# Algebra

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

 2. (30 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 $$a^6+b^6$$.

 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.) Find the minimum value of $$\frac{9x^2\sin^2x+4}{x\sin x}$$ for $$0< x< \pi$$.

 5. (10 p.) Let $$a$$, $$b$$, and $$c$$ be non-real roots of the polynimal $$x^3+x-1$$. Find $\frac{1+a}{1-a}+ \frac{1+b}{1-b}+ \frac{1+c}{1-c}.$

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