MathDB

25

Part of 2008 AMC 12/AHSME

Problems(2)

Sequence of Points

Source: AMC 12 2008A #25

2/17/2008
A sequence (a1,b1) (a_1,b_1), (a2,b2) (a_2,b_2), (a3,b3) (a_3,b_3), … \ldots of points in the coordinate plane satisfies (a_{n \plus{} 1}, b_{n \plus{} 1}) \equal{} (\sqrt {3}a_n \minus{} b_n, \sqrt {3}b_n \plus{} a_n)\hspace{3ex}\text{for}\hspace{3ex} n \equal{} 1,2,3,\ldots. Suppose that (a_{100},b_{100}) \equal{} (2,4). What is a_1 \plus{} b_1?
<spanclass=′latex−bold′>(A)</span>minus1297<spanclass=′latex−bold′>(B)</span>minus1299<spanclass=′latex−bold′>(C)</span> 0<spanclass=′latex−bold′>(D)</span> 1298<spanclass=′latex−bold′>(E)</span> 1296 <span class='latex-bold'>(A)</span>\\minus{} \frac {1}{2^{97}} \qquad <span class='latex-bold'>(B)</span>\\minus{} \frac {1}{2^{99}} \qquad <span class='latex-bold'>(C)</span>\ 0 \qquad <span class='latex-bold'>(D)</span>\ \frac {1}{2^{98}} \qquad <span class='latex-bold'>(E)</span>\ \frac {1}{2^{96}}
analytic geometrygeometrygeometric transformationdilationrotationtrigonometrygeometric sequence
Trapezoid + Bisectors

Source: AMC 12 2008B Problem 25

2/28/2008
Let ABCD ABCD be a trapezoid with AB∥CD AB\parallel{}CD, AB\equal{}11, BC\equal{}5, CD\equal{}19, and DA\equal{}7. Bisectors of ∠A \angle A and ∠D \angle D meet at P P, and bisectors of ∠B \angle B and ∠C \angle C meet at Q Q. What is the area of hexagon ABQCDP ABQCDP? <spanclass=′latex−bold′>(A)</span> 283<spanclass=′latex−bold′>(B)</span> 303<spanclass=′latex−bold′>(C)</span> 323<spanclass=′latex−bold′>(D)</span> 353<spanclass=′latex−bold′>(E)</span> 363 <span class='latex-bold'>(A)</span>\ 28\sqrt{3}\qquad <span class='latex-bold'>(B)</span>\ 30\sqrt{3}\qquad <span class='latex-bold'>(C)</span>\ 32\sqrt{3}\qquad <span class='latex-bold'>(D)</span>\ 35\sqrt{3}\qquad <span class='latex-bold'>(E)</span>\ 36\sqrt{3}
geometrytrapezoidrectangletrigonometrycircumcirclesymmetryratio