Virginia Tech Regional Math Contest (1987)

Problem 1

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

Problem 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.

Problem 3

Let $a_{1},a_{2},…,a_{n}$ be an arbitrary rearrangement of $1,2,…,n$. Prove that if $n$ is odd, then $(a_{1}-1)(a_{2}-2)…(a_{n}-n)$ is even.

Problem 4

Let $p(x)$ be given by $p(x) = a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n$ and let $|p(x)| \le |x|$ on $[-1, 1]$.

Problem 5

A sequence of integers $\lbrace n_1, n_2, … \rbrace$ is defined as follows: $n_1$ is assigned arbitrarily and, for $k>1$, $$n_k=\sum_{j=1}^{j=k-1}z(n_j),$$ 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 $n_1=9$, is $\lbrace 9, 2, 3, 3, 3, … \rbrace$, or in binary, $\lbrace 1001, 10, 11, 11, 11, … \rbrace$.

Problem 6

A sequence of polynomials is given by $p_{n}(x)=a_{n+2}x^{2}+a_{n+1}x-a_{n}$, for $n\ge0$, where $a_{0}=a_{1}=1$ and, for $n\ge0$, $a_{n+2}=a_{n+1}+a_{n}$. Denote by $r_{n}$ and $s_{n}$ the roots of $p_{n}(x)=0,$ with $r_{n}\le s_{n}$. Find $\displaystyle{\lim_{n \to \infty} r_n}$ and $\displaystyle{\lim_{n \to \infty} s_n}$.

Problem 7

Let $A=\lbrace a_{ij} \rbrace$ and $B=\lbrace b_{ij} \rbrace$ be $n \times n$ matrices such that $A^{-1}$ exists.

Define $A(t)=\lbrace a_{ij}(t) \rbrace$ and $B(t)=\lbrace b_{ij}(t) \rbrace$ 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 $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \\ \end{bmatrix}$, then $A(t) = \begin{bmatrix} 1 & 2 \\ 3t & 4t \\ \end{bmatrix}$.

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

Problem 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 $t\ge0$ 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.