# Algebra

 1. (9 p.) Let $$a$$, $$b$$, and $$c$$ be non-real roots of the polynimal $$x^3+x-1$$. Find $\frac{1+a}{1-a}+ \frac{1+b}{1-b}+ \frac{1+c}{1-c}.$

 2. (22 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 $$p+q$$.

 3. (24 p.) Consider the polynomial $P(x)=(1 + x + x^2 + \dots + x^{17})^2 - x^{17}.$ Assume that the roots of $$P$$ are $$x_k=r_k \cdot e^{i2\pi a_k}$$, for $$k = 1, 2, ... , 34$$, $$0 < a_1 \leq a_2 \leq \dots \leq a_{34} < 1$$, and some positive real numbers $$r_k$$. The sum $$a_1 + a_2 + a_3 + a_4 + a_5$$ is equal to $$p/q$$ for two coprime integers $$p$$ and $$q$$. Determine $$p+q$$.

 4. (15 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. (28 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 $$u+v$$.

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