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 $\sqrt{ab}$ < (a + b)/2 where a and b are positive real numbers.

2. Find the remainder r, 1 < 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 {c_k}_{k = 0}^$\infty$ of circles. The largest, C₀, is centered at (1, 1) and is tangent to both the x and y-axes. 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 x-axis. Denote the diameter of C_n by d_n for n = 0, 1, 2,.... Find

   (a)

   d₁

   (b)

   S_{n = 0}^$\infty$d_n

5. Find the function f = f (x), defined and continuous on R⁺ = {x | 0 < x < $\infty$}, that satisfies f (x + 1) = f (x) + x on R⁺ and f (1) = 0.

6. 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.

7. 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).

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