# Algebra

 1. (5 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. (13 p.) Real numbers $$x,y,z$$ are real numbers greater than 1 and $$w$$ is a positive real number. If $$\log_xw=24$$, $$\log_yw=40$$ and $$\log_{xyz}w=12$$, find $$\log_zw$$.

 3. (54 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 $$a_{513}$$.

 4. (18 p.) Let $$a$$ be the coefficient of $$x^2$$ in the polynomial $(1-x)(1+2x)(1-3x)\dots (1+14x)(1-15x).$ Determine $$|a|$$

 5. (8 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).

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