MathDB

12

Part of 2023 AMC 12/AHSME

Problems(2)

cube cube cube cube cube

Source: 2023 AMC 12A #12

11/9/2023
What is the value of 23−12+43−33+63−53+⋯+183−173? 2^3 - 1^2 + 4^3 - 3^3 + 6^3 - 5^3 + \dots + 18^3 - 17^3?
<spanclass=′latex−bold′>(A)</span>2023<spanclass=′latex−bold′>(B)</span>2679<spanclass=′latex−bold′>(C)</span>2941<spanclass=′latex−bold′>(D)</span>3159<spanclass=′latex−bold′>(E)</span>3235<span class='latex-bold'>(A) </span> 2023 \qquad<span class='latex-bold'>(B) </span> 2679 \qquad<span class='latex-bold'>(C) </span> 2941 \qquad<span class='latex-bold'>(D) </span> 3159 \qquad<span class='latex-bold'>(E) </span> 3235
AMCAMC 122023 AMC2023 AMC 12Apattern
Operation Complex Numbers

Source: 2023 AMC 12B #12

11/15/2023
For complex numbers u=a+biu=a+bi and v=c+div=c+di, define the binary operation ⊗\otimes by u⊗v=ac+bdi.u\otimes v=ac+bdi. Suppose zz is a complex number such that z⊗z=z2+40z\otimes z=z^{2}+40. What is ∣z∣|z|?
<spanclass=′latex−bold′>(A)</span> 10<spanclass=′latex−bold′>(B)</span> 32<spanclass=′latex−bold′>(C)</span> 26<spanclass=′latex−bold′>(D)</span> 6<spanclass=′latex−bold′>(E)</span> 52<span class='latex-bold'>(A)</span>~\sqrt{10}\qquad<span class='latex-bold'>(B)</span>~3\sqrt{2}\qquad<span class='latex-bold'>(C)</span>~2\sqrt{6}\qquad<span class='latex-bold'>(D)</span>~6\qquad<span class='latex-bold'>(E)</span>~5\sqrt{2}
AMCAMC 12complex numbers2023 AMC2023 AMC 12B