# 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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