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, 1492=22201 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 ff be a function defined along the rational numbers such that f(mn)=1nf(\tfrac mn)=\tfrac1n for all relatively prime positive integers mm and nn. The product of all rational numbers 0<x<10<x<1 such that f(xf(x)1f(x))=f(x)+952f\left(\dfrac{x-f(x)}{1-f(x)}\right)=f(x)+\dfrac9{52} can be written in the form pq\tfrac pq for positive relatively prime integers pp and qq. Find p+qp+q.

Let ab\displaystyle \frac ab be the smallest of the fractions that are greater than 1315\displaystyle \frac {13}{15} where a,b<500 a,b < 500 are positive integers.

What is the value of a+b a+b?

Let aka_k represent the repeating decimal 0.133k0.\overline{133}_k for k4k \geq 4. The product a4a5a99a_4 a_5 \cdots a_{99} can be expressed as mn!\frac{m}{n!} where m,nm, n are positive integers and nn is as small as possible. mn\frac{m}{n} can be expressed as pq \frac{p}{q} where p,qp, q are coprime integers. What is p+qp+q?

Note: 0.133k0.\overline{133}_k refers to the repeating decimal 0.133133133 0.133133133\ldots evalauted in base kk.


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