# Algebra

 1. (5 p.) Let $$P$$ be the product of the non-real roots of the polynomial $$x^4-4x^3+6x^2-4x=2008$$. Evaluate $$[ P]$$.

 2. (34 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. (25 p.) A sequence $$x_n$$ of real numbers satisfies $$x_0=0$$ and $$|x_{n}|=|x_{n-1}+1|$$ for $$n\geq 1$$. Find the minimal value of $$|x_1+x_2+\dots+ x_{2008}|$$.

 4. (25 p.) Let $$f:\mathbb N\rightarrow\mathbb R$$ be the function defined by $$f(1) = 1$$, $$f(n) = n/10$$ if $$n$$ is a multiple of 10 and $$f(n) = n+1$$ otherwise. For each positive integer $$m$$ define the sequence $$x_1$$, $$x_2$$, $$x_3$$, ... by $$x_1 = m$$, $$x_{n+1} = f(x_n)$$. Let $$g(m)$$ be the smallest $$n$$ such that $$x_n = 1$$. (Examples: $$g(100) = 3$$, $$g(87) = 7$$.) Denote by $$N$$ be the number of positive integers $$m$$ such that $$g(m) = 20$$. The number of distinct prime factors of $$N$$ is equal to $$2^u\cdot v$$ for two non-negative integers $$u$$ and $$v$$ such that $$v$$ is odd. Determine $$u+v$$.

 5. (8 p.) Real numbers $$x,y,z$$ are real numbers greater than 1 and $$w$$ is a positive real number. If $$\log_xw=24$$, $$\log_yw=40$$ and $$\log_{xyz}w=12$$, find $$\log_zw$$.

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