2001 SMT/Algebra Problems

Problem 1

Find the result of adding seven to the result of forty divided by one-half.

Problem 2

Each valve $A$ , $B$ , and $C$ , when open, releases water into a tank at its own constant rate. With all three valves open, the tank fills in 1 hour, with only valves $A$ and $C$ open it takes 1.5 hours, and with only valves $B$ and $C$ open it takes 2 hours. How many hours will it take to fill the tank with only valves $A$ and $B$ open?

Solution

Problem 3

Julie has a 12 foot by 20 foot garden. She wants to put fencing around it to keep out the neighbour’s dog. Normal fenceposts cost $$$$ 2 each while strong ones cost $$$$ 3 each. If Julie needs one fencepost for every 2 feet and has $$$$ 70 to spend on fenceposts, what is the greatest number of strong fenceposts she can buy?

Solution

Problem 4

$p(x)$ is a real polynomial of degree at most 3. Suppose there are four distinct solutions to the equation $p(x)$ = 7. What is $p(0)$ ?

Solution

Problem 5

Let $f \colon \mathbb{N} \to \mathbb{N}$ be defined by $f(n) =\begin{cases}2 & \text{if }x = 0, \\(f(x-1))^2 & \text{if }x \neq 0 \end{cases}$ . What is $\log_2f(11)$ ?

Solution

Problem 6

If for three distinct positive numbers $x$ , $y$ , and $z$ , $\frac{y}{x+z} = \frac{x+y}{z} = \frac{x}{y}$ Then find the numerical value of $\frac{x}{y}$ .