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

3. (30 p.)
The sequence of complex numbers \( z_0,z_1,z_2,\dots \) is defined by \( z_0=1+i/211 \) and \( z_{n+1}=\frac{z_n+i}{z_n-i} \). 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 \( 2^{333x-2} + 2^{111x+2} = 2^{222x+1} + 1 \) 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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