brain

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 $R, W, B$.

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.

The motion can be rewritten in coordinates.

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.

We are given a pile of stones and two players who alternate turns, with GG moving first. On each move, a player removes either one or two stones from the pile. The player who cannot make a move loses.

We are given a static array of integers. For each query, we are also given two numbers $a$ and $b$, which define a function on any value $x$: $$f(x) = (a oplus x) - b$$ where $oplus$ is bitwise XOR.

We are given a grid where each cell contains a symbol describing how it participates in a tiling made of L-shaped triominoes. Each L-shape consists of a center cell marked C and three adjacent arms extending in the four cardinal directions, labeled U, D, L, and R.

We are given five types of spirits, indexed from 1 to 5, and we have a certain number of copies of each type. Each spirit of type i has a fixed intrinsic value i. A single operation, called a “summon”, chooses some subset of available spirits.

We are given a 3D grid of cells with coordinates $(i, j, k)$. Each cell initially has a color, either black or white, and we are allowed to flip its color at a given cost.

We are given a permutation of size $n$, and we look at every inversion pair in it. An inversion is a pair of positions $i < j$ where the value at $i$ is larger than the value at $j$.

Codeforces 105223N: Larger but smaller!

Codeforces 105223M: Minimize Sum

We are asked to decide whether we can place $n$ distinct lattice points in the plane and connect them in a cycle so that all edges have equal Euclidean length, the polygon is simple (no self-intersections), and no three consecutive vertices lie on a single straight line.

Substituting special values is the first constraint.

Codeforces 105223F: Fire Kings

Codeforces 105223E: Lazy Fouad

We are given a rooted tree where each node represents a one-liter tank. Water can be poured into any chosen tank, but the filling does not stay local.

We are given an axis-aligned square and inside it several straight segments, each of which is either horizontal or vertical. The segments are strictly inside the square, but they may overlap or even coincide.

We are given several independent strings, and for each one we must decide whether it contains a specific pattern, the word “bitset”, as a contiguous block inside it.

We are given several independent test cases. Each test case describes a game state consisting of several piles of stones.

We are given an array that changes over time. After each update, we are asked to compute a global quantity built in two layers. First, take any non-empty subsequence of the current array.

We are given several test cases. In each test case, we receive an array of integers. Our task is to count how many pairs of positions $(i, j)$ with $i < j$ satisfy a very specific algebraic relationship between the values $ai$ and $aj$.

We start with an existing tree on $n$ vertices. We are allowed to add new vertices and connect them with edges, but we are not allowed to create cycles, so the final structure must still be a tree. After these additions, the resulting tree must satisfy two structural conditions.

We are given an array and for every position $i$, we want to count how many segments $[l, r]$ that include $i$ have a special property tied to bitwise AND.

We are given a line of students, each with a fixed initial height. The school is unhappy with any student who is strictly shorter than both of their immediate neighbors. The first and last students are exempt because they only have one neighbor.

We are given a tree, and only some nodes matter as “items to distribute”: the leaves. Every leaf must be assigned to exactly one of K groups.

We are given a one-dimensional board of positions from 1 to n. Each position i contains a jump length t[i]. If a ball is dropped at position i, it repeatedly performs deterministic jumps: from i it moves to i + t[i], then from that new position j it moves to j + t[j], and so…

We are maintaining an array that changes over time, and after each change we may be asked to compute an optimal score based on splitting the array into contiguous parts.

We are given a single string made of lowercase letters. From this string we may delete characters while preserving order, producing any subsequence.

We are given a number of cards and asked to repeatedly construct the tallest possible “card pyramid”, remove the cards used, and continue until no further pyramid can be built. The final answer is how many pyramids were constructed across all iterations.