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2017-09-04 Advanced

         

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.x.

Find the maximum value of xx.

Consider the following model:

A raindrop falls from rest with a small initial mass, m0.m_0. As it falls, it accumulates mass by absorbing the smaller water drops in its path, which we approximate by the uniform density ρatm.\rho_\textrm{atm}.

Throughout the fall, it maintains the same characteristic shape defined by its width, 2r,2r, which increases as it picks up mass at the rate m˙=ρatmπr2v\dot{m} = \rho_\textrm{atm}\pi r^2 v (in time dtdt, it sweeps out a vertical cylinder of height vdtvdt).

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=gKa=\frac gK (convince yourself of this!).

What is the value of K?K?


Details and Assumptions:

  • Neglect the effects of air resistance, wind speed, the density of air, or any other atmospheric factor except those stated in the problem.

Inspired by a similar problem by Tapas Mazumdar

How many real xx satisfying 0x20170 \leq x \leq 2017 are there such that xsin(πx)x \sin(\pi x) is an integer?


Bonus: Can you come up with a simple formula for how many 0xn0 \leq x \leq n there are such that xsin(πx)x \sin(\pi x) is an integer, where nn is a positive integer?

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,2017^\text{th}, 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 2016th2016^\text{th} 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 MM and mm be the maximum and minimum possible values, respectively, of the single box on the top. Then M+m=a×2b,M + m = a \times 2^b, where aa and bb are positive integers such that bb is as large as possible. Find the value of a+b.a + b.


Note: Below is an example of how a pyramid of 4 rows could be filled.

Let σ\sigma be a permutation of {1,2,3,,100}\{1, 2, 3, \ldots, 100\}. Then there exists a smallest positive integer f(σ)f(\sigma) such that σf(σ)=id\sigma^{f(\sigma)} = \text{id}, where id\text{id} is the identity permutation.

What is the largest value of f(σ)f(\sigma) over all σ?\sigma?


Note: σk\sigma^k is σ\sigma applied kk times in succession.

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