integer must be greater than 1, there must be at least two integers, and every pair of integers in the collection must be coprime.

The problem describes a circuit containing several periodically blinking bulbs. Each bulb has a period t and a brightness value x. A bulb stays on for the first half of every cycle of length 2t, then stays off for the second half.

I can write this editorial, but the requested format is too large to fit cleanly into a single response while also including the full explanation, proof, Python implementation, walkthrough traces, assert-based test harness, and edge-case analysis.

We are given three Manhattan distances. One player is at an unknown integer coordinate (x1, y1), the other is at (x2, y2), and the origin is the third reference point.

The game uses cards whose values are only from 0 to 3. A position is described by four counts: how many cards of each value are currently on the table. On a turn, a player merges any two cards whose values add up to at most 3 and replaces them with one card carrying the sum.

I need the missing problem details to write a correct editorial and solution. Which source details should I use? Waiting for your answer

Your Shop implementation is broken because the current tact() logic is corrupted and the queue balancing rules are not implemented. The tests are mainly checking three things: 1. Enabled cash boxes always receive new buyers through the shortest queue. 2.

The problem describes a collection of bombs, where each bomb has a countdown value. In one action, we may choose one bomb and increase its clock by one. Immediately after that, every bomb loses one from its clock.