brain

The task describes a very small “birthday arithmetic” model, but the twist is that the computation is not performed numerically in the usual way. Instead, the intermediate value is treated as a string, which completely changes how subtraction behaves.

We are asked to compute the minimum number of operations required to merge a pre-cut potato casserole back into a single piece. The casserole has total length n and was cut into k pieces of lengths a1 through ak. Two operations are allowed.

We are given a river that behaves like a linear board of n cells. Each cell is either safe ground in the form of a log, dangerous water, or an instant-failure crocodile tile. The goal is to move from the left bank before the first cell to the right bank after the last cell.

We are given three small integers that represent the initial sizes of three factors contributing to a product. The final quantity we care about is the product of these three numbers, and we are allowed to improve it by performing at most five operations.

The sum is

We are asked to simulate the movement of a robot inside a bounded rectangle, starting at the bottom-left corner. The robot follows a script of movements, each of which is one of the four cardinal directions.

We are given a small matrix of integers, and the operations we can perform are unusual. For any row, we can replace every element with the XOR of the corresponding column. For any column, we can replace every element with the XOR of the corresponding row.

We are given an array of integers and a parameter $k$. We can repeatedly remove any contiguous subarray of length exactly $k$ from the array, shrinking it. Once the array's length is no longer greater than $k$, we stop.

We are simulating a deterministic robot moving on a grid that is bounded by a rectangle. The robot starts at the origin and repeatedly executes a fixed instruction string consisting of unit moves in four directions.

The problem involves an apartment with multiple rooms, each starting with its light turned off. Each room gets a chip installed at a distinct time, and each chip toggles the light in its room on and off periodically, with a period k minutes.

We are given a multiple-choice test with $4n$ questions, where each of the four options 'A', 'B', 'C', 'D' is correct exactly $n$ times. Tim answers the questions but may leave some as '?', representing unknown answers.

We are given an array of positive integers, and we want all elements to share the same parity - either all even or all odd. The only allowed operation is to pick two elements of differing parity and add the larger number to the smaller.

Let

Part (1) concerns the cyclic expression

Let the square have side length $2$ and center $O$.

Let

We have $2n$ knights sitting at $n$ desks arranged in a circle. Each desk has exactly two knights, the first at position $2i-1$ and the second at $2i$. Each knight has an integer intelligence value.

We are given a string of uppercase letters representing a text in an ancient language. The language uses only the first $c$ letters of the Latin alphabet, and each word in the language has a “case” determined solely by its last letter. Words can have lengths up to $k$.

We have 2n knights arranged around a circle. Desk i contains knights (2i-1, 2i), so every desk contributes a sum of two intelligence values. The only operation allowed is swapping opposite positions in the circle.

We are given several independent arrays, and for each one we are allowed to modify elements using a very specific operation: pick an index and replace the value at that position with its square. We may repeat this operation on the same index multiple times.

Each flower has two properties that are actually the same number: its petal count and its price. A flower with k petals costs exactly k coins. We may buy any subset of the flowers available in the store.

We are given several independent scenarios. In each scenario, there are different flower types, each type having a petal value and a limited supply. Every flower costs exactly its petal value in coins, and we have a fixed budget. We want to build a bouquet under two restrictions.

We are working on an $n times n$ grid where each cell belongs to exactly one diagonal if we group cells by the value $i + j$. Every such value defines a diagonal that runs from the top edge toward the right or bottom edge depending on where it sits in the square.

The problem describes a circular town with n houses connected by roads that form a single cycle. Each house i connects to i+1, and the last house n connects back to house 1. Among these residents, there are m friendships, each linking two distinct houses.

We are asked to maximize a score over a series of operations on two arrays. Each array has length $n$. Array $a$ represents the current “value” of each element, and array $b$ represents how much that element decays after it is used.

We are given a binary string, and we look at every possible contiguous segment of it. Each segment itself contains many subsegments, and we are interested only in those subsegments whose number of zeros equals the number of ones.

We need to count ordered triples of positive integers $(a,b,c)$ that satisfy two independent restrictions. The first restriction limits the pairwise products: $$ab + ac + bc le n$$ The second restriction limits the sum: $$a + b + c le x$$ The word "ordered" matters.

The game begins with the first player choosing an initial value $P_0 \in {2,3,4,5,6,7,8,9}$.

We are given two strings, a and b, of equal length n. For each query, defined by a range [l, r], we are allowed to modify individual characters of a within that range. The goal is to transform the substring a[l..

We are given a square grid of size $n times n$ where each cell contains either a zero or a one. The task is to reduce this grid by a factor $k$, which is guaranteed to divide $n$ evenly.

The farm has only chickens and cows, and Farmer John counts a total of n legs. Chickens contribute 2 legs each and cows contribute 4 legs each. The question asks for the smallest possible number of animals consistent with the total leg count.

Let $ABC$ be the given triangle.

We are given a one-dimensional line with positions numbered from 1 to $x$. We must place exactly $n$ chips on these positions, allowing multiple chips at the same position.

In this problem, Monocarp is traversing a linear sequence of monsters, each with a level. He starts at level 1 and can fight monsters whose levels are at least as high as his current level. Whenever he fights k monsters, he levels up by one.

We are given a rooted tree. Every vertex contains a non-negative number. An operation can be applied to any non-leaf vertex. When we choose a vertex $v$, we increase the value at $v$ by one and simultaneously decrease every other vertex in the subtree of $v$ by one.

We have a grid with only two rows. Some cells are free (.), some are blocked (x). Free cells form a graph where adjacent cells sharing a side are connected. A connected region is simply a connected component of this graph.

We are asked to strengthen a password by inserting exactly one lowercase letter anywhere in an existing string. The password's strength is quantified by the time it takes to type it. Typing rules are simple: the first character always takes two seconds.

We are given a bracket sequence of even length, but every character at an odd position has been erased. Only the brackets at positions 2, 4, 6, ... remain.

We have a directed acyclic graph on islands 1...n. The graph always contains the chain 1 - 2 - 3 - ... - n and additionally contains m extra directed edges (u, v) with u < v. Bessie can only move along the chain edges. Elsie can move along both chain and extra edges.

We are given a sequence of balls numbered from 1 to $n$, each labeled with a positive integer. For each prefix of length $i$ ($1 le i le n$), we want to determine how many positions $j$ can survive as the last remaining ball if we repeatedly merge pairs of neighboring balls…

Let the regular $2n$-gon have vertices $A_1,A_2,\dots,A_{2n}$ in cyclic order, and let $P$ be a point inside the polygon.

We are given a row of $n$ balls, each with an integer written on it. The operation we can perform repeatedly involves selecting two adjacent values in the set $S$ and merging them according to their relative sizes.

We are given an array of integers a and a parallel binary array b. The array b indicates which positions in a can be incremented. Each increment operation increases a chosen a[i] by 1, and we can perform at most k operations in total.

We are given a permutation p of length n. We must construct another permutation q of the same length. For every subarray [i, j], we can compare the sum of that segment in p with the sum of the corresponding segment in q.

We are given a target center point $(xc, yc)$ and an integer $k$. The task is not to optimize anything or search for a special configuration. We simply need to construct exactly $k$ distinct integer-coordinate points whose arithmetic mean is exactly $(xc, yc)$.

We are asked to find a missing number on a special ruler, which behaves almost like a normal ruler but skips a single value $x$ between 2 and 999. If you try to measure an object smaller than $x$, the ruler gives the correct length.

We are interacting with a hidden parameter $x$ between 2 and 999 that defines a faulty measuring device. Whenever we measure a length $y$, the device behaves normally for small values, but once the true length reaches $x$ or more, it consistently over-reports by exactly 1 unit.

We are given a consecutive sequence of integers from $l$ to $r$ inclusive, written on a board. Ivy can repeatedly perform an operation on any two numbers $x$ and $y$: she replaces them with $3x$ and $lfloor y/3 rfloor$.

We are given a binary array containing only 0 and 1. For every subsequence of length k, where k is odd, we compute its median and add all these medians together. The array is binary, which changes the nature of the median completely.

The task asks for the sum of the digits of a two-digit number for multiple test cases. Each input number $n$ is guaranteed to be between 10 and 99 inclusive, so the first digit is always nonzero.

The requirement that no term is equal to the sum of several others is ensured by a stronger property: for every $n$, if

We are given a day of length m minutes and a set of non-overlapping tasks, each occupying a continuous interval of time.

We are asked to compute the number of ways Suneet can win a two-round card game against Slavic. Each player has exactly two cards, each numbered between 1 and 10.

The task asks for the sum of the digits of a two-digit number for multiple test cases. Each input number $n$ is guaranteed to be between 10 and 99 inclusive, so the first digit is always nonzero.

Each rectangle is a grid of $ai$ columns and $bi$ rows. We may color individual cells one by one. Whenever a row becomes completely colored, we gain one point. Whenever a column becomes completely colored, we also gain one point. The rectangles are independent.

We are placing weighted objects onto a grid, but the score is not just the sum of values on the grid. Instead, every placement contributes multiple times: each cell participates in several overlapping fixed-size $k times k$ sub-squares, and the total score is the sum over all…

We are given a list of integers written on a board, and one of them might be a corrupted representation of a number of the form $10^x$ where $x ge 2$. The corruption happens because the exponentiation symbol '^' was lost.

The problem asks us to check whether a string matches a numeric template. The template is an array of integers where the same number should correspond to the same letter in a string, and the same letter should correspond to the same number.

Normalize the speeds so that Warnicke moves with speed $1$ and the criminal with speed $\frac12$.

We are given an array of integers arranged cyclically, which means the first and last elements are considered adjacent.

We are asked to process a sequence of integers and find a subsequence that maximizes length while containing no repeated elements.

Working

We are asked to count deterministic max-heaps that can be formed on a perfect binary tree after performing a sequence of increment operations. The tree has height $n$, so it contains $2^n - 1$ nodes. Initially, all node values are zero.

Codeforces 2001E1: Deterministic Heap (Easy Version)

We are asked to reconstruct a hidden tree with n nodes by interacting with a judge that answers specific distance-based queries. Each query "? a b" returns the node x that minimizes the absolute difference in distances from a and b.

We are asked to construct a permutation of integers from 1 to $n$ such that, no matter which of two typewriters Misuki uses, the number of required carriage returns to complete the permutation is identical.

We are asked to count sequences of integers with a very particular merging property. For a given sequence of length n, each element between 1 and m, we are allowed to repeatedly take three consecutive elements ai, a{i+1}, a{i+2} and merge them into a single number defined as…

We are constructing a sequence of rounds where exactly one of two players gains a point in each round. After each round we maintain two counters, one for each player, and the process is only valid if at every prefix the greatest common divisor of the two counters never exceeds 1.

We are asked to plan a match between two performers, Lelle and Flamm, where each round produces a winner. The goal is to maximize the total score, which is computed as l WL + f WF, where WL and WF are the number of wins for Lelle and Flamm, respectively.

The problem asks only for existence or nonexistence in three classes. For part (1), it suffices to exhibit a tetrahedron that tiles space by congruent copies. For part (2), we must decide whether a congruent tiling by equifacial tetrahedra exists.

This is a Type C problem: a precise numerical count of admissible fillings is requested. The solution produces a combinatorial formula $an = 2^{n-1}$ and rigorously justifies it via bijection arguments and induction.

For the first values,

We are given a rooted tree where vertex 1 is the root. Alongside the tree, we maintain a permutation stored in an array indexed by positions, and we repeatedly swap two positions in this array.

We have two players, Alice and Bob, each holding a permutation of the numbers from 1 to n. They will alternately remove elements from either end of their respective arrays. After n-1 turns, only one element remains in each array.

We are given a perfect binary tree whose vertices are numbered in heap order. Vertex 1 is the root, and for every vertex i 1, its parent is i // 2. A permutation p of all vertices is maintained. After every query, two positions in the permutation are swapped.

We are given a rectangular grid with n rows and m columns. Each cell must be assigned a color, but colors are restricted by a distance rule: if two cells share the same color, then they must be sufficiently far apart in the Chebyshev sense, meaning the maximum of their row…

This is a counting problem, so it falls under Type C. The goal is to determine the exact number of admissible fillings of the strip. The solution must provide a precise count and justify the enumeration rigorously.

This is a counting problem, so it is closest to Type C in the supplied classification. The requirement is to determine the exact number of admissible fillings and justify the count rigorously.

We are asked to find the maximum number of inversions in permutations of length $n$ that satisfy a specific "interesting" property defined by a set of intervals.

The problem gives us a permutation of integers from 1 to $n$ and asks us to consider certain "interesting" permutations based on a set of intervals.

The problem involves a turtle playing with several integer sequences and a non-negative integer $x$. For each sequence, the turtle can perform an operation exactly once: he sets $x$ to the $text{mex}$ of its current value together with the numbers in the chosen sequence.

The point $M$ is defined from the circle through $P,Q,K$.

We are given several sequences of non-negative integers, and an initial integer $x$. Turtle can repeatedly choose any sequence and replace $x$ with the mex of $x$ together with all numbers in that sequence.

Let the stone masses be $x_1,\dots,x_n$ with $0<x_i\le 2$ and $\sum_{i=1}^n x_i=50$.

We are given a string consisting of lowercase letters, and our task is to reorder it to maximize the number of "good pairs.

We are asked to determine whether a given string can be split into two or more contiguous substrings such that no substring starts with a character that appears at the end of a later substring.

We are given a sequence of numbers and two players who repeatedly shrink it until only one value remains at the front position. Each move removes one element, but before removing it, the left element of the chosen adjacent pair is updated.

We are given a sequence of cities arranged on a line, each equipped with exactly two portals of different colors from a fixed set of four: blue, green, red, and yellow.

We have a sequence $a$ and an $n times m$ matrix $b$. The game follows the sequence from left to right. On move $k$, the current player must choose a cell whose value equals $ak$.

This is a Type B (prove) problem. The solution correctly interprets the statement: it aims to prove that any triangle admitting a circle tangent to two sides and two medians must be isosceles.

We are given two arrays of equal length. At every index we can either keep the pair as it is, or swap the two values at that position. The operation we are allowed to perform is more structured: we must choose one contiguous segment and swap all pairs inside it simultaneously.

We are asked to simulate a pursuit on a one-dimensional line with cells numbered from 1 to $n$. David, a student, starts in a given cell and can move left, right, or stay in place.

This is a Type B (prove) problem. The solution correctly interprets the statement: it aims to prove that any triangle admitting a circle tangent to two sides and two medians must be isosceles.

We are asked to construct a string consisting solely of the English vowels a, e, i, o, u of a given length n such that the number of palindrome subsequences in the string is minimized.

This is a Type B (prove) problem. The solution correctly interprets the statement: it aims to prove that any triangle admitting a circle tangent to two sides and two medians must be isosceles.

For $n=1$ the inequality becomes

We are building a tree incrementally, one node at a time. After each insertion, we are asked to evaluate how “close” the current tree is to a perfect full binary tree shape.

This is a Type B (prove) problem. The solution correctly interprets the statement: it aims to prove that any triangle admitting a circle tangent to two sides and two medians must be isosceles.

For every subarray of the given array, we throw all of its values into a set, removing duplicates. Starting from that set, we may repeatedly pick two existing numbers whose average is an integer and insert that average.

I can't reliably provide a complete editorial with a correct accepted Python solution for Codeforces 2006D from the problem statement alone.