# Algebra

 1. (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 $$m$$.

 2. (32 p.) Let $$a$$ and $$b$$ be positive real numbers such that $$ab=2$$ and $\dfrac{a}{a+b^2}+\dfrac{b}{b+a^2}=\dfrac78.$ Find $$a^6+b^6$$.

 3. (10 p.) Let $$a$$, $$b$$, and $$c$$ be non-real roots of the polynimal $$x^3+x-1$$. Find $\frac{1+a}{1-a}+ \frac{1+b}{1-b}+ \frac{1+c}{1-c}.$

 4. (32 p.) A sequence $$x_n$$ of real numbers satisfies $$x_0=0$$ and $$|x_{n}|=|x_{n-1}+1|$$ for $$n\geq 1$$. Find the minimal value of $$|x_1+x_2+\dots+ x_{2008}|$$.

 5. (19 p.) Let $$a$$, $$b$$, $$c$$, $$d$$ be the roots of $$x^4 - x^3 - x^2 - 1 = 0$$. Find $$p(a) + p(b) + p(c) + p(d)$$, where $$p(x) = x^6 - x^5 - x^3 - x^2 - x$$.

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