Math Is Fun Forum

Problem 1:  A rectangular box $P$ is inscribed in a sphere of radius $r$. The surface area of $P$ is 384, and the sum of the lengths of its 12 edges is 112. What is  $r$?

Problem 2:  The equation of the line passing through $(1,8)$ and $(5,6)$ can be expressed in the form
\[\frac{x}{a} + \frac{y}{b} = 1.\]
Find $a$.

Problem 3:  Let $P = (5,1)$, and let $Q$ be the reflection of $P$ over the line $y = \frac{1}{2} x + 2$. Find the coordinates of $Q$.

Problem 4:  The vertices of a triangle are the points of intersection of the line $y = -x-1$, the line $x=2$, and $y = \frac{1}{5}x+\frac{13}{5}$. Find an equation of the circle passing through all three vertices.

Problem 5:  In triangle $PQR$, we have $\angle P = 90^\circ$, $QR = 15$, and $\tan R = 5\cos Q$. What is $PQ$?

Problem 6:  Two circles of radius 1 are externally tangent at $Q$. Let $\overline{PQ}$ and $\overline{QR}$ be diameters of the two circles. From $P$ a tangent is drawn to the circle with diameter $\overline{QR}$, and from $R$ a parallel tangent is drawn to the circle with diameter $\overline{PQ}$. Find the distance between these two tangent lines.

Two circles are externally tangent at point $P$, as shown. Segment $\overline{CPD}$ is parallel to common external tangent $\overline{AB}$. Prove that the distance between the midpoints of $\overline{AB}$ and $\overline{CD}$ is $AB/2$.

Here is the diagram:  http://latex.artofproblemsolving.com/f/0/f/f0fccd0e6e157a2063fd2c451a6812eb1a4fa7f9.png