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2018-03-12 Advanced

         

A hemispherical bubble is placed on top of a spherical bubble of radius 11.

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

Find the maximum possible height of such a tower with 8080 hemispheres on top of the sphere.

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

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

Given that the function ff is continuous and the areas RR and SS are equal, which of the following statements is true?

Let (an)(a_n) be a sequence of integers defined as a0=0;a1=1;an+1=the next integer that shares no digits with an.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 a2018a_{2018} have?

Hints:

  1. Here are a few more terms in the sequence: a2=2, a3=3, ..., a9=9, a10=10, a11=22, a12=30, a13=41,.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 (a8k+2)k3.(a_{8k+2})_{k \geq 3}.

The polynomial p(x)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,x= x_0, the computer will output the value p(x0)p(x_0) at a cost of $1.

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

Let DnD_n be the product of all positive divisors of a positive integer n.n. (\big(For example, D1=1D_1 = 1 and D4=1×2×4=8.)D_4 = 1 \times 2 \times 4 = 8.\big)

What is the smallest positive integer nn for which DnD_n can be written as Dn=pa×qb×rc×sd,D_n = p^a \times q^b \times r^c \times s^d, where p,q,r,sp, q, r, s are four distinct prime numbers and a,b,c,da,b,c,d are four distinct positive integers?


Bonus 1: Can you solve this for prime numbers pp , qq , rr , ss , tt and integers a,b,c,d,e?a,b,c,d,e?
Bonus 2: Can you generalize for p1,...,pn,p_{1}, ...,p_{n}, where all primes in the prime factorization are raised to different powers?

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