# Problem Set 2

In this problem set we present four questions:

(Middle School Students)
The irrational number $0.1234012340012340001234\dots$ is formed
by using alternating blocks of $1234$ and zeros, where the $n$th block
of zeros following the decimal contains $n$ zeros. What is the digit
in the 2550th place following the decimal?

(High School Students) Consider the function $f : \mathbb{R} \rightarrow \mathbb{R}$ given by

$f(x) = \left \{ \begin{array}{ll} 1-\frac{\sin(x)}{x} & x \neq 0 \cr 0 & \mbox{otherwise} \end{array} \right.$

Find the maximum and minimum value of the function.

(Undergraduate Students) Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be an infinitely differentiable function, such that

1. $f(x) >0$ whenever $x \neq 0$
2. $f(0)=0$ and $f^{(n)}(0) = 0$ where $n \geq 1$ and $f^{(n)}$ is the $n$-th derivative of $f.$

Let $g : \mathbb{R} \rightarrow \mathbb{R}$ is given by

$g(x) = \sqrt{f(x)},\,\,\forall x \in {\mathbb R}$

Decide whether $g$ is infinitely differentiable on $\mathbb{R}$.

(Masters Students and above) If $A$ is a measurable subset of $\mathbb{R}$ such that

$a\in A,\,\,\,b\in A,\,\,\, a\neq b\,\,\, \Rightarrow \,\,\, \frac{a+b}{2}\notin A$

then $A$ has measure $0.$