Virginia Tech Regional Mathematics Contest

- Prove that
∫
_{0}^{1}∫_{x2}^{1}*e*^{y3/2}*dydx*= (2*e*- 2)/3. - Prove that if
*f*:**R**-->**R**is continuous and*f*(*x*) = ∫_{0}^{x}*f*(*t*)*dt*, then*f*(*x*) is identically zero. - Let
*f*_{1}(*x*) =*x*and*f*_{n+1}(*x*) =*x*^{fn(x)}, for*n*= 1, 2.... Prove that*f*_{n}'(1) = 1 and*f*_{n}''(1) = 2, for all*n*≥2. - Prove that a triangle in the plane whose vertices have integer
coordinates cannot be equilateral.
- Find
∑
_{n=1}^{∞}3^{-n}/*n*. - Let
*f*:**R**^{2}-->**R**^{2}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 feet.

- 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