IMOmath

Number Theory

1. (1 p.)
\( n \) is an integer between 100 and 999 inclusive, and \( n^{\prime} \) is the integer formed by reversing the digits of \( n \). How many possible values are for \( |n-n^{\prime}| \)?

2. (47 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 \( 1000 \).

3. (18 p.)
Find the largest possible integer \( n \) such that \( \sqrt n + \sqrt{n+60} = \sqrt m \) for some non-square integer \( m \).

4. (3 p.)
How many pairs of integers \( (x,y) \) are there such that \( x^2-y^2=2400^2 \)?

5. (28 p.)
It is given that \( 181^2 \) can be written as the difference of the cubes of two consecutive positive integers. Find the sum of these two integers.





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