15th Annual
Virginia Tech Regional Mathematics Contest
From 9:00a.m. to 11:30a.m., October 30, 1993

Fill out the individual registration form

1. Prove that 01x21ey3/2 dydx = (2e - 2)/3.

2. Prove that if f : R --> R is continuous and f (x) = ∫0xf (tdt, then f (x) is identically zero.

3. Let f1(x) = x and fn+1(x) = xfn(x), for n = 1, 2.... Prove that fn'(1) = 1 and fn''(1) = 2, for all n≥2.

4. Prove that a triangle in the plane whose vertices have integer coordinates cannot be equilateral.

5. Find n=13-n/n.

6. Let 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.

7. 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 feet.

8. 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 is possible.

Peter Linnell 2007-06-15