Virginia Tech Regional Mathematics Contest
From 9:00a.m. to 11:30a.m., October 30, 1993
Fill out the individual registration form
- Prove that
∫01∫x21ey3/2 dydx = (2e - 2)/3.
- Prove that if
f : R --> R is continuous
f (x) = ∫0xf (t) dt, then f (x) is identically zero.
f1(x) = x and
fn+1(x) = xfn(x), for
n = 1, 2.... Prove that
fn'(1) = 1 and
fn''(1) = 2, for all
- Prove that a triangle in the plane whose vertices have integer
coordinates cannot be equilateral.
f : R2 --> R2 be a surjective
map with the property that if the points A, B and C
are collinear, then so are f (A), f (B) and f (C).
Prove that f is bijective.
- On a small square billiard table with sides of length 2ft., a ball
is played from the center and after rebounding off the sides several
times, goes into a cup at one of the corners. Prove that the total
distance travelled by the ball is not an integer number of
- A popular Virginia Tech logo looks something like
Suppose that wire-frame copies of this logo are constructed of 5
equal pieces of wire welded at three places as shown:
If bending is allowed, but no re-welding, show clearly how to cut the
maximum possible number of ready-made copies of such a logo from the
piece of welded wire mesh shown. Also, prove that no larger number