2018-03-12 Advanced

         

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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