IMOmath

Number Theory

1. (55 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. (2 p.)
The square \( \begin{array}{|c|c|c|} \hline x&20&151 \\\hline 38 & & \\ \hline & & \\ \hline\end{array} \) is magic, i.e. in each cell there is a number so that the sums of each row and column and of the two main diagonals are all equal. Find \( x \).

3. (33 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.

4. (2 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 \( |n-n^{\prime}| \)?

5. (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 \( a + b + c \).





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