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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 \( nn^{\prime} \)?
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 twoelement 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 \( 100m/n \leq m1 \). What is the smallest possible value of \( n \)?
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