© | Dror Bar-Natan: Classes: 2014-15: Math 475 - Problem Solving Seminar: | (12) |
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**Reading.** Section 1.3 of Larson's textbook.

**Next Quiz.** On Tuesday January 27, mostly problems from Larson's
Section 1.3.

**Problem 1** (Larson's 1.3.1). Find a general formula for the $n$th derivative
of $f(x)=1/(1-x^2)$.

**Problem 2** (Larson's 1.3.2). Find all solutions of $x^4+x^3+x^2+x+1=0$.

**Problem 3** (Larson's 1.3.3). $P$ is a point inside a given
triangle $ABC$, and $D$, $E$, and $F$ are the points closest to $P$
on $BC$, $CA$, and $AB$ respectively. Find all $P$ for which

**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)?

**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 $0\leq
a\leq b\leq c\leq d\leq n$.

**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?

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