IMOmath

Number Theory

1. (36 p.)
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 \( p+q \).

2. (7 p.)
Find the sum of all positive integers of the form \( n = 2^a3^b \) \( (a, b \geq 0) \) such that \( n^6 \) does not divide \( 6^n \).

3. (36 p.)
Suppose \( m \) and \( n \) are positive integers with \( m> 1 \) such that the domain of the function \( f(x) = \text{arcsin}(\log_{m}(nx)) \) is a closed interval of length \( \frac{1}{2013} \). Let \( S \) be the smallest possible value of \( m+n \). Find the remainder when \( S \) is divided by \( 1000 \).

4. (11 p.)
If the corresponding terms of two arithmetic progressions are multiplied we get the sequence 1440, 1716, 1848, ... . Find the eighth term of this sequence.

5. (8 p.)
Determine the number of positive integers with exactly three proper divisors each of which is less than 50. (1 is a proper divisor of every integer greater than 1)





2005-2017 IMOmath.com | imomath"at"gmail.com | Math rendered by MathJax
Home | Olympiads | Book | Training | IMO Results | Forum | Links | About | Contact us