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?

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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