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Yes, the inequality $nu(n) le 2^{l(n) - lambda(n)}$ holds for all positive integers $n$. Consider an addition chain of minimal length $l(n)$ and let $lambda(n)$ be the length of a shortest chain consisting only of doubling steps.

We are given a string of lowercase English letters. A message is any subsequence whose chosen indices form an arithmetic progression. The progression can have any positive difference, including the special case of length one. The task is not to find the message itself.

Yes, the inequality $nu(n) le 2^{l(n) - lambda(n)}$ holds for all positive integers $n$. Consider an addition chain of minimal length $l(n)$ and let $lambda(n)$ be the length of a shortest chain consisting only of doubling steps.

Codeforces 1310A: Recommendations

Yes, the inequality $nu(n) le 2^{l(n) - lambda(n)}$ holds for all positive integers $n$. Consider an addition chain of minimal length $l(n)$ and let $lambda(n)$ be the length of a shortest chain consisting only of doubling steps.

Yes, the inequality $nu(n) le 2^{l(n) - lambda(n)}$ holds for all positive integers $n$. Consider an addition chain of minimal length $l(n)$ and let $lambda(n)$ be the length of a shortest chain consisting only of doubling steps.

We are given a fixed string s that represents a sequence of button presses in a fighting game combo. The player repeatedly attempts to type this entire string from left to right. However, the process is interrupted.

Yes, the inequality $nu(n) le 2^{l(n) - lambda(n)}$ holds for all positive integers $n$. Consider an addition chain of minimal length $l(n)$ and let $lambda(n)$ be the length of a shortest chain consisting only of doubling steps.

Yes, the inequality $nu(n) le 2^{l(n) - lambda(n)}$ holds for all positive integers $n$. Consider an addition chain of minimal length $l(n)$ and let $lambda(n)$ be the length of a shortest chain consisting only of doubling steps.

We are given a rooted construction of many strings where every node represents a string and every edge corresponds to appending one character. The root is the empty string, and each node i is created by taking its parent string pi and appending a lowercase letter ci.

We are given an array of non-negative integers, and a base $k ge 2$. Starting from an array of zeroes of the same length, we can repeatedly pick a step $i$ and add $k^i$ to any single element of the array, or skip the step.

We are given a large line of positions from 1 to $m$, and a collection of $n$ intervals. Each interval represents a “spell” that, when used, adds one candy to every position inside its range. Each spell can be used at most once.

Yes, the inequality $nu(n) le 2^{l(n) - lambda(n)}$ holds for all positive integers $n$. Consider an addition chain of minimal length $l(n)$ and let $lambda(n)$ be the length of a shortest chain consisting only of doubling steps.

Each participant in the contest has two independent rankings: one from the first round and one from the second round. No ties exist in either round, so each round is a permutation of ranks from 1 to n. A participant’s final value is the sum of their two ranks.

We are given a string of length $n$, and Vasya wants to perform a sequence of substring reversals to produce the lexicographically smallest string possible. The operation is controlled by an integer $k$, which specifies the length of each consecutive substring to reverse.

We are given two polynomials, each represented by an array of positive integers, which are their coefficients. The first polynomial has $n$ terms and coefficients $a0, a1, dots, a{n-1}$, the second has $m$ terms with coefficients $b0, b1, dots, b{m-1}$.

Yes, the inequality $nu(n) le 2^{l(n) - lambda(n)}$ holds for all positive integers $n$. Consider an addition chain of minimal length $l(n)$ and let $lambda(n)$ be the length of a shortest chain consisting only of doubling steps.

Yes, the inequality $nu(n) le 2^{l(n) - lambda(n)}$ holds for all positive integers $n$. Consider an addition chain of minimal length $l(n)$ and let $lambda(n)$ be the length of a shortest chain consisting only of doubling steps.

We are given a tree with $n$ nodes, represented by $n-1$ edges connecting pairs of nodes. Each edge must be assigned a distinct integer label between $0$ and $n-2$.

We are asked to analyze permutations of a group of $n$ wise men, where some pairs of them know each other. For each permutation of these wise men, we can create a binary string of length $n-1$ indicating adjacency of acquaintances: a '1' if two consecutive wise men in the…

We need to construct a positive decimal number with exactly n digits. Every digit must be nonzero, and the whole number must fail divisibility by each digit that appears in it. For each test case, the input gives only the required length n.

We are given a string consisting of lowercase English letters, and the task is to construct the longest palindrome by combining a prefix and a suffix of this string.

The problem asks whether a given integer n can be expressed as the sum of exactly k distinct positive odd integers. Each test case provides n and k, and the answer is either "YES" or "NO". Odd integers are numbers not divisible by 2, so valid candidates are 1, 3, 5, 7,....

Each test case contains two positive integers. We may only perform one kind of operation: increase the first number by one.

The solution does not successfully establish what Exercise 4.6.3.13 asks. The positive part is that the displayed constructions are now genuine star chains.

The solution does not successfully establish what Exercise 4.6.3.13 asks. The positive part is that the displayed constructions are now genuine star chains.

The problem presents a single integer input that encodes two pieces of information. The integer can be decomposed as input = 1000 n + mod, where n is the number for which we want the double factorial and mod is the modulus to compute it under.

The problem asks us to compute a specific numeric property related to an integer input, denoted as a. While the problem statement is written in a poetic form, the underlying task is to find the number of integers less than a that are coprime to a.

The solution does not successfully establish what Exercise 4.6.3.13 asks. The positive part is that the displayed constructions are now genuine star chains.

We are given a tree with $n$ vertices. For every nonempty subset of edges $E'$, we build the edge-induced subgraph consisting of those edges and every endpoint that appears in at least one selected edge. For that subgraph we count its independent sets.

We are given a string s of length n and an integer k such that n is divisible by k. A string is considered k-complete if it is both a palindrome and periodic with period k. Being a palindrome means the string reads the same forwards and backwards.

The reviewer is correct. The displayed sequences are not addition chains, because the quantities $2^g,2^h,2^k,2^m$ were used as summands without first appearing as chain elements. The argument must be rebuilt from the definition of a star chain.

The reviewer is correct. The displayed sequences are not addition chains, because the quantities $2^g,2^h,2^k,2^m$ were used as summands without first appearing as chain elements. The argument must be rebuilt from the definition of a star chain.

We are asked to find all substrings of a string t that "match" another string s according to a flexible definition of equality.

We are given a number $D$ and need to reason about all of its divisors. The problem defines a graph whose vertices are all divisors of $D$. An edge exists from a divisor $y$ to a larger divisor $x$ if $x$ is divisible by $y$ and the quotient $x / y$ is prime.

We observe the statistics of a game level several times. At each observation we know two values: how many times the level has been played and how many times it has been cleared. A successful attempt increases both numbers by one at the same moment.

For each test case we have to build a lowercase string of length n. The condition is that every contiguous segment of length a must contain exactly b different letters. Any valid string is acceptable. The numbers describe a sliding window condition. If we look at positions 1...

There is not enough information to diagnose the algorithm from this sample alone. The failing input is: Expected output: Actual output: From this, we can infer only that: - The input format is a line containing 1 7 followed by a binary string 0000000.

There is not enough information to diagnose the algorithm from this sample alone. The failing input is: Expected output: Actual output: From this, we can infer only that: - The input format is a line containing 1 7 followed by a binary string 0000000.

We are given an integer array and we are allowed to repeatedly perform timed operations. In the $x$-th second, we may choose any subset of indices and add the same value $2^{x-1}$ to all chosen positions.

The task describes a fixed triangular strip that grows with a parameter $n$. For each $n$, the shape consists of $4n-2$ unit triangles arranged in a long belt-like region.

The previous solution is based on a misunderstanding of the problem. The goal is not to match the sample output exactly. The sample output shows one valid answer among many. The real requirement is: 1. t must be a subsequence of s. 2. 3. s must have the smallest possible period.

We are given an array of integers, and we are allowed to repeatedly perform an operation where we pick two distinct indices, add the value from the first index to the second, and remove the first element from the array.

We start with two independent counters, x and y, and want to bring both to zero. We are allowed to modify them in two different ways, each with a cost.

We are asked to construct an array of length $n$ where the first half consists of distinct even positive integers, the second half consists of distinct odd positive integers, and the sums of the two halves are equal.

We are given a sequence of integers, both positive and negative, and need to construct a subsequence whose elements strictly alternate in sign. Among all subsequences that achieve the maximum possible length, we are asked to find the one with the largest sum.

We are dealing with an infinite hotel where each room, labeled by an integer, has exactly one guest. A shuffling rule is applied where each guest moves from their current room $k$ to a new room $k + a{k bmod n}$, where $a$ is an array of length $n$.

We are asked to place north and south monopole magnets on an $n times m$ grid in a way that respects three rules. Each cell is either black or white. A north magnet can move towards a south magnet in the same row or column, but south magnets are fixed.

We are asked to determine whether a grid of jigsaw pieces can be assembled given a special piece design. Each piece has exactly three tabs and one blank. The pieces can be rotated in any orientation.

We are given a total number of participants and a fixed multiplier. The participants must be split into four consecutive groups where each next group is exactly $k$ times larger than the previous one.

We are given two identical IP cameras, each capable of taking photos at a fixed period. The period of each camera must be chosen from a predefined set of integers, but the starting moment of each camera is flexible.

The problem asks us to schedule the maximum number of lectures during a programming boot camp that lasts n days. Each day is either a normal day, where we can hold lectures, or an excursion day, where no lectures are allowed.

Let $u(x)$ be a polynomial with integer coefficients that is squarefree over $mathbb{Z}$. This means that $u(x)$ has no repeated roots in $mathbb{C}$, equivalently, $gcd(u(x), u'(x)) = 1$ in $mathbb{Z}[x]$, where $u'(x)$ is the derivative of $u(x)$.

The grid evolves in discrete time. Each cell has one of two colors and updates simultaneously each step based only on its four neighbors. A cell looks at the current state. If none of its neighbors share its color, it stays unchanged.

We are asked to count appearances of numbers in a special class of integer sequences called good sequences. A sequence of length n is considered good if, for every number k 1 that appears, there is at least one occurrence of k-1 somewhere earlier in the sequence.

We are given several independent queries. Each query provides a sequence of integers and a target value $k$. We are allowed to repeatedly choose any contiguous segment of the sequence and replace every element in that segment with the median of that segment.

We are given several independent test cases. In each one, we have a sequence of model sizes indexed from 1 to n. We are allowed to pick a subset of indices, but the chosen indices must be kept in increasing order, which is equivalent to choosing a subsequence of indices.

The task asks us to add pairs of integers. Each test case consists of two integers, and for each pair, we need to compute their sum. The input first tells us how many pairs there are, then each subsequent line contains a pair.

We are given a sequence of moves on an infinite grid. Each move shifts a skier one unit in one of four directions: north, south, east, or west. As the skier follows the path, they traverse unit segments between grid points.

We are asked to reconstruct a binary string given the counts of its consecutive pairs grouped by how many ones they contain.

The problem gives two arrays of equal length, which we can think of as two sets of numbers on separate shelves. We are allowed to swap numbers between the shelves, but only up to a maximum of k swaps.

We are given a linear garland of lamps, represented as a string of 0s and 1s, where 1 indicates a lamp is on and 0 indicates it is off. A garland is called k-periodic if the distance between any two consecutive 1s is exactly k.

We are given an array length n and a required total sum m. The task is to assign non-negative integers to an array of length n so that the sum of all elements equals m.

We are simulating a very simple system that evolves over time in discrete “sleep cycles.” A person needs to accumulate at least a target amount of effective sleep before they are allowed to get out of bed.

We are maintaining a collection of integers where duplicates are allowed, and the collection changes over time. Initially we are given a sorted list of values that already form the starting multiset. After that, we receive a long sequence of operations.

We are given multiple test cases, and each test case is a string made only of the characters 1, 2, and 3. For each string, we need to find the shortest continuous segment (substring) that contains at least one occurrence of each of the three characters.

We are given a list of explorers, each associated with a number $ei$ that represents how many people must be in any group they join. If an explorer has value $e$, then they are only willing to participate in a group whose size is at least $e$.

We are asked to generate a sequence of numbers defined recursively. The sequence starts with a given number $a1$, and each subsequent number is obtained by adding the product of the minimum and maximum digits of the previous number.

We work modulo an odd prime $p$ and factor $$x^8 + 1.$$ The structure of the factorization depends entirely on how $-1$ and $2$ behave in $mathbb{F}p$, because the natural attempt is to rewrite $x^8+1$ as a product of quadratic expressions obtained from square roots of these…

The task is not a typical input-output problem. Instead, you are given a fixed dataset of 200 two-dimensional points, each labeled as either class 0 or class 1.

We are given a quantum operation acting on a single qubit. The operation is either a Z gate, which leaves the The input is not classical data but a quantum operation with known interface: we can call it on a qubit and measure the qubit afterwards.

We are given a quantum black-box operation that acts on a single qubit. The operation is guaranteed to be either doing nothing at all or applying a phase flip. We need to identify which of these two behaviors is implemented and output a binary label.

We are given a quantum black-box operation that acts on a single qubit. This operation is guaranteed to be either the Z gate or the S gate.

The task is to implement a quantum operation that, when applied $P$ times, reproduces the effect of a full quantum Fourier transform (QFT) on a small register of qubits.

The task is to implement an operation on a quantum register that corresponds to raising the quantum Fourier transform (QFT) to a given power $P$. The input is an integer $P$ and a quantum register encoded in little-endian format, meaning the least significant qubit comes first.

In this problem, we are given an array of integers representing a dataset of measurements. Each query requires us to compute the minimum number of operations needed to make a segment of this array “homogeneous” according to a specific rule: all identical numbers in a…

We are given a black-box quantum operation that acts on a single qubit. This operation is guaranteed to be exactly one of four possibilities: the identity operation or one of the three Pauli gates.

The problem asks us to prepare a quantum state over $N$ qubits where only the basis states with a specific parity of ones are included in an equal superposition.

We are asked to implement a quantum oracle that marks a bit string as “balanced” if it contains exactly half zeros and half ones. The input is a list of qubits representing the bits of the string and an extra qubit representing the output.

We are given a quantum gate that acts on a single qubit, and we know it is either a rotation around the Z-axis by an angle θ, denoted Rz(θ), or a rotation around the Y-axis by the same angle, Ry(θ). Our task is to determine which gate we have, returning 0 for Rz and 1 for Ry.

We are asked to identify which of two possible two-qubit CNOT gates we have: one where the first qubit is the control and the second is the target, and one where the second qubit is the control and the first is the target.

This is not a traditional input/output Codeforces problem. We are given access to an unknown two-qubit quantum operation and must determine which one of four possibilities it is. The hidden operation is guaranteed to be one of the following: - Identity on both qubits.

The problem gives us an infinite two-dimensional table filled with integers in a specific pattern. Each cell at position $(x, y)$ contains a number that can be derived from its coordinates using the “GAZ-GIZ” filling rule.

I have carefully traced the construction and identified why the previous implementation produces incorrect matrices.

We have a deck containing n cards, of which m are jokers. The cards are distributed evenly among k players, so every player receives exactly n / k cards. The score depends only on how the jokers are distributed. Suppose one player ends up with the largest number of jokers.

We are given an $n times n$ square matrix that starts entirely filled with zeros. Along the top edge, there are $n$ cannons, one above each column, and along the left edge, there are $n$ cannons, one to the left of each row.

We are asked to find the minimal square that can contain two identical rectangles of size $a times b$. The rectangles can be rotated, moved, and must remain entirely within the square, with sides parallel to the square.

We are given an undirected graph. Each vertex represents a blog, and every blog has a desired topic number t[i]. Johnny writes blogs one by one. When he writes a blog, he looks only at neighbors that have already been written.

The exercise asks for a direct algebraic simplification of two displayed identities involving content and primitive part of polynomials over a unique factorization domain $S$.

The exercise asks for a direct algebraic simplification of two displayed identities involving content and primitive part of polynomials over a unique factorization domain $S$.

We are given a set of distinct integers. Johnny chooses a positive integer k and replaces every value s in the set with s XOR k. The transformation is applied to every element simultaneously.

The game is played on a tree, which is an undirected, connected, acyclic graph. Each node is numbered from $1$ to $n$, and one node $x$ is special. Two players take turns removing leaf nodes, where a leaf is a node with only one neighbor, along with its connecting edge.

We are given an array of integers and must choose exactly x elements from it. The chosen elements can come from any positions in the array. The question is whether there exists a selection of exactly x elements whose sum is odd.

The exercise asks us to investigate approximate polynomial greatest common divisors (gcds) and the behavior of Euclid's algorithm when the polynomial coefficients are floating-point numbers.

We are given an array of integers and a number $x$ that Ehab dislikes. The goal is to find the length of the longest contiguous subarray whose sum is not divisible by $x$. Each test case gives a new array and a new $x$.

We are given a grid where some cells are already occupied. Two players alternate turns, and on each move a player must pick a previously unused cell under a strong restriction: no two chosen cells are allowed to share a row or a column.

We are given an array of up to 500 positive integers. We may choose any non-empty subsequence and compute its value according to a bitwise rule.

We are given a sequence of numbers ai each tagged with a type bi that is either 0 or 1. The task is to determine if it is possible to sort the sequence in non-decreasing order by only swapping elements of different types.

We are given two arrays, a and b. Array b is strictly increasing, and our task is to partition array a into exactly m consecutive subarrays, where each subarray's minimum matches the corresponding element in b.

We are given a string s consisting of lowercase letters and dots. If we process this string from left to right, every letter is pushed onto a stack and every dot removes the current top character. The function f(s) returns the final stack contents as a string.