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 © | Dror Bar-Natan: Classes: 2014-15: Math 475 - Problem Solving Seminar: (53) Next: Blackboards for Thursday March 26 Previous: Quiz #10 - "Generalize"

# Handout for March 26, "Generalize"

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).

1. Prove that there are infinitely many primes of the form $6n-1$.
Hint. Consider $(p_1p_2\cdots p_k)^2-2$.
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

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

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 $\sqrt[3]{x}+\sqrt[3]{y}$.