7th Annual
Virginia Tech Regional Mathematics Contest
From 9:30a.m. to noon November 2, 1985
Fill out the individual registration form
 Prove that
< (a + b)/2 where a and b are positive
real numbers.
 Find the remainder r,
1 < r < 13, when 2^{1985} is divided
by 13.
 Find real numbers c_{1} and c_{2} so that

I + c_{1}M + c_{2}M^{2} = 
( 

) 
where
and I is the identity matrix.
 Consider an infinite sequence
{c_{k}}_{k = 0}^{} of circles.
The largest, C_{0}, is centered at (1, 1) and is tangent to both the
x and yaxes. Each smaller circle C_{n} is centered on the line
through (1, 1) and (2, 0) and is tangent to the next larger circle
C_{n  1} and to the xaxis. Denote the diameter of C_{n} by d_{n}
for
n = 0, 1, 2,.... Find
 (a)
 d_{1}
 (b)

S_{n = 0}^{}d_{n}
 Find the function f = f (x), defined and continuous on
R^{+} = {x  0 < x < }, that satisfies
f (x + 1) = f (x) + x on
R^{+} and f (1) = 0.
 Find an expression for 3/5 as a finite sum of distinct reciprocals of
positive integers. (For example:
2/7 = 1/7 + 1/8 + 1/56.)
 Prove that any positive rational number can be so expressed.
 Let f = f (x) be a real function of a real variable which has
continuous third derivative and which satisfies, for a given c and
all real x,
x =/= c,
(f (x)  f (c))/(x  c) = (f'(x) + f'(c))/2.
Show that
f''(x) = f'(x  f'(c))/(x  c).
 Let
p(x) = a_{0} + a_{1}x + ... + a_{n}x^{n}, where the coefficients
a_{i} are real. Prove that p(x) = 0 has at least one root in the
interval
0 < x < 1 if
a_{0} + a_{1}/2 + ... + a_{n}/(n + 1) = 0.
Peter Linnell
20011003