One hundred concentric circles with radii are drawn in a plane. The interior of the circle of radius 1 is colored red, and each region bounded by consecutive circles is colored either red or green, with no two adjacent regions the same color. The ratio of the total area of the green regions to the area of the circle of radius 100 can be expressed as where and are relatively prime positive integers. Find
Let the set Susan makes a list as follows: for each two-element subset of she writes on her list the greater of the set's two elements. Find the sum of the numbers on the list.
Given that and that find
Consider the set of points that are inside or within one unit of a rectangular parallelepiped (box) that measures 3 by 4 by 5 units. Given that the volume of this set is where and are positive integers, and and are relatively prime, find
The sum of the areas of all triangles whose vertices are also vertices of a 1 by 1 by 1 cube is where and are integers. Find
Point is on with and Point is not on so that and and are integers. Let be the sum of all possible perimeters of Find
In an increasing sequence of four positive integers, the first three terms form an arithmetic progression, the last three terms form a geometric progression, and the first and fourth terms differ by Find the sum of the four terms.
An integer between and inclusive, is called balanced if the sum of its two leftmost digits equals the sum of its two rightmost digits. How many balanced integers are there?
Triangle is isosceles with and Point is in the interior of the triangle so that and Find the number of degrees in
An angle is chosen at random from the interval Let be the probability that the numbers and are not the lengths of the sides of a triangle. Given that where is the number of degrees in and and are positive integers with find
In convex quadrilateral and The perimeter of is 640. Find (The notation means the greatest integer that is less than or equal to )
Let be the number of positive integers that are less than or equal to 2003 and whose base-2 representation has more 1's than 0's. Find the remainder when is divided by 1000.
The decimal representation of where and are relatively prime positive integers and contains the digits 2, 5, and 1 consecutively, and in that order. Find the smallest value of for which this is possible.
In and Let be the midpoint of and let be the point on such that bisects angle Let be the point on such that Suppose that meets at The ratio can be written in the form where and are relatively prime positive integers. Find