Date modified: Thu 2023-07-13 . 10:53 PM
Date created: Sat 2023-10-07 . 09:02 PM
# A problem list. Here is a list of interesting problems we discussed / considered. === Show if $f:[0,1]\to[0,1]$ is a continuous function, then $f$ has a fixed point. That is there exists some $p\in[0,1]$ such that $f(p) = p$. === Does the series $\sum_{n \text{ has no 7's}} \frac{1}{n}$ converge? === Take an equilateral triangle, and trisect each edge. The cut off the corners from the third position to the next third position. Repeat. What is the area of the resulting shape? Is it a circle? === The game "RUN/JUMP". === Consider three positive numbers $a,b,c > 0$ such that $a^{3}+b^{3}=c^{3}$. (1) Show that $a,b,c$ forms the sides of a triangle. (2) Denote $\gamma$ to be the angle opposing the side $c$. What are the possible values of the angle $\gamma$? === Given $\gamma$ an irrational number, show that the set $A=\{\langle n\gamma:n\in\mathbb{N}\rangle\}$ is dense in $[0,1]$. === Consider the powers of $2$: $2,4,8,16,32,64,128,512,\ldots$. Let us focus on the left-most digit (leading digit). Does $7,8,$ or $9$ ever show up as the leading digit (in base 10)? Does your phone number show up in some power of $2$? === Which of the follow sets are dense in $[0,1]$ ? $A=\{\langle\log(n)\rangle:n\in\mathbb{N}\}$ $B=\{\langle|\sin(n)|\rangle:n\in\mathbb{N}\}$ === Is the following statement true: For any positive irrational $\gamma > 0$, the sequence $\langle n!\gamma\rangle$ has a range that is dense in $[0,1]$ === Is the following statement true: For any positive irrational $\gamma > 0$, the sequence $\langle 2^{n}\gamma\rangle$ has a range that is dense in $[0,1]$ === Show if $f:X\to Y$ is a continuous function, and $D$ is dense in $X$, then the set $f(D)$ is dense in the image $f(X)$. === Show the following: Let $\mathbb{F}$ be an ordered field that is **Dedekind complete**. Then (1) $\mathbb{F}$ is monotone complete: Every bounded monotone sequence is convergent in $\mathbb{F}$ (2) $\mathbb{F}$ is nested complete: If $I_{n}=[a_{n},b_{n}]$ are closed intervals such that $I_{1}\supset I_{2}\supset I_{3}\supset\cdots$, then their common intersection $\bigcap I_{n}$ is not empty. Furthermore, if $b_{n}-a_{n}\to 0$, then $\bigcap I_{n} = \{\ast\}$ is a singleton set. (3) $\mathbb{F}$ is Bolzano complete: Every bounded sequence $(a_{n})$ has a convergent subsequence $(a_{n(k)})$. (4) $\mathbb{F}$ is Cauchy complete: Every Cauchy sequence is convergent (5) $\mathbb{F}$ is topologically connected (with respect to the order topology) === Consider a subset $A$ of some metric space $X$. And consider all possible finite sequence of applications of closure or complement to $A$. That is, the sets $$A,A^{c},(A^{c})^{k},((A^{c})^{k})^{c},\ldots,A^{k}, (A^{k})^{c}, ((A^{k})^{c})^{k},\ldots$$ How many different sets can we get? === If we remove $\mathbb{Q}$ from $\mathbb{R}$, then the resulting set $\mathbb{R}-\mathbb{Q}$ is disconnected. How about $\mathbb{R}^{2}-\mathbb{Q}^{2}$ === Snow falls before noon, falling at a constant rate. You start cleaning at noon, cleaning at a constant rate. The first hour, you clean one mile, but the second hour, only half a mile. When did snow fall begin? === Why is the shape of a hanging chain (catenary) a hyperbolic cosine?