Number Theory

Rational Numbers

Rational Numbers: Level 4 Challenges


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 \(f\) be a function defined along the rational numbers such that \(f(\tfrac mn)=\tfrac1n\) for all relatively prime positive integers \(m\) and \(n\). The product of all rational numbers \(0<x<1\) such that \[f\left(\dfrac{x-f(x)}{1-f(x)}\right)=f(x)+\dfrac9{52}\] can be written in the form \(\tfrac pq\) for positive relatively prime integers \(p\) and \(q\). Find \(p+q\).

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 \(k\).


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