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%TCIDATA{<META NAME="Title" CONTENT="Math Club - Set 1">}
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\begin{document}


\begin{enumerate}
\item The angles of a triangle $ABC$ are $50^{\circ }$, \ $50^{\circ }$, and 
$80^{\circ }$, as shown on the picture below.\FRAME{dtbpF}{3.2716in}{2.1188in%
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From $A$, we measure an angle of $30^{\circ }$. \ From $B$, we measure an
angle of $10^{\circ }$. The two lines intersect at point $D$. Find the
measure of the angle $ACD$.\FRAME{dtbpF}{3.614in}{2.1257in}{0pt}{}{}{%
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\item A stairway has $10$ steps. At any point, we can step up one or two
steps. \ We never step downwards. How many different ways are there to walk
up the stairs?

\item Let $m$ and $n$ be any two natural numbers. Define the set $A$ as 
\begin{equation*}
A=\left\{ \alpha n+\beta m:\text{ \ where \ }\alpha \text{ \ \ and \ }\beta 
\text{ \ are integers}\right\} 
\end{equation*}%
Let $a$ be the smallest positive element in $A$. \ Prove that $a$ is a
divisor of every element of $A$.
\end{enumerate}

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