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\homework{1: Introduction / Russian Problems}{Due at the start of the next class.}

Solve as many as you can!  Instructions are in the syllabus.  Prove any claims you make.

\begin{enumerate}
\item Is $\sin(10^\circ)$ a rational number?  
\item Which number is larger: $\log_2(3)$ or $\log_3(5)$?  (For this problem, simply using Mathematica is not enough; you need a proof that doesn't use any computer-aided algebra systems.)
\item Take an arbitrary quadrilateral.  In how many ways can you represent it as the union of two triangles?  (Hints: break your quadrilateral into two cases, depending on whether or not your polygon is convex.)
\item Can you find an equilateral triangle $T$ in the plane such that all three vertices of $T$ have integer co\"ordinates?  
\item Find all of the functions $F: \mathbb{R} \to \mathbb{R}$ that satisfy the following property:
\begin{align*}
\textrm{For all }x,y\textrm{ in }\mathbb{R}, \quad F(x) - F(y) \leq (x-y)^2
\end{align*}
\item Call a pair of integers $a,b$ \textbf{intermingled}\footnote{I made up this term.} if they satisfy the following properties:
\begin{enumerate}
\item $a$ and $b$ are distinct.
\item The prime decompositions of $a$ and $b$ share the same primes.  In other words: if $p$ is a prime that divides $a$, then it divides $b$.  As well, if $p$ is a prime that divides $b$, then it divides $a$ as well.
\item The prime decompositions of $a+1$ and $b+1$ also share the same primes.
\end{enumerate}
For example, $(2, 8)$ satisfies this property; 2 and 8 both have only 2's in their prime factorizations, while 3 and 9 both only have 3's.  $(6, 48)$ is another pair.

Are there infinitely many such pairs?
\end{enumerate}





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