# Five Hundred Mathematical Challenges

Edward J. Barbeau
Murray S. Klamkin
William O. J. Moser
Series: Spectrum
Edition: 1
Pages: 238
https://www.jstor.org/stable/10.4169/j.ctt13x0n6v

1. Front Matter
(pp. i-vi)
2. PREFACE
(pp. vii-viii)
E. Barbeau, M. Klamkin and W. Moser
(pp. ix-x)
4. PROBLEMS
(pp. 1-46)

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...

5. SOLUTIONS
(pp. 47-210)

Problem 1. The three lengths are of the formad,a,a+dwith (ad)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...

6. THE TOOL CHEST
(pp. 211-226)

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...

7. Index of Problems
(pp. 227-227)