Virginia Tech Regional Mathematics Contest
From 9:30a.m. to 12:00 noon, October 19, 1991
Fill out the individual registration form
- An isosceles triangle with an inscribed circle is labeled as shown in
the figure. Find an expression, in terms of the angle
the length a, for the area of the curvilinear triangle bounded by
sides AB and AC and the arc BC.
- Find all differentiable functions f which satisfy
f (x)3 = ∫0xf (t)2 dt for all real x.
- Prove that if
α is a real root of
(1 - x2)(1 + x + x2 + ... + xn) - x = 0 which lies in (0, 1), with
n = 1, 2,..., then
α is also a root of
(1 - x2)(1 + x + x2 + ... + xn+1) - 1 = 0.
- Prove that if x > 0 and n > 0, where x is real and n is
an integer, then
xn/((x + 1)n+1)≤nn/((n + 1)n+1).
f (x) = x5 -5x3 + 4x. In each part (i)-(iv), prove or
disprove that there exists a real number c for which
f (x) - c = 0
has a root of multiplicity
(i) one, (ii) two, (iii) three, (iv) four.
- Let a0 = 1 and for n > 0, let an be defined by
an = - ∑k=1nan-k/k!.
an = (- 1)n/n!, for
n = 0, 1, 2,....
- A and B play the following money game, where an and bn denote
the amount of holdings of A and B, respectively, after the nth
round. At each round a player pays one-half his holdings to the bank,
then receives one dollar from the bank if the other player had
less than c dollars at the end of the previous round. If
a0 = .5 and b0 = 0, describe the behavior of an and bn
when n is large, for
(i) c = 1.24 and (ii) c = 1.26.
- Mathematical National Park has a collection of trails. There are
designated campsites along the trails, including a campsite at each
intersection of trails. The rangers call each stretch of trail
between adjacent campsites a ``segment". The trails have been laid
out so that it is possible to take a hike that starts at any
campsite, covers each segment exactly once, and ends at the beginning
campsite. Prove that it is possible to plan a collection
C of hikes with all of the following properties:
- Each segment is covered exactly once in one hike
and never in any of the other hikes of
h∈C has a base campsite that is its beginning
and end, but which is never passed in the middle of the hike.
(Different hikes of
C may have different base
- Except for its base campsite at beginning and end, no hike in
C passes any campsite more than once.