© | Dror Bar-Natan: Classes: 2014-15: Math 475 - Problem Solving Seminar: | (53) |
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**Reading.** Section 1.12 of Larson's textbook.

**Last Quiz.** Thursday April 2 on that section and this handout.

**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.

Hint. Consider the triangle with least height whose vertices.

**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 $(p_1p_2\cdots p_k)^2-2$. - 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 $1\in A$.

**New Problem 1** (From Larson's 1.12.1, 1.12.4, 5.1.9, 5.4.4). Evaluate the
sum $\sum_{k=1}^\infty k^2/2^k$, and then also the sums

- $\displaystyle\sum_{k=1}^n k^2/2^k$.
- $\displaystyle\sum_{k=1}^n (2k+1)\binom{n}{k}$.
- $\displaystyle\sum_{k=1}^n k(k-1)\binom{n}{k}$.
- $\displaystyle\sum_{k=1}^n k^2\binom{n}{k}$.
- $\displaystyle\sum_{k=1}^n 3^k\binom{n}{k}$.
- $\displaystyle\sum_{k=1}^n \frac{1}{k+1}\binom{n}{k}$.
- $\displaystyle\sum_{k=1}^n \frac{(-1)^k}{k+1}\binom{n}{k}$.
- $\displaystyle\sum_{k=0}^\infty \frac{(k+1)^2}{k!}$.

**New Problem 2** (Larson's 1.12.2 and 1.12.5). Compute the Vandermonde determinant:
\[ \det\begin{pmatrix}
1 & a_1 & a_1^2 & \cdots & a_1^{n-1} \\
1 & a_2 & a_2^2 & \cdots & a_2^{n-1} \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
1 & a_n & a_n^2 & \cdots & a_n^{n-1}
\end{pmatrix}.
\]
When done, consider the variant
\[ \det\begin{pmatrix}
1 & a & a^2 & a^4 \\
1 & b & b^2 & b^4 \\
1 & c & c^2 & c^4 \\
1 & d & d^2 & d^4
\end{pmatrix}.
\]

**New Problem 3** (Larson's 4.1.4). Prove that none of the following integers is prime:
\[ 1,\ 10001,\ 100010001,\ 1000100010001,\ \ldots \]

**New Problem 4** (Larson's 2.4.1, off topic). Prove that if $\alpha_1+\alpha_2+\ldots+\alpha_n=\pi$
and $\alpha_i\geq 0$, then $\sin\alpha_1+\sin\alpha_2+\ldots+\sin\alpha_n\leq n\sin(\pi/n)$.

**New Problem 5** (Larson's 2.4.3, modified). Let $F_n$ denote the Fibonacci numbers, defined by
$F_0=F_1=1$ and $F_n=F_{n-1}+F_{n-2}$ for $n\geq 2$. Prove that $F_{2n}=(F_n)^2+(F_{n-1})^2$.

**New Problem 6** (Larson's 2.4.3, reworded). Let $S$ denote an
$n\times n$ lattice of equally-spaced points, for $n\geq 3$. Prove that
there exists a polygonal path made of $2n-2$ straight segments which
passes through all $n^2$ points of $S$.

**New Problem 7** (Larson's 1.12.7, reworded). Which is larger, $\sqrt[3]{60}$ or $2+\sqrt[3]{7}$?

Hint. First compare $\sqrt[3]{4(x+y)}$ and $\sqrt[3]{x}+\sqrt[3]{y}$.