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 nth 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.