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.
by
Digvijay Singh
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?
by
Chan Lye Lee
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:
- Here are a few more terms in the sequence: \[a_2=2,\ a_3=3,\ ...,\ a_9=9,\ a_{10}=10,\ a_{11}=22,\ a_{12}=30,\ a_{13}=41,\, \ldots.\]
- Try finding the pattern for the subsequence \((a_{8k+2})_{k \geq 3}.\)
by
Romain Bouchard
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?
by
Daniel Xiang
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?
by
Mike Lust