IMOmath

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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