7th Annual
Virginia Tech Regional Mathematics Contest
From 9:30a.m. to noon November 2, 1985

Fill out the individual registration form

1. Prove that  < (a + b)/2 where a and b are positive real numbers.

2. Find the remainder r, < r < 13, when 21985 is divided by 13.

3. Find real numbers c1 and c2 so that

I + c1M + c2M2 = (
 0 0 0 0
)

where

M = (
 1 3 0 2
)
and I is the identity matrix.

4. Consider an infinite sequence {ck}k = 0 of circles. The largest, C0, is centered at (1, 1) and is tangent to both the x and y-axes. Each smaller circle Cn is centered on the line through (1, 1) and (2, 0) and is tangent to the next larger circle Cn - 1 and to the x-axis. Denote the diameter of Cn by dn for n = 0, 1, 2,.... Find
(a)
d1
(b)
Sn = 0dn

5. 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.

1. 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.)

2. Prove that any positive rational number can be so expressed.

6. 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).

7. Let p(x) = a0 + a1x + ... + anxn, where the coefficients ai are real. Prove that p(x) = 0 has at least one root in the interval < x < 1 if a0 + a1/2 + ... + an/(n + 1) = 0.

Peter Linnell
2001-10-03