Virginia Tech Regional Mathematics Contest
From 9:00a.m. to 11:30a.m., October 28, 1995
Fill out the individual registration form
∫03∫021/(1 + (max(3x, 2y))2 dxdy).
R2 denote the xy-plane, and define
θ : R2 --> R2 by
θ(x, y) = (4x - 3y + 1, 2x - y + 1). Determine
θ100(1, 0), where
θ 100 times.
n≥2 be a positive integer and let f (x) be the
1 - (x + x2 + ... + xn) + (x + x2 + ... + xn)2 - ... + (- 1)n(x + x2 + ... + xn)n.
If r is an integer such that
2≤r≤n, show that the
coefficient of xr in f (x) is zero.
τ = (1 + √5)/2. Show that
[τ2n] = [τ[τn] + 1] for every positive integer n. Here [r]
denotes the largest integer that is not larger than r.
R denote the real numbers, and let
θ : R --> R be a map with the property that
x > y implies
(θ(x))3 > θ(y). Prove that
θ(x) > - 1 for all x, and that
0≤θ(x)≤1 for at most one
value of x.
- A straight rod of length 4 inches has ends which are allowed to
slide along the perimeter of a square whose sides each have length 12
inches. A paint brush is attached to the rod so that it can slide
between the two ends of the rod. Determine the total possible area
of the square which can be painted by the brush.
- If n is a positive integer larger than 1, let
n = ∏pkii be the unique prime factorization of n, where the pi's
are distinct primes, 2, 3, 5, 7, 11, ..., and define f (n) by
f (n) = ∑kipi and g(n) by
g(n) = limm--> ∞fm(n), where fm is meant the m-fold
application of f. Then n is said to have property H
n/2 < g(n) < n.
- Evaluate g(100) and
- Find all positive odd integers larger than 1 that have property