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Difference between revisions of "2001 SMT/Algebra Problems"

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Latest revision as of 12:26, 22 January 2020

Problem 1

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

Solution

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 $\frac{x}{y}$.

Solution

Problem 7

If $\log_A B + \log_B A = 3$ and $A<B$, find $\log_B A$

Solution

Problem 8

Determine the value of $1+\frac{1}{2+\frac{1}{1+\frac{1}{2+\frac{1}{1+\cdots}}}}$

Solution

Problem 9

Find all solutions to $(x - 3)(x - 1)(x + 3)(x + 5) = 13$

Solution


Problem 10

Suppose $x$, $y$, $z$ satisfy \begin{align*} x+y+z &= 3\\ x^2+y^2+z^2 &= 5\\ x^3+y^3+z^3 &= 7\\ \end{align*} Find $x^4+y^4+z^4$

Solution


See Also

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