9th Annual
Virginia Tech Regional Mathematics Contest
From 9:30a.m. to 12:00 noon, October 31, 1987

Fill out the individual registration form

1. A path zig-zags from (1, 0) to (0, 0) along line segments $\overline{{P_nP_{n+1}}}$, where P₀ is (1, 0) and P_n is (2^-n,(- 2)^-n), for n > 0. Find the length of the path.

2. A triangle with sides of lengths a, b, and c is partitioned into two smaller triangles by the line which is perpendicular to the side of length c and passes through the vertex opposite that side. Find integers a < b < c such that each of the two smaller triangles is similar to the original triangle and has sides of integer lengths.

3. Let a₁, a₂,..., a_n be an arbitrary rearrangement of 1, 2,..., n. Prove that if n is odd, then (a₁ -1)(a₂ -2)...(a_n - n) is even.

4. Let p(x) be given by p(x) = a₀ + a₁x + a₂x² + ... + a_nxⁿ and let | p(x)|≤| x| on [- 1, 1].

   (a) Evaluate a₀.     (b) Prove that | a₁|≤1.

5. A sequence of integers {n₁, n₂,...} is defined as follows: n₁ is assigned arbitrarily and, for k > 1,
   n_k = ∑_j=1^j=k-1z(n_j),
   where z(n) is the number of 0's in the binary representation of n (each representation should have a leading digit of 1 except for zero which has the representation 0). An example, with n₁ = 9, is {9, 2, 3, 3, 3,...}, or in binary, {1001, 10, 11, 11, 11,...}.

   (a)

   Find n₁ so hat lim_{k-> ∞}n_k = 31, and calculate n₂, n₃,..., n₁₀.

   (b)

   Prove that, for every choice of n₁, the sequence {n_k} converges.

6. A sequence of polynomials is given by p_n(x) = a_n+2x² + a_n+1x - a_n, for n≥ 0, where a₀ = a₁ = 1 and, for n≥ 0, a_n+2 = a_n+1 + a_n. Denote by r_n and s_n the roots of p_n(x) = 0, with r_n≤s_n. Find lim_{n-> ∞}r_n and lim_{n-> ∞}s_n.

7. Let A = {a_ij} and B = {b_ij} be n×n matrices such that A^-1 exists. Define A(t) = {a_ij(t)} and B(t) = {b_ij(t)} by a_ij(t) = a_ij for i < n, a_nj(t) = ta_nj, b_ij(t) = b_ij for i < n, and b_nj(t) = tb_nj. For example, if

   A =

   [

   	1	2
   	3	4

   ]

   then

   A(t) =

   [

   	1	2
   	3t	4t

   ]

   Prove that A(t)^-1B(t) = A^-1B for t > 0 and any n. (Partial credit will be given for verifying the result for n = 3.)

8. On Halloween, a black cat and a witch encounter each other near a large mirror positioned along the y-axis. The witch is invisible except by reflection in the mirror. At t = 0, the cat is at (10, 10) and the witch is at (10, 0). For t≥ 0, the witch moves toward the cat at a speed numerically equal to their distance of separation and the cat moves toward the apparent position of the witch, as seen by reflection, at a speed numerically equal to their reflected distance of separation. Denote by (u(t), v(t)) the position of the cat and by (x(t), y(t)) the position of the witch.

   (a)

   Set up the equations of motion of the cat and the witch for t≥ 0.

   (b)

   Solve for x(t) and u(t) and find the time when the cat strikes the mirror. (Recall that the mirror is a perpendicular bisector of the line joining an object with its apparent position as seen by reflection.)