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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.
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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=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 (an) be a sequence of integers defined as a0=0;a1=1;an+1=the next integer that shares no digits with an. How many digits does the term a2018 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=x0, the computer will output the value p(x0) 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 Dn be the product of all positive divisors of a positive integer n. (For example, D1=1 and D4=1×2×4=8.)
What is the smallest positive integer n for which Dn can be written as Dn=pa×qb×rc×sd, 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 p1,...,pn, where all primes in the prime factorization are raised to different powers?
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