2006 SMT/Geometry Problems
Contents
[hide]Problem 1
Given a cube, determine the ratio of the volume of the octahedron formed by connecting the centers of each face of the cube to the volume of the cube.
Problem 2
Given square of side length , with on and in the interior of the square so that and , find the area of quadrilateral .
Problem 3
Circle is centered at with radius . Circle is externally tangent to circle and tangent to the axis. Find an equation, solved for if possible, for the locus of possible centers of circle .
Problem 4
The distance is . Find the area of the locus of points such that and is on the same side of line as a given point .
Problem 5
Let denote a set of points . We define the shadow of to be the set of points for which there exists a real number such that is in . For example, the shadow of a sphere with radius centered on the axis is a circle in the plane centered at the origin with radius . Suppose a cube has a shadow consisting of a regular hexagon with area . What is the side length of the cube?
Problem 6
A circle of radius is placed tangent to two perpendicular lines. Another circle is placed tangent to the same two lines and the first circle. In terms of , what is the radius of a third circle that is tangent to one line and tangent to both other circles?
Problem 7
A certain by pool table has pockets, denoted as shown. A pool player strikes a ball at point , of the way up side , aiming for a point up the opposite side of the table. He makes his mark, and the ball ricochets around the edges of the table until it finally lands in one of the pockets. How many times does it ricochet before it falls into a pocket, and which pocket? Write your answer in the form .
Problem 8
In triangle , the altitudes from and measure and , respectively. Find .
Problem 9
Poles , and are vertical line segments with bases on the axis. The tops of poles and are and , respectively. A string contains and and intersects another string connecting and at point . Pole is constructed with as its top point. For each integer , pole is constructed so that its top point is the intersection of and the line segment connecting the base of (on the axis) and the top of pole . Find the height of pole .
Problem 10
In triangle , points and lie on sides and , respectively, so that . If the area of is , determine the area of the triangle formed by the points of intersection of lines and .