Weekly Contest #54

(Ended)

Standings

Q1.

Older television screens have an aspect ratio of $4: 3$. That is, the ratio of the width to the height is $4: 3$. The aspect ratio of many movies is not $4: 3$, so they are sometimes shown on a television screen by "letterboxing" - darkening strips of equal height at the top and bottom of the screen, as shown. Suppose a movie has an aspect ratio of $2: 1$ and is shown on an older television screen with a $27$-inch diagonal. What is the height, in inches, of each darkened strip?

[asy] unitsize(1mm); filldraw((0,0)--(21.6,0)--(21.6,2.7)--(0,2.7)--cycle,grey,black); filldraw((0,13.5)--(21.6,13.5)--(21.6,16.2)--(0,16.2)--cycle,grey,black); draw((0,0)--(21.6,0)--(21.6,16.2)--(0,16.2)--cycle); [/asy]

Answer must be a floating-point or integer value and precision error less than 10^-6 is allowed.


Editorial
Tags: Algebra Probability

Q2.

Triangle $ABC$, with sides of length $5$, $6$, and $7$, has one vertex on the positive $x$-axis, one on the positive $y$-axis, and one on the positive $z$-axis. Let $O$ be the origin. What is the volume of tetrahedron $OABC$?

Answer must be a floating-point or integer value and precision error less than 10^-6 is allowed.


Editorial
Tags: Algebra Geometry

Q3.

What is the reciprocal of $\frac{1}{2}+\frac{2}{3}$?

Answer must be a floating-point or integer value and precision error less than 10^-6 is allowed.


Editorial
Tags: Algebra Probability

Q4.

While Steve and LeRoy are fishing 1 mile from shore, their boat springs a leak, and water comes in at a constant rate of 10 gallons per minute. The boat will sink if it takes in more than 30 gallons of water. Steve starts rowing towards the shore at a constant rate of 4 miles per hour while LeRoy bails water out of the boat. What is the slowest rate, in gallons per minute, at which LeRoy can bail if they are to reach the shore without sinking?

Answer must be a floating-point or integer value and precision error less than 10^-6 is allowed.


Editorial
Tags: Algebra Probability