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

3. (18 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)

4. (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 \( a + b + c + d \).

5. (24 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.





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