IMOmath

Number Theory

1. (52 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 \( 1000 \).

2. (6 p.)
Let \( a \), \( b \), \( c \) be positive integers forming an increasing geometric sequence such that \( b-a \) is a square. If \( \log_6a + \log_6b + \log_6c = 6 \), find \( a + b + c \).

3. (20 p.)
Find the largest possible integer \( n \) such that \( \sqrt n + \sqrt{n+60} = \sqrt m \) for some non-square integer \( m \).

4. (4 p.)
How many pairs of integers \( (x,y) \) are there such that \( x^2-y^2=2400^2 \)?

5. (16 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.





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