Problems(2)
Two distinct lines pass through the center of three concentric circles of radii 3, 2, and 1. The area of the shaded region in the diagram is 8/13 of the area of the unshaded region. What is the radian measure of the acute angle formed by the two lines? (Note: π radians is 180 degrees.)[asy]
defaultpen(linewidth(0.8));
pair O=origin;
fill(O--Arc(O, 2, 20, 160)--cycle, mediumgray);
fill(O--Arc(O, 1, 20, 160)--cycle, white);
fill(O--Arc(O, 2, 200, 340)--cycle, mediumgray);
fill(O--Arc(O, 1, 200, 340)--cycle, white);
fill(O--Arc(O, 3, 160, 200)--cycle, mediumgray);
fill(O--Arc(O, 2, 160, 200)--cycle, white);
fill(O--Arc(O, 1, 160, 200)--cycle, mediumgray);
fill(O--Arc(O, 3, -20, 20)--cycle, mediumgray);
fill(O--Arc(O, 2, -20, 20)--cycle, white);
fill(O--Arc(O, 1, -20, 20)--cycle, mediumgray);
draw(Circle(origin, 1));draw(Circle(origin, 2));draw(Circle(origin, 3));
draw(5*dir(200)--5*dir(20)^^5*dir(160)--5*dir(-20));[/asy]<spanclass=′latex−bold′>(A)</span>8π<spanclass=′latex−bold′>(B)</span>7π<spanclass=′latex−bold′>(C)</span>6π<spanclass=′latex−bold′>(D)</span>5π<spanclass=′latex−bold′>(E)</span>4π Let 1,4,⋯ and 9,16,⋯ be two arithmetic progressions. The set S is the union of the fi rst 2004 terms of each sequence. How many distinct numbers are in S?
<spanclass=′latex−bold′>(A)</span> 3722<spanclass=′latex−bold′>(B)</span> 3732<spanclass=′latex−bold′>(C)</span> 3914<spanclass=′latex−bold′>(D)</span> 3924<spanclass=′latex−bold′>(E)</span> 4007 modular arithmetic