## Tape Around a Pipe

A long strip of insulating tape is to be wrapped around a hot water pipe. If the width of the strip is w and the diameter of the pipe is d, at what angle must the strip be placed at the edge of the pipe so that no overlapping and no spaces occur?

## Triangle Rotating in Square

Equilateral triangle EKP with side EK of length 2 inches is placed inside square EAMI with side of length 4 inches so that K is on side EA. The triangle is rotated clockwise about K, then P, and so on along the sides of the square until E, K, and P return to their original positions. The length of the path in inches traversed by vertex P is equal to:

1. 20/3
2. 32/3
3. 12
4. 40/3
5. 15

## Tangent Circles, Area of Shaded Region

A circle centered at A of radius r is externally tangent to a circle centered at B of radius 2r. Tangent lines are drawn from A to the larger circle with points of tangency at D and E respectively. Find the area of the shaded portion of the diagram.

## Yaks, Blats, and Blobs

Given the two hypotheses:

1. Some yaks are not blats and
2. No blats are blobs,

if “some” means “at least one,” we can conclude that:

1. Some yaks are not blobs.
2. Some blobs are not yaks.
3. No yak is a blob.
4. Some yaks are blobs.
5. I, II, III, and IV are not deducible from the given statements.

## Wind Farm

Denmark is a major manufacturer of energy-producing wind turbines. Suppose both the following statements are true:

1. Some wind turbines in Ohio are not from Denmark, but are American-built.
2. All of the wind turbines in Lorain County are indeed from Denmark.

From these two statements, which of the following can be concluded?

1. Some American-built wind turbines are in Lorain County.
2. All Lorain county wind turbines are not American-built.
3. Some American-built wind turbines are not in Lorain County.
4. No Lorain County wind turbine is American-built.
5. Some wind turbines in Ohio, but not in Lorain County, were built in Denmark.