IMOmath

Number Theory

1. (10 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 \( 6^n \).

2. (53 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 \( p+q \).

3. (21 p.)
Find the largest possible integer \( n \) such that \( \sqrt n + \sqrt{n+60} = \sqrt m \) for some non-square integer \( m \).

4. (12 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)

5. (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 \( x \).





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