Polynomial Interpolation

Polynomial Interpolation: Level 5 Challenges


Consider the following system of linear equations:

{1+a+b+c+d+e=132+16a+8b+4c+2d+e=2243+81a+27b+9c+3d+e=31024+256a+64b+16c+4d+e=43125+625a+125b+25c+5d+e=5.\begin{cases} \begin{array}{rcrcrcrcrcrcrl} 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{array} \end{cases}


8bcd125ae. \left|\dfrac{8bcd}{125ae}\right|.

A polynomial f(x)f(x) has degree 88 and f(i)=2if(i)=2^i for i=0,1,2,3,4,5,6,7,8.i=0,1,2,3,4,5,6,7,8.

Find f(9).f(9).

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

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

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

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

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

Find the number of positive integers nn such that 2n1002 \leq n \leq 100 and there exists a polynomial f(x)f(x) with real coefficients and degree <n <n such that for all integers x,x, f(x)f(x) is an integer if and only if xx is not a multiple of n.n.

Suppose f(x)f(x) is a degree-88 polynomial such that f(2i)=12if(2^i)=\frac{1}{2^i} for all integers 0i8 0 \leq i \leq 8. If f(0)=abf(0)= \frac{a}{b}, where aa and b b are coprime positive integers, what is the value of a+b?a+b?


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