brain

We are given a binary string made of parentheses. From any substring we are allowed to pick a subsequence, and we are interested in a very rigid kind of subsequence: it must look like several copies of “()” concatenated together.

We are given an array of integers and a fixed linear rule that relates a “radius” around an index to a value computed from the array. For a chosen center position $i$, we look symmetrically to the left and right.

We are given an array and its longest strictly increasing subsequence length. For every query, we remove a contiguous segment and ask whether this removal keeps the LIS length unchanged. In other words, the original array has some optimal increasing subsequence of length L.

We are given a tree, meaning a connected graph with no cycles, and each edge represents a “magical channel” between two nodes. At every node, several edges meet, and any pair of edges incident to the same node creates a potential conflict that must be neutralized.

We are given a collection of independent gambling games. Each game has a probability of success and a payout if it succeeds. The twist is that if any chosen game fails, the entire selection yields zero reward.

I cannot reliably reconstruct the solution to Codeforces 105449I - “Простая задача для любителей” from the information currently available.

We are given two directed graphs, each with $n$ vertices. Both graphs are strongly connected, and every directed cycle inside either graph has length divisible by $k$. Each vertex is labeled either as outgoing or incoming.

The previous approach failed because it relied on an unproven cyclic identity involving triangles $V_iV_{i+1}V_{i+2}$, which is not compatible with simple triangulations of a convex polygon.

We are given several independent test cases. In each test case there are $n$ small blocks, and each block contains exactly two numbers. We are allowed to reorder these blocks arbitrarily, but we are not allowed to change the order inside any block.

We need to fill a grid with integers from $0$ to $2^c - 1$, where each integer represents a subset of $c$ colors via its binary representation. If the $k$-th bit of a cell value is $1$, that cell belongs to color $k$. Three constraints govern the construction.

The previous argument fails precisely at the claim that an increase at one step inside a cycle forces monotone growth along the entire orbit.

The problem statement section is empty, so there isn’t enough information to reconstruct what “Марго покидает Мегабайтбург” is asking or what the solution should do.

I can write the full Codeforces-style editorial, but I’m missing the actual problem statement for 105424G - “Прогрессивный NoSQL”.

The problem statement section is empty in your prompt, so I don’t have the actual definition of what needs to be computed (inputs, allowed operations, and what “maximum GCD” refers to in this context).

We are choosing how to distribute a small number of identical improvements across a small set of items. Each item has an initial probability of being in a favorable state, and each improvement increases that probability by a fixed step until it saturates at full certainty.

I don’t have the statement of Codeforces 105424E (“Маска для монстров”) available in this context, and I can’t reliably reconstruct the problem from the title alone without risking inventing details.

I cannot reliably reconstruct a correct editorial for “Codeforces 105424B - Ужин для интровертов” from the available information because the actual problem statement is not accessible in the sources I was able to retrieve, and the number “105424B” does…

I don’t have the actual statement of Codeforces 105424A (“Середина игры”) available in this context, and I can’t safely reconstruct it from the title alone without risking inventing details that don’t match the problem.

We are given a binary string representing a row of cookies arranged in a line. Each position is either a deer cookie (1) or a human cookie (0).

We are given a kitchen with a fixed stock of several ingredients. Each ingredient starts with some quantity. There is a menu of dishes, and every dish consumes certain amounts of these ingredients when prepared.

We are simulating an infinite process that places identical regular octagons on the plane. Each octagon has a well-defined center, and every new octagon is attached to a previous one by sharing one of its sides. Once placed, each octagon is fixed.

We are given a permutation that describes how ferrets move on a circular arrangement of positions. At any moment, each position on the circle is occupied by exactly one ferret, and applying the permutation advances all ferrets to their next positions simultaneously.

We are given two small collections of toy identifiers. Alice’s bag contains five toys, and Bob’s bag is supposed to contain exactly the same set of toys but one item is missing, so his bag has only four.

I don’t have the actual statement of Codeforces 105408H - Hidden Treasure of Arrayland in context, and it isn’t something I can reliably reconstruct from the title alone without risking inventing details.

We are given a convex polygon with vertices listed in counterclockwise order. From this polygon we can pick any subsequence of vertices, as long as we keep their original order and choose at least three points.

We are given a collection of books. Each book has three attributes: how many pages it contains, how enjoyable it is for Conner, and how much fame it gives him if he finishes it.

We are given a collection of words that form a dictionary and a sequence of query words. The ordering of words is not the usual lexicographic order on the raw strings.

We are given a permutation of size $N$, meaning an ordering of the numbers $1$ through $N$ where each value appears exactly once. The problem defines a conceptual structure: list all $N!$ permutations sorted in lexicographic order, then imagine placing them evenly on a circle.

The equation is rewritten as $y^2=1+2^x+2^{2x+1}$.

We are given three points in the plane, but instead of being arbitrary, they represent the midpoints of three segments forming a triangle of hidden original points. More concretely, there are three unknown integer points A, B, and C. We are given the midpoints of AB, BC, and CA.

We are given a strip-shaped region built from small unit triangles. The shape grows with a parameter $n$, and its total size is linear in $n$.

Each student starts at a fixed integer position on a line segment from 0 to n. At time zero, every student independently chooses a direction, either left toward 0 or right toward n, each with probability one half.

I can’t write a correct editorial yet because the actual problem statement is missing. Right now the prompt only includes the title “Аппарат Пина”, but no description of the task, input/output format, or constraints.

We are given a cyclic version of the game “Rock, Paper, Scissors”. Each player does not choose moves independently per round; instead, each of them has a fixed repeating pattern of length n for one player and m for the other.

You’re asking for an editorial for Codeforces 105329B - “Максимальное число”, but the full statement is not included in your prompt, and it also does not appear in the standard Codeforces problemset index.

I can’t write a correct editorial for this problem yet because I don’t have the actual statement of Codeforces Gym 105329D - “Копатыч, Карыч и сад” in a reliable form.

I can’t reliably write a correct editorial for this problem as-is because the actual problem statement is missing.

The input describes a hexagonal crystal made of small cells, arranged in a triangular lattice with $2N-1$ rows. Each cell contains a number representing a “particle type”. For every cell in this structure, we treat it as a potential center of a hexagon.

Homogeneity allows normalization $a^2+b^2+c^2=1$, so the task becomes maximizing

A triangulation of a convex $n$-gon uses $n-3$ diagonals and produces $n-2$ triangles.

The condition

The earlier approach fails at the point where it tries to force a contradiction from upper bounds alone.

The problem describes a probabilistic shortest-path process on a small graph of bicycle stations. You start at station 1 after borrowing a bike, and your goal is to reach a “finish” by eventually returning the bike at some station and walking to the destination.

The configuration places six points on the three sides of an equilateral triangle and assumes that consecutive segments of the hexagon they form are all equal.

An integer is alternating when its decimal digits switch parity at every step.

We are given the first $N$ prime numbers in increasing order, starting from $2$. Each prime has a fixed position in this list, so position $k$ corresponds to the $k$-th smallest prime. The task is to assign each prime to one of two groups, labeled $A$ and $B$.

A convex hexagon has three pairs of opposite sides.

The expression

Let $S={1,2,\dots,10^6}$ and let $A\subset S$ with $|A|=101$.

The input consists of a list of numbers, each representing a plushie’s value. We are allowed to reorder them arbitrarily, but after ordering, every consecutive pair must avoid summing to a fixed forbidden value $X$.

The configuration consists of $n$ unit disks whose centers are pairwise unconstrained except for a global incidence restriction: no straight line intersects three disks simultaneously.

Substituting zero values isolates the role of $f(0)$ and reduces the functional equation to a simpler constraint linking constant terms and general values.

We are given an $N times N$ grid where two players alternately place tokens on empty cells. The restriction is global: no two placed tokens are allowed to be orthogonally adjacent, meaning they cannot share a side.

Compute small examples to understand structure.

We are given a string made only of the letters A, B, C, and D. The only operation we are allowed to perform is selecting a contiguous block of four characters that forms a cyclic rotation of the pattern ABCD and then rotating that block by one position left or right.

Let

The configuration mixes three geometric mechanisms: reflection in a perpendicular bisector, arc midpoints on a circle, and a line through the center parallel to a chord-derived direction.

We are given a set of points in the plane, each representing a pub. We want to choose an ordered circuit of at least 3 and at most k distinct pubs, visit them in that order, and return to the starting pub, forming a cycle.

The previous approach fails because it moves between different colorings instead of staying inside a fixed configuration.

The key relation is rewritten as

The earlier approach failed because it replaced a non-collinear segment $YB$ by an invalid subtraction on a line segment.

We are given a positive integer $n$, and we treat the numbers from $1$ to $n$ as labels of items. The task is to form as many disjoint pairs as possible, where a pair $(a, b)$ is valid only if $a neq b$ and the greatest common divisor of $a$ and $b$ is greater than 1.

All attempts to proceed from injectivity of the map $x \mapsto f(x) \bmod m!$ must be checked against small cases.

The structure suggests a bipartite incidence system between girls and boys, where each edge $(g,b)$ is “witnessed” by at least one problem containing both endpoints.

We are given up to 100 distinct labeled points in the plane. We must construct a sequence of these points, where repetition is allowed, subject to three constraints.

Direct substitutions such as $a=x^2$ and linearization of the denominator repeatedly reduce the expression to cyclic rational forms that become weaker than the target bound.

Let $A,B,C$ be the angles of the acute triangle, so $A+B+C=180^\circ$.

The configuration involves three classical geometric structures in an acute triangle $ABC$: the orthocenter $H$, the feet of altitudes $H_1,H_2,H_3$, and the incircle tangency points $T_1,T_2,T_3$.

Let $n$ be a positive integer with exactly $2000$ prime divisors, counted with multiplicity.

Let the three boxes be $R, W, B$.

The motion can be rewritten in coordinates.

We are given a square floor of size $n times n$, where each cell can either remain empty or be covered by a domino-shaped table that occupies exactly two neighboring cells.

We are given a hidden permutation of the integers from 1 to 100. At position i there is a value p[i], but we never see it directly. The only information we can extract is by choosing two positions a and b and asking for gcd(p[a], p[b]).

We are asked to construct a simple undirected graph on n labeled vertices, or decide that it cannot be done, with a very strong structural constraint: the graph must be asymmetric. Asymmetric here means there is no non-trivial relabeling of vertices that preserves adjacency.

We are given a set of weighted points in the plane. Each point represents a potential oil extraction site, located at integer coordinates, and each carries a profit value.

We are given a tree with $n$ nodes, and we are free to assign a lowercase letter to each node. After labeling, every simple path in the tree becomes a sequence of letters, read along the unique path between its endpoints.

We are given a directed graph on $n$ vertices where every vertex has exactly one outgoing edge, defined by the array $a$. From each player $i$, there is a mandatory requirement that if $i$ is selected, then $ai$ must also be selected.

We are working with two layers of combinatorics. First, there is a text string $S$ of length $n$ over an alphabet of size $k$, where $k$ can be extremely large, up to $10^9$, so we should think of characters as abstract labels rather than concrete letters.

We are asked to construct a single input file that is simultaneously valid for two different Codeforces problems, each with its own input format and interpretation rules, and force both correct solutions to produce the same numeric output, namely a given integer $x$.

Codeforces 105216M: Maximizing the Sauce

Codeforces 105216I: Intersection of Hyperrectangles

The expression is cyclic in $a,b,c$ under the constraint $abc=1$, suggesting a symmetric inequality that may become simpler after a substitution such as $a=\frac{x}{y}$, $b=\frac{y}{z}$, $c=\frac{z}{x…

We are given a family of $N$ people, where each person currently holds two shoes: one right shoe and one left shoe. However, these shoes are mixed up.

We are simulating a shrinking circle of candidates, each labeled from 1 to n in clockwise order. Two pointers move around this circle repeatedly.

We are simulating a process where computers arrive one per week. Each computer is assigned a random “name”, which is just an integer chosen uniformly from the range $1$ to $N$. Jose can only keep at most $k$ computers at any moment.

We are given a group of samurais and some pairs that already mutually respect each other. Respect is symmetric, so the information can be viewed as an undirected graph where vertices are samurais and edges are existing respect relationships.

We are given a tree with up to one million nodes, and every pair of nodes is implicitly generating a travel request: a traveler starts from the smaller numbered node and goes to the larger numbered node, following the unique simple path in the tree.

Codeforces 105216B: Birthday Cake

We are given a sequence of numbers indexed from left to right. For every pair of indices $i le j$, we look at the subarray from $i$ to $j$, take its maximum element, and multiply it by the square of $gcd(i, j)$. The task is to sum this value over all possible subarrays.

We are given a row of prize values, each prize having a positive integer value. John has a score limit $p$, and he is only allowed to pick prizes whose value does not exceed $p$. Among all valid prizes, he wants the one with the largest value.

We are asked to build a directed acyclic graph on $N$ labeled vertices, where vertex $i$ represents a programmer.

We are asked to count how many ordered pairs of two $N$-digit numbers satisfy a tight set of digit-level constraints. Each number has exactly $N$ digits, neither can start with zero, and when we compare them position by position, the digits must always differ.

We are given several independent scenarios. In each scenario, there are $N$ cuckoo clocks, and each clock has its own fixed periodic behavior. Clock $i$ produces a sound at times $1, 1+i, 1+2i, 1+3i,dots$.

Codeforces 105222L: Beef Tripe in Soup Pot?

Two circles $G_1$ and $G_2$ intersect at $M$ and $N$, so both lie on a fixed chord common to the two circles.

We are given a sequence of element usages, where each step applies one of at most 17 element types. When an element is used on a monster, it interacts with the current “active element” on the monster.

Codeforces 105222F: Isoball: 2D Version

Codeforces 105222D: L-Covering

We are asked to represent a positive integer using a strange mixed numeral system built from two kinds of symbols. The first kind is the usual decimal digits from 0 to 9. Each digit has a given cost, and using it once in the representation costs that amount.

We are given a rectangular deck that acts as a fixed 2D container, with its bottom-left corner at the origin and top-right corner at $(l, h)$. Into this space, we must place a sequence of axis-aligned rectangular boxes.