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Algebra
1.
(17 p.)
Let \( a \) be the coefficient of \( x^2 \) in the polynomial \[ (1x)(1+2x)(13x)\dots (1+14x)(115x).\] Determine \( a \)
2.
(7 p.)
Let \( P \) be the product of the nonreal roots of the polynomial \( x^44x^3+6x^24x=2008 \). Evaluate \( [ P] \).
3.
(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 \( p+q \).
4.
(23 p.)
Let \( a \), \( b \), \( c \), \( d \) be the roots of \( x^4  x^3  x^2  1 = 0 \). Find \( p(a) + p(b) + p(c) + p(d) \), where \( p(x) = x^6  x^5  x^3  x^2  x \).
5.
(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 \( m+n \).
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