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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 \(x.\)

Find the maximum value of \(x\).

Consider the following model:

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

Throughout the fall, it maintains the same characteristic shape defined by its width, \(2r,\) which increases as it picks up mass at the rate \(\dot{m} = \rho_\textrm{atm}\pi r^2 v\) (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 \(a=\frac gK\) (convince yourself of this!).

What is the value of \(K?\)

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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 \(x\) satisfying \(0 \leq x \leq 2017\) are there such that \(x \sin(\pi x)\) is an integer?


Bonus: Can you come up with a simple formula for how many \(0 \leq x \leq n\) there are such that \(x \sin(\pi x) \) is an integer, where \(n\) 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 \(2017^\text{th},\) 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 \(2016^\text{th}\) 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 \(M\) and \(m\) be the maximum and minimum possible values, respectively, of the single box on the top. Then \(M + m = a \times 2^b,\) 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.

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

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

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Note: \(\sigma^k\) is \(\sigma\) applied \(k\) times in succession.

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