Algebra

 1. (3 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.) 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 $$p+q$$.

 3. (5 p.) Let $$P$$ be the product of the non-real roots of the polynomial $$x^4-4x^3+6x^2-4x=2008$$. Evaluate $$[ P]$$.

 4. (38 p.) Define a function $$f:\mathbb{Z}\to\mathbb{Z}$$ such that $$f(k)=k^2+k+1$$ for every integer $$k$$. Find the largest positive integer $$n$$ such that $2015f(1^2)f(2^2)\cdots f(n^2)\geq \Big(f(1)f(2)\cdots f(n)\Big)^2.$

 5. (38 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 $$a_{513}$$.

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