5th Annual
Virginia Tech Regional Mathematics Contest
From 9:30a.m. to 12:00 noon, November 5, 1983

Fill out the individual registration form

1. In the expansion of (a + b)ⁿ, where n is a natural number, there are n + 1 dissimilar terms. Find the number of dissimilar terms in the expansion of (a + b + c)ⁿ.

2. A positive integer N (in base 10) is called special if the operation C of replacing each digit d of N by its nine's-complement 9 - d, followed by the operation R of reversing the order of the digits, results in the original number. (For example, 3456 is a special number because R[(C3456)] = 3456.) Find the sum of all special positive integers less than one million which do not end in zero or nine.

3. Let a triangle have vertices at O(0, 0), A(a, 0), and B(b, c) in the (x, y)-plane.

   (a)

   Find the coordinates of a point P(x, y) in the exterior of DOAB satisfying area(OAP) = area(OBP) = area(ABP).

   (b)

   Find a point Q(x, y) in the interior of DOAQ satisfying area(OAQ) = area(OBQ) = area(ABQ).

4. A finite set of roads connect n towns T₁, T₂,..., T_n where n$\ge$2. We say that towns T_i and T_j (i$\ne$j) are directly connected if there is a road segment connecting T_i and T_j which does not pass through any other town. Let f (T_k) be the number of other towns directly connected to T_k. Prove that f is not one-to-one.

5. Find the function f (x) such that for all L$\ge$ 0, the area under the graph of y = f (x) and above the x-axis from x = 0 to x = L equals the arc length of the graph from x = 0 to x = L. (Hint: recall that d /dx cosh^-1x = 1/$\sqrt{x^2 - 1}$.)

6. Let f (x) = 1/x and g(x) = 1 - x for x $\in$ (0, 1). List all distinct functions that can be written in the form fogofogo...ofogof where o represents composition. Write each function in the form (ax + b)/(cx + d ), and prove that your list is exhaustive.

7. If a and b are real, prove that x⁴ + ax + b = 0 cannot have only real roots.

8. A sequence f_n is generated by the recurrence formula
   f_{n + 1} = (f_nf_{n - 1} + 1)/f_{n - 2},
   for n = 2, 3, 4,..., with f₀ = f₁ = f₂ = 1. Prove that f_n is integer-valued for all integers n$\ge$ 0.