Forgot password? New user? Sign up
Existing user? Log in
Write one number in each red circle so that the sum of the numbers along each blue, diagonal line is always the same number x.
Find the maximum value of x.
Are you sure you want to view the solution?
Consider the following model:
A raindrop falls from rest with a small initial mass, m0. As it falls, it accumulates mass by absorbing the smaller water drops in its path, which we approximate by the uniform density ρatm.
Throughout the fall, it maintains the same characteristic shape defined by its width, 2r, which increases as it picks up mass at the rate m˙=ρatmπr2v (in time dt, it sweeps out a vertical cylinder of height vdt).
How will its velocity depend on time? Surprisingly, this complicated problem has a simple solution: after a short time, the raindrop will accelerate at the constant rate a=Kg (convince yourself of this!).
What is the value of K?
Details and Assumptions:
Inspired by a similar problem by Tapas Mazumdar
Are you sure you want to view the solution?
How many real x satisfying 0≤x≤2017 are there such that xsin(πx) is an integer?
Bonus: Can you come up with a simple formula for how many 0≤x≤n there are such that xsin(πx) is an integer, where n is a positive integer?
Are you sure you want to view the solution?
Consider a pyramid with 2017 rows made up of empty boxes. The first (top) row has 1 box, and each successive row has an additional box so that the 2017th, bottom row has 2017 boxes.
First, place each of the first 2017 positive integers into the boxes in the bottom row. Then, each empty box in the 2016th row is filled with the sum of the two numbers beneath it. Then, each successively higher row of boxes is filled in the same way.
Let M and m be the maximum and minimum possible values, respectively, of the single box on the top. Then M+m=a×2b, where a and b are positive integers such that b is as large as possible. Find the value of a+b.
Note: Below is an example of how a pyramid of 4 rows could be filled.
Are you sure you want to view the solution?
Let σ be a permutation of {1,2,3,…,100}. Then there exists a smallest positive integer f(σ) such that σf(σ)=id, where id is the identity permutation.
What is the largest value of f(σ) over all σ?
Note: σk is σ applied k times in succession.
Are you sure you want to view the solution?
Problem Loading...
Note Loading...
Set Loading...