2552. Count Increasing Quadruplets
Given a 0-indexed integer array nums of size n containing all numbers from 1 to n, return the number of increasing quadruplets.
A quadruplet (i, j, k, l) is increasing if:
- 0 <= i < j < k < l < n, and
- nums[i] < nums[k] < nums[j] < nums[l].
You have k bags. You are given a 0-indexed integer array weights where weights[i] is the weight of the ith marble. You are also given the integer k.
Divide the marbles into the k bags according to the following rules:
- No bag is empty.
- If the ith marble and jth marble are in a bag, then all marbles with an index between the ith and jth indices should also be in that same bag.
- If a bag consists of all the marbles with an index from i to j inclusively, then the cost of the bag is weights[i] + weights[j].
The score after distributing the marbles is the sum of the costs of all the k bags.
Return the difference between the maximum and minimum scores among marble distributions.
2550. Count Collisions of Monkeys on a Polygon
There is a regular convex polygon with n vertices. The vertices are labeled from 0 to n - 1 in a clockwise direction, and each vertex has exactly one monkey. The following figure shows a convex polygon of 6 vertices.
Each monkey moves simultaneously to a neighboring vertex. A neighboring vertex for a vertex i can be:
- the vertex (i + 1) % n in the clockwise direction, or
- the vertex (i - 1 + n) % n in the counter-clockwise direction.
A collision happens if at least two monkeys reside on the same vertex after the movement.
Return the number of ways the monkeys can move so that at least one collision happens. Since the answer may be very large, return it modulo 109 + 7.
Note that each monkey can only move once.
2549. Count Distinct Numbers on Board
You are given a positive integer n, that is initially placed on a board. Every day, for 109 days, you perform the following procedure:
- For each number x present on the board, find all numbers 1 <= i <= n such that x % i == 1.
- Then, place those numbers on the board.
Return the number of distinct integers present on the board after 109 days have elapsed.
Note:
- Once a number is placed on the board, it will remain on it until the end.
- % stands for the modulo operation. For example, 14 % 3 is 2.
