Number Theory

1. (17 p.)
Let \( a \), \( b \), and \( c \) be non-real roots of the polynimal \( x^3+x-1 \). Find \[ \frac{1+a}{1-a}+ \frac{1+b}{1-b}+ \frac{1+c}{1-c}.\]

2. (28 p.)
Let \( 0 < a < b < c < d \) be integers such that \( a \), \( b \), \( c \) is an arithmetic progression, \( b \), \( c \), \( d \) is a geometric progression, and \( d - a = 30 \). Find \( a + b + c + d \).

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

4. (21 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)

5. (28 p.)
If the corresponding terms of two arithmetic progressions are multiplied we get the sequence 1440, 1716, 1848, ... . Find the eighth term of this sequence.





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