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Algebra

# Polynomial Interpolation: Level 5 Challenges

Consider the following system of linear equations: $\begin{cases} 1&+&a&+&b&+&c&+&d&+&e&=&1,\\ 32&+&16a&+&8b&+&4c&+&2d&+&e&=&2,\\ 243&+&81a&+&27b&+&9c&+&3d&+&e&=&3,\\ 1024&+&256a&+&64b&+&16c&+&4d&+&e&=&4,\\ 3125&+&625a&+&125b&+&25c&+&5d&+&e&=&5.\\ \end{cases}$

Evaluate $\left|\dfrac{8bcd}{125ae}\right|.$

A polynomial $$f(x)$$ has degree $$8$$ and $$f(i)=2^i$$ for $$i=0,1,2,3,4,5,6,7,8.$$

Find $$f(9).$$

Let $$a,b,c$$ be complex numbers satisfying

$$(a+1)(b+1)(c+1) = 1$$

$$(a+2)(b+2)(c+2) = 2$$

$$(a+3)(b+3)(c+3) = 3$$

Find $$(a+4)(b+4)(c+4)$$.

Find the number of positive integers $$n$$ such that $$2 \leq n \leq 100$$ and there exists a polynomial $$f(x)$$ with real coefficients and degree $$<n$$ such that for all integers $$x,$$ $$f(x)$$ is an integer if and only if $$x$$ is not a multiple of $$n.$$

Suppose $$f(x)$$ is a degree $$8$$ polynomial such that $$f(2^i)=\frac{1}{2^i}$$ for all integers $$0 \leq i \leq 8$$. If $$f(0)= \frac{a}{b}$$, where $$a$$ and $$b$$ are coprime positive integers, what is the value of $$a+b$$?

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