2nd Annual
Virginia Tech Regional Mathematics Contest
From 9:30a.m. to 12:00 noon, November 8, 1980

Fill out the individual registration form

1. Let * denote a binary operation on a set S with the property that

(w*x)*(y*z) = w*z    for allw, x, y, z S.

Show
(a)
If a*b = c, then c*c = c.

(b)
If a*b = c, then a*x = c*x for all x S.

2. The sum of the first n terms of the sequence

1,    (1 + 2),    (1 + 2 + 22),...,(1 + 2 + ... + 2k - 1),...

is of the form 2n + R + Sn2 + Tn + U for all n > 0. Find R, S, T and U.

3. Let an = (1·3·5·...·(2n - 1))/(2·4·6·...·2n).
(a)
Prove that limn - > an exists.

(b)
Show that an = ((1 - (1/2)2)(1 - (1/4)2)...(1 - (1/2n)2))/((2n + 1)an).

(c)

4. Let P(x) be any polynomial of degree at most 3. It can be shown that there are numbers x1 and x2 such that P(x) dx = P(x1) + P(x2), where x1 and x2 are independent of the polynomial P.
(a)
Show that x1 = - x2.

(b)
Find x1 and x2.

5. For x > 0, show that ex < (1 + x)1 + x.

6. Given the linear fractional transformation of x into f1(x) = (2x - 1)/(x + 1), define fn + 1(x) = f1(fn(x)) for n = 1, 2, 3,.... It can be shown that f35 = f5. Determine A, B, C, and D so that f28(x) = (Ax + B)/(Cx + D).

7. Let S be the set of all ordered pairs of integers (m, n) satisfying m > 0 and n < 0. Let < be a partial ordering on S defined by the statement: (m, n) < (m', n') if and only if mm' and nn'. An example is (5, - 10) < (8, - 2). Now let O be a completely ordered subset of S, i.e. if (a, b) O and (c, d ) O, then (a, b) < (c, d ) or (c, d ) < (a, b). Also let O denote the collection of all such completely ordered sets.
(a)
Determine whether an arbitrary O O is finite.

(b)
Determine whether the cardinality || O|| of O is bounded for O O.

(c)
Determine whether || O|| can be countably infinite for any O O.

8. Let z = x + iy be a complex number with x And y rational and with | z| = 1.
(a)
Find two such complex numbers.

(b)
Show that | z2n - 1| = 2| sinnq|, where z = eiq.

(c)
Show that | z2n - 1| is rational for every n.

Peter Linnell
2001-08-17