$ \def\bbN{{\mathbb N}} \def\bbQ{{\mathbb Q}} \def\bbR{{\mathbb R}} \def\bbZ{{\mathbb Z}} $

(12)

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Handout for January 22, "Formulate an Equivalent Problem"

Class Photo. If you did not yet, please send me an email identifying yourself in the class photo page. Also let me know if you agree that your name will appear on that page, which is publically accessible. My email remains drorbn@math.toronto.edu.

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

$\displaystyle \frac{BC}{PD}+\frac{CA}{PE}+\frac{AB}{PF}$ is minimal.

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".