IMOmath

Algebra

1. (34 p.)
Let \( f:\mathbb N\rightarrow\mathbb R \) be the function defined by \( f(1) = 1 \), \( f(n) = n/10 \) if \( n \) is a multiple of 10 and \( f(n) = n+1 \) otherwise. For each positive integer \( m \) define the sequence \( x_1 \), \( x_2 \), \( x_3 \), ... by \( x_1 = m \), \( x_{n+1} = f(x_n) \). Let \( g(m) \) be the smallest \( n \) such that \( x_n = 1 \). (Examples: \( g(100) = 3 \), \( g(87) = 7 \).) Denote by \( N \) be the number of positive integers \( m \) such that \( g(m) = 20 \). The number of distinct prime factors of \( N \) is equal to \( 2^u\cdot v \) for two non-negative integers \( u \) and \( v \) such that \( v \) is odd. Determine \( u+v \).

2. (11 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. (27 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. (6 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. (18 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 \( m+n \).





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