MathDB

17

Part of 2020 AMC 12/AHSME

Problems(2)

Naturally Logarithmic Quadrilateral

Source: 2020 AMC 12A #17

1/31/2020
The vertices of a quadrilateral lie on the graph of y=ln⁡xy = \ln x, and the xx-coordinates of these vertices are consecutive positive integers. The area of the quadrilateral is ln⁡9190\ln \frac{91}{90}. What is the xx-coordinate of the leftmost vertex?
<spanclass=′latex−bold′>(A)</span> 6<spanclass=′latex−bold′>(B)</span> 7<spanclass=′latex−bold′>(C)</span> 10<spanclass=′latex−bold′>(D)</span> 12<spanclass=′latex−bold′>(E)</span> 13<span class='latex-bold'>(A)</span>\ 6\qquad<span class='latex-bold'>(B)</span>\ 7\qquad<span class='latex-bold'>(C)</span>\ 10\qquad<span class='latex-bold'>(D)</span>\ 12\qquad<span class='latex-bold'>(E)</span>\ 13
2020 AMC 12A2020 AMCquadrilateralcoordinateAMC 12AMC
cis(120) * r is also a root

Source: 2020 AMC 12B #17

2/6/2020
How many polynomials of the form x5+ax4+bx3+cx2+dx+2020x^5 + ax^4 + bx^3 + cx^2 + dx + 2020, where aa, bb, cc, and dd are real numbers, have the property that whenever rr is a root, so is −1+i32⋅r\frac{-1+i\sqrt{3}}{2} \cdot r? (Note that i=−1i=\sqrt{-1})
<spanclass=′latex−bold′>(A)</span>0<spanclass=′latex−bold′>(B)</span>1<spanclass=′latex−bold′>(C)</span>2<spanclass=′latex−bold′>(D)</span>3<spanclass=′latex−bold′>(E)</span>4<span class='latex-bold'>(A) </span> 0 \qquad <span class='latex-bold'>(B) </span>1 \qquad <span class='latex-bold'>(C) </span> 2 \qquad <span class='latex-bold'>(D) </span> 3 \qquad <span class='latex-bold'>(E) </span> 4
algebrapolynomial2020 AMC 12B2020 AMCAMC 12AMC 12 BAMC