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.
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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?
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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:
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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?
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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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