instruction_seed stringlengths 13 12.7k | response_seed stringlengths 13 12.7k |
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In the diagram, $ABCD$ is a rectangle, $E$ is a point on $\overline{AB}$, and $\triangle DEF$ is a right triangle. The length of $\overline{ED}$ is twice the length of $\overline{BF}$. If the area of $\triangle DEF$ is $\frac{3}{4}$ of the area of $\triangle FAB$, what is the ratio of the length of $\overline{FA}$ to t... | In the diagram, $ABCD$ is a rectangle, $E$ is a point on $\overline{AB}$, and $\triangle DEF$ is a right triangle. The length of $\overline{ED}$ is twice the length of $\overline{BF}$. If the area of $\triangle DEF$ is $\frac{3}{4}$ of the area of $\triangle FAB$, what is the ratio of the length of $\overline{FA}$ to t... |
Let \(f(x)\) be a function such that
$$f(x)=\left\{
\begin{array}{cl}
3x+2 & \mbox{if } x\le -1, \\
5x-7 & \mbox{if } x>-1.
\end{array}
\right.$$
Find the range of $f(x)$. | Let \(f(x)\) be a function such that
$$f(x)=\left\{
\begin{array}{cl}
3x+2 & \mbox{if } x\le -1, \\
5x-7 & \mbox{if } x>-1.
\end{array}
\right.$$
Find the range of $f(x)$. |
Let $\mathbf{a}$ and $\mathbf{b}$ be vectors, and let $\mathbf{m}$ be the midpoint of $\mathbf{a}$ and $\mathbf{b}.$ Given $\mathbf{m} = \begin{pmatrix} 4 \\ 9 \end{pmatrix}$ and $\mathbf{a} \cdot \mathbf{b} = 8,$ find $\|\mathbf{a}\|^2 + \|\mathbf{b}\|^2.$ | Let $\mathbf{a}$ and $\mathbf{b}$ be vectors, and let $\mathbf{m}$ be the midpoint of $\mathbf{a}$ and $\mathbf{b}.$ Given $\mathbf{m} = \begin{pmatrix} 4 \\ 9 \end{pmatrix}$ and $\mathbf{a} \cdot \mathbf{b} = 8,$ find $\|\mathbf{a}\|^2 + \|\mathbf{b}\|^2.$ |
An ellipse has its foci at $(0,7)$ and $(0,-7)$. One endpoint of the minor axis is $(3,0)$. Determine the coordinates of the vertices. | An ellipse has its foci at $(0,7)$ and $(0,-7)$. One endpoint of the minor axis is $(3,0)$. Determine the coordinates of the vertices. |
In a triangle, the length of the median $AD$ is given by the equation $AD^2 = \frac{1}{4}(2AB^2 + 2AC^2 - BC^2)$, where $A$, $B$, and $C$ are the vertices of the triangle. If the lengths of the sides $AB$ and $AC$ are both equal to 5, and the length of the side $BC$ is equal to 6, then what is the value of $AD^2$? | In a triangle, the length of the median $AD$ is given by the equation $AD^2 = \frac{1}{4}(2AB^2 + 2AC^2 - BC^2)$, where $A$, $B$, and $C$ are the vertices of the triangle. If the lengths of the sides $AB$ and $AC$ are both equal to 5, and the length of the side $BC$ is equal to 6, then what is the value of $AD^2$? |
A group of five friends - Alex, Ben, Charlie, David, and Emily - want to take a group photo. They decide to stand in a row, but with certain restrictions: Charlie wants to stand next to David, and Alex wants to stand next to Ben. In how many ways can they arrange themselves for the photo? | A group of five friends - Alex, Ben, Charlie, David, and Emily - want to take a group photo. They decide to stand in a row, but with certain restrictions: Charlie wants to stand next to David, and Alex wants to stand next to Ben. In how many ways can they arrange themselves for the photo? |
If $\mathbf{A} = \begin{pmatrix} a & b \\ c & d \end{pmatrix},$ then its transpose is given by
\[\mathbf{A}^T = \begin{pmatrix} a & c \\ b & d \end{pmatrix}.\]Given that $\mathbf{A}^T = \mathbf{A}^{-1},$ find $a^2 + b^2 + c^2 + d^2.$ | If $\mathbf{A} = \begin{pmatrix} a & b \\ c & d \end{pmatrix},$ then its transpose is given by
\[\mathbf{A}^T = \begin{pmatrix} a & c \\ b & d \end{pmatrix}.\]Given that $\mathbf{A}^T = \mathbf{A}^{-1},$ find $a^2 + b^2 + c^2 + d^2.$ |
Expand the following expression: $(9x+4)\cdot 2x^2$ | Expand the following expression: $(9x+4)\cdot 2x^2$ |
Find all solutions to the equation:
\[ \sqrt[5]{x} = \frac{15}{8 - \sqrt[5]{x}}. \] | Find all solutions to the equation:
\[ \sqrt[5]{x} = \frac{15}{8 - \sqrt[5]{x}}. \] |
Determine the smallest positive integer that is divisible by the sum of its digits, the product of its digits, and each of its distinct digits, with no repeated digits allowed. | Determine the smallest positive integer that is divisible by the sum of its digits, the product of its digits, and each of its distinct digits, with no repeated digits allowed. |
Painters like to work with tints and shades of colors by adding white or black to mixtures. If a painter has five jars of different shades of yellow paint, how many distinct shades can she produce if she can add up to three different jars together? | Painters like to work with tints and shades of colors by adding white or black to mixtures. If a painter has five jars of different shades of yellow paint, how many distinct shades can she produce if she can add up to three different jars together? |
Two cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is a King and the second card is a $\heartsuit$? | Two cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is a King and the second card is a $\heartsuit$? |
A rectangular parking lot 80 feet long and 15 feet wide is enclosed by a fence. To make the parking lot larger, while using the same fence, its shape is changed to a square. By how many square feet does this enlarge the parking lot? | A rectangular parking lot 80 feet long and 15 feet wide is enclosed by a fence. To make the parking lot larger, while using the same fence, its shape is changed to a square. By how many square feet does this enlarge the parking lot? |
In a circus performance, 5 acrobats stand on a balance beam in a row, with the requirement that the two heaviest acrobats must be on either end of the beam. If the 5 acrobats have different weights, how many ways can they be arranged on the beam? | In a circus performance, 5 acrobats stand on a balance beam in a row, with the requirement that the two heaviest acrobats must be on either end of the beam. If the 5 acrobats have different weights, how many ways can they be arranged on the beam? |
Given that $x = \frac{5}{7}$ is a solution to the equation $56 x^2 + 27 = 89x - 8,$ what is the other value of $x$ that will solve the equation? Express your answer as a common fraction. | Given that $x = \frac{5}{7}$ is a solution to the equation $56 x^2 + 27 = 89x - 8,$ what is the other value of $x$ that will solve the equation? Express your answer as a common fraction. |
The roots of the equation $2x^2 - 5x - 4 = 0$ can be written in the form $x = \frac{m \pm \sqrt{n}}{p}$, where $m$, $n$, and $p$ are positive integers with a greatest common divisor of 1. What is the value of $n$? | The roots of the equation $2x^2 - 5x - 4 = 0$ can be written in the form $x = \frac{m \pm \sqrt{n}}{p}$, where $m$, $n$, and $p$ are positive integers with a greatest common divisor of 1. What is the value of $n$? |
A department store offers two promotions. Promotion A says, "Buy one pair of shoes, get the second pair for half the price." Promotion B says, "Buy one pair of shoes, get $\$10$ off the second pair." Jane wants to buy two pairs of shoes that cost $\$30$ each. She can only use one of the promotions, A or B. Jane decides... | A department store offers two promotions. Promotion A says, "Buy one pair of shoes, get the second pair for half the price." Promotion B says, "Buy one pair of shoes, get $\$10$ off the second pair." Jane wants to buy two pairs of shoes that cost $\$30$ each. She can only use one of the promotions, A or B. Jane decides... |
Find the least common multiple of the polynomials $x^2 + 5x + 6$ and $x^2 + 3x + 2$. | Find the least common multiple of the polynomials $x^2 + 5x + 6$ and $x^2 + 3x + 2$. |
Find the largest value of $d$ such that $3$ is in the range of $g(x)=x^2-5x+d$. | Find the largest value of $d$ such that $3$ is in the range of $g(x)=x^2-5x+d$. |
If $a=4$ and $b=5$, express the value of the following as a common fraction:
$$
\frac
{~\frac{2}{b}~}
{\frac{3}{a}}
$$ | If $a=4$ and $b=5$, express the value of the following as a common fraction:
$$
\frac
{~\frac{2}{b}~}
{\frac{3}{a}}
$$ |
A school play has 12 students, of which only 4 are skilled enough to play lead roles, while all other roles can be played by anyone. In how many ways can we choose a cast consisting of a protagonist, an antagonist, a lead actor, and a supporting actor? | A school play has 12 students, of which only 4 are skilled enough to play lead roles, while all other roles can be played by anyone. In how many ways can we choose a cast consisting of a protagonist, an antagonist, a lead actor, and a supporting actor? |
Consider the cubic polynomial:
$$Q(x) = 3x^3 + px^2 + qx + r$$
Given that the sum of its zeros, the product of its zeros, and the sum of the coefficients are all equal. Additionally, the $y$-intercept of the graph of $y = Q(x)$ is 6. Find the value of $q$. | Consider the cubic polynomial:
$$Q(x) = 3x^3 + px^2 + qx + r$$
Given that the sum of its zeros, the product of its zeros, and the sum of the coefficients are all equal. Additionally, the $y$-intercept of the graph of $y = Q(x)$ is 6. Find the value of $q$. |
In the figure shown, segment $AB$ is parallel to segment $YZ$. If $AZ = 42$ units, $BQ = 12$ units, and $QY = 24$ units, what is the length of segment $QZ$? [asy]
import olympiad; import geometry; size(150); defaultpen(linewidth(0.8));
pair Y = (0,0), Z = (16,0), A = (0,8), B = (6,8);
draw(A--B--Y--Z--cycle);
label("$A... | In the figure shown, segment $AB$ is parallel to segment $YZ$. If $AZ = 42$ units, $BQ = 12$ units, and $QY = 24$ units, what is the length of segment $QZ$? [asy]
import olympiad; import geometry; size(150); defaultpen(linewidth(0.8));
pair Y = (0,0), Z = (16,0), A = (0,8), B = (6,8);
draw(A--B--Y--Z--cycle);
label("$A... |
Let $\log_{4}3=x$. Then $\log_{2}27=kx$. Find $k$. | Let $\log_{4}3=x$. Then $\log_{2}27=kx$. Find $k$. |
How many three-digit positive integers are congruent to 2 (mod 5)? | How many three-digit positive integers are congruent to 2 (mod 5)? |
Tom drives from city A to city B along a straight road. He stops at a rest stop halfway through his journey to take a break. The rest stop is 0.25 miles away from the road, and he has to drive there and back before continuing his journey. If the total distance from city A to city B is 150 miles, and the rest stop is di... | Tom drives from city A to city B along a straight road. He stops at a rest stop halfway through his journey to take a break. The rest stop is 0.25 miles away from the road, and he has to drive there and back before continuing his journey. If the total distance from city A to city B is 150 miles, and the rest stop is di... |
A particle of mass 100 kg is projected vertically upwards with initial velocity 50 m/s from the surface of the Earth. Assume that there is no air resistance and the acceleration due to gravity is constant throughout the motion. Determine the total energy of the particle at its highest point. | A particle of mass 100 kg is projected vertically upwards with initial velocity 50 m/s from the surface of the Earth. Assume that there is no air resistance and the acceleration due to gravity is constant throughout the motion. Determine the total energy of the particle at its highest point. |
A ball is thrown upwards from the ground with an initial velocity of 50 feet per second. Each time it bounces back, its velocity is reduced by 20%. After how many bounces will the ball first reach a maximum height less than 1 foot? | A ball is thrown upwards from the ground with an initial velocity of 50 feet per second. Each time it bounces back, its velocity is reduced by 20%. After how many bounces will the ball first reach a maximum height less than 1 foot? |
Let $g$ be a function that takes a string of letters as input and returns the number of consecutive pairs of letters that are in alphabetical order. For example, $g("abc") = 2$ and $g("bac") = 0$. Define a new function $h$ such that $h(s) = g(g(s))$ for any string $s$. Find $h("abcdefg")$. | Let $g$ be a function that takes a string of letters as input and returns the number of consecutive pairs of letters that are in alphabetical order. For example, $g("abc") = 2$ and $g("bac") = 0$. Define a new function $h$ such that $h(s) = g(g(s))$ for any string $s$. Find $h("abcdefg")$. |
Consider a square pyramid with a base length of 6 units. The cross-section of the pyramid, perpendicular to the base and passing through the apex, forms a triangle with the base as one side and the slant height as the other. If the area of this triangular cross-section is 18 square units, find the height of the pyramid... | Consider a square pyramid with a base length of 6 units. The cross-section of the pyramid, perpendicular to the base and passing through the apex, forms a triangle with the base as one side and the slant height as the other. If the area of this triangular cross-section is 18 square units, find the height of the pyramid... |
A company offers two different plans for renting cars. Plan A charges a daily fee of $\$40$ plus $\$0.25$ per mile driven, while Plan B charges a daily fee of $\$20$ plus $\$0.35$ per mile driven. What is the minimum number of miles that must be driven in a day for Plan A to be cheaper than Plan B? | A company offers two different plans for renting cars. Plan A charges a daily fee of $\$40$ plus $\$0.25$ per mile driven, while Plan B charges a daily fee of $\$20$ plus $\$0.35$ per mile driven. What is the minimum number of miles that must be driven in a day for Plan A to be cheaper than Plan B? |
Given that a cylinder has a height of 15 meters and a volume of 1200 cubic meters, and a circular base with an area of 80 square meters, find the radius of the cylinder. | Given that a cylinder has a height of 15 meters and a volume of 1200 cubic meters, and a circular base with an area of 80 square meters, find the radius of the cylinder. |
Find the matrix $\mathbf{A}$ such that
\[\mathbf{A} \mathbf{v} = 2 \mathbf{v}\]for all vectors $\mathbf{v}.$ | Find the matrix $\mathbf{A}$ such that
\[\mathbf{A} \mathbf{v} = 2 \mathbf{v}\]for all vectors $\mathbf{v}.$ |
Find the number of 10-tuples $(x_1, x_2, \dots, x_{10})$ of real numbers such that
\[(1 - x_1)^2 + (x_1 - x_2)^2 + (x_2 - x_3)^2 + \dots + (x_9 - x_{10})^2 + x_{10}^2 = \frac{1}{11}\]
and
\[(1 + x_1)^2 + (x_1 + x_2)^2 + (x_2 + x_3)^2 + \dots + (x_9 + x_{10})^2 + x_{10}^2 = \frac{1}{11}.\] | Find the number of 10-tuples $(x_1, x_2, \dots, x_{10})$ of real numbers such that
\[(1 - x_1)^2 + (x_1 - x_2)^2 + (x_2 - x_3)^2 + \dots + (x_9 - x_{10})^2 + x_{10}^2 = \frac{1}{11}\]
and
\[(1 + x_1)^2 + (x_1 + x_2)^2 + (x_2 + x_3)^2 + \dots + (x_9 + x_{10})^2 + x_{10}^2 = \frac{1}{11}.\] |
A sequence of positive integers is arranged in a series of rows, with the first row having 1 integer, the second row having 2 integers, the third row having 3 integers, and so on. The sequence starts with 2, and each subsequent integer is 1 more than the previous one. The integers are arranged in such a way that the fi... | A sequence of positive integers is arranged in a series of rows, with the first row having 1 integer, the second row having 2 integers, the third row having 3 integers, and so on. The sequence starts with 2, and each subsequent integer is 1 more than the previous one. The integers are arranged in such a way that the fi... |
Find the curve defined by the equation
\[r = \frac{1}{\sin \theta - \cos \theta}.\](A) Line
(B) Circle
(C) Parabola
(D) Ellipse
(E) Hyperbola
Enter the letter of the correct option. | Find the curve defined by the equation
\[r = \frac{1}{\sin \theta - \cos \theta}.\](A) Line
(B) Circle
(C) Parabola
(D) Ellipse
(E) Hyperbola
Enter the letter of the correct option. |
Find the cubic polynomial, with integer coefficients, which has $1 - i\sqrt{2}$ as a root, and where the coefficient of $x^2$ is $-7$. | Find the cubic polynomial, with integer coefficients, which has $1 - i\sqrt{2}$ as a root, and where the coefficient of $x^2$ is $-7$. |
Compute the remainder when $2^{1024}+7^{1024}$ is divided by 8. | Compute the remainder when $2^{1024}+7^{1024}$ is divided by 8. |
Consider the triangle with vertices (0,0), (6,0), and (6,8). A rectangle with sides parallel to the x and y axes is to be inscribed within this triangle. Find the maximum possible area of the rectangle.
[asy]
size(59);
draw((0,0)--(6,0)--(6,8)--(0,0));
[/asy] | Consider the triangle with vertices (0,0), (6,0), and (6,8). A rectangle with sides parallel to the x and y axes is to be inscribed within this triangle. Find the maximum possible area of the rectangle.
[asy]
size(59);
draw((0,0)--(6,0)--(6,8)--(0,0));
[/asy] |
The sum of the first $n$ terms in the infinite geometric sequence $\left\{\frac{1}{2},\frac{1}{6},\frac{1}{18},\dots \right\}$ is $\frac{17}{24}$. Find $n$. | The sum of the first $n$ terms in the infinite geometric sequence $\left\{\frac{1}{2},\frac{1}{6},\frac{1}{18},\dots \right\}$ is $\frac{17}{24}$. Find $n$. |
Let $a,$ $b,$ $c$ be distinct complex numbers such that
\[a + b + c = k(ab + bc + ac)\]
and
\[abc = k^2.\]
Find the sum of all possible values of $k.$ | Let $a,$ $b,$ $c$ be distinct complex numbers such that
\[a + b + c = k(ab + bc + ac)\]
and
\[abc = k^2.\]
Find the sum of all possible values of $k.$ |
Find the integer $n$, $0 \le n \le 9$, that satisfies \[n \equiv -2187 \pmod{10}.\] | Find the integer $n$, $0 \le n \le 9$, that satisfies \[n \equiv -2187 \pmod{10}.\] |
A bookshelf has 5 shelves, and the number of books on each shelf increases by 2 for every shelf, starting from the top. If the total number of books on the bookshelf is 75, how many books are on the middle shelf? | A bookshelf has 5 shelves, and the number of books on each shelf increases by 2 for every shelf, starting from the top. If the total number of books on the bookshelf is 75, how many books are on the middle shelf? |
The Bookworm is a bookstore that sells a variety of books and gifts. The store's sales for June are broken down as follows: What percentage of its sales were not fiction or non-fiction books? \begin{tabular}{|l|c|}
\multicolumn{2}{c}{}\\\hline
\textbf{Item}&\textbf{$\%$~of June Sales}\\\hline
Fiction Books&42\\\hline
N... | The Bookworm is a bookstore that sells a variety of books and gifts. The store's sales for June are broken down as follows: What percentage of its sales were not fiction or non-fiction books? \begin{tabular}{|l|c|}
\multicolumn{2}{c}{}\\\hline
\textbf{Item}&\textbf{$\%$~of June Sales}\\\hline
Fiction Books&42\\\hline
N... |
Find all solutions to the equation $\sqrt{5+2z} = 11$.
Another problem inspired by this one:
Find all solutions to the equation $\sqrt{5+2z} = \sqrt{z^2 + 1}$.
This problem requires a different approach to solve, as it involves equating two square roots and solving for z. It also introduces a quadratic term, which a... | Find all solutions to the equation $\sqrt{5+2z} = 11$.
Another problem inspired by this one:
Find all solutions to the equation $\sqrt{5+2z} = \sqrt{z^2 + 1}$.
This problem requires a different approach to solve, as it involves equating two square roots and solving for z. It also introduces a quadratic term, which a... |
Find all possible values of the base $b$ if $271_b \cdot 60_b \cdot 2002_b = 77_b \cdot 21_b \cdot 1700_b$ | Find all possible values of the base $b$ if $271_b \cdot 60_b \cdot 2002_b = 77_b \cdot 21_b \cdot 1700_b$ |
If $\log_9 (x-2)=\frac{1}{2}$, find $\log_{625} x$. | If $\log_9 (x-2)=\frac{1}{2}$, find $\log_{625} x$. |
The trip from Greenville to Springfield requires $2\frac 34$ hours when traveling at an average speed of 55 miles per hour. How many hours does the trip require when traveling at an average speed of 50 miles per hour? Express your answer as a decimal to the nearest hundredth. | The trip from Greenville to Springfield requires $2\frac 34$ hours when traveling at an average speed of 55 miles per hour. How many hours does the trip require when traveling at an average speed of 50 miles per hour? Express your answer as a decimal to the nearest hundredth. |
A standard deck of cards has 52 cards divided into 4 suits, each of which has 13 cards. Two of the suits ($\heartsuit$ and $\diamondsuit$, called 'hearts' and 'diamonds') are red, the other two ($\spadesuit$ and $\clubsuit$, called 'spades' and 'clubs') are black. The cards in the deck are placed in random order (usu... | A standard deck of cards has 52 cards divided into 4 suits, each of which has 13 cards. Two of the suits ($\heartsuit$ and $\diamondsuit$, called 'hearts' and 'diamonds') are red, the other two ($\spadesuit$ and $\clubsuit$, called 'spades' and 'clubs') are black. The cards in the deck are placed in random order (usu... |
A bookstore sells two types of books: fiction and non-fiction. The price of a fiction book is $\$15$, and the price of a non-fiction book is $\$7$. If a customer buys a total of 22 books and spends $\$251$, and it is known that the number of fiction books is 4 more than the number of non-fiction books, how many non-fic... | A bookstore sells two types of books: fiction and non-fiction. The price of a fiction book is $\$15$, and the price of a non-fiction book is $\$7$. If a customer buys a total of 22 books and spends $\$251$, and it is known that the number of fiction books is 4 more than the number of non-fiction books, how many non-fic... |
The expression $| \frac{3\pi}{2} - | \frac{\pi}{2} - 2 | |$ can be simplified to $a \pi + b$ for some integers $a$ and $b$. Find $a+b$. | The expression $| \frac{3\pi}{2} - | \frac{\pi}{2} - 2 | |$ can be simplified to $a \pi + b$ for some integers $a$ and $b$. Find $a+b$. |
A spinner has 6 equal sections, labeled 1 through 6. If the spinner is spun twice, what is the probability that the product of the two numbers spun is a multiple of 4? Express your answer as a common fraction. | A spinner has 6 equal sections, labeled 1 through 6. If the spinner is spun twice, what is the probability that the product of the two numbers spun is a multiple of 4? Express your answer as a common fraction. |
If the fraction $\frac{42}{7^4\cdot 3^2}$ is expressed as a decimal, how many non-zero digits are to the left of the decimal point? | If the fraction $\frac{42}{7^4\cdot 3^2}$ is expressed as a decimal, how many non-zero digits are to the left of the decimal point? |
A point $P$ is initially at the origin of a coordinate system. It is first reflected across the line $y = x$, then rotated $90^\circ$ counterclockwise about the origin, then translated $3$ units in the positive $x$-direction, and finally reflected across the $x$-axis. Find the coordinates of the point after all these t... | A point $P$ is initially at the origin of a coordinate system. It is first reflected across the line $y = x$, then rotated $90^\circ$ counterclockwise about the origin, then translated $3$ units in the positive $x$-direction, and finally reflected across the $x$-axis. Find the coordinates of the point after all these t... |
What is the smallest positive integer $n$ such that $\left(\dfrac{3}{4}\right)^n$ is less than 0.01? | What is the smallest positive integer $n$ such that $\left(\dfrac{3}{4}\right)^n$ is less than 0.01? |
Let $\omega$ be a nonreal root of $x^3 = 1.$ Compute
\[(1 - \omega + \omega^2)^4 + (1 + \omega - \omega^2)^4.\] | Let $\omega$ be a nonreal root of $x^3 = 1.$ Compute
\[(1 - \omega + \omega^2)^4 + (1 + \omega - \omega^2)^4.\] |
Determine a positive integer $k$ such that $3^{100} + 1$ is divisible by $k^2$. Find the remainder when $3^{200} - 1$ is divided by $k$. | Determine a positive integer $k$ such that $3^{100} + 1$ is divisible by $k^2$. Find the remainder when $3^{200} - 1$ is divided by $k$. |
Let $n$ be the smallest real solution to the equation
\[\dfrac{2}{x-2} + \dfrac{7}{x-7} + \dfrac{13}{x-13} + \dfrac{15}{x-15} = x^2 - 10x - 3\]There are positive integers $p, q,$ and $r$ such that $n = p - \sqrt{q - \sqrt{r}}$. Find $p+q+r$. | Let $n$ be the smallest real solution to the equation
\[\dfrac{2}{x-2} + \dfrac{7}{x-7} + \dfrac{13}{x-13} + \dfrac{15}{x-15} = x^2 - 10x - 3\]There are positive integers $p, q,$ and $r$ such that $n = p - \sqrt{q - \sqrt{r}}$. Find $p+q+r$. |
Let $f(x)$ be a function such that $f(x+1)-f(x) = 2x+1$ for all real numbers $x$. If $f(0) = 1$, find $f(-1)+f(1)$. | Let $f(x)$ be a function such that $f(x+1)-f(x) = 2x+1$ for all real numbers $x$. If $f(0) = 1$, find $f(-1)+f(1)$. |
Given that |2x - 1| = p, where x > 1/2, find x + p in terms of p. | Given that |2x - 1| = p, where x > 1/2, find x + p in terms of p. |
What is the height of Jack's house, in feet, if the house casts a shadow 56 feet long at the same time a 21-foot tree casts a shadow that is 24 feet long? Express your answer to the nearest whole number. | What is the height of Jack's house, in feet, if the house casts a shadow 56 feet long at the same time a 21-foot tree casts a shadow that is 24 feet long? Express your answer to the nearest whole number. |
Here is a new problem:
A deck of cards contains 52 cards, with 4 suits (hearts, diamonds, clubs, and spades) and 13 cards in each suit. If a card is randomly selected from the deck, what is the probability that it is either a face card (king, queen, or jack) or a card of the suit hearts? Express your answer as a commo... | Here is a new problem:
A deck of cards contains 52 cards, with 4 suits (hearts, diamonds, clubs, and spades) and 13 cards in each suit. If a card is randomly selected from the deck, what is the probability that it is either a face card (king, queen, or jack) or a card of the suit hearts? Express your answer as a commo... |
Find the integer $k$ such that the polynomial $x^3 - 2x + k$ divides the polynomial $(x-1)^{10} + (x+1)^{10}$. | Find the integer $k$ such that the polynomial $x^3 - 2x + k$ divides the polynomial $(x-1)^{10} + (x+1)^{10}$. |
Find the largest integer value of $y$ for which $8-3y>22$. | Find the largest integer value of $y$ for which $8-3y>22$. |
The value of $y$ varies inversely as $\sqrt x$ and when $x=2$, $y=4$. What is $x$ when $y=1$? | The value of $y$ varies inversely as $\sqrt x$ and when $x=2$, $y=4$. What is $x$ when $y=1$? |
A bookshelf has 5 shelves, and each shelf can hold 8 boxes of pens. If each box of pens weighs 2 pounds, how many pounds of pens can the bookshelf hold in total? | A bookshelf has 5 shelves, and each shelf can hold 8 boxes of pens. If each box of pens weighs 2 pounds, how many pounds of pens can the bookshelf hold in total? |
In an isosceles triangle, one of the angles opposite an equal side is $40^{\circ}$. How many degrees are in the measure of the triangle's largest angle? [asy] draw((0,0)--(6,0)--(3,2)--(0,0)); label("$\backslash$",(1.5,1)); label("{/}",(4.5,1));
label("$40^{\circ}$",(.5,0),dir(45));
[/asy] | In an isosceles triangle, one of the angles opposite an equal side is $40^{\circ}$. How many degrees are in the measure of the triangle's largest angle? [asy] draw((0,0)--(6,0)--(3,2)--(0,0)); label("$\backslash$",(1.5,1)); label("{/}",(4.5,1));
label("$40^{\circ}$",(.5,0),dir(45));
[/asy] |
A tetrahedron has four triangular faces, and each face has three edges. If we select two distinct edges at random from a tetrahedron, what is the probability that they form an edge of one of the triangular faces? | A tetrahedron has four triangular faces, and each face has three edges. If we select two distinct edges at random from a tetrahedron, what is the probability that they form an edge of one of the triangular faces? |
An infinite arithmetic-geometric series has the form:
$$S=a+(a+d)r+(a+2d)r^2+(a+3d)r^3+\dots$$
where $a$ is the first term, $r$ is the common ratio, and $d$ is the common difference. Find the sum $S$ of the series, given that $a=2$, $r=\frac{1}{3}$, and $d=-\frac{1}{2}$. | An infinite arithmetic-geometric series has the form:
$$S=a+(a+d)r+(a+2d)r^2+(a+3d)r^3+\dots$$
where $a$ is the first term, $r$ is the common ratio, and $d$ is the common difference. Find the sum $S$ of the series, given that $a=2$, $r=\frac{1}{3}$, and $d=-\frac{1}{2}$. |
A cylindrical container with a height of 12 cm and a radius of 5 cm is filled with water. A hole in the shape of a hemisphere with a radius of 2 cm is drilled into the side of the cylinder, 4 cm from the bottom. What percentage of the original volume of water is lost when the hole is drilled? | A cylindrical container with a height of 12 cm and a radius of 5 cm is filled with water. A hole in the shape of a hemisphere with a radius of 2 cm is drilled into the side of the cylinder, 4 cm from the bottom. What percentage of the original volume of water is lost when the hole is drilled? |
Two integers, $m$ and $n$, are randomly selected from the set of positive integers less than 100, with replacement. If the probability that the sum of the two integers is even is $\frac{3}{5}$, what is the probability that the sum of the two integers is a multiple of 3? | Two integers, $m$ and $n$, are randomly selected from the set of positive integers less than 100, with replacement. If the probability that the sum of the two integers is even is $\frac{3}{5}$, what is the probability that the sum of the two integers is a multiple of 3? |
How many integer bases greater than or equal to 2 and less than or equal to 12 have a representation of the decimal number 210 that ends in a 0? | How many integer bases greater than or equal to 2 and less than or equal to 12 have a representation of the decimal number 210 that ends in a 0? |
Let $\mathbf{v}$ be a non-zero vector such that
$\mathbf{v}\cdot\begin{pmatrix}1\\-1\\1\end{pmatrix}=0$. Find $\mathbf{v}$ such that
\[\frac{\mathbf{v}\cdot\begin{pmatrix}1\\2\\1\end{pmatrix}}{\mathbf{v}\cdot\begin{pmatrix}1\\0\\1\end{pmatrix}}=\frac{5}{3}.\] | Let $\mathbf{v}$ be a non-zero vector such that
$\mathbf{v}\cdot\begin{pmatrix}1\\-1\\1\end{pmatrix}=0$. Find $\mathbf{v}$ such that
\[\frac{\mathbf{v}\cdot\begin{pmatrix}1\\2\\1\end{pmatrix}}{\mathbf{v}\cdot\begin{pmatrix}1\\0\\1\end{pmatrix}}=\frac{5}{3}.\] |
Solve for $w$ in the following equation: $\frac{2-w}{2+w} = i$, where $i^2 = -1$. Simplify your answer as much as possible. | Solve for $w$ in the following equation: $\frac{2-w}{2+w} = i$, where $i^2 = -1$. Simplify your answer as much as possible. |
What is the sum of all the numbers between 1 and 10 that are not prime? | What is the sum of all the numbers between 1 and 10 that are not prime? |
In a music festival, there are 5 stages, each featuring a different genre of music: rock, pop, jazz, classical, and folk. A music enthusiast wants to attend exactly 3 stages, with the condition that they must visit at least one stage featuring a genre that starts with the letter "J" (jazz) and at least one stage featur... | In a music festival, there are 5 stages, each featuring a different genre of music: rock, pop, jazz, classical, and folk. A music enthusiast wants to attend exactly 3 stages, with the condition that they must visit at least one stage featuring a genre that starts with the letter "J" (jazz) and at least one stage featur... |
Find the greatest common divisor (GCD) of 12, 15, and 21. | Find the greatest common divisor (GCD) of 12, 15, and 21. |
In a school play, there are 8 students playing the lead roles, but 4 of them have a 3/7 chance of getting stage fright and dropping out. What is the probability that at most 2 students will drop out, ensuring the play can still go on? | In a school play, there are 8 students playing the lead roles, but 4 of them have a 3/7 chance of getting stage fright and dropping out. What is the probability that at most 2 students will drop out, ensuring the play can still go on? |
Consider a sequence where each term is the product of the prime factors of the previous term. The first term of the sequence is 2023. What is the smallest prime number that will never appear as a term in this sequence? | Consider a sequence where each term is the product of the prime factors of the previous term. The first term of the sequence is 2023. What is the smallest prime number that will never appear as a term in this sequence? |
In triangle $ABC$, $AB=AC$ and angle $BAC$ measures $36^\circ$. The angle bisectors of angles $B$ and $C$ intersect at a point $D$. What is the measure of obtuse angle $BDC$?
[asy]
import geometry;
import olympiad;
unitsize(0.8inch);
dotfactor = 3;
defaultpen(linewidth(1pt)+fontsize(10pt));
pair A,B,C,D,E,F;
A = (0,... | In triangle $ABC$, $AB=AC$ and angle $BAC$ measures $36^\circ$. The angle bisectors of angles $B$ and $C$ intersect at a point $D$. What is the measure of obtuse angle $BDC$?
[asy]
import geometry;
import olympiad;
unitsize(0.8inch);
dotfactor = 3;
defaultpen(linewidth(1pt)+fontsize(10pt));
pair A,B,C,D,E,F;
A = (0,... |
Five consecutive two-digit positive integers, each less than 30, are not prime. What is the largest of these five integers? | Five consecutive two-digit positive integers, each less than 30, are not prime. What is the largest of these five integers? |
A bakery sells two types of bread: whole wheat and white. The cost of producing one loaf of whole wheat bread is $0.75, and the cost of producing one loaf of white bread is $0.50. If the bakery sells a total of 250 loaves of bread per day, and the total cost of production is $175, how many loaves of each type of bread ... | A bakery sells two types of bread: whole wheat and white. The cost of producing one loaf of whole wheat bread is $0.75, and the cost of producing one loaf of white bread is $0.50. If the bakery sells a total of 250 loaves of bread per day, and the total cost of production is $175, how many loaves of each type of bread ... |
Given that the equation
$k\ds\sqrt {\ds\frac{x-1}{x+1}}+2\sqrt{\ds\frac{x+1}{x-1}}=3$
is correct,
determine $k$. | Given that the equation
$k\ds\sqrt {\ds\frac{x-1}{x+1}}+2\sqrt{\ds\frac{x+1}{x-1}}=3$
is correct,
determine $k$. |
Find the coefficient of the $x^3$ term in the expansion of $(x^2+2)^{15}$. | Find the coefficient of the $x^3$ term in the expansion of $(x^2+2)^{15}$. |
What is the smallest five-digit whole number that is both a perfect square and a perfect fourth power? | What is the smallest five-digit whole number that is both a perfect square and a perfect fourth power? |
In a round-robin chess tournament, 12 players are seated around a circular table. Each player plays a game against the player sitting opposite them, then moves to the next seat in a clockwise direction and plays against the player sitting opposite their new seat. If this pattern continues, how many total games will eac... | In a round-robin chess tournament, 12 players are seated around a circular table. Each player plays a game against the player sitting opposite them, then moves to the next seat in a clockwise direction and plays against the player sitting opposite their new seat. If this pattern continues, how many total games will eac... |
A box contains 12 different colored balls: 4 red, 4 blue, and 4 green. You are to draw a ball from the box, then flip a coin, and finally roll a fair six-sided die. In how many ways can you perform these actions such that the color of the ball and the number on the die have the same parity (i.e., both are odd or both a... | A box contains 12 different colored balls: 4 red, 4 blue, and 4 green. You are to draw a ball from the box, then flip a coin, and finally roll a fair six-sided die. In how many ways can you perform these actions such that the color of the ball and the number on the die have the same parity (i.e., both are odd or both a... |
The ratio of boys to girls in Mr. Johnson's science class is 5:4. If there are a total of 72 students, how many boys are in Mr. Johnson's science class? | The ratio of boys to girls in Mr. Johnson's science class is 5:4. If there are a total of 72 students, how many boys are in Mr. Johnson's science class? |
A set of rectangular tiles are used to completely cover a square table. The tiles are 2x2 feet, 3x3 feet, and 4x4 feet. What is the largest size of the square table that can be covered using only these tiles, with no tile overlapping or extending beyond the table's edges? | A set of rectangular tiles are used to completely cover a square table. The tiles are 2x2 feet, 3x3 feet, and 4x4 feet. What is the largest size of the square table that can be covered using only these tiles, with no tile overlapping or extending beyond the table's edges? |
What is the largest positive integer $n$ such that every interval of length $n$ contains a prime or a number that can be written as the sum of two primes? | What is the largest positive integer $n$ such that every interval of length $n$ contains a prime or a number that can be written as the sum of two primes? |
A function f is defined on the interval [0, 7] as follows:
\[f(x) = \begin{cases}
x^2-3 & 0\le x\le 4 \\
-x+7 & 4 < x\le 7
\end{cases}\]
How many values of $x$ satisfy $f(f(x)) = x$? | A function f is defined on the interval [0, 7] as follows:
\[f(x) = \begin{cases}
x^2-3 & 0\le x\le 4 \\
-x+7 & 4 < x\le 7
\end{cases}\]
How many values of $x$ satisfy $f(f(x)) = x$? |
Find the smallest integer $n$ such that $2^n > \pi \cdot 10^{12}$. | Find the smallest integer $n$ such that $2^n > \pi \cdot 10^{12}$. |
Solve for $x$: $\sqrt[3]{20x + \sqrt[3]{20x + 13}} = 13.$ | Solve for $x$: $\sqrt[3]{20x + \sqrt[3]{20x + 13}} = 13.$ |
Let's consider a function with a similar trigonometric theme and domain restriction.
Find the smallest value of the positive integer $k$ such that the function $g(x) = \arccos(\sqrt{kx})$ has a domain that includes the interval $[0, \frac{1}{2}]$. | Let's consider a function with a similar trigonometric theme and domain restriction.
Find the smallest value of the positive integer $k$ such that the function $g(x) = \arccos(\sqrt{kx})$ has a domain that includes the interval $[0, \frac{1}{2}]$. |
In triangle $ABC$, a line parallel to $BC$ intersects the extensions of sides $AB$ and $AC$ at points $D$ and $E$, respectively. If the ratio of the area of triangle $ADE$ to the area of triangle $ABC$ is $2:5$, find the ratio of the length of $DE$ to the length of $BC$. | In triangle $ABC$, a line parallel to $BC$ intersects the extensions of sides $AB$ and $AC$ at points $D$ and $E$, respectively. If the ratio of the area of triangle $ADE$ to the area of triangle $ABC$ is $2:5$, find the ratio of the length of $DE$ to the length of $BC$. |
A group of friends went to a restaurant to have dinner together. They ordered a plate of ribs, a serving of mashed potatoes, a vegetable salad, and a round of drinks. The total cost of the meal was $80. If the serving of mashed potatoes was $5 more expensive than the vegetable salad, and the plate of ribs cost $10 more... | A group of friends went to a restaurant to have dinner together. They ordered a plate of ribs, a serving of mashed potatoes, a vegetable salad, and a round of drinks. The total cost of the meal was $80. If the serving of mashed potatoes was $5 more expensive than the vegetable salad, and the plate of ribs cost $10 more... |
When $m$ is divided by 4, the remainder is 3. What is the remainder when $2m$ is divided by 4? | When $m$ is divided by 4, the remainder is 3. What is the remainder when $2m$ is divided by 4? |
A line is parameterized by a parameter $t,$ so that the vector on the line at $t = 2$ is $\begin{pmatrix} 1 \\ 4 \end{pmatrix},$ and the vector on the line at $t = 3$ is $\begin{pmatrix} 3 \\ -4 \end{pmatrix}.$ Find the vector on the line at $t = -7.$ | A line is parameterized by a parameter $t,$ so that the vector on the line at $t = 2$ is $\begin{pmatrix} 1 \\ 4 \end{pmatrix},$ and the vector on the line at $t = 3$ is $\begin{pmatrix} 3 \\ -4 \end{pmatrix}.$ Find the vector on the line at $t = -7.$ |
If $x$ is a number between 0 and 1, which of the following represents the smallest value?
A). $x$
B). $x^2$
C). $2x$
D). $\sqrt{x}$
E). $\frac{1}{x}$
Express your answer as A, B, C, D, or E. | If $x$ is a number between 0 and 1, which of the following represents the smallest value?
A). $x$
B). $x^2$
C). $2x$
D). $\sqrt{x}$
E). $\frac{1}{x}$
Express your answer as A, B, C, D, or E. |
Find the least positive integer $y$ that satisfies $y+2743 \equiv 1876 \pmod{8}$. | Find the least positive integer $y$ that satisfies $y+2743 \equiv 1876 \pmod{8}$. |
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