MathDB

24

Part of 2002 AMC 12/AHSME

Problems(3)

Binomial raised to a large power

Source: AMC 12 A 2002 #24

8/1/2007
Find the number of ordered pairs of real numbers (a,b) (a,b) such that (a \plus{} bi)^{2002} \equal{} a \minus{} bi. <spanclass=′latex−bold′>(A)</span> 1001<spanclass=′latex−bold′>(B)</span> 1002<spanclass=′latex−bold′>(C)</span> 2001<spanclass=′latex−bold′>(D)</span> 2002<spanclass=′latex−bold′>(E)</span> 2004 <span class='latex-bold'>(A)</span>\ 1001\qquad <span class='latex-bold'>(B)</span>\ 1002\qquad <span class='latex-bold'>(C)</span>\ 2001\qquad <span class='latex-bold'>(D)</span>\ 2002\qquad <span class='latex-bold'>(E)</span>\ 2004
trigonometrygeometryAMCAIMEcalculusintegrationmodular arithmetic
quadrilateral area

Source:

2/26/2008
A convex quadrilateral ABCD ABCD with area 2002 2002 contains a point P P in its interior such that PA \equal{} 24, PB \equal{} 32, PC \equal{} 28, and PD \equal{} 45. FInd the perimeter of ABCD ABCD. (A)\ 4\sqrt {2002}\qquad (B)\ 2\sqrt {8465}\qquad (C)\ 2\left(48 \plus{} \sqrt {2002}\right) (D)\ 2\sqrt {8633}\qquad (E)\ 4\left(36 \plus{} \sqrt {113}\right)
geometryperimeterinequalitiestrigonometry
Distances from a point inside a Tetrahedron

Source:

4/6/2013
Let ABCDABCD be a regular tetrahedron and let EE be a point inside the face ABCABC. Denote by ss the sum of the distances from EE to the faces DABDAB, DBCDBC, DCADCA, and by SS the sum of the distances from EE to the edges ABAB, BCBC, CACA. Then sS\dfrac sS equals
<spanclass=′latex−bold′>(A)</span>2<spanclass=′latex−bold′>(B)</span>223<spanclass=′latex−bold′>(C)</span>62<spanclass=′latex−bold′>(D)</span>2<spanclass=′latex−bold′>(E)</span>3<span class='latex-bold'>(A) </span>\sqrt2\qquad<span class='latex-bold'>(B) </span>\dfrac{2\sqrt2}3\qquad<span class='latex-bold'>(C) </span>\dfrac{\sqrt6}2\qquad<span class='latex-bold'>(D) </span>2\qquad<span class='latex-bold'>(E) </span>3
geometry3D geometrytetrahedronprism