# Algebra

 1. (9 p.) Find the product of the real roots of the equation $$x^2+18x+30=2\sqrt{x^2+18x+45}$$ (the answer is an integer).

 2. (9 p.) Let $$P$$ be the product of the non-real roots of the polynomial $$x^4-4x^3+6x^2-4x=2008$$. Evaluate $$[ P]$$.

 3. (48 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 $$|x_1+x_2+\dots+ x_{2008}|$$.

 4. (25 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. (6 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 $$m$$.

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