instruction_seed stringlengths 13 12.7k | response_seed stringlengths 13 12.7k |
|---|---|
Let $f(x) = x^4 - 4x^3 + 6x^2 - 4x + 1$. Find the minimum value of $f(x)$ on the interval $[0, 2]$. | Let $f(x) = x^4 - 4x^3 + 6x^2 - 4x + 1$. Find the minimum value of $f(x)$ on the interval $[0, 2]$. |
A bookshelf has 5 shelves, and the number of books on each shelf forms a geometric sequence. If the top shelf has 12 books and the total number of books on all 5 shelves is 228, how many books will be on the 10th shelf if the bookshelf is extended to hold a total of 10 shelves? | A bookshelf has 5 shelves, and the number of books on each shelf forms a geometric sequence. If the top shelf has 12 books and the total number of books on all 5 shelves is 228, how many books will be on the 10th shelf if the bookshelf is extended to hold a total of 10 shelves? |
Express $0.2\overline{718}$ as a common fraction. | Express $0.2\overline{718}$ as a common fraction. |
Rationalize the denominator of $\frac{3}{1-\sqrt{5}}$. The answer can be written as $\frac{A\sqrt{B}+C}{D}$, where $A$, $B$, $C$, and $D$ are integers, $D$ is positive, and $B$ is not divisible by the square of any prime. If the greatest common divisor of $A$, $C$, and $D$ is 1, find $A+B+C+D$. | Rationalize the denominator of $\frac{3}{1-\sqrt{5}}$. The answer can be written as $\frac{A\sqrt{B}+C}{D}$, where $A$, $B$, $C$, and $D$ are integers, $D$ is positive, and $B$ is not divisible by the square of any prime. If the greatest common divisor of $A$, $C$, and $D$ is 1, find $A+B+C+D$. |
Find the modular inverse of $4$, modulo $21$.
Express your answer as an integer from $0$ to $20$, inclusive. | Find the modular inverse of $4$, modulo $21$.
Express your answer as an integer from $0$ to $20$, inclusive. |
If $27^8=9^q$, what is $q$? | If $27^8=9^q$, what is $q$? |
The complex number $z = 2 - 3i$ is expressed in polar form as $r(\cos\theta + i\sin\theta)$, where $r > 0$ and $0^\circ \le \theta < 360^\circ$. If $z$ is raised to the power of $2$, the result is a complex number that can be expressed in the form $a + bi$, where $a$ and $b$ are real numbers. Determine the value of $a ... | The complex number $z = 2 - 3i$ is expressed in polar form as $r(\cos\theta + i\sin\theta)$, where $r > 0$ and $0^\circ \le \theta < 360^\circ$. If $z$ is raised to the power of $2$, the result is a complex number that can be expressed in the form $a + bi$, where $a$ and $b$ are real numbers. Determine the value of $a ... |
What is the sum of all the positive divisors of the prime numbers between 1 and 20? | What is the sum of all the positive divisors of the prime numbers between 1 and 20? |
Find the area of the triangle with vertices $(6,5,3),$ $(3,3,1),$ and $(15,11,9).$ | Find the area of the triangle with vertices $(6,5,3),$ $(3,3,1),$ and $(15,11,9).$ |
In a trapezoid $ABCD$, the bases $AB$ and $CD$ are parallel, and the diagonals $AC$ and $BD$ intersect at point $E$. If the coordinates of points $A$, $B$, $C$, and $D$ are $(-3, 0)$, $(3, 0)$, $(5, 4)$, and $(-1, 4)$, respectively, find the coordinates of point $E$. | In a trapezoid $ABCD$, the bases $AB$ and $CD$ are parallel, and the diagonals $AC$ and $BD$ intersect at point $E$. If the coordinates of points $A$, $B$, $C$, and $D$ are $(-3, 0)$, $(3, 0)$, $(5, 4)$, and $(-1, 4)$, respectively, find the coordinates of point $E$. |
A box contains 15 identical blue balls, 20 identical red balls, and 25 identical green balls. A boy takes 5 balls from the box and puts them in his pocket. How many distinct possible combinations of colored balls could he have in his pocket? | A box contains 15 identical blue balls, 20 identical red balls, and 25 identical green balls. A boy takes 5 balls from the box and puts them in his pocket. How many distinct possible combinations of colored balls could he have in his pocket? |
Simplify $(9 \times 10^8) \div (3 \times 10^3)$. (Your answer should be a single number, with no products or quotients.) | Simplify $(9 \times 10^8) \div (3 \times 10^3)$. (Your answer should be a single number, with no products or quotients.) |
One root of the equation $3x^2+mx=1$ is 1. What is the other? | One root of the equation $3x^2+mx=1$ is 1. What is the other? |
Five points $A, B, C, D, E$ are marked on a circle. How many distinct convex polygons can be formed by connecting 3 or more of these points? | Five points $A, B, C, D, E$ are marked on a circle. How many distinct convex polygons can be formed by connecting 3 or more of these points? |
A square and four congruent right triangles are arranged as shown. The square has a side length of $12$ meters. If the right triangles are isosceles, what is the length in meters of one of the legs of the right triangles?
[asy]
draw((0,0)--(12,0)--(12,12)--(0,12)--cycle);
draw((12,12)--(18,12)--(12,0));
draw((12,... | A square and four congruent right triangles are arranged as shown. The square has a side length of $12$ meters. If the right triangles are isosceles, what is the length in meters of one of the legs of the right triangles?
[asy]
draw((0,0)--(12,0)--(12,12)--(0,12)--cycle);
draw((12,12)--(18,12)--(12,0));
draw((12,... |
Given that $a$, $b$, $c$, $d$, $e$, and $f$ are integers such that $729x^3 + 64 = (ax^2 + bx + c)(dx^2 + ex + f)$ for all $x$, find $a^2 + b^2 + c^2 + d^2 + e^2 + f^2$. | Given that $a$, $b$, $c$, $d$, $e$, and $f$ are integers such that $729x^3 + 64 = (ax^2 + bx + c)(dx^2 + ex + f)$ for all $x$, find $a^2 + b^2 + c^2 + d^2 + e^2 + f^2$. |
In a triangle, the measure of angle C is twice the measure of angle A, and the measure of angle B is 15 degrees more than the measure of angle A. What is the measure of angle C? | In a triangle, the measure of angle C is twice the measure of angle A, and the measure of angle B is 15 degrees more than the measure of angle A. What is the measure of angle C? |
There are four even integers in the top five rows of Pascal's Triangle. How many even integers are in the top 10 rows of the triangle? | There are four even integers in the top five rows of Pascal's Triangle. How many even integers are in the top 10 rows of the triangle? |
A company has a strict hierarchy with a president, two vice presidents, and two teams of employees reporting to each vice president. However, there's an additional constraint: the president and vice presidents must be chosen from different departments, and each team of employees must have exactly one person from the pr... | A company has a strict hierarchy with a president, two vice presidents, and two teams of employees reporting to each vice president. However, there's an additional constraint: the president and vice presidents must be chosen from different departments, and each team of employees must have exactly one person from the pr... |
Tom has been saving money for a new bike and has $120 in his piggy bank. Each week, he deposits an amount that is 20% more than what he deposited the previous week. If he saves for 8 weeks, what will be the total amount of money he has at the end of the 8th week? | Tom has been saving money for a new bike and has $120 in his piggy bank. Each week, he deposits an amount that is 20% more than what he deposited the previous week. If he saves for 8 weeks, what will be the total amount of money he has at the end of the 8th week? |
A school has three types of classrooms: lecture halls, discussion rooms, and labs. The number of discussion rooms is $\frac{1}{4}$ the total number of classrooms, and the number of labs is $\frac{3}{5}$ the total number of classrooms. If the school has 18 lecture halls, what is the total number of classrooms? | A school has three types of classrooms: lecture halls, discussion rooms, and labs. The number of discussion rooms is $\frac{1}{4}$ the total number of classrooms, and the number of labs is $\frac{3}{5}$ the total number of classrooms. If the school has 18 lecture halls, what is the total number of classrooms? |
What is the largest number that can be expressed as a sum of distinct powers of 3 using each power at most once? | What is the largest number that can be expressed as a sum of distinct powers of 3 using each power at most once? |
The diagonals of rectangle $PQRS$ intersect at point $X$. If $PS = 10$ and $RS=24$, then what is $\cos \angle PXS$? | The diagonals of rectangle $PQRS$ intersect at point $X$. If $PS = 10$ and $RS=24$, then what is $\cos \angle PXS$? |
A car travels from city A to city B at an average speed of 60 miles per hour, and then returns from city B to city A at an average speed of 45 miles per hour. If the total travel time for the round trip is 7 hours, what is the distance between city A and city B? | A car travels from city A to city B at an average speed of 60 miles per hour, and then returns from city B to city A at an average speed of 45 miles per hour. If the total travel time for the round trip is 7 hours, what is the distance between city A and city B? |
What is the largest positive integer $n$ such that $n!$ can be expressed as a product of $n$ consecutive integers? | What is the largest positive integer $n$ such that $n!$ can be expressed as a product of $n$ consecutive integers? |
In Mr. Thompson's class, 12 of the 20 students scored above 90 on the latest test. If the same proportion of students scored above 90 on Ms. Patel's latest test, and if Ms. Patel has 30 students total, how many students in Ms. Patel's class scored above 90? | In Mr. Thompson's class, 12 of the 20 students scored above 90 on the latest test. If the same proportion of students scored above 90 on Ms. Patel's latest test, and if Ms. Patel has 30 students total, how many students in Ms. Patel's class scored above 90? |
Consider the functions
$$f(x) = 2 + \sqrt {2 - x^2}$$
$$g(x) = -8 + \sqrt {5 + x^2}$$
The function
$$h(x) = g \circ f (x)$$
is defined only for
$x \in [a, b]$.
What is the value of $b - a$? | Consider the functions
$$f(x) = 2 + \sqrt {2 - x^2}$$
$$g(x) = -8 + \sqrt {5 + x^2}$$
The function
$$h(x) = g \circ f (x)$$
is defined only for
$x \in [a, b]$.
What is the value of $b - a$? |
What is the digit in the tens place when $7^{2005}$ is expressed in decimal notation? | What is the digit in the tens place when $7^{2005}$ is expressed in decimal notation? |
The equation $x^4-4x^2+4=0$ has two real solutions and two non-real solutions. What is the positive difference between the two real solutions? | The equation $x^4-4x^2+4=0$ has two real solutions and two non-real solutions. What is the positive difference between the two real solutions? |
Compute the number of ways to arrange 5 red balls and 3 blue balls in a row such that no two blue balls are adjacent. | Compute the number of ways to arrange 5 red balls and 3 blue balls in a row such that no two blue balls are adjacent. |
Find the remainder when $3^{1999}$ is divided by $13$. | Find the remainder when $3^{1999}$ is divided by $13$. |
A circle has an area of $\pi$ square units. What is the length of the circle's diameter, in units? | A circle has an area of $\pi$ square units. What is the length of the circle's diameter, in units? |
Find the sum of all positive integer divisors of 100 that are not multiples of 3. | Find the sum of all positive integer divisors of 100 that are not multiples of 3. |
While watching a circus show, I counted out the number of acrobats and elephants. I counted 40 legs and 15 heads. How many acrobats did I see in the show? | While watching a circus show, I counted out the number of acrobats and elephants. I counted 40 legs and 15 heads. How many acrobats did I see in the show? |
A plane flew from city A to city B at a constant speed of x miles per hour. On the return trip, the plane flew at a constant speed of y miles per hour. If the total distance between the two cities is 1000 miles and the total flying time was 2 hours and 30 minutes, what is the average speed of the plane for the entire t... | A plane flew from city A to city B at a constant speed of x miles per hour. On the return trip, the plane flew at a constant speed of y miles per hour. If the total distance between the two cities is 1000 miles and the total flying time was 2 hours and 30 minutes, what is the average speed of the plane for the entire t... |
A Ferris wheel has 16 cars, each of which can hold 4 people. If the ride starts with 3 people in the first car, 4 people in the second car, 5 people in the third car, and so on, and each person rides for 5 complete revolutions, how many people will be in the first car after 2008 revolutions? | A Ferris wheel has 16 cars, each of which can hold 4 people. If the ride starts with 3 people in the first car, 4 people in the second car, 5 people in the third car, and so on, and each person rides for 5 complete revolutions, how many people will be in the first car after 2008 revolutions? |
Sets $X$ and $Y$, shown in the Venn diagram, are such that the total number of elements in set $X$ is thrice the total number of elements in set $Y$. Altogether, there are 1824 elements in the union of $X$ and $Y$, and their intersection has 405 elements. What is the total number of elements in set $X$?
[asy]
label("$... | Sets $X$ and $Y$, shown in the Venn diagram, are such that the total number of elements in set $X$ is thrice the total number of elements in set $Y$. Altogether, there are 1824 elements in the union of $X$ and $Y$, and their intersection has 405 elements. What is the total number of elements in set $X$?
[asy]
label("$... |
Determine $b$ if the equation
$$\frac{z^3-3z^2+2z+b}{z^2+1}=1$$
has only one real root. | Determine $b$ if the equation
$$\frac{z^3-3z^2+2z+b}{z^2+1}=1$$
has only one real root. |
A fair 6-sided die is rolled twice. What is the probability that the sum of the two rolls is a prime number? | A fair 6-sided die is rolled twice. What is the probability that the sum of the two rolls is a prime number? |
If $x^2 - 3x + 7$ is a factor of $x^3 + ax^2 + bx + 21,$ find $a^2 - 2b.$ | If $x^2 - 3x + 7$ is a factor of $x^3 + ax^2 + bx + 21,$ find $a^2 - 2b.$ |
What is the number of degrees in the smaller angle formed by the hour and minute hands of a clock at 8:15? Express your answer as a decimal to the nearest tenth.
[asy]
size(200);
draw(Circle((0,0),5),linewidth(1.2));
pair[] mins;
for(int i = 0; i < 60; ++i){
mins[i] = 4.5*dir(-6*i + 90);
dot(mins[i]);
}
for(int i = 1... | What is the number of degrees in the smaller angle formed by the hour and minute hands of a clock at 8:15? Express your answer as a decimal to the nearest tenth.
[asy]
size(200);
draw(Circle((0,0),5),linewidth(1.2));
pair[] mins;
for(int i = 0; i < 60; ++i){
mins[i] = 4.5*dir(-6*i + 90);
dot(mins[i]);
}
for(int i = 1... |
Let $\mathbf{v}$ be a vector such that
\[\left\| \mathbf{v} + \begin{pmatrix} 3 \\ -1 \end{pmatrix} \right\| = 8.\]Find the smallest possible value of $\|\mathbf{v}\|.$ | Let $\mathbf{v}$ be a vector such that
\[\left\| \mathbf{v} + \begin{pmatrix} 3 \\ -1 \end{pmatrix} \right\| = 8.\]Find the smallest possible value of $\|\mathbf{v}\|.$ |
The measure of angle $XYZ$ is 25 degrees. If ray $YX$ is rotated 270 degrees about point $Y$ in a counterclockwise direction, what will be the positive measure of the new acute angle $XYZ$, in degrees? | The measure of angle $XYZ$ is 25 degrees. If ray $YX$ is rotated 270 degrees about point $Y$ in a counterclockwise direction, what will be the positive measure of the new acute angle $XYZ$, in degrees? |
A convergent geometric series has a sum of 100 and a second term of 20. Find the value of the eighth term of the sequence. | A convergent geometric series has a sum of 100 and a second term of 20. Find the value of the eighth term of the sequence. |
Calculate the sum of the geometric series $1 + \left(\frac{2}{3}\right) + \left(\frac{2}{3}\right)^2 + \left(\frac{2}{3}\right)^3 + \dots$. Express your answer as a common fraction. | Calculate the sum of the geometric series $1 + \left(\frac{2}{3}\right) + \left(\frac{2}{3}\right)^2 + \left(\frac{2}{3}\right)^3 + \dots$. Express your answer as a common fraction. |
What is the value of $x$ for which $|2x-3|$ is not positive? Express your answer as a common fraction. | What is the value of $x$ for which $|2x-3|$ is not positive? Express your answer as a common fraction. |
Find the polynomial of minimal degree, in $x,$ which has rational coefficients, leading coefficient $1$, and roots $2+\sqrt{5}$ and $3-\sqrt{2}.$ (Write the terms in decreasing order of degree.) | Find the polynomial of minimal degree, in $x,$ which has rational coefficients, leading coefficient $1$, and roots $2+\sqrt{5}$ and $3-\sqrt{2}.$ (Write the terms in decreasing order of degree.) |
Let $x$ and $y$ be positive real numbers such that $x^2 + y^2 = 2xy.$ Simplify
\[\frac{x^2}{y^2} + \frac{y^2}{x^2} - \frac{2}{xy}.\] | Let $x$ and $y$ be positive real numbers such that $x^2 + y^2 = 2xy.$ Simplify
\[\frac{x^2}{y^2} + \frac{y^2}{x^2} - \frac{2}{xy}.\] |
In how many ways can four people sit in a row of five chairs? | In how many ways can four people sit in a row of five chairs? |
Find the length of the segment $AB$ to the nearest tenth in the triangle $ABC$ below.
Note: In this triangle, $\angle BAC = 45^\circ$, $AC = 10$, and $AB \perp CD$.
[asy]
import graph;
size(8cm);
real lsf=0.5;
pen dps=linewidth(0.7)+fontsize(10);
defaultpen(dps);
pen ds=black;
real xmin=-4.8,xmax=4.8,ymin=-2.1,ymax=6... | Find the length of the segment $AB$ to the nearest tenth in the triangle $ABC$ below.
Note: In this triangle, $\angle BAC = 45^\circ$, $AC = 10$, and $AB \perp CD$.
[asy]
import graph;
size(8cm);
real lsf=0.5;
pen dps=linewidth(0.7)+fontsize(10);
defaultpen(dps);
pen ds=black;
real xmin=-4.8,xmax=4.8,ymin=-2.1,ymax=6... |
At a community event, 300 people participated in a survey about their interests in reading books, watching movies, and playing video games. The results showed that 120 people enjoyed reading books, 180 enjoyed watching movies, and 100 enjoyed playing video games. However, only 50 people were interested in both reading ... | At a community event, 300 people participated in a survey about their interests in reading books, watching movies, and playing video games. The results showed that 120 people enjoyed reading books, 180 enjoyed watching movies, and 100 enjoyed playing video games. However, only 50 people were interested in both reading ... |
Let $a$ and $b$ be real numbers. If $z$ is a complex number such that $|z-a|=2$ and $|z-b|=4$, determine $|a-b|$. | Let $a$ and $b$ be real numbers. If $z$ is a complex number such that $|z-a|=2$ and $|z-b|=4$, determine $|a-b|$. |
If $x$ satisfies $x^2 + 3x + \frac{3}x + \frac{1}{x^2} = 26$ and $x$ can be written as $a + \sqrt{b}$ where $a$ and $b$ are positive integers, then find $a + b$. | If $x$ satisfies $x^2 + 3x + \frac{3}x + \frac{1}{x^2} = 26$ and $x$ can be written as $a + \sqrt{b}$ where $a$ and $b$ are positive integers, then find $a + b$. |
The units digit of $7^n$ is 7 for certain values of $n$. Determine these values. | The units digit of $7^n$ is 7 for certain values of $n$. Determine these values. |
The equation $y=-4.9t^2+3.5t+5$ describes the height (in meters) of a ball thrown upward at $3.5$ meters per second from $5$ meters above the ground, where $t$ is the time in seconds. In how many seconds will the ball hit the ground? Express your answer as a common fraction. | The equation $y=-4.9t^2+3.5t+5$ describes the height (in meters) of a ball thrown upward at $3.5$ meters per second from $5$ meters above the ground, where $t$ is the time in seconds. In how many seconds will the ball hit the ground? Express your answer as a common fraction. |
Let $\mathbf{a}$ and $\mathbf{b}$ be two vectors such that
\[\|\mathbf{a} + \mathbf{b}\| = \|\mathbf{b}\|.\]Find the angle between the vectors $\mathbf{a} + 2 \mathbf{b}$ and $\mathbf{a},$ in degrees | Let $\mathbf{a}$ and $\mathbf{b}$ be two vectors such that
\[\|\mathbf{a} + \mathbf{b}\| = \|\mathbf{b}\|.\]Find the angle between the vectors $\mathbf{a} + 2 \mathbf{b}$ and $\mathbf{a},$ in degrees |
Find the number of positive integers $n$ such that $n^3+2n^2-3n+1$ is between $\sqrt{50}$ and $\sqrt{500}$. | Find the number of positive integers $n$ such that $n^3+2n^2-3n+1$ is between $\sqrt{50}$ and $\sqrt{500}$. |
Find the product of all constants $m$ such that the quadratic $x^2 + mx + 6$ can be factored in the form $(x+c)(x+d)$, where $c$ and $d$ are integers. | Find the product of all constants $m$ such that the quadratic $x^2 + mx + 6$ can be factored in the form $(x+c)(x+d)$, where $c$ and $d$ are integers. |
The diagonal of a rectangle is $5$ inches long and the width is $3$ inches. How many inches long is the length of the rectangle? | The diagonal of a rectangle is $5$ inches long and the width is $3$ inches. How many inches long is the length of the rectangle? |
Find the area of the region bounded by the graphs of $r = 2\cos \theta$ and $r = 2\sin \theta,$ in the first quadrant. | Find the area of the region bounded by the graphs of $r = 2\cos \theta$ and $r = 2\sin \theta,$ in the first quadrant. |
Emily drove 45 miles at 20 miles per hour and 25 miles at 40 miles per hour. What was her average speed, in miles per hour, for the entire trip? | Emily drove 45 miles at 20 miles per hour and 25 miles at 40 miles per hour. What was her average speed, in miles per hour, for the entire trip? |
A regular polygon has interior angles of 120 degrees. How many sides does the polygon have? | A regular polygon has interior angles of 120 degrees. How many sides does the polygon have? |
Factor the expression $x^6+64y^6$ completely. | Factor the expression $x^6+64y^6$ completely. |
How many digits are there in the base-7 representation of $956$? | How many digits are there in the base-7 representation of $956$? |
The parabolas defined by the equations $y = (x - 2)^2$ and $x = (y - 2)^2$ intersect at four points $(x_1,y_1),(x_2,y_2),(x_3,y_3), (x_4,y_4)$. Find $x_1 + y_1 + x_2 + y_2 + x_3 + y_3 + x_4 + y_4$. | The parabolas defined by the equations $y = (x - 2)^2$ and $x = (y - 2)^2$ intersect at four points $(x_1,y_1),(x_2,y_2),(x_3,y_3), (x_4,y_4)$. Find $x_1 + y_1 + x_2 + y_2 + x_3 + y_3 + x_4 + y_4$. |
Let $\mathbf{M}$ be a matrix, and let $\mathbf{v}$ and $\mathbf{w}$ be vectors, such that
\[\mathbf{M} \mathbf{v} = \begin{pmatrix} 1 \\ -5 \end{pmatrix} \quad \text{and} \quad \mathbf{M} \mathbf{w} = \begin{pmatrix} 7 \\ 2 \end{pmatrix}.\]Compute $\mathbf{M} (-2 \mathbf{v} + \mathbf{w}).$ | Let $\mathbf{M}$ be a matrix, and let $\mathbf{v}$ and $\mathbf{w}$ be vectors, such that
\[\mathbf{M} \mathbf{v} = \begin{pmatrix} 1 \\ -5 \end{pmatrix} \quad \text{and} \quad \mathbf{M} \mathbf{w} = \begin{pmatrix} 7 \\ 2 \end{pmatrix}.\]Compute $\mathbf{M} (-2 \mathbf{v} + \mathbf{w}).$ |
A cone is inscribed in a right circular cylinder so that the radius of the base of the cone is one third of the height of the cone. If the height of the cylinder is three times that of the cone, what is the ratio of the volume of the cone to the volume of the cylinder? | A cone is inscribed in a right circular cylinder so that the radius of the base of the cone is one third of the height of the cone. If the height of the cylinder is three times that of the cone, what is the ratio of the volume of the cone to the volume of the cylinder? |
For real numbers $t \neq 0,$ the point
\[(x,y) = \left( \frac{t + 1}{t}, \frac{t - 1}{t} \right)\]is plotted. All the plotted points lie on what kind of curve?
(A) Line
(B) Circle
(C) Parabola
(D) Ellipse
(E) Hyperbola
Enter the letter of the correct option. | For real numbers $t \neq 0,$ the point
\[(x,y) = \left( \frac{t + 1}{t}, \frac{t - 1}{t} \right)\]is plotted. All the plotted points lie on what kind of curve?
(A) Line
(B) Circle
(C) Parabola
(D) Ellipse
(E) Hyperbola
Enter the letter of the correct option. |
In $\triangle XYZ$, we have $\angle X = 90^\circ$ and $\tan Z = 7$. If $YZ = 100$, then what is $XY$? | In $\triangle XYZ$, we have $\angle X = 90^\circ$ and $\tan Z = 7$. If $YZ = 100$, then what is $XY$? |
Let $x$ be a real number. What is the range of the function $$f(x) = \left| \left|x\right| - x\right|~?$$Express your answer in interval notation. | Let $x$ be a real number. What is the range of the function $$f(x) = \left| \left|x\right| - x\right|~?$$Express your answer in interval notation. |
Let $f(x)= \frac{2}{x} + x$. Find $f(f(f(2)))$. | Let $f(x)= \frac{2}{x} + x$. Find $f(f(f(2)))$. |
A right pyramid with a square base has total surface area 432 square units. The area of each triangular face is half the area of the square face. What is the volume of the pyramid? | A right pyramid with a square base has total surface area 432 square units. The area of each triangular face is half the area of the square face. What is the volume of the pyramid? |
When $-2 + 2i$ is converted to the exponential form $re^{i \theta}$, what is $\theta$? | When $-2 + 2i$ is converted to the exponential form $re^{i \theta}$, what is $\theta$? |
Suppose $f(x) = 4x - 2$ and $g(x) = \frac{x}{2} + 1$. Find $f(g(x)) - g(f(x))$. | Suppose $f(x) = 4x - 2$ and $g(x) = \frac{x}{2} + 1$. Find $f(g(x)) - g(f(x))$. |
The function $f(x) = 2x^3 - 15x^2 - 36x + 7$ has a local maximum. Find the largest value of $a$ for which the function $g(x) = f(x) + ax$ has no local maximum. | The function $f(x) = 2x^3 - 15x^2 - 36x + 7$ has a local maximum. Find the largest value of $a$ for which the function $g(x) = f(x) + ax$ has no local maximum. |
Find the value(s) of $x$ such that the inequality $2xy - 3y + 5x - 2 > 0$ is true for all values of $y$. | Find the value(s) of $x$ such that the inequality $2xy - 3y + 5x - 2 > 0$ is true for all values of $y$. |
In triangle $ABC,$ $M$ is the midpoint of $\overline{BC},$ $AB = 12,$ and $AC = 16.$ Let $E$ be on $\overline{AC},$ and $F$ be on $\overline{AB},$ and let $G$ be the intersection of $\overline{EF}$ and $\overline{AM}.$ If $AE = 2AF,$ then find $\frac{EG}{GF}.$
[asy]
unitsize(0.3 cm);
pair A, B, C, E, F, G, M;
real ... | In triangle $ABC,$ $M$ is the midpoint of $\overline{BC},$ $AB = 12,$ and $AC = 16.$ Let $E$ be on $\overline{AC},$ and $F$ be on $\overline{AB},$ and let $G$ be the intersection of $\overline{EF}$ and $\overline{AM}.$ If $AE = 2AF,$ then find $\frac{EG}{GF}.$
[asy]
unitsize(0.3 cm);
pair A, B, C, E, F, G, M;
real ... |
Among all pairs of real numbers $(x, y)$ such that $\cos \cos x = \cos \cos y$ with $-5\pi \le x, y \le 5\pi$, Alex randomly selects a pair $(X, Y)$. Compute the probability that $X = Y$. | Among all pairs of real numbers $(x, y)$ such that $\cos \cos x = \cos \cos y$ with $-5\pi \le x, y \le 5\pi$, Alex randomly selects a pair $(X, Y)$. Compute the probability that $X = Y$. |
Let $\beta \neq -1$ be a complex number such that the distance from $\beta^3$ to -1 is three times the distance from $\beta$ to -1, while the distance from $\beta^6$ to -1 is six times the distance from $\beta$ to -1. Enter all possible values of $\beta,$ separated by commas. | Let $\beta \neq -1$ be a complex number such that the distance from $\beta^3$ to -1 is three times the distance from $\beta$ to -1, while the distance from $\beta^6$ to -1 is six times the distance from $\beta$ to -1. Enter all possible values of $\beta,$ separated by commas. |
A right triangle has one leg of length $x$, while the other leg is twice the length of the first leg. The perimeter of the triangle is 90 units. What is the length of the hypotenuse? | A right triangle has one leg of length $x$, while the other leg is twice the length of the first leg. The perimeter of the triangle is 90 units. What is the length of the hypotenuse? |
Compute
\[
\begin{vmatrix} \sin 1 & \sin 2 & \sin 3 \\ \sin 4 & \sin 5 & \sin 6 \\ \sin 7 & \sin 8 & \sin 9 \end{vmatrix}
.\]All the angles are in radians. | Compute
\[
\begin{vmatrix} \sin 1 & \sin 2 & \sin 3 \\ \sin 4 & \sin 5 & \sin 6 \\ \sin 7 & \sin 8 & \sin 9 \end{vmatrix}
.\]All the angles are in radians. |
Ten students are taking both algebra and drafting. There are 24 students taking algebra. There are 11 students who are taking drafting only. How many students are taking algebra or drafting but not both? | Ten students are taking both algebra and drafting. There are 24 students taking algebra. There are 11 students who are taking drafting only. How many students are taking algebra or drafting but not both? |
Let \[f(x) =
\begin{cases}
9x+4 &\text{if }x\text{ is an integer}, \\
\lfloor{x}\rfloor+5 &\text{if }x\text{ is not an integer}.
\end{cases}
\]Find $f(\sqrt{29})$. | Let \[f(x) =
\begin{cases}
9x+4 &\text{if }x\text{ is an integer}, \\
\lfloor{x}\rfloor+5 &\text{if }x\text{ is not an integer}.
\end{cases}
\]Find $f(\sqrt{29})$. |
Find the value of $x$ that satisfies the equation:
$$x^3 - 9x^2 + 27x - 27 = (x - 3)^3$$
This problem requires a different approach than the original, as it involves factoring and manipulating a cubic equation rather than a quadratic difference of squares. | Find the value of $x$ that satisfies the equation:
$$x^3 - 9x^2 + 27x - 27 = (x - 3)^3$$
This problem requires a different approach than the original, as it involves factoring and manipulating a cubic equation rather than a quadratic difference of squares. |
A hotel has two types of rooms: single and double. The daily rate for a single room is $5 less than that of a double room. During a certain period, the hotel had an average occupancy of 80% for its single rooms and 90% for its double rooms. If the total revenue from single rooms was $1200 and the total revenue from dou... | A hotel has two types of rooms: single and double. The daily rate for a single room is $5 less than that of a double room. During a certain period, the hotel had an average occupancy of 80% for its single rooms and 90% for its double rooms. If the total revenue from single rooms was $1200 and the total revenue from dou... |
What is the degree measure of the supplement of the complement of a 42-degree angle? | What is the degree measure of the supplement of the complement of a 42-degree angle? |
Determine the product of the non-real roots of the equation $x^4 - 2x^3 + 4x^2 + 1 = 0$, given that they have the same imaginary part. | Determine the product of the non-real roots of the equation $x^4 - 2x^3 + 4x^2 + 1 = 0$, given that they have the same imaginary part. |
How many perfect cubes are between 100 and 900? | How many perfect cubes are between 100 and 900? |
In quadrilateral ABCD, AB = BC = CD = 12 cm, angle BAD = 30°, angle ABD = 90°, and AD = 20 cm. Find the measure of angle BCD. | In quadrilateral ABCD, AB = BC = CD = 12 cm, angle BAD = 30°, angle ABD = 90°, and AD = 20 cm. Find the measure of angle BCD. |
A $3\times 3\times 3$ cube is made up of 27 unit cubes, and one unit cube is removed from each corner of the large cube. If a "diagonal segment" is a line segment connecting two vertices that do not belong to the same face of the large cube, how many such diagonal segments are there in the resulting figure? | A $3\times 3\times 3$ cube is made up of 27 unit cubes, and one unit cube is removed from each corner of the large cube. If a "diagonal segment" is a line segment connecting two vertices that do not belong to the same face of the large cube, how many such diagonal segments are there in the resulting figure? |
If $x+7$ is a factor of $cx^3 + 19x^2 - 3cx + 35$, find the constant $c$. | If $x+7$ is a factor of $cx^3 + 19x^2 - 3cx + 35$, find the constant $c$. |
Alice and Bob are playing a game where they take turns drawing a card from a standard deck of 52 cards without replacement. Alice draws the first card, and then Bob draws a card. What is the probability that the first card drawn by Alice is a heart and the first card drawn by Bob is a diamond? Express your answer as a ... | Alice and Bob are playing a game where they take turns drawing a card from a standard deck of 52 cards without replacement. Alice draws the first card, and then Bob draws a card. What is the probability that the first card drawn by Alice is a heart and the first card drawn by Bob is a diamond? Express your answer as a ... |
What is the remainder when the product of the first 20 odd positive integers is divided by 10? | What is the remainder when the product of the first 20 odd positive integers is divided by 10? |
Given that $13^{-1} \equiv 29 \pmod{47}$, find $34^{-1} \pmod{47}$, as a residue modulo 47. (Give a number between 0 and 46, inclusive.) | Given that $13^{-1} \equiv 29 \pmod{47}$, find $34^{-1} \pmod{47}$, as a residue modulo 47. (Give a number between 0 and 46, inclusive.) |
Find $\tan Y$ in the right triangle shown below.
[asy]
pair X,Y,Z;
X = (0,0);
Y = (24,0);
Z = (0,7);
draw(X--Y--Z--X);
draw(rightanglemark(Y,X,Z,23));
label("$X$",X,SW);
label("$Y$",Y,SE);
label("$Z$",Z,N);
label("$25$",(Y+Z)/2,NE);
label("$24$",Y/2,S);
[/asy] | Find $\tan Y$ in the right triangle shown below.
[asy]
pair X,Y,Z;
X = (0,0);
Y = (24,0);
Z = (0,7);
draw(X--Y--Z--X);
draw(rightanglemark(Y,X,Z,23));
label("$X$",X,SW);
label("$Y$",Y,SE);
label("$Z$",Z,N);
label("$25$",(Y+Z)/2,NE);
label("$24$",Y/2,S);
[/asy] |
Let $p$ and $q$ be positive real numbers such that $\sqrt{p} + \sqrt{q} = 5$ and $\frac{1}{\sqrt{p}} + \frac{1}{\sqrt{q}} = 2.$ Find $\frac{p+q}{pq}$. | Let $p$ and $q$ be positive real numbers such that $\sqrt{p} + \sqrt{q} = 5$ and $\frac{1}{\sqrt{p}} + \frac{1}{\sqrt{q}} = 2.$ Find $\frac{p+q}{pq}$. |
The equation of the circle passing through the complex numbers $1 + 2i$, $2 + 3i$, and $3 + 4i$ can be expressed in the form $|z - a|^2 = b$ for some complex number $a$ and real number $b$. Find the value of $a$. | The equation of the circle passing through the complex numbers $1 + 2i$, $2 + 3i$, and $3 + 4i$ can be expressed in the form $|z - a|^2 = b$ for some complex number $a$ and real number $b$. Find the value of $a$. |
The parametric equations of a curve are given by:
x = 2cosθ - sinθ, y = 3sinθ - 2cosθ
Determine the equation of the curve in the rectangular coordinates (x, y).
Is the curve a line, a circle, a parabola, an ellipse, or a hyperbola? | The parametric equations of a curve are given by:
x = 2cosθ - sinθ, y = 3sinθ - 2cosθ
Determine the equation of the curve in the rectangular coordinates (x, y).
Is the curve a line, a circle, a parabola, an ellipse, or a hyperbola? |
A boat is moving on the surface of a lake, creating a circular wave. The distance of the boat from a fixed point on the shore is 20 meters, and the angle between the boat and the point is 60 degrees. If the radius of the circular wave is 15 meters, find the rectangular coordinates of the boat relative to the fixed poin... | A boat is moving on the surface of a lake, creating a circular wave. The distance of the boat from a fixed point on the shore is 20 meters, and the angle between the boat and the point is 60 degrees. If the radius of the circular wave is 15 meters, find the rectangular coordinates of the boat relative to the fixed poin... |
A right triangle has an area of 120 square units, and a leg length of 24 units. What is the perimeter of the triangle, in units?
Another problem inspired by this one:
A right triangle has an area of 120 square units, and a leg length of 24 units. What is the length of the other leg of the triangle, in units?
Solutio... | A right triangle has an area of 120 square units, and a leg length of 24 units. What is the perimeter of the triangle, in units?
Another problem inspired by this one:
A right triangle has an area of 120 square units, and a leg length of 24 units. What is the length of the other leg of the triangle, in units?
Solutio... |
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