# 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. (40 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. (24 p.) The sequence of complex numbers $$z_0,z_1,z_2,\dots$$ is defined by $$z_0=1+i/211$$ and $$z_{n+1}=\frac{z_n+i}{z_n-i}$$. If $$z_{2111}=\frac ab+\frac cdi$$ for positive integers $$a,b,c,d$$ with $$\gcd(a,b)=\gcd(c,d)=1$$, find $$a+b+c+d$$.

 4. (18 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 $$p(x) = x^6 - x^5 - x^3 - x^2 - x$$.

 5. (14 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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