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Name (Last, First): $\underline{\hspace{3in}}$ \hfill Student ID: $\underline{\hspace{2in}}$

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{\small
  \href{\myurl}{Dror Bar-Natan}:
  \href{\myurl/classes/}{Classes}:
  \href{\myurl/classes/\#1415}{2014-15}:
  \href{\myurl/classes/15-475-ProblemSolving}{MAT 475 Problem Solving Seminar}:
  \hfill\url{http://drorbn.net/15-475}
}

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{\large\bf Quiz 10} on March 26, 2015: ``Consider Extreme Cases''. You have 25 minutes to solve two of the three problems below. Please write on both sides of the page. \hfill {\bf Good Luck!}

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\noindent{\bf Marking Comment.} My decision remains to simplify the management of this course and mark the quizzes myself, though at a delay of one week, in a symbolic acknowledgement of the ongoing TA strike.

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\noindent{\bf Problem 1} (Larson's 1.11.4). Prove that the
product of $n$ successive integers is always divisible by $n!$.

\vskip 2mm\noindent{\bf Problem 2} (Larson's 1.11.2, reworded). Let $A$ be a set of $2n$ points in the plane, no three of them on the same line. Suppose that $n$ of them are coloured red and $n$ are coloured blue. Show that you can choose a pairing of the reds and the blues such the straight line segments between the pairs do not intersect.

\vskip 2mm\noindent{\bf Problem 3} (Larson's 1.11.7). Show that there exists a rational number $c/d$, with $d<100$, such that $\lfloor k\frac{c}{d}\rfloor = \lfloor k\frac{73}{100}\rfloor$ for $k=1,2,\ldots,99$.

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