# General Practice Test

 1. (24 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)

 2. (44 p.) In a tournament club $$C$$ plays 6 matches, and for each match the probabilities of a win, draw and loss are equal. If the probability that $$C$$ finishes with more wins than losses is $$\frac pq$$ with $$p$$ and $$q$$ coprime $$(q>0)$$, find $$p+q$$.

 3. (4 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 $$|n-n^{\prime}|$$?

 4. (24 p.) Let $$X$$ be a square of side length 2. Denote by $$S$$ the set of all segments of length 2 with endpoints on adjacent sides of $$X$$. The midpoints of the segments in $$S$$ enclose a region with an area $$A$$. Find $$[100A]$$.

 5. (4 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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