Problems(3)
Each of the small circles in the figure has radius one. The innermost circle is tangent to the six circles that surround it, and each of those circles is tangent to the large circle and to its small-circle neighbors. Find the area of the shaded region.
[asy]unitsize(.3cm);
defaultpen(linewidth(.8pt));
path c=Circle((0,2),1);
filldraw(Circle((0,0),3),grey,black);
filldraw(Circle((0,0),1),white,black);
filldraw(c,white,black);
filldraw(rotate(60)*c,white,black);
filldraw(rotate(120)*c,white,black);
filldraw(rotate(180)*c,white,black);
filldraw(rotate(240)*c,white,black);
filldraw(rotate(300)*c,white,black);[/asy]<spanclass=′latex−bold′>(A)</span> π<spanclass=′latex−bold′>(B)</span> 1.5π<spanclass=′latex−bold′>(C)</span> 2π<spanclass=′latex−bold′>(D)</span> 3π<spanclass=′latex−bold′>(E)</span> 3.5π geometry
Circles of radius 2 and 3 are externally tangent and are circumscribed by a third circle, as shown in the figure. Find the area of the shaded region.
[asy]unitsize(3mm);
defaultpen(linewidth(0.7)+fontsize(8));filldraw(Circle((0,0),5),grey,black);
filldraw(Circle((-2,0),3),white,black);
filldraw(Circle((3,0),2),white,black);
dot((-2,0));
dot((3,0));
draw((-2,0)--(1,0));
draw((3,0)--(5,0));
label("3",(-0.5,0),N);
label("2",(4,0),N);[/asy]
<spanclass=′latex−bold′>(A)</span> 3π<spanclass=′latex−bold′>(B)</span> 4π<spanclass=′latex−bold′>(C)</span> 6π<spanclass=′latex−bold′>(D)</span> 9π<spanclass=′latex−bold′>(E)</span> 12π geometry
Let (an)n≥1 be a sequence such that a1=1 and 3an+1−3an=1 for all n≥1. Find a2002.<spanclass=′latex−bold′>(A)</span>666<spanclass=′latex−bold′>(B)</span>667<spanclass=′latex−bold′>(C)</span>668<spanclass=′latex−bold′>(D)</span>669<spanclass=′latex−bold′>(E)</span>670 induction