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question:You are given a grid of size n x n filled with integers. Starting from the top-left corner of the grid, you can only move right or down at each step until you reach the bottom-right corner of the grid. Along the path, you collect the integers in the cells you visit. The goal is to calculate the maximum sum of numbers you can collect on such a path. Input: The first line contains an integer n (1 ≤ n ≤ 50) — the size of the grid. Each of the next n lines contains n integers, representing the grid. Output: Print a single integer — the maximum sum of numbers you can collect. Examples: Input: 3 1 3 1 1 5 1 4 2 1 Output: 12 Input: 2 1 2 1 2 Output: 5 Note: In the first example, the path 1 -> 3 -> 5 -> 2 -> 1 gives a maximum sum of 12. In the second example, the path 1 -> 2 -> 2 gives a maximum sum of 5.
answer:def max_path_sum(n, grid): Calculate the maximum sum of numbers collected from the top-left corner to the bottom-right corner only moving right or down. Parameters: n (int): The size of the grid. grid (List[List[int]]): The grid of integers. Returns: int: The maximum sum of numbers collected. # Create a DP table to store the maximum sum at each cell dp = [[0] * n for _ in range(n)] # Initialize the top-left corner with the first cell value dp[0][0] = grid[0][0] # Fill the first row (can only come from the left) for j in range(1, n): dp[0][j] = dp[0][j-1] + grid[0][j] # Fill the first column (can only come from above) for i in range(1, n): dp[i][0] = dp[i-1][0] + grid[i][0] # Fill the rest of the dp table for i in range(1, n): for j in range(1, n): dp[i][j] = max(dp[i-1][j], dp[i][j-1]) + grid[i][j] return dp[n-1][n-1]
question:Julia discovered an enchanted forest where magical coordinates are hidden in the form of integer points on the Cartesian plane. Each set of these points forms a triangle, often with unique properties. She is particularly interested in right-angled triangles, which have one angle exactly 90 degrees. Julia wants to catalog sets of four such points where every subset of three points forms a unique right-angled triangle. In other words, given four points, every possible combination of three points should form a right-angled triangle. Your task is to help Julia verify if a given set of four points has this property. Input The input consists of four lines, each containing a pair of integer coordinates xi and yi (-1000 ≤ xi, yi ≤ 1000). It's guaranteed that all four points are distinct. Output Print "YES" if all subsets of three points form a right-angled triangle. Otherwise, print "NO". Example Input 0 0 3 0 0 4 3 4 Output YES Input 0 0 1 1 2 2 3 3 Output NO Explanation In the first example, every subset of three points forms a right-angled triangle: (0,0), (3,0), (0,4) form a right-angled triangle; (0,0), (3,0), (3,4) form a right-angled triangle, and so on. Therefore, the output is "YES". In the second example, the points (0,0), (1,1), (2,2), and (3,3) are collinear, and thus cannot form right-angled triangles. Therefore, the output is "NO".
answer:def is_right_angle(x1, y1, x2, y2, x3, y3): Checks if the triangle formed by (x1, y1), (x2, y2), (x3, y3) is a right-angled triangle. # Calculate squares of the lengths of the sides a2 = (x2 - x1) ** 2 + (y2 - y1) ** 2 b2 = (x3 - x1) ** 2 + (y3 - y1) ** 2 c2 = (x3 - x2) ** 2 + (y3 - y2) ** 2 # Check if any permutation of these squares satisfies Pythagorean Theorem return a2 + b2 == c2 or a2 + c2 == b2 or b2 + c2 == a2 def are_all_subsets_right_angled(points): Verifies if all subsets of three points out of the four provided points form a right-angled triangle. :param points: List of tuples containing four points (x, y) :return: "YES" if all subsets of three points form a right-angled triangle, otherwise "NO" # Extract individual points (x1, y1), (x2, y2), (x3, y3), (x4, y4) = points # Check all subsets of 3 points to see if they form a right-angled triangle if ( is_right_angle(x1, y1, x2, y2, x3, y3) and is_right_angle(x1, y1, x2, y2, x4, y4) and is_right_angle(x1, y1, x3, y3, x4, y4) and is_right_angle(x2, y2, x3, y3, x4, y4) ): return "YES" return "NO"
question:A company wants to distribute its annual bonuses to its employees based on their individual performances. The performance of each employee is measured on a scale from 0 to 100. The company has decided to give out the bonuses such that: 1. Each employee gets at least one unit of bonus. 2. Employees with a higher performance rating receive more bonuses than their immediate neighbors (i.e., the employees ranked directly before and after them). Given the performance ratings of all employees, determine the minimum total number of bonuses the company needs to give out. Input The first line of input contains an integer n (1 ≤ n ≤ 105), the number of employees. The second line contains n integers, where the i-th integer pi (0 ≤ pi ≤ 100) is the performance rating of the i-th employee. Output Output the minimum total number of bonuses the company needs to give out. Examples Input 3 1 2 2 Output 4 Input 5 10 20 30 20 10 Output 9 Input 6 1 2 3 4 5 6 Output 21 Note In the first example, the minimum number of bonuses can be distributed as follows: [1, 2, 1]. Thus, the total number of bonuses will be 4. In the second example, bonuses can be distributed as [1, 2, 3, 2, 1]. The total number of bonuses will be 9. In the third example, bonuses can be distributed as [1, 2, 3, 4, 5, 6], since the ratings are strictly increasing. The total number of bonuses will be 21.
answer:def min_bonuses(n, ratings): # Initialize an array to store bonuses with at least one bonus for each employee bonuses = [1] * n # Traverse from left to right for i in range(1, n): if ratings[i] > ratings[i - 1]: bonuses[i] = bonuses[i - 1] + 1 # Traverse from right to left for i in range(n - 2, -1, -1): if ratings[i] > ratings[i + 1]: bonuses[i] = max(bonuses[i], bonuses[i + 1] + 1) return sum(bonuses)
question:There is a game which involves arranging blocks to form a tower. You are given an array of integers where each integer represents the height of a block. The only rule in this game is that you cannot place a taller block on top of a shorter block. Given the heights of the blocks, determine the maximum height of a tower that can be formed. You are required to output this height. Note that you can only use each block once. Input The first line contains integer n (1 ≤ n ≤ 1000). The second line contains n integers representing the heights of the blocks (1 ≤ height ≤ 10⁶). Output Output a single integer — the maximal height of a tower that can be formed. Examples Input 5 5 3 8 6 2 Output 18 Note In the test example, one way to form the tallest tower is by arranging the blocks with heights 5, 6, and 8. This gives a total height of 5 + 6 + 8 = 19. The tower with blocks 2 and 3 has a total height of only 5. Thus, the maximum height is 18
answer:def max_tower_height(n, heights): Returns the maximum height of a tower that can be formed with the given blocks, obeying the rule that taller blocks cannot be placed on top of shorter blocks. :param n: int : Number of blocks :param heights: list : List of integers representing the heights of the blocks :return: int : Maximum tower height # Sort the blocks in descending order sorted_heights = sorted(heights, reverse=True) # Initialize tower height tower_height = 0 # Iterate through the sorted heights and construct the tower for height in sorted_heights: tower_height += height return tower_height