IMOmath

Number Theory

1. (3 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 \( x \).

2. (24 p.)
Let \( 0 < a < b < c < d \) be integers such that \( a \), \( b \), \( c \) is an arithmetic progression, \( b \), \( c \), \( d \) is a geometric progression, and \( d - a = 30 \). Find \( a + b + c + d \).

3. (36 p.)
Let \( \tau (n) \) denote the number of positive divisors of \( n \), including 1 and \( n \). Define \( S(n) \) by \( S(n)=\tau(1)+ \tau(2) + \dots + \tau(n) \). Let \( a \) denote the number of positive integers \( n \leq 2008 \) with \( S(n) \) odd, and let \( b \) denote the number of positive integers \( n \leq 2008 \) with \( S(n) \) even. Find \( |a-b| \).

4. (6 p.)
How many pairs of integers \( (x,y) \) are there such that \( x^2-y^2=2400^2 \)?

5. (30 p.)
Find the largest possible integer \( n \) such that \( \sqrt n + \sqrt{n+60} = \sqrt m \) for some non-square integer \( m \).





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