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# 2018-05-21 Advanced

$\displaystyle \lim_{n\to\infty} \dfrac{n + n^2+ n^3 + \cdots + n^n}{1^n + 2^n + 3^n + \cdots + n^n}$

What is the value of this limit?

A well-known physics problem asks whether the weight of an hourglass sitting on a scale would be the same whether all of the sand is at rest, or whether it is flowing. Usually, the answer given is yes, because of the exact balance between the "missing weight" of the sand in free fall, and the force of the sand hitting the bottom.

This well-known physics result is wrong. The hourglass above has a total mass $M$, constructed with a constant cross sectional area $A$ on the top and bottom, and a very thin section connecting them which allows the sand with density $\rho$ to flow through at a constant rate of mass flow $q$. The total amount of sand fills the bottom of the hourglass once all the sand has fallen.

What is the weight of the hourglass at steady state when the sand is flowing?

Note: You may assume that the mass of the falling sand at any given time is negligible compared to the masses at rest above and below, and that once the sand lands, it rapidly spreads out onto a flat surface. 3 great circles cut the sphere into at most 8 regions.

What is the greatest number of regions that 30 great circles can cut the sphere into?

Note: A great circle on a sphere is the intersection of a sphere with a plane that contains the sphere's center.

There are 100 people on an island; some people always tell the truth, and the others always lie.

A visitor asks them, "How many of you tell the truth?"

For $n=1, 2, ..., 100,$ the $n^\text{th}$ person replies, "$f(n)$ of us tell the truth," where $f(n)$ is the last two digits of $n^2.$

How many of them always tell the truth?

Note: All 100 islanders replied. For example, the $3^\text{rd}$ person replied, "9 of us tell the truth" because $3^2={\color{#D61F06}09}.$ and the $17^\text{th}$ person replied, "89 of us tell the truth" because $17^2=2{\color{#D61F06}89}.$ A pen of unit length is lying tilted on a rough table.

The lower end of the pen is dragged in the horizontal direction until the pen becomes vertical. During this process, the upper end of the pen travels a unit distance.

How many units does the lower end travel?

Assume that the velocity of the upper end of the pen is always parallel to the pen.

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