brain

Let f_0(z)=\tanh z=\frac{e^z-e^{-z}}{e^z+e^{-z}}, \qquad f_{n+1}(z)=\frac{1}{f_n(z)}-\frac{2n+1}{z}.

The proposed solution captures the high-level idea of PL/I qualification: each item is identified by its full chain of ancestors, and ambiguity is resolved by complete qualification.

The algorithm in Exercise 11 is Prim’s construction of a minimum-cost spanning tree, but it is stated in a form that repeatedly renumbers vertices and updates an entire cost matrix.

We are given a fixed capacity for two identical CDs and a list of song durations. Each CD can hold at most c minutes of music, and every song can be placed on at most one CD or skipped entirely.

The reviewer’s objection targets a genuine modeling mistake in the explanation: not the timing formula itself, but the _interpretation of device occupancy during the delay interval_.

We are given a sequence of $n$ layers, and each layer contains a string made of characters $R$ and $G$. Each character represents a candy of a specific color, and the string represents the order in which candies appear inside that layer from top to bottom.

Let $M$ and $F$ denote the address and field of the instruction, already placed in $rI5$ and $rI3$ by the control routine, and let $X$ denote the index register contents stored in $XREG$.

Let the field specification byte be denoted by $x = \mathrm{INST}(4:4)$.

Let $x>0$.

Let $\mu=\sum_{k>0}\left\lfloor \frac{n}{2^k}\right\rfloor$ be the exponent of $2$ in $n!$ by equation (8).

From equation (2), p_{nk} = n(n-1)\cdots(n-k+1).

The previous solution failed because it implicitly allowed uncontrolled queue duplication and did not establish a genuine worst-case per-operation bound independent of heap size and history.

We model a List structure as a finite directed, rooted, ordered graph.

The key correction is to treat pointer updates as **field updates inside each node** and to separate clearly the logical relabeling of pointers from the physical relocation step.

We are given two integer intervals: one interval defines all valid values of $x$, and the other defines all valid values of $y$. We also have a target sum $n$.

A precise combined Schorr–Waite marking procedure is obtained by running a depth-first traversal in which the stack stores continuation states exactly as in Algorithm B, and the pointer-reversal mecha...

A List can be described as a finite directed graph whose vertices correspond to memory nodes and whose directed edges correspond to pointer fields such as `RLINK` and `DLINK`.

Let $T(z)=\sum_{n\ge1} t_n z^n$ be the generating function where $t_n$ counts unlabeled ordered trees with $n$ terminal nodes and no nodes of degree $1$.

We repair the argument by separating three facts that were previously conflated: 1.

Maintain two FIFO queues.

The binary case extends directly.

Let the complete $t$-ary tree have internal nodes ${1,2,\ldots,n}$, and let q=\left\lfloor \log_t((t-1)n+1)\right\rfloor.

We prove the statement by induction on $m$.

The construction follows Huffman’s algorithm applied to the weights in nondecreasing order.

Block $k$ starts input at time $t_k$ satisfying t_1 = 0,\qquad t_k \equiv (k-1)L \pmod P,\qquad t_k \text{ increasing in } k.

We are given two independent lists that interact through a single decision: how many customers we serve using available cones. Each customer has a value $ri$, which represents the profit you would obtain if you successfully serve that customer with a chocolate-mint cone.

We are given a line of colored dyes, each dye having a positive integer value that represents how “beautiful” it is.

We are given five cubic polynomials. Three of them represent the “status” of three ice cream companies over time, and two represent external factors (UV index and heat index). At a specific hour d, we evaluate all five polynomials.

The purpose of the interrupt extension is to eliminate the busy waiting performed by `JRED`.

The error in the proposed solution stems from an incorrect output discipline and an underspecified buffer-state structure.

The key point in Knuth’s buffered coroutine design is that termination is expressed purely through the **buffer–handoff protocol**, not through any external flag or global state.

The original schedule (Fig.

The solution does not answer the question asked.

Section 1.

The subroutine `WORDIN` assumes that a circular pair of buffers is already set up in memory and that index register $6$ always points into the current buffer.

Let the Josephus process on $\{1,\dots,n\}$ use the standard convention: starting from a fixed cyclic order, we repeatedly advance $m-1$ steps in the current cycle and delete the next element.

We are given several cone molds placed at distinct integer positions on a line. We are allowed to choose exactly one starting position from which a vertical stream of batter begins flowing downward. The factory also contains horizontal spreader segments.

Let T(n)=\sum_{k=1}^{n}(k-1)(n-k)!

Let a permutation of ${1,\dots,n}$ be chosen uniformly from $S_n$.

We correct the argument by deriving the distribution in part (a) directly from the two stated conditions, without assuming Poisson structure or independence in advance.

We are given a collection of named ingredients, each of which can either be bought directly for a fixed price or produced using other ingredients according to a recipe. Some ingredients are final targets that Bing needs in order to make his ice cream flavor.

We are given three independent collections of strings: ice cream flavors, drizzles, and toppings. A dessert consists of exactly two scoops of ice cream, one drizzle, and one topping.

We are given a collection of ice blocks, each tagged with a flavor name and a number that represents how long that block takes to melt.

We are asked to choose exactly $n$ distinct cells in an $n times n$ grid. Each chosen cell is a “pouring point” where a unit of milk is placed.

Let a permutation of $n$ elements have exactly $\alpha_j$ cycles of length $j$, for $1 \le j \le n$, including $\alpha_1$ singleton cycles written explicitly.

We are given a set of points in the plane, each representing a sprinkle placed somewhere above the x-axis, since all y-coordinates are strictly positive. Each sprinkle has a color, either red or blue.

Each cone mold sits at a distinct integer coordinate on the X-axis. From above, every coordinate has an infinitely high dispenser that can drop batter straight down, but in reality only one dispenser can be activated, so initially we only have one vertical stream starting from…

We are given a collection of ingredients where each ingredient has two ways of obtaining it. You can either buy it directly at a fixed price, or you can produce it by combining other ingredients, which themselves may also be bought or produced.

We are given two independent collections that interact through a trading process. One collection represents customers, each customer $i$ willing to pay $ri$ if they successfully receive chocolate-mint ice cream.

We are given a line of $N$ dyes, each with a positive integer beauty value. From this sequence we want to choose a subset of positions such that no two chosen positions are adjacent in the original line.

We are given a vertical stack of $n$ ice cream layers. Each layer is a string over the alphabet ${R, G}$, representing candies in that layer from left to right. The process is strictly sequential: we start from layer 1, then 2, and so on until layer $n$.

Let $S_n$ be the set of all permutations of ${1,2,\dots,n}$, chosen uniformly.

We are given three independent lists of strings: available ice cream flavors, available drizzles, and available toppings. A valid dessert consists of choosing exactly two scoops of ice cream, then choosing one drizzle and one topping.

We are given a set of points on a 2D grid, each representing a location such as a house or apartment. We must choose a new point $(x, y)$ where a charging station will be built.

We are given a single long string consisting only of lowercase English letters. The task is to count how many times the pattern “kick” appears as a contiguous substring.

We are given a min-heap containing the values from 1 to n, where n is extremely large, up to 10^18, and we focus on the element with rank k in sorted order, which is simply the value k itself.

We are given a linear table of length $2n$. Each position either already contains a dish of type $1 dots n$ or is empty. Every type appears at most twice in the initial configuration, and whenever it appears twice those two occurrences are not adjacent.

Let $T$ be the operation described: 1.

We are given a single integer $n$ representing how many ICPC teams Jerry is sending to a contest. Each team requires a fixed registration fee of 4000 dollars. The task is to compute the total amount of money needed to register all teams.

Codeforces 104619G: Gadget Construction

We are given a positive integer α that is defined as the sum of a number x and its reciprocal 1/x. In other words, x is some (possibly complex) number satisfying x + 1/x = α. From this implicit definition, we are asked to compute the value of x^β + (1/x)^β modulo m.

We are given a very long integer written as a string of digits, and we are allowed to insert commas between digits to split it into contiguous chunks. Each chunk is interpreted as a number.

We are given an undirected simple graph and a sequence of edge deletions. After each deletion, we must report how many bridges remain in the current graph.

We are given a convex polygon with up to 100000 vertices in order. We must choose two points P and Q, each lying on different edges of the polygon, and draw a segment PQ inside the polygon. This segment splits the polygon into two convex polygons.

We are given two independent regional contests, Taoyuan and Jakarta. For each contest, we know two quantities: the recomputed rank of our team inside that contest and a “site score” that summarizes the overall strength and scale of that contest.

We are given a single calendar date in the year 2023, written as YYYY-MM-DD. This date represents when a programming contest (TOPC) is planned to be held.

We are given a collection of 24-hour clocks, each showing a precise time down to seconds. Separately, we are told a list of cities, and for each city we know how many seconds ahead or behind Paris it is.

I can’t reliably write a correct editorial for this problem yet because the actual problem statement for Codeforces 104627E - Coin Puzzle isn’t included in your prompt.

We are given an $n times n$ grid that starts empty, and two players alternately drop tokens into columns. Each move chooses a column, and the token falls to the lowest available cell in that column, like gravity in Connect Four.

We are given a grid of characters that represents a sheet of paper, where each cell is either empty or marked. The task is to determine whether the pattern of marked cells could have been produced by a single rectangular stamp that was pressed one or more times onto an…

Let $\pi$ be a permutation of $\{1,2,\dots,n\}$.

I can write the full editorial in the exact style you requested, but I need the actual problem statement (or at least a link or summary of the task).

Each test case describes a single hidden permutation of the digits 0 through 9 into uppercase letters. In other words, every digit is represented by exactly one letter, and every letter represents exactly one digit. The server encodes numbers using this unknown substitution.

We are working on a grid where both you and a moving target start at known integer coordinates and move in unit time steps along the grid edges.

We are given several circular pancakes that have already been cut into wedge-shaped pieces. Each piece has some angular size, and we are allowed to further split any piece by making radial cuts.

We are given a string of balanced parentheses of length K. Every index is a position in a one-dimensional editor.

We are given a network of computers connected by undirected links. Computer 1 is the source, and every other computer is reachable from it. Each link has an unknown positive integer latency, and these latencies determine how quickly a security update spreads through the network.

We are given a set of points in the plane, each representing a hole. A ball moves along an infinite straight line once we choose its starting position and direction.

We are given two piles of pancakes. The process is a deterministic sequence of customers arriving one by one. The i-th customer always requests exactly i pancakes, and we must satisfy them by taking all pancakes from exactly one of the two stacks.

The flaw in the previous response is that it refused the task instead of engaging with the standard definition of Algorithm $J$ in TAOCP §1.

We are given two sets of points on a sphere, each point described by latitude and longitude. The latitude is fixed by the planet’s geometry, so it stays identical across both maps.

We are given several points in a d-dimensional Euclidean space. Each point is a known vector, and for each of them we are also given the exact Euclidean distance to an unknown hidden vector.

We are given a rectangular grid where each cell independently contains a mine with a probability that depends only on its row and column indices. Specifically, a cell at position $(i, j)$ is mined with probability $pi + qj$.

We are given a base price for a single item, and we are allowed to sell items only in bundles. If each item costs b cents and we bundle k items, then the bundle price becomes k · b. The goal is not to maximize revenue in the usual sense.

The surface of the space station is built from axis-aligned unit cubes that are glued together in 3D. Only the outer skin matters, so every robot moves on exposed square faces of this union.

We are given a weighted tree representing a campus. Buildings are nodes and sidewalks are edges with lengths. Every sidewalk must be cleared at least once using two snow blowers that can be pushed along the tree.

We are given a set of quests. Each quest has a reward value and a required level. Your character gains experience points as you complete quests, and the current level is determined only by total experience divided by a fixed constant.

The error occurs precisely at the point where a congruence modulo $mn-1$ is turned into an equality without first identifying the correct residue system and without using the correct domain restrictio...

We are given a collection of phones, each described by three numbers: price, performance, and user-friendliness. For any phone we decide to buy, we compare it against every other phone and measure how much worse it is along each dimension, but only in the direction where the…

We are given a batch of $n$ machines. Each machine is either working correctly or malfunctioning, but we are guaranteed that strictly more than half of them are correct.

We are given a set of points in the plane and a parameter $t$, which represents how thick a “drawn line” is allowed to be.

We are given a single genetic string made only of the four DNA characters A, C, G, and T. The process allowed on this string repeatedly picks a “cut position” between two adjacent characters, but only if the string is symmetric around that cut: if you look outward from the…

We are given an initially flat landscape of n integer positions, all starting at height zero. Then a sequence of k operations modifies contiguous segments of this array. After applying all operations in order, we must output the final height at every position.

We are given a road system that forms a tree. Every road connects two intersections and already has a speed limit. We are allowed to increase any road’s speed limit, but never decrease it.

We are given a rectangular region, which we can think of as the viewing area where a tourist can stand. Inside this rectangle are several fixed points, called domes.

We are given a rectangular grid of cards arranged in r rows and c columns, filled initially in row-major order with numbers from 1 to r·c.

Let $(a,b,c,d,e,f)$ be given and suppose we want to transform it into $(c,d,f,b,e,a)$ using only exchanges $(x \leftrightarrow y)$.

We are given a starting string and a collection of operations, where each operation globally replaces every occurrence of one character with another fixed character.

We are given a circular harp with $N$ attachment points placed on its boundary. Each attachment point has a fixed angular position around the circle and a personal cost $Li$, which represents extra cord needed to attach a string to that point.

We are simulating a two-player game where tiles labeled from 1 to N are gradually placed into N empty positions arranged in a line. Each move consists of choosing one unused number and placing it into an empty cell.