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Algebra
1.
(30 p.)
Consider the set \( S\subseteq(0,1]^2 \) in the coordinate plane that consists of all points \( (x,y) \) such that both \( [\log_2(1/x)] \) and \( [\log_5(1/y)] \) are even. The area of \( S \) can be written in the form \( p/q \) for two relatively prime integers \( p \) and \( q \). Evaluate \( p+q \).
2.
(12 p.)
Let \( a \), \( b \), and \( c \) be nonreal roots of the polynimal \( x^3+x1 \). Find \[ \frac{1+a}{1a}+ \frac{1+b}{1b}+ \frac{1+c}{1c}.\]
3.
(30 p.)
The sequence of complex numbers \( z_0,z_1,z_2,\dots \) is defined by \( z_0=1+i/211 \) and \( z_{n+1}=\frac{z_n+i}{z_ni} \). If \( z_{2111}=\frac ab+\frac cdi \) for positive integers \( a,b,c,d \) with \( \gcd(a,b)=\gcd(c,d)=1 \), find \( a+b+c+d \).
4.
(20 p.)
The equation \( 2^{333x2} + 2^{111x+2} = 2^{222x+1} + 1 \) has three real roots. Assume that their sum is expressed in the form \( \frac mn \) where \( m \) and \( n \) are relatively prime positive integers. Find \( m+n \).
5.
(7 p.)
Find the product of the real roots of the equation \( x^2+18x+30=2\sqrt{x^2+18x+45} \) (the answer is an integer).
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