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[hide]AIME 2000 II
Problem 13
The equation has exactly two real roots, one of which is , where , and are integers, and are relatively prime, and . Find .
Problem 14
Every positive integer has a unique factorial base expansion , meaning that , where each is an integer, , and . Given that is the factorial base expansion of , find the value of .
Problem 15
Find the least positive integer such that
AIME 2001 II
Problem 13
In quadrilateral , and , , , and . The length may be written in the form , where and are relatively prime positive integers. Find .
Problem 14
There are complex numbers that satisfy both and . These numbers have the form , where and angles are measured in degrees. Find the value of .
Problem 15
Let , , and be three adjacent square faces of a cube, for which , and let be the eighth vertex of the cube. Let , , and , be the points on , , and , respectively, so that . A solid is obtained by drilling a tunnel through the cube. The sides of the tunnel are planes parallel to , and containing the edges, , , and . The surface area of , including the walls of the tunnel, is , where , , and are positive integers and is not divisible by the square of any prime. Find .
AIME 2002 II
Problem 13
In triangle , point is on with and , point is on with and , , and and intersect at . Points and lie on so that is parallel to and is parallel to . It is given that the ratio of the area of triangle to the area of triangle is , where and are relatively prime positive integers. Find .
Problem 14
The perimeter of triangle is , and the angle is a right angle. A circle of radius with center on is drawn so that it is tangent to and . Given that where and are relatively prime positive integers, find .
Problem 15
Circles and intersect at two points, one of which is , and the product of the radii is . The x-axis and the line , where , are tangent to both circles. It is given that can be written in the form , where , , and are positive integers, is not divisible by the square of any prime, and and are relatively prime. Find .