# Algebra

 1. (16 p.) Let $$a$$ be the coefficient of $$x^2$$ in the polynomial $(1-x)(1+2x)(1-3x)\dots (1+14x)(1-15x).$ Determine $$|a|$$

 2. (47 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 $2015f(1^2)f(2^2)\cdots f(n^2)\geq \Big(f(1)f(2)\cdots f(n)\Big)^2.$

 3. (11 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. (7 p.) Let $$P$$ be the product of the non-real roots of the polynomial $$x^4-4x^3+6x^2-4x=2008$$. Evaluate $$[ P]$$.

 5. (16 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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