Number Theory
# Rational Numbers

Find the smallest positive integer whose square consists of 5 identical leading digits.

As an explicit example, $149^2 = 22201$ would be the answer for 3 identical leading digits.

You have to place dots along a strip such that:

- The first two dots are in the different halves of the strip
- The first three dots are in the different thirds of the strip
- The first four dots are in the different fourths of the strip, and so on.

What is the maximum number of dots you can place on the strip?

Let $\displaystyle \frac ab$ be the smallest of the fractions that are greater than $\displaystyle \frac {13}{15}$ where $a,b < 500$ are positive integers.

What is the value of $a+b$?

Let $a_k$ represent the repeating decimal $0.\overline{133}_k$ for $k \geq 4$. The product $a_4 a_5 \cdots a_{99}$ can be expressed as $\frac{m}{n!}$ where $m, n$ are positive integers and $n$ is as small as possible. $\frac{m}{n}$ can be expressed as $\frac{p}{q}$ where $p, q$ are coprime integers. What is $p+q$?

Note: $0.\overline{133}_k$ refers to the repeating decimal $0.133133133\ldots$ evalauted in base $k$.