SAT1000 - P923

Geometry Level 4

Find the value of ∑ k = 1 2 n ( sin ⁡ k π 2 n + 1 ) − 2 \displaystyle \sum_{k=1}^{2n} \left(\sin \dfrac{k \pi}{2n+1}\right)^{-2} at n = 2020 n=2020 .

Let A A denote the value. Submit ⌊ A ⌋ \lfloor A \rfloor .

Have a look at my problem set: SAT 1000 problems


The answer is 5443226.

Alice Smith by Alice Smith , 20, China

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1 solution

2 n = 4040 = 2 + 673 × 6 2n=4040=2+673\times 6

⟹ ⌊ A ⌋ = ( 2 × 673 + 1 ) ( 6 × 673 + 2 ) + 2 × 673 = 5443226 \implies \lfloor A\rfloor =(2\times 673+1)(6\times 673+2)+2\times 673=\boxed {5443226} .

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