# General Practice Test

 1. (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)

 2. (3 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 $$|n-n^{\prime}|$$?

 3. (33 p.) A circle of radius 1 is randomly placed inside a $$15 \times 36$$ rectangle $$ABCD$$. The probability that it does not intersect the diagonal $$AC$$ can be expressed as $$p/q$$ where $$p$$ and $$q$$ are relatively prime integers. Find $$p+q$$.

 4. (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 $$S$$.

 5. (36 p.) There are 27 candidates in elections and $$n$$ citizens that vote for them. If a candidate gets $$m$$ votes, then $$100m/n \leq m-1$$. What is the smallest possible value of $$n$$?

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