Virginia Tech Regional Math Contest (1989)

Problem 1

A square of side $a$ is inscribed in a triangle of base $b$ and height $h$ as shown. Prove that the area of the square cannot exceed one-half the area of the triangle.

Problem 2

Let $A$ be a $3 \times 3$ matrix in which each element is either $0$ or $1$ but is otherwise arbitrary.

Problem 3

The system of equations

$$\begin{align*} a_{11} x_1 + a_{12} x_2 + a_{13} x_3 &= b_1 \newline a_{21} x_1 + a_{22} x_2 + a_{23} x_3 &= b_2 \newline a_{31} x_1 + a_{32} x_2 + a_{33} x_3 &= b_3 \end{align*}$$

has the solution $x_1 = -1$, $x_2 = 3$, $x_3 = 2$ when $b_1 = 1$, $b_2 = 0$, $b_3 = 1$ and it has the solution $x_1 = 2$, $x = -2$, $x_3 = 1$ when $b_1 = 0$, $b_2 = -1$, $b_3 = 1$. Find a solution of the system when $b_1 = 2$, $b_2 = -1$, $b_3 = 3$.

Problem 4

Let $a, b, c, d$ be distinct integers such that the equation $$(x - a)(x - b)(x - c)(x - d) - 9 = 0$$ has an integer root $r$. Show that $4r = a + b + c + d$. (This is essentially a problem from the 1947 Putnam examination.)

Problem 5

Problem 6

Let $g$ be defined on $(1, \infty)$ by $g(x) = x/(x - 1)$, and let $f^k(x)$ be defined by $f^0(x) = x$ and for $k > 0$, $f^k(x) = g(f^{k-1}(x))$.

Evaluate $\displaystyle{\sum_{k=0}^{\infty} 2^{-k} f^k(x)}$ in the form $\dfrac{ax^2 + bx + c}{dx + e}$.

Problem 7

Three farmers sell chickens at a market. One has 10 chickens, another has 16, and the third has 26. Each farmer sells at least one, but not all, of his chickens before noon, all farmers selling at the same price per chicken. Later in the day each sells his remaining chickens, all again selling at the same reduced price. If each farmer received a total of $35 from the sale of his chickens, what was the selling price before noon and the selling price after noon? (From “Math Can Be Fun” by Ya Perelman.)

Problem 8

The integer sequence $\lbrace a_0, a_1, \dots, a_{n-1} \rbrace$ is such that, for each $i$ ($0 \le i \le n - 1$), $a_i$ is the number of $i$’s in the sequence. (Thus for $n = 4$ we might have the sequence $\lbrace 1, 2, 1, 0 \rbrace$.)

Hint: show that the sum of all the terms is $n$, and that there are $n - a_0 - 1$ nonzero terms other than $a_0$ which sum to $n - a_0$. (This problem is slightly modified from one on the Cambridge Men’s Colleges Joint Awards and Entrance Examination, 24 November 1970.)