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?
Are you sure you want to view the solution?
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?
Are you sure you want to view the solution?
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?
Are you sure you want to view the solution?
If \(a + a^2 + a^3 + a^4 + a^5 + \cdots \) is a positive number, then which of the following is larger,
\[a+a^3+a^5+a^7+\cdots\quad \text{or}\quad a^2+a^4+a^6+a^8+\cdots\, ?\]
Are you sure you want to view the solution?
\[\large \left ( 1 + \dfrac{1}{x} \right )^{x+1} = \left ( 1 + \dfrac{1}{1999} \right )^{1999}\]
Find the sum of all real \(x\) such that the above equation is true.
Are you sure you want to view the solution?
Problem Loading...
Note Loading...
Set Loading...