Virginia Tech Regional Mathematics Contest
From 9:00a.m. to 11:30a.m., October 24, 2009
Fill out the individual registration form
- A walker and a jogger travel along the same straight line in the same
direction. The walker walks at one meter per second, while the jogger
runs at two meters per second. The jogger starts one meter in front
of the walker. A dog starts with the walker, and then runs back and
forth between the walker and the jogger with constant speed of
three meters per second.
Let f (n) meters denote the total distance travelled by the dog when
it has returned to the walker for the nth time (so f (0) = 0).
Find a formula for f (n).
- Given that 40! =
abc def 283 247 897 734 345 611 269 596 115 894 272 pqr stu
p, q, r, s, t, u, v, w, x, and then find
a, b, c, d, e, f.
f (x) = ∫0x∫0xeu2v2 dudv.
2f''(2) + f'(2) (here
f'(x) = df /dx).
- Two circles
α,β touch externally at the point X.
Let A, P be two distinct points on
α different from X, and
let AX and PX meet
β again in the points B and Q
respectively. Prove that AP is parallel to QB.
C denote the complex numbers and let
M3(C) denote the 3 by 3 matrices with entries in
B≠ 0, and
AB = 0 (where 0
denotes the 3 by 3 matrix with all entries zero). Prove that there
0≠D∈M3(C) such that
AD = DA = 0.
- Let n be a nonzero integer. Prove that
n4 -7n2 + 1 can never
be a perfect square (i.e. of the form m2 for some integer m).
- Does there exist a twice differentiable function
f : R -> R such that
f'(x) = f (x + 1) - f (x) for all x
f''(0)≠ 0? Justify your answer.
R denotes the real numbers and f' denotes
the derivative of f.)