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Problems of the Week

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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 $$K?$$


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 $$\sigma?$$


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

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