brain

Let $x_1,\dots,x_{20}$ denote the observations, and let $x_{(1)} \le \cdots \le x_{(20)}$ be the ordered sample.

Let $V(z)=\sum_{m\ge1} V_m z^m,$ and define $v_{nk}=\frac{1}{n}[z^k]V(z)^n,\qquad V_n(x)=\sum_{k=0}^n v_{nk}x^k.$ From the multinomial expansion of a power series,

Let $f$ be a random function on a (typically infinite) set, meaning that for each input $x$, the value $f(x)$ is chosen according to a fixed distribution on outputs, and the random variables $\{f(x)\}...

Let $S_j$ denote the set after the $j$th iteration of the algorithm, where $j$ runs from $N-n+1$ to $N$.

The solution does not correctly answer the problem.

The city is modeled as an undirected weighted graph. Each intersection is a vertex and each road is an edge whose weight is its length.

The function $((x))$ is $1$-periodic and defined on $0 \le x < 1$ by $((x)) = x - \frac12,$ since $\lfloor x \rfloor = 0$ and $\lceil x \rceil = 1$ for $0 < x < 1$ in (7).

Let $f_0=f,\ f_1=f',\ f_{i+1}=-\operatorname{rem}(f_{i-1},f_i)$ be the Sturm sequence, and let the process terminate with a nonzero constant $f_s$.

A correct solution must repair three fundamental issues in the previous attempt: 1.

Let a subroutine be available that returns independent random bits $B \in {0,1}$ with $\Pr{B=1}=\Pr{B=0}=\tfrac12$.

Let $X$ have continuous distribution function $F(x)=\Pr{X\le x}$ and define $Y=cX$, where $c$ is a constant.

Let $u = (u_{n-1}\ldots u_1u_0)_b$.

Let $U = (u_{n-1}\ldots u_0)_b,\quad V = (v_{n-1}\ldots v_0)_b,\quad W = (w_{n-1}\ldots w_0)_b.$ Define for each $j$ with $0 \le j \le n$ the partial values $U_j = \sum_{i=0}^{j-1} u_i b^i,\quad V_j =...

Let $p$ be prime and let f(x)=x^{2}-cx-a\in\mathbb{F}_{p}[x],\qquad R=\mathbb{F}_{p}[x]/(f).

We are given a terrain described by a polyline that is monotone in x, so it is a chain of straight segments from left to right. A camera sits at a fixed point on this terrain, at a specified x-coordinate, meaning its y-coordinate is determined by the terrain at that x.

We are given two very large integers written as digit strings. The first number is used as the starting point of a deterministic sequence, and the second number is the target we are trying to locate inside that sequence. The sequence evolves in a very specific way.

We are asked to design a simplified “leap-year system” for a fictional planet whose year length is not exactly an integer number of local days. From physics, the input gives enough information to compute how long the planet takes to complete one orbit around its star.

The maze is not given explicitly. Instead, it is generated by repeatedly expanding a symbolic string that behaves like a growing fractal instruction system.

We are given a rectangular maze and two robots placed on different cells with initial directions. The maze is a grid where movement is blocked by internal walls and the outer boundary, except for a single exit located on the southern border of one specific cell.

We are given a multiset of cards, each card described only by its rank. Suits do not matter. The task is to compute a single score based on three independent scoring rules applied over the entire collection, not just five cards.

Let $m = 2^e$ with $e \ge 4$.

In MIX arithmetic each word consists of several bytes, and operations such as multiplication and addition propagate carries from less significant positions toward more significant positions.

Let ${U_n}_{n \ge 0}$ be a binary sequence, so each $U_n \in {0,1}$.

Let $(Y_n)_{n\ge 1}$ be an i.

Let $U_0, U_1, \ldots$ be a $[0,..,1)$ sequence with $U_0 = 0$.

We fix the construction precisely before proving any properties.

Define 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}.

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.