MathDB

22

Part of 2011 AMC 10

Problems(2)

Different colorings of pentagon

Source: 2011 AMC A Problem 22

6/25/2011
Each vertex of convex pentagon ABCDEABCDE is to be assigned a color. There are 66 colors to choose from, and the ends of each diagonal must have different colors. How many different colorings are possible?
<spanclass=′latex−bold′>(A)</span> 2520<spanclass=′latex−bold′>(B)</span> 2880<spanclass=′latex−bold′>(C)</span> 3120<spanclass=′latex−bold′>(D)</span> 3250<spanclass=′latex−bold′>(E)</span> 3750 <span class='latex-bold'>(A)</span>\ 2520\qquad<span class='latex-bold'>(B)</span>\ 2880\qquad<span class='latex-bold'>(C)</span>\ 3120\qquad<span class='latex-bold'>(D)</span>\ 3250\qquad<span class='latex-bold'>(E)</span>\ 3750
geometrygeometric transformationrotationsymmetryAMC
Volume of cube inscribed in pyramid

Source: AMC 10 2011 b Problem 22

2/24/2011
A pyramid has a square base with sides of length 1 and has lateral faces that are equilateral triangles. A cube is placed within the pyramid so that one face is on the base of the pyramid and its opposite face has all its edges on the lateral faces of the pyramid. What is the volume of this cube?
<spanclass=′latex−bold′>(A)</span> 52−7<spanclass=′latex−bold′>(B)</span> 7−43<spanclass=′latex−bold′>(C)</span> 2227<spanclass=′latex−bold′>(D)</span> 29<spanclass=′latex−bold′>(E)</span> 39 <span class='latex-bold'>(A)</span>\ 5\sqrt{2}-7 \qquad <span class='latex-bold'>(B)</span>\ 7-4\sqrt{3} \qquad <span class='latex-bold'>(C)</span>\ \frac{2\sqrt{2}}{27} \qquad <span class='latex-bold'>(D)</span>\ \frac{\sqrt{2}}{9} \qquad <span class='latex-bold'>(E)</span>\ \frac{\sqrt{3}}{9}
geometry3D geometrypyramidPythagorean Theoremsimilar trianglesAMC