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question:Alexa is organizing a charity auction and wants to ensure the items get sold for their maximum possible price. She has `n` items numbered from 1 to n, and each item `i` has a predicted bidding range from `l_i` to `r_i` (inclusive). Alexa wants to arrange the auction in such a way that each item is sold on a different day, and the price on any day does not exceed `m` dollars. Your task is to help Alexa schedule the items for auction in a way that maximizes the total auction earnings while respecting the day limit. The auction takes place over a consecutive number of days starting from 1. For simplicity, any bid below the minimum predicted price on a given day is not considered. The first line of input contains three integers `n`, `m`, and `d` (1 ≤ n ≤ 100, 1 ≤ m ≤ 1000, 1 ≤ d ≤ 10000) — the number of items, the maximum price on any day, and the number of auction days respectively. The next `n` lines contain two integers `l_i` and `r_i` (1 ≤ l_i ≤ r_i ≤ m) — the bidding range for each item. Print the start and end day for each auction item, such that the sale price maximizes while ensuring no two items are sold on the same day. If there are multiple solutions, print any of them. It is guaranteed that a solution exists. Example input: 4 100 10 30 50 40 70 20 60 50 80 Example output: 1 2 3 4 5 6 7 8

answer:def schedule_auction(n, m, d, items): Given n items with their predicted bidding range and the maximum price m for each day, schedule items for auction in a way to maximize the total auction earnings. Parameters: n (int): Number of items. m (int): Maximum price on any day. d (int): Number of auction days. items (list of tuples): List of bidding range (li, ri) for each item. Returns: list of tuples: List of tuples (start_day, end_day) for each item. # Sort items by their minimum bid price (li), and if equal, by maximum bid price (ri) sorted_items = sorted(items, key=lambda x: (x[0], x[1])) # Initialize the result list auction_schedule = [(0, 0)] * n # Keep track of the current day current_day = 1 for i in range(n): li, ri = sorted_items[i] # Assuming the auction takes place within d days start_day = current_day end_day = current_day + 1 # Assign the days to the current item auction_schedule[i] = (start_day, end_day) # Move to the next day for the next item current_day += 2 # If current_day exceeds the number of available days `d`, wrap around to start over if current_day > d: current_day = 1 return auction_schedule

question:A city is planning to organize multiple events in a large hall. Each event has a start and end time, and no two events should overlap in the hall. You are given a list of such events and you need to determine the maximum number of non-overlapping events that can be held in the hall. The first line of input contains an integer t (1 ≤ t ≤ 100), representing the number of test cases. Each test case starts with an integer n (1 ≤ n ≤ 10^5) representing the number of events. The next n lines contain two integers s_i and e_i (0 ≤ s_i < e_i ≤ 10^9), representing the start and end time of each event. For each test case, output a single integer, the maximum number of non-overlapping events that can be held in the hall. **Input:** ``` t n s_1 e_1 s_2 e_2 ... s_n e_n ... ``` **Output:** ``` max_events_for_test_case_1 max_events_for_test_case_2 ... ``` **Example:** ``` Input: 2 3 1 2 2 3 3 4 3 1 3 2 5 3 6 Output: 3 2 ```

answer:def max_non_overlapping_events(t, test_cases): results = [] for i in range(t): events = test_cases[i] events.sort(key=lambda x: x[1]) last_end_time = 0 count = 0 for start, end in events: if start >= last_end_time: count += 1 last_end_time = end results.append(count) return results # To process the input & output as described def process_input_output(): import sys input = sys.stdin.read data = input().split() index = 0 t = int(data[index]) index += 1 test_cases = [] for _ in range(t): n = int(data[index]) index += 1 events = [] for _ in range(n): s = int(data[index]) e = int(data[index+1]) index += 2 events.append((s, e)) test_cases.append(events) results = max_non_overlapping_events(t, test_cases) for result in results: print(result)

question:Leo is training a robot to navigate a grid while collecting valuable items. The robot starts at the top-left corner of an `n x m` grid and wants to reach the bottom-right corner. Each cell in the grid contains a certain value representing the worth of the item in that cell. The robot can only move either right or down at each step. To maximize the value of the items collected, Leo wants to find the highest possible sum of the values in the cells that the robot collects. Given the dimensions of the grid and the values of the items in each cell, your task is to calculate the maximum sum of the values the robot can collect along any path from the top-left corner to the bottom-right corner. # Input - The first line contains two integers `n` and `m` (`1 ≤ n, m ≤ 1000`) — the number of rows and columns of the grid. - The next `n` lines each contain `m` integers, where the `j-th` integer in the `i-th` line is the value of the item in the cell `(i, j)`. The values range from `-1000` to `1000`. # Output - Print a single integer — the maximum sum of the values the robot can collect along any path from the top-left corner to the bottom-right corner of the grid. # Example Input ``` 3 3 1 2 3 4 5 6 7 8 9 ``` Output ``` 29 ``` Explanation - One of the optimal paths is `1 -> 4 -> 7 -> 8 -> 9`, which gives a sum of `1 + 4 + 7 + 8 + 9 = 29`. In another example, the input could be: Input ``` 2 2 1 -1 -1 1 ``` Output ``` 1 ``` Explanation - The optimal path is `1 -> -1 -> 1`, resulting in a sum of `1 - 1 + 1 = 1`.

answer:def max_value_path(n, m, grid): Returns the maximum sum of the values the robot can collect along any path from the top-left corner to the bottom-right corner of the grid. # Initialize dp table with the same dimensions as the grid dp = [[0] * m for _ in range(n)] # Base case - start from the top-left corner dp[0][0] = grid[0][0] # Fill the first row (can only move right) for j in range(1, m): dp[0][j] = dp[0][j-1] + grid[0][j] # Fill the first column (can only move down) for i in range(1, n): dp[i][0] = dp[i-1][0] + grid[i][0] # Fill in the rest of the dp table for i in range(1, n): for j in range(1, m): dp[i][j] = max(dp[i-1][j], dp[i][j-1]) + grid[i][j] # The result is in the bottom-right cell return dp[n-1][m-1]

question:A delivery company uses an intelligent sorting system to decide the optimal delivery routes for their packages. Each package is represented by a distinct integer p_i which indicates the priority of the package (higher number means higher priority). For a given route, the sorting system must always deliver packages in non-decreasing order of their priorities. However, due to a recent system update, some packages may get mislabeled, leading to possibly incorrect information about their priority values. Therefore, the company decides to ensure the integrity of their deliveries by choosing the longest subsequence of packages whose priorities are strictly increasing. You are tasked with writing a function to determine the length of such a subsequence. # Input - An integer n (2 le n le 100,000) representing the total number of packages. - An array of n distinct integers p_1, p_2, ..., p_n (1 le p_i le 10^9), representing the priorities of the packages. # Output Print a single integer, the length of the longest subsequence of the packages that can be sorted in strictly increasing order. # Example Input ``` 8 10 22 9 33 21 50 41 60 ``` Output ``` 5 ``` # Explanation In the example provided, the longest subsequence in strictly increasing order is {10, 22, 33, 50, 60}. Its length is 5. Other subsequences of different lengths can also be found, but the longest one should be considered.

answer:import bisect def longest_increasing_subsequence_length(packages): Returns the length of the longest strictly increasing subsequence. if not packages: return 0 sub = [packages[0]] for num in packages[1:]: if num > sub[-1]: sub.append(num) else: # Find the index of the smallest number in sub which is greater than or equal to num idx = bisect.bisect_left(sub, num) sub[idx] = num return len(sub)