# General Practice Test

 1. (46 p.) It is given that $$181^2$$ can be written as the difference of the cubes of two consecutive positive integers. Find the sum of these two integers.

 2. (6 p.) Let $$S$$ be the set of vertices of a unit cube. Find the number of triangles whose vertices belong to $$S$$.

 3. (34 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 $$p+q$$.

 4. (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 $$x$$.

 5. (9 p.) Let $$S = \{1, 2, 3, 5, 8, 13, 21, 34\}$$. Find the sum $$\sum \max(A)$$ where the sum is taken over all 28 two-element subsets $$A$$ of $$S$$.

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