© | Dror Bar-Natan: Classes: 2014-15: Math 475 - Problem Solving Seminar: | (57) |
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**Exam Preparation.** Very simply, you should solve
everything, and the best for you is if you manage to form
study groups to do it together. I will continue holding my
normal office hours throughout the examination period until the
day of the final (though before coming you are advised to check http://drorbn.net/?title=Office_Hours
for possible last-minute changes). In addition, I will have an "open
office" the day before the final, on Monday April 20, from 9:30AM until
5PM (excluding washroom and lunch breaks...). You are encouraged to work
in the small lounge by my office throughout this period and drop in to
ask questions as you please. My office remains at the Bahen building,
room 6178.

**Jan 15 Problem 6** (Larson's 7.1.14). In a convex quadrilateral, prove that
the sum of the lengths of the diagonal lies between the perimeter and half
the perimeter.

**Jan 15 Problem 7** (Larson's 7.4.19). Noting that the function
$f(x)=\sqrt{x}$ is concave, show that if $a,b,c$ are positive and satisfy
$c>a\cos^2\theta+b\sin^2\theta$, then
$\sqrt{c}>\sqrt{a}\cos^2\theta+\sqrt{b}\sin^2\theta$.

**Jan 22 Problem 5** (Larson's 1.3.5). On a circle $n$ different points
are selected and the chords joining them in pairs are drawn. Assuming no
three of these chords pass through the same point, how many intersetion
points will there be (inside the circle)?

**Jan 22 Problem 6** (Larson's 1.3.6). Given a positive integer $n$,
find the number of quadruples of integers $(a,b,c,d)$ such that $0\leq
a\leq b\leq c\leq d\leq n$.

**Jan 22 Problem 7** (Larson's 1.3.7). The number $5$ can be expressed
as a sum of $3$ natural numbers, taking order into account, in $6$
ways: $5=1+1+3=1+2+2=1+3+1=2+1+2=2+2+1=3+1+1$. Let $k\leq n$ be natural
numbers. In how many ways can $n$ be written as a sum of $k$ natural
numbers, minding the order?

**Jan 22 Problem 8.** Same as the previous question, but with "natural
numbers" replaced with "non-negative integers".

**Feb 5 Problem 5** (Larson's 1.6.4). Let $P$ be a point on
the graph $G$ of $y=f(x)$, where $f$ is a cubic polynomial. Assume the
tangent to the curve at $P$ intersects $G$ again at a point $Q$. Let $A$
be the area bound by $G$ and the segment $PQ$, and let $B$ be the area
defined in exactly the same manner, except starting with $Q$ rather than
with $P$. What is the relationship between $A$ and $B$?

**Feb 24 Problem 4** (Larson's 1.8.3). In the figure on the right, everything
is as it seems: $O$ is the centre, $AB$ is a diameter, $CE$ and $BC$ are
tangents, and all lines are straight. Show that $BC=CD$.

**Feb 26 Problem 6** (Larson's 2.6.3, modified). The points on a $3\times 7$
rectangular grid are coloured $Q$ and $M$. Show that you can find a rectangle
among these points, whose sides are parallel to the grid axes, and all of
whose corners are coloured the same way.

**Feb 26 Problem 7** (off topic, but fun). A rectangle is said to be "part
whole" if the length of at least one of its sides is a whole number. Prove
that
if a rectangle $R$ can be partitioned into part whole subrectangles, then $R$
is part whole.

**Hint.** Consider $\int e^{2\pi i(x+y)}dx\,dy$. (!?!!)

**Not so new!** I just learned of an article
that has 14 proofs of this result! *Fourteen
Proofs of a Result About Tiling a Rectangle* by Stan Wagon,
The American Mathematical Monthly **94-7** (1987) 601-617.

**Feb 26 Problem 8** (Larson's 2.6.6). If 20 integers are chosen from within
the elements of the arithmetic progression 1,4,7,...,100, show that you can
find two of them whose sum is 104.

**March 5 Problem 10** (Larson's 3.2.1 and 3.2.11, combined) Prove that any
subset of size 55 of the set $\{1,2,3,\ldots,100\}$ must contains two numbers
differing by 9, 10, 12, and 13, but not need not contain a pair of numbers
differing by 11.

**March 5 Problem 11** (Larson's 3.2.17). Let $S$ be a set of primes such that
$a,b\in S$ ($a$ and $b$ may be the same) implies that $ab+4\in S$. Prove that
$S$ is empty.

**Hint.** It's nice to work mod 7.

**March 5 Problem 12** (Larson's 5.4.1). Prove that
$e=\sum_{k=0}^\infty\frac{1}{k!}$ is an irrational number.

**March 12 Problem 11** (not for marks). Complete your understanding
of Ramsey's theorem!

**March 19 Problem 1** (Larson's 1.11.1, hinted). Given a finite
number of points in the plane, not all of them on the same line, prove
that there is a straight line that passes through exactly two of them.

**March 19 Problem 8** (Larson's 3.3.28, off topic, modified).

- Prove that there are infinitely many primes of the form $6n-1$.

Hint. Consider $(p_1p_2\cdots p_k)^2-1$. - Prove that there are infinitely many primes of the form $4n-1$.

**March 19 Problem 9.** Let $A$ be a subset of $[0,1]$ which is both
open and closed, and assume that $0\in A$. Prove that also $1\in A$.