Vol. 10 No 6 - Pi Mu Epsilon

498 PI MU EPSILON JOURNAL
PROBLEMS AND SOLUTIONS 499
Solution by Heath Schulterman, student, Fort Smith Northside High
School, Barling, Arkansas.
From the quadratic formula we know that the solutions to the above
equation are
We cube these roots to see that
a± Ja 2 - 4
X= •
2
3 a 3 - 3a a 2 1 ~-=---
x = ± - ya 2 - 4.
2 2
If a is odd, then both 03 - 3a and a 2 - 1 are even numbers. Thus we may
take p = (03 - 3a)/2, q = (a 2 - 1)/2, and r = a 2 - 4, and p, q, and rare all
integers. If a is even, then p = (03 - 3a)/2 is still an integer. Now a 2 is
divisible by 4, so we take r = (a2 - 4)/4 and q = a 2 - 1, and both q and r
are integers. In either case we have shown that X3 can be written in the
desired form.
Also solved by Avraham Adler, Miguel Amengual Covas, Frank P.
Battles, Paul S. Bruckman, James Campbell, William Cbau, Lisa M. Croft,
Charles R. Diminnie, Russell Euler, George P. Evanovicb, Mark Evans,
Victor G. Feser, Jayanthi Ganapatby, Robert C. Gebhardt, S. Gendler,
Richard I. Hess, Joe Howard, Thomas C. Leong, Peter A. Lindstrom,
David E. Manes, Yosbinobu Murayoshi, William H. Peirce, Bob Prielipp,
John F. Putz, H.-J. Seiffert, Skidmore College Problems Group, Kenneth
M. Wilke, Rex H. Wu, Monte J. Zerger, and the Proposer.
879. [Spring 1996} Proposed by Barton L. Willis, University of
Nebraska at Kearney, Kearney, Nebraska.
A Mystery Space. LetS be a set of ordered pairs of elements. Define
binary operations + , *, and + on S by
and
(a, b) + (c, d) = (a + c, b + d),
(a, b)*(c, d) = (ac, ad + be),
(a, b) + (c, d) = (a + c, b + c - ad+ ~).
Although it might be fun to deduce properties of space S (commutativity,
associativity, etc.), the problem is to find an application for S.
I. Solution by PaulS. Bruclanan, Highwood, lllinois.
Let the ordered pair (a, b) denote the fraction bla. A problem occurs if
a = O, which may be circumvented by including a point at infinity [(0, b)
= oo if b ¢. 0, and (0, 0) is undefined} or by requiring that a be a positive
integer.
One more relation needs to be defined:
(ka, kb) = (a, b) for all k '#- 0.
The binary operation "+" is then defined as the operation whereby the
numerators and denominators of the two fractions, respectively, are added.
The result is called the mediant of the two given fractions, one application
of which is found in baseball batting averages. A batter with 10 hits in 40
times at bat bas a batting average of 1000 X 10/40 = 250. If in the next
game that batter gets 2 bits in 4 times at bat, his or her average for the
game is 1000 x 214 = 500 and the new accumulative batting ave~ge
becomes the mediant 1000(10 + 2)/(40 + 4) = 273 of the two prevtous
fractions.
The binary operations "*" and "+" represent respectively the sum and
difference of the two fractions. That is,
and
(a, b)"*" (c, d) = bla + die = (ad + bc)lac = (ac, ad+ be)
(a, b)"+" (c, d) = bla -die = (be- ad)lac = (b/c - adl~)l(alc)
= (ale, b/c- ad/~).
Of course, there is no unit under "+" (unless we allow (0, 0)), the unit _
under "*" is (1, 0), and since we have (a, b) "+" (1, 0) = (a, b), then (1, 0) •
is the right hand unit under "+ ".