# 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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