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2017-06-12 Intermediate

         

You trace a circle (the red one) using a pen with a thick circular end. Both the outside and inside borders of the trace are also circles (in blue):

If you trace any ellipse using the same kind of pen, will the outside and inside borders also be ellipses?

Imagine two stars \(A\) and \(B\) forming a binary star system, where they revolve around a common point in space under their mutual gravitational attraction alone. If \(A\) is heavier than \(B\) and they both revolve in circular orbits, then which star will have a smaller orbital radius?

The first few powers of 101 all begin with 1, as highlighted by red colors below: \[ \begin{eqnarray} 101^1 &=& {\color{red}{1}}01 \\ 101^2 &=& {\color{red}{1}}0201 \\ 101^3 &=& {\color{red}{1}}030301 \\ 101^4 &=& {\color{red}{1}}04060401 \\ &\vdots.& \end{eqnarray} \] Is that always the case for all powers of 101?

If \(a + a^2 + a^3 + a^4 + a^5 + \cdots \) is a positive number, then which of the following is larger,

\[a+a^3+a^5+a^7+\cdots\quad \text{or}\quad a^2+a^4+a^6+a^8+\cdots\, ?\]

\[\large \left ( 1 + \dfrac{1}{x} \right )^{x+1} = \left ( 1 + \dfrac{1}{1999} \right )^{1999}\]

Find the sum of all real \(x\) such that the above equation is true.

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