IMOmath

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. (37 p.)
Let \( a \) and \( b \) be positive real numbers such that \( ab=2 \) and \[\dfrac{a}{a+b^2}+\dfrac{b}{b+a^2}=\dfrac78.\] Find \( a^6+b^6 \).

3. (12 p.)
Real numbers \( x,y,z \) are real numbers greater than 1 and \( w \) is a positive real number. If \( \log_xw=24 \), \( \log_yw=40 \) and \( \log_{xyz}w=12 \), find \( \log_zw \).

4. (7 p.)
Let \( P \) be the product of the non-real roots of the polynomial \( x^4-4x^3+6x^2-4x=2008 \). Evaluate \( [ P] \).

5. (12 p.)
Let \( a \), \( b \), and \( c \) be non-real roots of the polynimal \( x^3+x-1 \). Find \[ \frac{1+a}{1-a}+ \frac{1+b}{1-b}+ \frac{1+c}{1-c}.\]





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