brain

Let $f(x)=e^{-x^{2}/n}, \qquad n>0.$ The Poisson summation formula in the form used in TAOCP states that for sufficiently rapidly decreasing $f$, $\sum_{k=-\infty}^{\infty} f(k)=\sum_{m=-\infty}^{\inf...

A partition of $n$ in part-count form is a vector $c1,dots,cn$ satisfying $c1+2c2+cdots+n cn=n,$ where $ck$ is the number of parts equal to $k$.

There are two enemy forces, each with its own health pool and a fixed per-round damage value. Time progresses in discrete rounds, and in round $i$, the player is forced to use exactly $i$ attack power in total, and all of it must be directed to exactly one of the two enemies.

The task is to render a command line progress indicator for multiple scenarios. Each scenario describes a total amount of work and how much of it is already completed.

Let P(q)=\prod_{k=1}^{\infty}(1-q^k)^{-1}, \qquad q=e^{-t}, \quad t>0.

Let E(z)=\prod_{k=1}^{\infty}(1-z^k), \qquad P(z)=\frac{1}{E(z)}=\sum_{n\ge 0} p(n)z^n.

We are given several independent arrays. Each array contains integers, and across all arrays the total number of elements is at most one hundred thousand. After reading these arrays, we process a sequence of queries.

We are given several independent building descriptions. Each building consists of multiple floors, and each floor is represented by a string. Characters in the string describe whether a position contains trash and what type it is.

We are given four large integers. Each integer is not directly the object of interest. Instead, each one encodes whether a particular sightseeing location has been visited.

We start with a collection of $n$ strings, all having the same length $m$, and only lowercase letters. The allowed operations let us repeatedly reduce the number of strings by either merging two strings in a few different ways or discarding one string entirely.

Let E(z)=\prod_{k=1}^{\infty}(1-z^k), \qquad P(z)=\frac{1}{E(z)}=\sum_{n\ge 0} p(n)z^n.

We are given a collection of coin types. Each type has a fixed denomination and a limited supply. From these coins, we want to know how many distinct total sums we can form using any combination of coins, but only considering totals from 1 up to m.

We are simulating an online programming contest scoreboard that evolves over time as submissions arrive. Each submission belongs to a team, targets a problem, arrives at a timestamp, and is either accepted or rejected.

We are given a starting year and an integer shift. The shift is applied to the year, but the result is not used directly.

We are given a rooted tree structure indirectly, not by edges, but by a list of depths for each node. Node 1 is fixed as the root, and every node i comes with an integer ai describing how far it is from the root in terms of number of nodes on the path, including itself.

We are given a collection of numbers, and we are asked to repeatedly combine them in a structured way until only one value remains.

Let $U$ denote the set of all multicombinations $dt ldots d2 d1$ satisfying (6), that is $$s ge dt ge cdots ge d2 ge d1 ge 0.$$ The complement operation described in the hint is the standard involution on $U$ induced by reversal and complementation with respect to $s$.

I can’t write a correct editorial for “Codeforces 102898D - Digidivisible Numbers” without the actual problem statement. The title alone isn’t enough to reliably reconstruct the task, and guessing would almost certainly produce a wrong solution and misleading reasoning.

I don’t have the statement of Codeforces 102898C - Garbage Robot available in my context, and I shouldn’t guess the problem details because that would produce a misleading editorial.

I’m missing the actual problem statement for Codeforces 102898B - Teacher Sorting. Right now you’ve provided the editorial format, but not the key part that defines: what the input is, what transformation is required, and what must be output.

We are given multiple test cases. Each test case describes a binary tree only through two traversal orders: a preorder sequence and a postorder sequence. All node values are distinct, so each sequence is a permutation of the same set of integers.

Let $U$ denote the set of all multicombinations $dt ldots d2 d1$ satisfying (6), that is $$s ge dt ge cdots ge d2 ge d1 ge 0.$$ The complement operation described in the hint is the standard involution on $U$ induced by reversal and complementation with respect to $s$.

We are given a fixed 4×4 matrix where every entry is constant except for one position that depends on an integer variable $x$. For every integer $x$ in the range $[l, r]$, we evaluate the determinant of this matrix and are asked to find the minimum value over the entire range.

We are given a single line of text representing a sentence that may contain the word “is” in different contexts.

Let $n ge m ge 1$ and let $a1 ge a2 ge cdots ge am ge 1$ be a partition of $n$ such that $ Indeed, if $a1$ is the maximum part and $am$ is the minimum part, the condition gives $a1 - am le 1$, hence $am in {a1, a1 - 1}$. Therefore every part equals either $a1$ or $a1 - 1$.

We are given a list of integers representing scores assigned by judges. All values are nonzero and all are distinct. We are allowed to remove exactly one of these numbers. After removing it, we multiply all remaining numbers together, and that product becomes the final score.

We are given a sequence of stock prices over n days. On each day i, the stock has a known price p[i]. KK is allowed to perform at most one complete transaction: he may buy once on some day and later sell once on a strictly later day.

We are given multiple test cases. Each test case describes a sequence of $n$ distinct integers, which is a permutation of $1 ldots n$. The sequence represents the order in which dancers pass in front of an observer.

We are given a sequence of days, and on each day kk records how many wrong answers (WA) he made. These values form an array a[1..n]. Starting from a score of zero, the score evolves day by day. When processing day i, we compare the WA count a[i] with every previous day j < i.

We are given two collections of heroes represented as cards. Each card has a short name, a string of lowercase letters, and a strength value written as a decimal number.

We are given several independent test cases. In each test case, there are multiple “defense sessions”. In every session, a list of students appears, and each student has a combined score computed as the sum of two components.

We are given a stream of geometric figures, each described by a type identifier followed by its dimensions. Every figure contributes an area, and the task is to accumulate the total area across all figures and output the final sum rounded to one decimal place.

The task is intentionally minimal: the program receives a single token on standard input and must respond with a fixed ASCII drawing of a pig.

Let $n ge m ge 1$ and let $a1 ge a2 ge cdots ge am ge 1$ be a partition of $n$ such that $ Indeed, if $a1$ is the maximum part and $am$ is the minimum part, the condition gives $a1 - am le 1$, hence $am in {a1, a1 - 1}$. Therefore every part equals either $a1$ or $a1 - 1$.

We are given a collection of axis-aligned 3D boxes in space. Each box represents a solid region defined by independent intervals on the x, y, and z axes. Boxes may overlap in any way.

I can’t write a correct editorial for “Codeforces 102900M - Gitignore” because the actual problem statement is missing from your prompt. Right now I don’t know what the input represents, what operations are allowed, or what the output should compute.

The problem statement is missing from your prompt, so I don’t have the actual rules for the grid world (movement rules, goals, constraints, etc.). Without that, any editorial would be fabricated and likely useless.

I can’t write a correct editorial for this yet because the actual problem statement for Codeforces 102900H - Rice Arrangement is missing from your prompt.

Let $n ge m ge 1$ and let $a1 ge a2 ge cdots ge am ge 1$ be a partition of $n$ such that $ Indeed, if $a1$ is the maximum part and $am$ is the minimum part, the condition gives $a1 - am le 1$, hence $am in {a1, a1 - 1}$. Therefore every part equals either $a1$ or $a1 - 1$.

I can’t write a correct editorial without the actual problem statement. Right now the “Problem Statement / Input / Output” sections are empty, and Codeforces solutions are entirely driven by the precise structure of the task.

The problem works with a sequence of values arranged on a line, where each value represents the “weight” of a unit segment. From these values we can compute the sum of any contiguous segment, which we will call its segment weight.

The problem statement is missing from your prompt, so I don’t actually know what Codeforces 102900G - Fibonacci is asking. Right now I only see the title, but no definition of the input, output, or task.

The problem statement is missing from your prompt (the “Problem Statement / Input / Output” sections are empty), so there’s no way to reconstruct Codeforces 102900E reliably.

We are given a one-dimensional segment from 0 to n. Two walkers start somewhere on this segment. Each walker has its own starting position and its own constant speed, and both are allowed to walk back and forth along the segment, but they can never step outside the interval.

The task describes a geometric race path made of straight segments connected end to end in the plane. A runner starts at the first point of a polyline and must follow the segments in order until the last point.

We are given two integers $X$ and $Y$, and we consider all pairs $(i, j)$ where $0 le i le X$ and $0 le j le Y$. For each pair, we only care about those where the bitwise AND of the two numbers is zero, meaning the two numbers never share a common set bit.

Let $E(q)=\prod_{m\ge 1}(1-q^m).$ Euler’s pentagonal number theorem gives $E(q)=\sum_{k\in \mathbb{Z}} (-1)^k q^{k(3k-1)/2}.$ Let $S(q)=\sum_{n\ge 1} \sigma(n) q^n.$

I can’t write a correct editorial for “Codeforces 102904D - Funny Cost” because the actual problem statement (definition of the input, cost function, and required output) is missing.

We are given a sequence of monetary requests that must be satisfied in order, and a fixed amount of available money that starts at zero. Each request either increases or decreases the available balance, and the process evolves step by step as we move through the sequence.

Let $T(m1,dots,mn)$ be the $n$-dimensional torus with cross order as in Section 7.2.1.3, and let Theorem W be the structural statement whose proof in Exercises 91-92 relies on the spread function $alpha$ behaving uniformly across coordinates.

We are given a string consisting of lowercase letters. The task is to construct the shortest possible string that cannot be obtained as a subsequence of the given string.

We are given a sequence of numbers, but instead of treating them as individual values, we should think of them as a multiset where we are allowed to repeatedly perform a very specific transformation.

We are given a sequence of integers and we want to count how many pairs of positions can be chosen so that the product of the two corresponding values does not contain any squared prime factor. Another way to phrase the condition is to look at prime factorizations.

We can reinterpret the input as a graph. Each vertex represents an item in the class, and each edge represents that two items are compatible.

I can’t write a correct editorial for this yet because the problem statement is missing. “Codeforces 102906B - Лестница из чисел” alone isn’t enough to reconstruct the task reliably, and guessing would very likely produce the wrong solution and reasoning.

We are given a list of philosophers, each associated with a positive integer value that represents their “strength of dominance.

We are given a set of students, and several student organizations. Each organization consists of a group of officers, and every officer must be physically present for their organization’s prom in order for the event to run.

I can’t write a correct Codeforces-style editorial for “102911I - Implementation Problem” without the actual problem statement.

We are given a sequence of distinct items, each item originally sitting in a fixed position from 1 to N. We must rearrange them into a new ordering.

Let $a1 ge a2 ge cdots ge am ge 1$ be a partition of $n$ into $m$ parts that is optimally balanced, meaning $ Let $t$ be the number of parts equal to $x$ and $m-t$ the number of parts equal to $x-1$. The partition has total sum $$n = tx + (m-t)(x-1) = mx - (m-t).

We are given a very specific chess endgame situation: white has only a king and a queen, while black has only a king. The white king starts on a fixed square c3 and the white queen starts on d4.

We are given an undirected tree with vertices labeled from 1 to n. We are allowed to choose any vertex as the root and then orient every edge away from it, turning the tree into a rooted structure.

We are given a sequence of length $n$, where every value lies between $1$ and $k$. We are allowed to remove some elements, and after removal we look at the longest strictly increasing subsequence of the remaining array.

We are given a character who survives over a timeline measured in seconds. At the start he has some maximum health cap and an initial amount of health. Every second that passes reduces his health by one unit, and if his health ever becomes zero he is considered dead.

We are organizing a complete round-robin chess tournament among n players, meaning every pair of players must meet exactly once. That creates a fixed set of n(n−1)/2 games, and the only flexibility we have is how to schedule them over time.

We are given a sequence of videobloggers in a fixed order. Each blogger has a threshold value a[i]. When we approach bloggers from left to right, a blogger will record a review automatically only if either they are explicitly convinced by the marketer, or the number of reviews…

Let $a1 ge a2 ge cdots ge am ge 1$ be a partition of $n$ into $m$ parts that is optimally balanced, meaning $ Let $t$ be the number of parts equal to $x$ and $m-t$ the number of parts equal to $x-1$. The partition has total sum $$n = tx + (m-t)(x-1) = mx - (m-t).

We are given a single string and a number k. From this string we are allowed to delete characters while preserving the relative order of the remaining characters.

Let F(a,b;u,v)=\sum_{k,l\ge 0} u^k v^l z^{kl} \frac{(z-az)(z-az^2)\cdots(z-az^k)}{(1-z)(1-z^2)\cdots(1-z^k)} \frac{(z-bz)(z-bz^2)\cdots(z-bz^l)}{(1-z)(1-z^2)\cdots(1-z^l)}.

We are given a collection of segments on a number line, and each segment spans between two even integers. The task is to place a set of points on the same line so that every segment contains exactly one chosen point, while also ensuring that every chosen point lies inside at…

We are given a collection of spells, each spell consumes some combination of three types of mana: blue, purple, and orange. A mage has a total pool of these three colors, and what matters is only the total amount of mana across all colors, not the individual distribution.

We are given a collection of treasures, each with a positive value. Two players take turns picking remaining items until none are left. One player is fully strategic and wants to maximize the total value he obtains.

We are given a day that spans a time segment from 0 to m. There are n coworkers, and each coworker i is present only during their own interval from ai to bi. Alex must choose a continuous working interval [x, y] inside the day.

Each step of the process is identical in structure. Pavel is repeatedly matched with a random character, chosen uniformly from a fixed set of $n$. When he meets character $i$, he must immediately pick exactly one of two quests.

We are given a shape on an $n times n$ grid, described by cells marked as belonging to a polyomino. The shape is connected, has no holes, and every cell has at least two neighboring cells inside the shape. So locally, nothing behaves like a leaf or dead-end.

We are given a sequence of building heights arranged in a straight line, each position having a building of width 1.

Let $a1 ge a2 ge cdots ge am ge 1$ be a partition of $n$ into $m$ parts that is optimally balanced, meaning $ Let $t$ be the number of parts equal to $x$ and $m-t$ the number of parts equal to $x-1$. The partition has total sum $$n = tx + (m-t)(x-1) = mx - (m-t).

We are given a system with the same number of switches and lights, and a binary connection matrix describing how switches influence lights.

We are given a sequence of daily stock fluctuation values. Each value represents the change from one day to the next, so any contiguous segment represents the total change over a continuous time window. For each query, we restrict ourselves to a subarray interval $[S, E]$.

The border between the two kingdoms is made of three horizontal lines, stacked one above another with equal vertical spacing. Each line has several “holes”, each located at an integer position along the horizontal axis.

We are given a line with $n$ robots, each starting at some real coordinate. We must relocate them so that after movement they form a perfectly regular chain: robot $i+1$ must be exactly distance $d$ to the right of robot $i$.

We are given a sequence of integers of length $n$, and we are told to imagine it as a derived statistic from a peculiar tournament on the set ${1,2,dots,n}$. In this tournament, every pair of distinct numbers is compared twice.

We are given a complete graph whose vertices are villages placed on a 2D grid. The cost of traveling between two villages is not fixed in advance as an edge weight in the usual sense, but instead comes from energy consumption: moving between two points consumes energy equal to…

Algorithm H generates all integer partitions $a1 ge cdots ge am ge 1$ of $n$ by maintaining a weakly decreasing sequence whose entries are positive and whose sum is always $n$.

We are given a weighted tree where each node is a candidate location for a café, and a subset of nodes are marked as apartment complexes. The distance between any two nodes is the sum of edge weights along their unique path in the tree.

We are given two cubes, each with six faces. Every face contains a positive integer, and for each cube the six values sum to 21. When the cube is rolled, each face is equally likely to appear, so each cube defines a uniform probability distribution over its six face values.

Algorithm H generates all integer partitions $a1 ge cdots ge am ge 1$ of $n$ by maintaining a weakly decreasing sequence whose entries are positive and whose sum is always $n$.

Let F(a,b;u,v)=\sum_{k,l\ge 0} u^k v^l z^{kl} \frac{(z-az)(z-az^2)\cdots(z-az^k)}{(1-z)(1-z^2)\cdots(1-z^k)} \frac{(z-bz)(z-bz^2)\cdots(z-bz^l)}{(1-z)(1-z^2)\cdots(1-z^l)}.

The problem statement section is empty, so I don’t actually have the rules of what “Антенна” is asking for. Writing a correct Codeforces editorial without the task definition would force me to guess the problem, which would make the solution unreliable.

I can’t write a correct Codeforces editorial for this yet because the actual problem statement is missing. Right now I only see the title “Изменённая ДНК” (“Modified DNA”), but no definition of the input, operations, or what needs to be computed.

I can absolutely write this editorial in the required Codeforces style, but I’m missing the actual problem statement for “1029351 - Два станка”.

Algorithm H generates all integer partitions $a1 ge cdots ge am ge 1$ of $n$ by maintaining a weakly decreasing sequence whose entries are positive and whose sum is always $n$.

The task is to read two integer values and compute their sum. The input consists of two numbers presented in a simple textual format, and the output is a single integer representing their arithmetic addition. There is no hidden structure such as graphs, sequences, or queries.

The problem statement is missing from the prompt, so there’s no way to reconstruct a correct editorial yet. “Codeforces 1029366 - Числа” is not enough to determine what is being asked (there are many standard “numbers” problems: digit DP, divisibility, counting…

The problem statement section is empty, so there isn’t enough information to write a correct editorial. For Codeforces editorials, I need at least the full statement (or a link / text describing inputs, outputs, and constraints).

Algorithm H generates all integer partitions $a1 ge cdots ge am ge 1$ of $n$ by maintaining a weakly decreasing sequence whose entries are positive and whose sum is always $n$.

The task is to navigate a robot across a rectangular grid representing a ski slope, moving from the left side of the grid to the right side. Each cell is either blocked, normal snow, or a ramp.

I can’t write a correct editorial for this yet because the actual problem content is missing. “Codeforces 102939F - Factory Batching” is referenced, but there’s no statement, no constraints, and no description of the input/output or rules.

Two robots stand at two fixed lattice points on a grid. They throw a ball back and forth, and the ball travels along the straight line segment connecting their positions. A third point, Eve, is trying to intercept the ball.

I can’t reliably write a correct editorial for Codeforces 102939E - Decryption Scheme without the actual problem statement.