Scary Sums and Products!

Number Theory Level 4

Given that f ( x ) = x 4 + 10 x 3 + 35 x 2 + 50 x + 24 f(x)=x^4+10 x^3+35 x^2+50 x+24 and that S = ∏ i = 1 50 ( ∏ j = 1 50 ( f ( i ) − f ( j ) ) ) , S=\prod_{i=1}^{50} \left( \prod_{j=1}^{50} (f(i)-f(j)) \right), find the value of ( S ⋅ ( ∑ k = 1 f ( 50 ) f ( k ) ) ∑ k = 1 f ( 49 ) f ( k ) + f ( 1 ) ) ( m o d 1000 ) \left(\frac{S \cdot (\sum_{k=1}^{f(50)} f(k))}{\sum_{k=1}^{f(49)}f(k)}+f(1) \right) \pmod{1000}


The answer is 120.

Minimario Minimario by minimario minimario , 23, USA

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1 solution

Note that S S contains the term f ( 1 ) − f ( 1 ) = 0 f(1)-f(1)=0 , so S = 0 S=0 . Therefore, the desired sum is equal to f ( 1 ) ( m o d 1000 ) = 120 f(1) \pmod{1000}=\boxed{120}

7 Helpful 0 Interesting 1 Brilliant 0 Confused

same here.

MOHD FARAZ - 7 years, 1 month ago

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