Problem 1. The length of the sides of a right triangle are three consecutive terms of an arithmetic progression. Prove that the lengths are in the ratio 3 : 4 : 5.

Problem 2. Consider all line segments of length 4 with one endpoint on the liney=xand the other endpoint on the liney= 2x. Find the equation of the locus of the midpoints of these line segments.

Problem 3. A rectangle is dissected as shown in Figure 1, with some of the lengths indicated. If the pieces are rearranged to form a square, what...

Problem 1. The three lengths are of the forma−d,a,a+dwith (a−d)²+a²= (a+d)². This reduces toa(a− 4d) = 0, ora= 4d. Thus the sides are 3d, 4d, 5d.

Problem 2. LetA(a, 2a) be an arbitrary point on the liney= 2x, and letB(b,b) be an arbitrary point on the liney=x. The midpoint of segmentABis the point (x,y) with$x=\frac{a+b}{2}$and$y=\frac{2a+b}{2}$, and the length of the segmentABis 4. Hence (a...

While the ingenuity of the solver alone will unravel many of the problems in this collection, many others require elementary knowledge in some mathematical area. Herewith is a brief list of “tools” that may be needed; your textbook should provide fuller details of those facts you do not wish to (or cannot) verify yourself.

A 1. “rfactorial”

r! =r(r− 1) · … · 2 · 1

is the number of ways of listing (in order)rdistinct objects.

A 2. “nchooser”\[\left( \begin{matrix} n \\ r \\ \end{matrix} \right)=\frac{n(n-1)\ldots (n-r-1)}{1\cdot 2\cdot \ldots \cdot r}\]is the number of ways of choosing r objects from amongndistinct...