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question:Sam is passionate about numbers and mathematical sequences. He recently became interested in a unique sequence that he calls the "Repeated Fibonacci." In the "Repeated Fibonacci" sequence, every element is the sum of the previous two elements, but each element can repeat a certain number of times before the sequence moves on to the next element. Your task is to help Sam generate the first n elements of this "Repeated Fibonacci" sequence, where the initial two numbers a and b and the repeat count r are given as input. # Problem: Given initial values a and b, a repeat count r, and a number n, generate the first n elements of the "Repeated Fibonacci" sequence. # Input: - The first line contains three integers a, b, and r (1 leq a, b leq 100), (1 leq r leq 50). - The second line contains an integer n (1 leq n leq 1000). # Output: - Print the first n elements of the "Repeated Fibonacci" sequence, each separated by a space. # Example Input 1: ``` 2 3 2 10 ``` # Example Output 1: ``` 2 2 3 3 5 5 8 8 13 13 ``` # Example Input 2: ``` 1 1 3 15 ``` # Example Output 2: ``` 1 1 1 1 1 1 2 2 2 3 3 3 5 5 5 ``` # Explanation: 1. For the first example: - The sequence starts with 2, which repeats twice; - Then 3, which repeats twice; - Then 2+3 = 5, which repeats twice; - Then 3+5 = 8, which repeats twice; - Then 5+8 = 13, which repeats twice, generating the first 10 elements as required. 2. For the second example: - The sequence starts with 1, which repeats three times; - Then 1 again, which repeats three times; - Then 1+1 = 2, which repeats three times; - Then 1+2 = 3, which repeats three times, and so on. Make sure to follow these rules to construct the sequence, taking into account the repeating behavior as specified.

answer:def repeated_fibonacci(a, b, r, n): Generates the first n elements of the Repeated Fibonacci sequence. Args: a: int - The first number of the sequence. b: int - The second number of the sequence. r: int - The repeat count for each element in the sequence. n: int - The number of elements to generate. Returns: List[int] - The first n elements of the Repeated Fibonacci sequence. if n == 0: return [] sequence = [] sequence.extend([a] * r) if n <= len(sequence): return sequence[:n] sequence.extend([b] * r) if n <= len(sequence): return sequence[:n] i, j = a, b while len(sequence) < n: i, j = j, i + j sequence.extend([j] * r) return sequence[:n]

question:------ Problem Statement ------ Alicia is fascinated by prime numbers and has devised a game that involves finding prime numbers in a certain range. Given a range [L, R], Alicia wants to find out the sum of all prime numbers that are present within the range. However, calculating primes in a large range can be challenging, so she needs your help. Your task is to write a function that takes L and R as input and returns the sum of all prime numbers between L and R, inclusive. ------ Input ------ The first line of the input contains an integer T, denoting the number of test cases. The next T lines contain two space-separated integers, L and R, denoting the range. ------ Output ------ For each test case, output the sum of all prime numbers between L and R, inclusive. If there are no prime numbers in the range, output 0. ------ Constraints ------ 1 ≤ T ≤ 100 1 ≤ L ≤ R ≤ 10^6 ------ Sample Input 1 ------ 3 10 20 1 10 14 14 ------ Sample Output 1 ------ 60 17 0 Note: The prime numbers between 10 and 20 are 11, 13, 17, 19, and their sum is 60. The prime numbers between 1 and 10 are 2, 3, 5, 7, and their sum is 17. There are no prime numbers in the range [14, 14], so the output is 0.

answer:def sum_of_primes(L, R): Returns the sum of all prime numbers between L and R, inclusive. def sieve(n): Return a list of primes up to n using Sieve of Eratosthenes is_prime = [True] * (n + 1) p = 2 while p * p <= n: if is_prime[p]: for i in range(p * p, n + 1, p): is_prime[i] = False p += 1 return [p for p in range(2, n + 1) if is_prime[p]] # Primes up to 10^6 primes = sieve(10**6) prime_set = set(primes) if L < 2: L = 2 # Compute the sum in the range return sum(p for p in primes if L <= p <= R) def process_input(T, ranges): Processes multiple ranges to find the sum of primes in each range. results = [] for L, R in ranges: results.append(sum_of_primes(L, R)) return results

question:A company is organizing a marathon in which participants run along a straight track. The track is marked with checkpoints at consistent intervals. Each participant's progress along the track is recorded as they reach certain checkpoints. Your job is to determine the number of participants who crossed at least one checkpoint. Each participant's progress is provided in distinct intervals. The intervals include the participant's starting and ending points along the track. A participant is considered to have crossed a checkpoint if their interval contains at least one of the checkpoints. -----Input:----- - First line will contain T, the number of test cases. Then the test cases follow. - The first line of each test case contains two space separated integers C and P: - C is the number of checkpoints. - P is the number of participants. - The second line of each test case contains C space separated integers representing the positions of the checkpoints. - The following lines contain two space separated integers a and b for each participant, representing the starting and ending points of their interval. -----Output:----- For each test case, output in a single line the number of participants who crossed at least one checkpoint. -----Constraints----- - 1 leq T leq 10^3 - 1 leq C leq 10^3 - 1 leq P leq 10^3 - 1 leq sum of all participants' start and end points over all test cases leq 10^6 - 1 leq checkpoint position, start point, end point leq 10^9 -----Sample Input:----- 2 3 4 1 5 8 0 3 4 6 7 9 2 10 2 2 5 10 3 6 11 15 -----Sample Output:----- 4 1 -----Explanation:----- Test Case 1: Participant 1's interval [0, 3] contains checkpoint 1. Participant 2's interval [4, 6] contains checkpoint 5. Participant 3's interval [7, 9] contains checkpoint 8. Participant 4's interval [2, 10] contains checkpoints 5 and 8. All 4 participants have crossed at least one checkpoint. Test Case 2: Participant 1's interval [3, 6] contains no checkpoints. Participant 2's interval [11, 15] contains checkpoint 10. Only participant 2 has crossed at least one checkpoint.

answer:def participants_crossed_checkpoints(T, test_cases): results = [] for idx in range(T): C, P = test_cases[idx][0] checkpoints = test_cases[idx][1] participant_intervals = test_cases[idx][2] crossed_count = 0 for start, end in participant_intervals: for checkpoint in checkpoints: if start <= checkpoint <= end: crossed_count += 1 break results.append(crossed_count) return results

question:Given an array of integers, write a function to find the maximum possible sum of a non-empty subarray that contains at most `k` distinct elements. A subarray is a contiguous part of the original array, and you are required to determine the subarray that meets the condition with the highest sum. Function Signature: ```python def max_sum_with_k_distinct(arr: List[int], k: int) -> int: ``` Example 1: ```plaintext Input: arr = [1, 2, 1, 2, 3], k = 2 Output: 6 Explanation: The subarray [1, 2, 1, 2] has the maximum sum of 6 with at most 2 distinct elements. ``` Example 2: ```plaintext Input: arr = [4, 1, 1, 3, 6], k = 1 Output: 6 Explanation: The subarray [6] has the maximum sum of 6 with at most 1 distinct element. ``` Constraints: - The length of the given array is at most 10000. - The value of each element in the array is between -10000 and 10000. - `k` is a positive integer less than or equal to the length of the array.

answer:from collections import defaultdict from typing import List def max_sum_with_k_distinct(arr: List[int], k: int) -> int: Returns the maximum possible sum of a subarray with at most k distinct elements. left = 0 current_sum = 0 max_sum = float('-inf') count = defaultdict(int) distinct_count = 0 for right in range(len(arr)): if count[arr[right]] == 0: distinct_count += 1 count[arr[right]] += 1 current_sum += arr[right] while distinct_count > k: count[arr[left]] -= 1 if count[arr[left]] == 0: distinct_count -= 1 current_sum -= arr[left] left += 1 if distinct_count <= k: max_sum = max(max_sum, current_sum) return max_sum