2nd Annual
Virginia Tech Regional Mathematics Contest
From 9:30a.m. to 12:00 noon, November 8, 1980
Fill out the individual registration form
- Let * denote a binary operation on a set S with the property that
(
w*
x)*(
y*
z) =
w*
z for all
w,
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.
- The sum of the first n terms of the sequence
1, (1 + 2), (1 + 2 + 2^{2}),...,(1 + 2 + ... + 2^{k - 1}),...
is of the form
2^{n + R} + Sn^{2} + Tn + U for all n > 0. Find
R, S, T and U.
- Let
a_{n} = (1·3·5·...·(2n - 1))/(2·4·6·...·2n).
- (a)
- Prove that
lim_{n - > }a_{n} exists.
- (b)
- Show that
a_{n} = ((1 - (1/2)^{2})(1 - (1/4)^{2})...(1 - (1/2n)^{2}))/((2n + 1)a_{n}).
- (c)
- Find
lim_{n - > }a_{n} and justify your answer.
- Let P(x) be any polynomial of degree at most 3. It can be shown
that there are numbers x_{1} and x_{2} such that
P(x) dx = P(x_{1}) + P(x_{2}), where x_{1} and x_{2} are independent
of the polynomial P.
- (a)
- Show that
x_{1} = - x_{2}.
- (b)
- Find x_{1} and x_{2}.
- For x > 0, show that
e^{x} < (1 + x)^{1 + x}.
- Given the linear fractional transformation of x into
f_{1}(x) = (2x - 1)/(x + 1), define
f_{n + 1}(x) = f_{1}(f_{n}(x)) for
n = 1, 2, 3,.... It can be shown that
f_{35} = f_{5}. Determine
A, B, C, and D so that
f_{28}(x) = (Ax + B)/(Cx + D).
- 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.
- 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
| z^{2n} - 1| = 2| sinnq|, where
z = e^{iq}.
- (c)
- Show that
| z^{2n} - 1| is rational for every n.
Peter Linnell
2001-08-17