# Algebra

 1. (14 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 $$p(x) = x^6 - x^5 - x^3 - x^2 - x$$.

 2. (19 p.) Consider the set $$S\subseteq(0,1]^2$$ in the coordinate plane that consists of all points $$(x,y)$$ such that both $$[\log_2(1/x)]$$ and $$[\log_5(1/y)]$$ are even. The area of $$S$$ can be written in the form $$p/q$$ for two relatively prime integers $$p$$ and $$q$$. Evaluate $$p+q$$.

 3. (11 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 $$p+q$$.

 4. (31 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. (23 p.) Let $$a$$ and $$b$$ be positive real numbers such that $$ab=2$$ and $\dfrac{a}{a+b^2}+\dfrac{b}{b+a^2}=\dfrac78.$ Find $$a^6+b^6$$.

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