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A rectangle has a side length of 6 cm and a diagonal of 10 cm. A larger rectangle similar to the first has a side length of 18 cm. What is the area of the larger rectangle?	A rectangle has a side length of 6 cm and a diagonal of 10 cm. A larger rectangle similar to the first has a side length of 18 cm. What is the area of the larger rectangle?
An octopus told me that his underwater cave is $245_{8}$ years old. How many years is this in base ten?	An octopus told me that his underwater cave is $245_{8}$ years old. How many years is this in base ten?
The fifth and eleventh terms of a geometric sequence are 16 and 256, respectively. What is the common ratio?	The fifth and eleventh terms of a geometric sequence are 16 and 256, respectively. What is the common ratio?
Find the $4 \times 4$ matrix $\mathbf{P}$ such that for a $4 \times 4$ matrix $\mathbf{Q},$ $\mathbf{P} \mathbf{Q}$ is the result of swapping the second row and third row of $\mathbf{Q},$ and tripling the first row of $\mathbf{Q}.$ In other words, \[\mathbf{P} \begin{pmatrix} a & b & c & d \\ e & f & g & h \\ i & j & ...	Find the $4 \times 4$ matrix $\mathbf{P}$ such that for a $4 \times 4$ matrix $\mathbf{Q},$ $\mathbf{P} \mathbf{Q}$ is the result of swapping the second row and third row of $\mathbf{Q},$ and tripling the first row of $\mathbf{Q}.$ In other words, \[\mathbf{P} \begin{pmatrix} a & b & c & d \\ e & f & g & h \\ i & j & ...
Let $f(x) = \log_{10} (x+2) - \log_{10} (x-1)$. Find the range of $f$ when the domain is restricted to $x > 1$.	Let $f(x) = \log_{10} (x+2) - \log_{10} (x-1)$. Find the range of $f$ when the domain is restricted to $x > 1$.
A bookshelf contains $n$ books, each with a thickness of 2 cm. After removing one book, the remaining books can be stacked into two piles of equal height, with no books left over. What is the smallest possible value of $n$ such that the height of each pile is a whole number of centimeters?	A bookshelf contains $n$ books, each with a thickness of 2 cm. After removing one book, the remaining books can be stacked into two piles of equal height, with no books left over. What is the smallest possible value of $n$ such that the height of each pile is a whole number of centimeters?
A box contains 18 marbles, some of which are blue and the rest of which are green. When two marbles are drawn simultaneously at random, the probability that both marbles are green is $\frac{1}{8}$. How many marbles in the box are green?	A box contains 18 marbles, some of which are blue and the rest of which are green. When two marbles are drawn simultaneously at random, the probability that both marbles are green is $\frac{1}{8}$. How many marbles in the box are green?
In right triangle $PQR$, $\tan{P}=\frac{3}{2}$, $QR=12$, and $\angle R = 90^\circ$. Find $PQ$. Express your answer in simplest radical form.	In right triangle $PQR$, $\tan{P}=\frac{3}{2}$, $QR=12$, and $\angle R = 90^\circ$. Find $PQ$. Express your answer in simplest radical form.
Evaluate $\|(4\sqrt{2}-4i)(\sqrt{3}+3i)\|$	Evaluate $\|(4\sqrt{2}-4i)(\sqrt{3}+3i)\|$
A complex number $z$ is expressed in polar form as $r \operatorname{cis} \theta$, where $r > 0$ and $0^\circ \le \theta < 360^\circ.$ If $z^2 = 16 \operatorname{cis} 120^\circ$, find the ordered pair $(r, \theta)$.	A complex number $z$ is expressed in polar form as $r \operatorname{cis} \theta$, where $r > 0$ and $0^\circ \le \theta < 360^\circ.$ If $z^2 = 16 \operatorname{cis} 120^\circ$, find the ordered pair $(r, \theta)$.
When the digits in the number $2005$ are reversed we obtain the number $5002,$ and $5002 = a \cdot b \cdot c$, such that $a$, $b$ and $c$ are three distinct primes. How many other positive integers are the products of exactly three distinct primes $p_1$, $p_2$ and $p_3$ such that $p_1 + p_2 + p_3 = a+b+c$?	When the digits in the number $2005$ are reversed we obtain the number $5002,$ and $5002 = a \cdot b \cdot c$, such that $a$, $b$ and $c$ are three distinct primes. How many other positive integers are the products of exactly three distinct primes $p_1$, $p_2$ and $p_3$ such that $p_1 + p_2 + p_3 = a+b+c$?
The 2007 AMC 10 will be scored by awarding $6$ points for each correct response, $0$ points for each incorrect response, and $1.5$ points for each problem left unanswered. After looking over the $25$ problems, Sarah has decided to attempt the first $22$ and leave only the last $3$ unanswered. How many of the first $22...	The 2007 AMC 10 will be scored by awarding $6$ points for each correct response, $0$ points for each incorrect response, and $1.5$ points for each problem left unanswered. After looking over the $25$ problems, Sarah has decided to attempt the first $22$ and leave only the last $3$ unanswered. How many of the first $22...
Given that $\lceil\log_{10}x\rceil=3$, determine the range of possible values for $x$.	Given that $\lceil\log_{10}x\rceil=3$, determine the range of possible values for $x$.
In a regular polygon, the number of diagonals that can be drawn from a single vertex is one less than the number of sides. If the number of diagonals from one vertex is multiplied by the number of sides, the product is 165. Find the number of sides of the polygon.	In a regular polygon, the number of diagonals that can be drawn from a single vertex is one less than the number of sides. If the number of diagonals from one vertex is multiplied by the number of sides, the product is 165. Find the number of sides of the polygon.
At a particular school with 43 students, each student takes chemistry, biology, or both. The chemistry class is three times as large as the biology class, and 5 students are taking both classes. How many people are in the chemistry class?	At a particular school with 43 students, each student takes chemistry, biology, or both. The chemistry class is three times as large as the biology class, and 5 students are taking both classes. How many people are in the chemistry class?
A cylindrical tank has a height of 15 meters and a radius of 8 meters. If 230 cubic meters of water are poured into the tank, what percentage of the tank's volume is occupied by water? Express your answer to the nearest tenth of a percent.	A cylindrical tank has a height of 15 meters and a radius of 8 meters. If 230 cubic meters of water are poured into the tank, what percentage of the tank's volume is occupied by water? Express your answer to the nearest tenth of a percent.
How many five-digit positive integers have the property that the sum of their digits is a multiple of 5?	How many five-digit positive integers have the property that the sum of their digits is a multiple of 5?
Express $\frac{240_{11}}{33_4} \cdot \frac{150_5}{25_6}$ in base 8.	Express $\frac{240_{11}}{33_4} \cdot \frac{150_5}{25_6}$ in base 8.
In a right triangle, the ratio of the lengths of the two legs is 3:4. If the length of the hypotenuse is 50 inches, what is the area of the triangle?	In a right triangle, the ratio of the lengths of the two legs is 3:4. If the length of the hypotenuse is 50 inches, what is the area of the triangle?
Compute the number of ways to arrange 8 distinct objects in a row such that the first 3 objects are in increasing order and the last 3 objects are in decreasing order.	Compute the number of ways to arrange 8 distinct objects in a row such that the first 3 objects are in increasing order and the last 3 objects are in decreasing order.
A square and four circles, each with a radius of 5 inches, are arranged as shown. What is the area, in square inches, of the square? [asy] unitsize(1mm); defaultpen(linewidth(0.7pt)); draw((0,0)--(20,0)--(20,20)--(0,20)--cycle); draw(Circle((5,5),5)); draw(Circle((15,5),5)); draw(Circle((5,15),5)); draw(Circle((15,15)...	A square and four circles, each with a radius of 5 inches, are arranged as shown. What is the area, in square inches, of the square? [asy] unitsize(1mm); defaultpen(linewidth(0.7pt)); draw((0,0)--(20,0)--(20,20)--(0,20)--cycle); draw(Circle((5,5),5)); draw(Circle((15,5),5)); draw(Circle((5,15),5)); draw(Circle((15,15)...
A committee consists of 12 members, 7 men and 5 women. Two members are selected randomly for a task. What is the probability that they are both men?	A committee consists of 12 members, 7 men and 5 women. Two members are selected randomly for a task. What is the probability that they are both men?
What is the least integer whose square is 48 more than its double?	What is the least integer whose square is 48 more than its double?
Find the projection of the vector $\begin{pmatrix} 4 \\ 5 \end{pmatrix}$ onto the vector $\begin{pmatrix} 2 \\ 0 \end{pmatrix}.$	Find the projection of the vector $\begin{pmatrix} 4 \\ 5 \end{pmatrix}$ onto the vector $\begin{pmatrix} 2 \\ 0 \end{pmatrix}.$
Convert the point $\left( 12, \frac{5 \pi}{4} \right)$ in polar coordinates to rectangular coordinates.	Convert the point $\left( 12, \frac{5 \pi}{4} \right)$ in polar coordinates to rectangular coordinates.
Let $5^a$ be the greatest power of $5$ that is a factor of $300$, and let $3^b$ be the greatest power of $3$ that is a factor of $300$. Evaluate the following expression: $$\left(\frac27\right)^{b - a}$$	Let $5^a$ be the greatest power of $5$ that is a factor of $300$, and let $3^b$ be the greatest power of $3$ that is a factor of $300$. Evaluate the following expression: $$\left(\frac27\right)^{b - a}$$
The students in Ms. Snow's math class are organizing a field trip. They need to rent buses, and each bus can hold 10 students. If the ratio of girls to boys is 3:2 and there are 45 students in total, how many buses will they need to rent so that no bus has more than 2 boys or 3 girls?	The students in Ms. Snow's math class are organizing a field trip. They need to rent buses, and each bus can hold 10 students. If the ratio of girls to boys is 3:2 and there are 45 students in total, how many buses will they need to rent so that no bus has more than 2 boys or 3 girls?
If $\tan a + \tan b = 12$ and $\cot a + \cot b = 15$, what is $\tan(a+b)$?	If $\tan a + \tan b = 12$ and $\cot a + \cot b = 15$, what is $\tan(a+b)$?
At a conference with $32$ participants, sixteen participants used laptops, and twelve participants used tablets. Seven participants used both laptops and tablets. How many participants used neither laptops nor tablets?	At a conference with $32$ participants, sixteen participants used laptops, and twelve participants used tablets. Seven participants used both laptops and tablets. How many participants used neither laptops nor tablets?
Let's determine if there exists a positive real number $x$ such that for any complex number $z$, if the point $x$ lies on the segment connecting $z$ and $\frac{1}{z}$ in the complex plane, then $$\|z\| + \left\|\frac{1}{z}\right\| \geq 2\sqrt{2}.$$	Let's determine if there exists a positive real number $x$ such that for any complex number $z$, if the point $x$ lies on the segment connecting $z$ and $\frac{1}{z}$ in the complex plane, then $$\|z\| + \left\|\frac{1}{z}\right\| \geq 2\sqrt{2}.$$
A gift box can be filled with red, green, or blue balls, and it can hold a maximum of 5 balls. Suppose that a gift box is filled with 5 balls, but we don't know how many balls of each color are present. If there are $n$ different ways to choose balls from the box such that at least one red ball and one green ball are s...	A gift box can be filled with red, green, or blue balls, and it can hold a maximum of 5 balls. Suppose that a gift box is filled with 5 balls, but we don't know how many balls of each color are present. If there are $n$ different ways to choose balls from the box such that at least one red ball and one green ball are s...
A bookshelf has 5 shelves, and each shelf can hold 8 rows of books. If the bookshelf is currently empty, what is the smallest number of books Xanthia needs to buy so that she can fill the bookshelf with an equal number of books on each shelf, with no books left over?	A bookshelf has 5 shelves, and each shelf can hold 8 rows of books. If the bookshelf is currently empty, what is the smallest number of books Xanthia needs to buy so that she can fill the bookshelf with an equal number of books on each shelf, with no books left over?
If $x$ is a real number and $x^2 = 16$, what is the sum of all possible values of $x$?	If $x$ is a real number and $x^2 = 16$, what is the sum of all possible values of $x$?
How many terms are there in the sequence $$\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \ldots, \frac{20}{21}, \frac{21}{22}$$ that have a denominator that is a multiple of 5?	How many terms are there in the sequence $$\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \ldots, \frac{20}{21}, \frac{21}{22}$$ that have a denominator that is a multiple of 5?
Find the area of the triangle with vertices at (-3, 4), (0, 0), and (5, -2).	Find the area of the triangle with vertices at (-3, 4), (0, 0), and (5, -2).
Find the equation of the line passing through the point (4,3) and perpendicular to the line whose slope is equal to the slope of the tangent line to the parabola y = x^2 at x = 2.	Find the equation of the line passing through the point (4,3) and perpendicular to the line whose slope is equal to the slope of the tangent line to the parabola y = x^2 at x = 2.
The value of $3x-5$ is twice the value of $x+10$. What is the value of $x$?	The value of $3x-5$ is twice the value of $x+10$. What is the value of $x$?
A water tank can be filled by two pipes, A and B, in 4 hours and 6 hours, respectively. How long would it take to fill the tank if both pipes are used simultaneously, but pipe A is only open for the first half of the time and pipe B is only open for the second half of the time?	A water tank can be filled by two pipes, A and B, in 4 hours and 6 hours, respectively. How long would it take to fill the tank if both pipes are used simultaneously, but pipe A is only open for the first half of the time and pipe B is only open for the second half of the time?
A firm produces two kinds of products: A and B. To produce each product, two operations are needed: the first operation can only be performed by machine $M_1$, and the second operation can only be performed by machine $M_2$. The time required for the first operation is the same for both products, but the time required ...	A firm produces two kinds of products: A and B. To produce each product, two operations are needed: the first operation can only be performed by machine $M_1$, and the second operation can only be performed by machine $M_2$. The time required for the first operation is the same for both products, but the time required ...
A bookstore has five shelves, each with a different genre of books (romance, mystery, sci-fi, fantasy, and non-fiction). If the bookstore wants to create bundles of three books from different genres, what is the maximum number of unique bundles that can be created?	A bookstore has five shelves, each with a different genre of books (romance, mystery, sci-fi, fantasy, and non-fiction). If the bookstore wants to create bundles of three books from different genres, what is the maximum number of unique bundles that can be created?