Problems of the Week

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In the domain $[ 0^ \circ, 360 ^ \circ ],$ how many solutions are there to $\frac {\tan 2x- \tan x}{1+\tan 2x\tan x}=1?$

How many values of $K$ are there, such that there exists distinct complex numbers $a, b,$ and $c$ which satisfy $\frac{a}{1-b} = \frac{b}{1-c} = \frac{c}{1-a} = K?$ Using 2 white dots, I mark 2 points uniformly at random on a rod. Then, I choose 9 points uniformly at random on the rod, and break the rod at those points to get 10 small pieces.

To 3 decimal places, what is the probability that the 2 dots will lie on the same piece?

A senior class has 56 students and 15 activity clubs. It is known that if any 7 clubs hold a combined meeting of all their members, then there are at least $n$ people in attendance.

What is the minimum value of $n$, such that we can conclude there must be (at least) one person who is in 3 or more clubs?

There are several planets other than Earth that are thought to be able to sustain life of the form we have on Earth, one of the basic requirements for which is to have liquid water on its surface.

One of the Sun-like stars is Tau Ceti. Its radius is $79 \%$ of Sun's radius, and its surface temperature is $T = \SI{5344}{\kelvin}$. What is the width of the spherical shell around Tau Ceti $($in $\si{\kilo\meter})$ in which planets can have liquid water?


Details and Assumptions:

• The radius of the Sun is $\SI{695500}{\kilo\meter}.$
• Planets and stars can be approximated to behave like black bodies.
• For the sake of simplicity, assume the atmospheric pressure on the surface of potentially inhabitable planets is the same as on Earth.
• Only one side of a planet is exposed to Tau Ceti's radiation.
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