MathDB

14

Part of 2021 AMC 12/AHSME Fall

Problems(2)

friendly reminder that equilateral hexagons are not regular

Source: 2021 Fall AMC 12A #14

11/11/2021
In the figure, equilateral hexagon ABCDEFABCDEF has three nonadjacent acute interior angles that each measure 30∘30^\circ. The enclosed area of the hexagon is 636\sqrt{3}. What is the perimeter of the hexagon? [asy] size(6cm); pen p=black+linewidth(1),q=black+linewidth(5); pair C=(0,0),D=(cos(pi/12),sin(pi/12)),E=rotate(150,D)*C,F=rotate(-30,E)*D,A=rotate(150,F)*E,B=rotate(-30,A)*F; draw(C--D--E--F--A--B--cycle,p); dot(A,q); dot(B,q); dot(C,q); dot(D,q); dot(E,q); dot(F,q); label("CC",C,2*S); label("DD",D,2*S); label("EE",E,2*S); label("FF",F,2*dir(0)); label("AA",A,2*N); label("BB",B,2*W); [/asy] (<spanclass=′latex−bold′>A</span>) 4(<spanclass=′latex−bold′>B</span>) 43(<spanclass=′latex−bold′>C</span>) 12(<spanclass=′latex−bold′>D</span>) 18(<spanclass=′latex−bold′>E</span>) 123(<span class='latex-bold'>A</span>)\: 4\qquad(<span class='latex-bold'>B</span>) \: 4\sqrt3\qquad(<span class='latex-bold'>C</span>) \: 12\qquad(<span class='latex-bold'>D</span>) \: 18\qquad(<span class='latex-bold'>E</span>) \: 12\sqrt3
AMCAMC 12AMC 12 A
Complex Roots

Source: 2021 Fall AMC 12B #14

11/17/2021
Suppose that P(z),Q(z)P(z), Q(z), and R(z)R(z) are polynomials with real coefficients, having degrees 22, 33, and 66, respectively, and constant terms 11, 22, and 33, respectively. Let NN be the number of distinct complex numbers zz that satisfy the equation P(z)⋅Q(z)=R(z)P(z) \cdot Q(z)=R(z). What is the minimum possible value of NN?
<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> 5<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> \: 5
AMCAMC 12AMC 12 B