# Problems of the Week

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A hemispherical bubble is placed on top of a spherical bubble of radius $$1$$.

A smaller, second hemispherical bubble is then placed on the first one. This process is continued until $$n$$ hemispheres are placed.

Find the maximum possible height of such a tower with $$80$$ hemispheres on top of the sphere.

Take a point $$A$$ on the graph of $$y=f(x),$$ and let $$B$$ be a point on the graph of $$y=\sqrt{x}$$ such that $$AB$$ is parallel to the $$y$$-axis. Call the area bounded by these two curves and the segment $$AB$$ as $$R.$$

Now, let $$C$$ be a point on the $$y$$-axis such that $$AC$$ is parallel to the $$x$$-axis. Call the area bounded by the curve $$y=f(x),$$ the $$y$$-axis, and the segment $$AC$$ as $$S.$$

Given that the function $$f$$ is continuous and the areas $$R$$ and $$S$$ are equal, which of the following statements is true?

Let $$(a_n)$$ be a sequence of integers defined as $a_0 = 0; \quad a_1 = 1; \quad a_{n+1}=\text{the next integer that shares no digits with }a_n.$ How many digits does the term $$a_{2018}$$ have?

Hints:

1. Here are a few more terms in the sequence: $a_2=2,\ a_3=3,\ ...,\ a_9=9,\ a_{10}=10,\ a_{11}=22,\ a_{12}=30,\ a_{13}=41,\, \ldots.$
2. Try finding the pattern for the subsequence $$(a_{8k+2})_{k \geq 3}.$$

The polynomial $$p(x)$$ is of degree 2017 and has non-negative integer coefficients which you don't know.

If you input a value like $$x= x_0,$$ the computer will output the value $$p(x_0)$$ at a cost of \$1.

If you want to determine all 2018 coefficients of $$p(x)$$ at a minimum cost using only positive integer inputs, what is your cost in dollars?

Let $$D_n$$ be the product of all positive divisors of a positive integer $$n.$$ $$\big($$For example, $$D_1 = 1$$ and $$D_4 = 1 \times 2 \times 4 = 8.\big)$$

What is the smallest positive integer $$n$$ for which $$D_n$$ can be written as $D_n = p^a \times q^b \times r^c \times s^d,$ where $$p, q, r, s$$ are four distinct prime numbers and $$a,b,c,d$$ are four distinct positive integers?

Bonus 1: Can you solve this for prime numbers $$p$$ , $$q$$ , $$r$$ , $$s$$ , $$t$$ and integers $$a,b,c,d,e?$$
Bonus 2: Can you generalize for $$p_{1}, ...,p_{n},$$ where all primes in the prime factorization are raised to different powers?

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