# Algebra

 1. (42 p.) Let $$a_1,a_2,...$$ be a sequence defined by $$a_1=1$$ and $a_{n+1}=\sqrt {a_n^2-2a_n+3}+1$ for $$n \ge 1$$. Find $$a_{513}$$.

 2. (10 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. (4 p.) The set $$A$$ consists of $$m$$ consecutive integers with sum $$2m$$. The set $$B$$ consists of $$2m$$ consecutive integers with sum $$m$$. The difference between the largest elements of $$A$$ and $$B$$ is 99. Find $$m$$.

 4. (27 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 $$p+q$$.

 5. (14 p.) The number $\frac1{2\sqrt1+1\sqrt 2}+\frac1{3\sqrt2+2\sqrt3}+\frac1{4\sqrt3+3\sqrt4} + \dots + \frac1{100\sqrt{99}+99\sqrt{100}}$ is a rational number. If it is expressed as $$\frac pq$$ for two relatively prime integers $$p$$ and $$q$$ evaluate $$p+q$$.

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