brain

We are given a town modeled as an undirected graph on n houses. Some pairs of houses are connected by roads, and every other pair is considered connected only by an implicit “non-road” relation, meaning Thomas must travel between them using skis.

We are given an undirected simple graph. The task is not to find just one cycle, but to determine how many cycles achieve the minimum possible length among all cycles in the graph.

We are given a hidden set of $n$ strings, one per football team, and every pair of distinct teams produces a recorded match string that is simply the concatenation of the two team names in order.

Let the ZDD represent a family $mathcal{F}$ of subsets of ${x1,dots,xn}$, ordered by the variable indices, and let each node $k$ be labeled by $V(k)in{1,dots,n}$.

We are given a circle centered at the origin in the plane, with radius $r$. Every point whose distance from the origin is strictly greater than $r$ is considered outside the circle.

We are given a chronological log of score snapshots from a ping pong game between two players. The game ends as soon as one player reaches 11 points, and no further points should exist beyond that moment.

We are given a sequence of n lectures arranged in order. Each lecture either has a coffee machine or does not. The student starts before the first lecture with no coffee cups, and during each lecture she must consume exactly one cup in order to stay awake.

We are given a chronological stream of events describing work requests arriving from different sites, and a single excavator that moves between sites and spends a fixed number of days working at each site it visits.

Let $A={i1,i2,ldots,iell}$ and let $F = e{i1}cupcdotscup e{iell}$. The ZDD for $F$ consists of a single decision chain ordered by indices, because each $e{it}$ contributes a node that tests membership of $it$ and reduces to $perp$ or $top$ as in the conventions of Exercise 7.1.4.

We are simulating a process where a person interacts repeatedly with cats of different breeds, and the only thing that matters is the order in which each breed was last seen or last interacted with. Each operation can be interpreted as a request involving a cat breed.

Two students solve a sequence of problems one after another. Kesha spends a fixed amount of time per problem, call it $tk$, and Melentiy spends $tm$. They both start at the same moment and work continuously until a decision moment called “going for a walk”.

The problem describes a device that consumes energy while operating for a fixed duration, but the cost per minute depends on which time-of-day tariff is active. There are two tariffs, each defined by a price per minute and a time interval during the day when it applies.

Let $M_m$ denote the majority function on $m$ Boolean inputs, defined in Section 7.

Let $A={i1,i2,ldots,iell}$ and let $F = e{i1}cupcdotscup e{iell}$. The ZDD for $F$ consists of a single decision chain ordered by indices, because each $e{it}$ contributes a node that tests membership of $it$ and reduces to $perp$ or $top$ as in the conventions of Exercise 7.1.4.

I’m missing the actual problem statement for Codeforces 1040334 - “Спираль”, and without it I can’t produce a correct editorial.

The problem statement for Codeforces 1040335 - “Надпись на табло” is not included in your message, and I don’t have enough information to reconstruct the task reliably.

The state of the system can be viewed through two positions: the frog position $f$ starting from 1 and increasing, and the grasshopper position $g$ starting from $N$ and decreasing. At every second, $f$ increases by either 2 or 3, while $g$ decreases by either 1 or 2.

I can’t reliably write an editorial for this yet because the actual problem statement is missing. “Codeforces 1040333 - Заказ в магазине” only gives a title, but the required solution, constraints, and even the meaning of input/output are not included in your…

We start with an array where each value behaves like a token sitting on a number line. The allowed moves are either a small step left by one unit costing 1 energy, or a large jump to floor(x/2) costing k energy.

I’m missing the actual problem statement for “Codeforces 1040345 - Лука и локальная сеть динозавров”.

The problem statement is missing from your prompt, so there isn’t enough information to write a correct editorial.

I can’t write a correct editorial for this yet because the actual problem statement is missing. Right now I only have the title “Лука в кинотеатре” (Luka in the cinema), but no description of what Luka is supposed to compute, what the inputs are, or what…

The problem statement is missing from the prompt, so there is no way to reconstruct the required reasoning, constraints, or solution correctly.

Algorithm D in Section 7.

I can’t reliably produce a correct editorial yet because the actual problem statement for “1040354 - Браслет” is not visible/complete from the provided material, and the Codeforces reference in your prompt does not uniquely identify a single stable statement (there…

The problem statement for “Codeforces 1040355 - Задача о числах” isn’t included in your message, so I don’t have enough information to write a correct editorial.

I can’t write a correct editorial for this yet because the actual problem statement is missing. Right now I only have the title “Длина числа” (“Length of a number”), but no input format, constraints, or task definition.

The problem statement section is empty, so there isn’t enough information to reconstruct what Codeforces 1040352 - “Разноэтажный дом” is actually asking.

I can’t reliably write a correct Codeforces-style editorial here because the actual problem statement for “1040351 - Время в школе” is missing from your prompt.

Let $B(f)$ denote the number of nodes in the reduced ordered BDD representing a family $f$, including the sink nodes $bot$ and $top$.

I can’t reliably write a correct editorial yet because the actual Codeforces problem statement for “1040364 - Платные музыкальные сервисы” is not available in the provided content, and it is not a standard public problem I can uniquely identify…

I can’t write a correct editorial yet because the actual problem statement for “Codeforces 1040365 - Линейный футбол” is not included in your prompt, and the identifier does not directly resolve to a standard Codeforces problem I can safely assume.

I don’t have the actual statement of Codeforces 1040362 “Раскрашенный куб” in your message, so I can’t reliably reconstruct the solution or write a correct editorial without risking inventing details.

I can’t reliably write a correct editorial for this without the actual problem statement. “Codeforces 1040363 - Антон и арбузы” isn’t enough to reconstruct the task, and guessing would very likely produce a completely wrong solution and reasoning.

Let $B(f)$ denote the number of nodes in the reduced ordered BDD representing a family $f$, including the sink nodes $bot$ and $top$.

I can write this editorial in exactly the style you want, but I’m missing the actual problem content. “Codeforces 104037D - Point” isn’t something I can reliably reconstruct from the title alone, and I don’t want to hallucinate constraints, operations, or intended…

I don’t have the statement of Codeforces 104037A - Power available in context, and this ID doesn’t correspond to a standard Codeforces problem I can reliably reconstruct.

I can write the full editorial in exactly that format, but I’m missing the actual problem statement for Codeforces 104037B - Decode.

Let $B(f)$ denote the number of nodes in the reduced ordered BDD representing a family $f$, including the sink nodes $bot$ and $top$.

We are given up to ten short phrases, each being a string of lowercase letters. The task is to construct a single combined phrase that contains every given phrase as a substring.

Let $f$ and $g$ be Boolean functions with respective BDDs.

I don’t have the actual statement for Codeforces 104048I - Meteoric Sword in your prompt, so I can’t write a correct editorial yet without guessing the problem structure.

We are given a single string representing an initial alloy recipe. Each character is a type of metal. The only allowed operation is to take a character that originally existed in the input and duplicate it exactly once, placing the copy adjacent to the original occurrence.

Let $B(f)$ denote the number of nodes in the reduced ordered BDD representing a family $f$, including the sink nodes $bot$ and $top$.

The problem statement for “Codeforces 104048G - Foil Folding” is missing from your prompt, so I don’t have the actual rules of the task, input format, or what needs to be computed.

I can write the full editorial in exactly that style, but I’m missing the actual problem content for Codeforces 104048F - Neodymium Gravity.

I can’t write a correct editorial yet because the actual problem content for “Codeforces 104048E - Steel Customs” is missing from your prompt.

I don’t have the statement of Codeforces 104048D - Fullmetal Alchemist I available in this context, and I can’t reliably reconstruct it from the title alone without risking inventing details.

The problem statement section is empty, so there isn’t enough information to write a correct editorial yet. For Codeforces problems, even small details like what “Foo Bar” actually does, how input is structured, and what transformations are required completely change the…

I can’t reliably reconstruct the full statement of Codeforces 104048C - Capturing Bronze from public sources, and I don’t want to invent a problem and then build a fake editorial around it.

We represent a family $f$ as a reduced ordered decision diagram over variables $x1,x2,dots,xn$, using the conventions of Section 7.1.4 and Exercise 203. A node $v$ has fields $$V(v),quad LO(v),quad HI(v),$$ and terminals $bot,top$.

I can’t write a correct editorial for that problem yet because the statement content is missing. “Codeforces 104049M - Knight in Shining Armor 2” isn’t available in your prompt, and without the actual rules (what the knight does, what the input/output represent…

I can’t reliably write a correct editorial for this yet because the actual problem statement is missing. Right now I only see the title “104049L - Loid Forger”, but none of the details about the input, output, or task.

Let $f$ and $g$ be reduced ordered BDDs over variables $x_1,\dots,x_n$ with fixed ordering.

I can’t write a correct editorial yet because the problem statement for Codeforces 104049J - Knight In Shining Armor 1 is missing from your prompt.

We represent a family $f$ as a reduced ordered decision diagram over variables $x1,x2,dots,xn$, using the conventions of Section 7.1.4 and Exercise 203. A node $v$ has fields $$V(v),quad LO(v),quad HI(v),$$ and terminals $bot,top$.

I can’t produce a correct editorial yet because the actual problem statement for Codeforces 104049I - Meteoric Sword is missing from your prompt.

We are given a rectangular sheet of foil represented as a grid of size $n times m$. Some cells contain imperfections marked as X, while others are clean. From this sheet, we want to extract a rectangular piece of metal that qualifies as an ingot.

The problem statement for Codeforces 104049H - Alluring Alloy is missing from your prompt, so I don’t have enough information to reconstruct the task or derive the algorithm.

I can write the full editorial in that format, but I’m missing the actual problem statement for Codeforces 104049F - Neodymium Gravity. Right now I only have the contest/problem identifier, not what the task is asking (input format, operations, constraints, etc.).

I can’t reliably write a correct editorial for Codeforces 104049D - Fullmetal Alchemist I without the actual problem statement.

I don’t have the statement of Codeforces 104049E - Steel Customs available from the prompt, and I can’t reliably reconstruct the problem from the title alone without risking inventing details.

We are given a binary string and we need to transform it into a string over a three-letter alphabet, typically A, B, and C, in such a way that the original binary string can still be uniquely reconstructed even after an adversarial deletion process that removes all occurrences…

Let $H$ denote the reduced ordered BDD for $h(x_1,\ldots,x_n)$.

We are given a bipartite structure where one side consists of students and the other side consists of slots, referred to as parallels. Each student is connected to some subset of these slots, and a connection means that the student is eligible to be assigned to that slot.

The solution answers all parts, but part (b) is incorrect and breaks subsequent reasoning. The error comes from an incorrect application of the quotient definition: it wrongly assumes that $alpha in f/e2$ must itself belong to $f$, which is not required.

We are given four non-negative integers describing how many items we have of four types, A, B, C, and D. From these items we want to assemble identical “sets”, and each set must be one of three fixed patterns: One pattern consumes two A items, one B item, and one C item.

The problem can be viewed as a sequence of levels, where each level contributes some number of elements, and also comes with an interval that describes where its influence applies.

We begin by making the construction in (37) explicit in the only way the proof can depend on it.

We are given a sequence of length $k$, where each element $ai$ is a non-negative integer bounded by $m$. For any such sequence, its value is defined as the sum over all pairs where the second index does not exceed the first, of the bitwise XOR of the pair: $$sum{i=1}^{k}…

The solution answers all parts, but part (b) is incorrect and breaks subsequent reasoning. The error comes from an incorrect application of the quotient definition: it wrongly assumes that $alpha in f/e2$ must itself belong to $f$, which is not required.

We are given a scenario where $n$ distinct people are to be arranged into $m$ labeled stations, and each station holds a queue. A queue here is not just a set, but an ordered list, so the internal ordering of people inside each station matters.

Each vertex of a connected undirected graph is turned into a random point in 3D space. The coordinates of a vertex are independent uniform real numbers in the unit cube, so every vertex gets a completely independent random position.

We are given a tree with n nodes. One node is chosen as the initial infection source, and that choice is random but weighted: node i is selected with probability proportional to ai.

Two players repeatedly build up numbers by multiplying chosen integers. Alice controls set $A$, Bob controls set $B$. Both start with values $alpha = 1$ and $beta = 1$. On every Alice move she picks any element from $A$ and multiplies it into $alpha$.

The solution answers all parts, but part (b) is incorrect and breaks subsequent reasoning. The error comes from an incorrect application of the quotient definition: it wrongly assumes that $alpha in f/e2$ must itself belong to $f$, which is not required.

We are given an initial set of distinct non-negative integers. Two players, Alice and Bob, take turns inserting arbitrary integers into this set. Each player makes exactly k moves, so in total 2k new numbers are added.

We are given a function defined for a modulus $m$: we look at the linear congruence $$a x equiv b pmod m$$ and define $f(a,b,m)$ as the smallest non-negative integer $x$ that satisfies it, or $0$ if no solution exists.

We are given several elevators, each starting from floor 1 but not at the same time. Every elevator moves upward at a constant speed of one floor per second, so once it starts at time $xi$, it reaches floor $f$ exactly at time $xi + (f-1)$ if nothing interferes.

Algorithm R builds a reduced ordered BDD by creating nodes and using an AVAIL stack to recycle nodes whose LO and HI pointers are found equal or whose subgraphs duplicate existing nodes.

We are given a rooted tree where vertex 1 is the starting point and every node represents a position the cat could have passed through. The cat moves from the root down the tree without revisiting any node, so its path is simply some root-to-leaf path.

We are working with an array a that assigns each position i a label a[i]. In addition, there are two auxiliary arrays b and c, both initially zero.

All operations are in the family algebra of Exercise 203. For families $f,g$, the quotient is $$f/g = {alpha mid forall beta in g,; alpha cup beta in f ;text{and}; alpha cap beta = varnothing},$$ and the remainder is $$f bmod g = f setminus (g sqcup (f/g)).

We are given a simple polygon whose edges alternate between horizontal and vertical segments, so the shape is an axis-aligned rectilinear loop. A laser starts from a boundary point and travels inside the polygon along a diagonal direction (1, 1).

We are given a rectangular grid with height ℓ and width w, and we need to partition it into exactly n disjoint regions. Each region must consist of whole grid cells, must form a rectangle aligned with the grid, and all n regions must have equal area.

All operations are in the family algebra of Exercise 203. For families $f,g$, the quotient is $$f/g = {alpha mid forall beta in g,; alpha cup beta in f ;text{and}; alpha cap beta = varnothing},$$ and the remainder is $$f bmod g = f setminus (g sqcup (f/g)).

A 2 × n bridge is built from n positions, where each position has two possible tiles: left and right. Exactly one of the two is safe at each position, and the safe side for position i is given by a string of length n, where each character tells whether the left or the right…

We are given a sequence of cards represented as a string, where each character is one of four types corresponding to suits.

We are planning the lifecycle of a single CPU over a timeline of n months. In each month i, there is a known purchase price for buying a fresh CPU.

We are interacting with a hidden string of lowercase English letters whose length can be up to 10,000. Our only way to obtain information is by asking queries of the form “given a set of letters, which positions in the hidden string contain any of these letters”.

The canteen starts with two “base soups” that we can think of as two pure ingredients: one is π-tato soup and the other is τ-mato soup. From day to day, the recipe evolves deterministically. On day 1 the soup is entirely π-tato, on day 2 it is entirely τ-mato.

We are given a geometric graph whose vertices are crossings in a flat map and whose edges are road segments. The roads are straight segments between given endpoints, and the crucial geometric guarantee is that any two segments only intersect at shared endpoints, so the drawing…

We are interacting with an unknown circular track of integer length $L$, where $1 le L le 10^{12}$. Perry starts at position 0 and moves forward at constant speed 1, so at time $t$ his position inside the current lap is exactly $t bmod L$.

We are given a rectangular LEGO base of size $a times b$. Normally, such a rectangle would only be placed axis-aligned on a grid of studs, but here the rule is different: the rectangle can be rotated freely in the plane.

We are given a collection of cicadas, each with a positive integer “periodicity” value. We are allowed to discard some of them, and we then consider only the remaining ones.

We are working with a cyclic street split into $n$ consecutive segments, where segment $i$ is adjacent to $i-1$ and $i+1$, and segment $1$ is also adjacent to segment $n$. Some segments can be closed over time, and a closed segment cannot be traversed.

Algorithm R builds a reduced ordered BDD by creating nodes and using an AVAIL stack to recycle nodes whose LO and HI pointers are found equal or whose subgraphs duplicate existing nodes.

We are tracking a single distinguished T-shirt inside a stack of $n$ shirts. The stack is always accessed from the top: every morning you wear the top shirt, and it goes into a laundry basket at the end of the day. At night, you occasionally do laundry.

We are given a collection of socks split into several groups. Each group describes socks of the same type, where a type is defined by a name and a fit category. The fit can be left, right, or any.

We work in the family algebra of Exercise 203. A family is a set of sets of positive integers, and all operations are defined elementwise at the level of these sets.