MathDB

12

Part of 2012 AMC 12/AHSME

Problems(2)

A Square Tangent to One Circle with Two Vertices On Another

Source: 2012 AMC 12A Problem #12

2/8/2012
A square region ABCDABCD is externally tangent to the circle with equation x2+y2=1x^2+y^2=1 at the point (0,1)(0,1) on the side CDCD. Vertices AA and BB are on the circle with equation x2+y2=4x^2+y^2=4. What is the side length of this square?
<spanclass=′latex−bold′>(A)</span> 10+510<spanclass=′latex−bold′>(B)</span> 255<spanclass=′latex−bold′>(C)</span> 223<spanclass=′latex−bold′>(D)</span> 219−45<spanclass=′latex−bold′>(E)</span> 9−175 <span class='latex-bold'>(A)</span>\ \frac{\sqrt{10}+5}{10}\qquad<span class='latex-bold'>(B)</span>\ \frac{2\sqrt{5}}{5}\qquad<span class='latex-bold'>(C)</span>\ \frac{2\sqrt{2}}{3}\qquad<span class='latex-bold'>(D)</span>\ \frac{2\sqrt{19}-4}{5}\qquad<span class='latex-bold'>(E)</span>\ \frac{9-\sqrt{17}}{5}
geometryAMC
A sequence of zeros and ones

Source: 2012 AMC 12B Problem 12

2/24/2012
How many sequences of zeros and ones of length 20 have all the zeros consecutive, or all the ones consecutive, or both?
<spanclass=′latex−bold′>(A)</span> 190<spanclass=′latex−bold′>(B)</span> 192<spanclass=′latex−bold′>(C)</span> 211<spanclass=′latex−bold′>(D)</span> 380<spanclass=′latex−bold′>(E)</span> 382 <span class='latex-bold'>(A)</span>\ 190\qquad<span class='latex-bold'>(B)</span>\ 192\qquad<span class='latex-bold'>(C)</span>\ 211\qquad<span class='latex-bold'>(D)</span>\ 380\qquad<span class='latex-bold'>(E)</span>\ 382
symmetryAMC