Log In
Register
IMOmath
Olympiads
Book
Training
IMO Results
Forum
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 \( ba \) 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 nonsquare integer \( m \).
4.
(4 p.)
How many pairs of integers \( (x,y) \) are there such that \( x^2y^2=2400^2 \)?
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.
20052017
IMOmath.com
 imomath"at"gmail.com  Math rendered by
MathJax
Home

Olympiads

Book

Training

IMO Results

Forum

Links

About

Contact us