instruction_seed stringlengths 13 12.7k | response_seed stringlengths 13 12.7k |
|---|---|
Given that $xy = 12$ and $\frac{x}{y} + \frac{y}{x} = 5$, find the value of $(x+y)^2$. | Given that $xy = 12$ and $\frac{x}{y} + \frac{y}{x} = 5$, find the value of $(x+y)^2$. |
Simplify: $$\frac{-(12\cdot 6 - 9\cdot 3)^2 + 36}{45}$$ | Simplify: $$\frac{-(12\cdot 6 - 9\cdot 3)^2 + 36}{45}$$ |
Find the smallest positive integer that can be expressed as the sum of three distinct prime numbers, each greater than 20. | Find the smallest positive integer that can be expressed as the sum of three distinct prime numbers, each greater than 20. |
Find the number of composite numbers between 40 and 70. | Find the number of composite numbers between 40 and 70. |
Let $n$ be a positive integer. A function $f:\mathbb{Z}\to\mathbb{Z}$ satisfies the following two properties:
(i) $f(m+n)=f(m)+f(n)+3mn$ for all integers $m$ and $n$.
(ii) $f(mn)=f(m)f(n)+m^2n^2$ for all integers $m$ and $n$.
Determine the number of possible values of $f(3)$. | Let $n$ be a positive integer. A function $f:\mathbb{Z}\to\mathbb{Z}$ satisfies the following two properties:
(i) $f(m+n)=f(m)+f(n)+3mn$ for all integers $m$ and $n$.
(ii) $f(mn)=f(m)f(n)+m^2n^2$ for all integers $m$ and $n$.
Determine the number of possible values of $f(3)$. |
A line passes through the points $(1,3,2)$ and $(4,5,7)$. Find the $y$-coordinate of the point on this line that has an $x$-coordinate of $2$. | A line passes through the points $(1,3,2)$ and $(4,5,7)$. Find the $y$-coordinate of the point on this line that has an $x$-coordinate of $2$. |
Find the number of points of intersection between the graph of the parabola $y=x^2$ and the graph of the ellipse $\frac{x^2}{4}+\frac{y^2}{9}=1$. | Find the number of points of intersection between the graph of the parabola $y=x^2$ and the graph of the ellipse $\frac{x^2}{4}+\frac{y^2}{9}=1$. |
Let $\mathbf{u}$, $\mathbf{v}$, and $\mathbf{w}$ be three vectors such that $\mathbf{u} + \mathbf{v} + \mathbf{w} = \mathbf{0}$, $\|\mathbf{u}\| = 3,$ $\|\mathbf{v}\| = 4,$ and $\|\mathbf{u} \times \mathbf{v}\| = 6.$ Find $\mathbf{w} \cdot (\mathbf{u} \times \mathbf{v}).$ | Let $\mathbf{u}$, $\mathbf{v}$, and $\mathbf{w}$ be three vectors such that $\mathbf{u} + \mathbf{v} + \mathbf{w} = \mathbf{0}$, $\|\mathbf{u}\| = 3,$ $\|\mathbf{v}\| = 4,$ and $\|\mathbf{u} \times \mathbf{v}\| = 6.$ Find $\mathbf{w} \cdot (\mathbf{u} \times \mathbf{v}).$ |
The average of the numbers in a set is 25. If the average of the squares of the numbers in the set is 650, what is the largest number in the set? | The average of the numbers in a set is 25. If the average of the squares of the numbers in the set is 650, what is the largest number in the set? |
The sides of this quadrilateral measure 11, 13, $5z-2$, and $3w+1$ units, consecutively. What is the value of $z+w$?
[asy]draw((0,0)--(25,0)--(30,20)--(5,20)--cycle);
label("$5z-2$",(10,0),S);
label("13",(22.5,10),E);
label("11",(17.5,20),N);
label("$3w+1$",(2.5,10),W);
[/asy] | The sides of this quadrilateral measure 11, 13, $5z-2$, and $3w+1$ units, consecutively. What is the value of $z+w$?
[asy]draw((0,0)--(25,0)--(30,20)--(5,20)--cycle);
label("$5z-2$",(10,0),S);
label("13",(22.5,10),E);
label("11",(17.5,20),N);
label("$3w+1$",(2.5,10),W);
[/asy] |
Solve for $x$: $\frac{2x-1}{x^2+4} + \frac{3}{x^2+4} = \frac{x+2}{x^2+4}$ | Solve for $x$: $\frac{2x-1}{x^2+4} + \frac{3}{x^2+4} = \frac{x+2}{x^2+4}$ |
In a semicircle with center $O$, the length of the arc $\widehat{AB}$ is 90 units. Segment $AC$ is perpendicular to segment $OB$, and intersects segment $OB$ at point $M$. If the measure of angle $\angle CMO$ is 60 degrees, what is the area of sector $AOB$? | In a semicircle with center $O$, the length of the arc $\widehat{AB}$ is 90 units. Segment $AC$ is perpendicular to segment $OB$, and intersects segment $OB$ at point $M$. If the measure of angle $\angle CMO$ is 60 degrees, what is the area of sector $AOB$? |
If $f(x)=f(2-x)$ for all $x$, then what line is necessarily an axis of symmetry of the graph of $y=f(x)$? (Give the simplest equation of this line.) | If $f(x)=f(2-x)$ for all $x$, then what line is necessarily an axis of symmetry of the graph of $y=f(x)$? (Give the simplest equation of this line.) |
Tom is planning a road trip from City A to City B. He knows that the average speed of his car is 60 miles per hour, but he also knows that he will have to stop for a 15-minute break every 2 hours. If the total distance between City A and City B is 480 miles, how many hours will the trip take, including breaks? Express ... | Tom is planning a road trip from City A to City B. He knows that the average speed of his car is 60 miles per hour, but he also knows that he will have to stop for a 15-minute break every 2 hours. If the total distance between City A and City B is 480 miles, how many hours will the trip take, including breaks? Express ... |
A right cone has a base with a circumference of $20\pi$ inches and a height of 48 inches. The height of this cone is reduced while the circumference stays the same. The volume of the shorter cone is $280\pi$ cubic inches. What is the ratio of the shorter height to the original height? Express your answer as a common fr... | A right cone has a base with a circumference of $20\pi$ inches and a height of 48 inches. The height of this cone is reduced while the circumference stays the same. The volume of the shorter cone is $280\pi$ cubic inches. What is the ratio of the shorter height to the original height? Express your answer as a common fr... |
A fair coin is flipped 7 times and the sequence of heads and tails is recorded. What is the minimum number of terms of the sequence a gambler needs to know in order to have a half chance of guessing correctly the entire sequence? | A fair coin is flipped 7 times and the sequence of heads and tails is recorded. What is the minimum number of terms of the sequence a gambler needs to know in order to have a half chance of guessing correctly the entire sequence? |
Four prime numbers are randomly selected without replacement from the first ten prime numbers. What is the probability that the sum of the four selected numbers is odd? Express your answer as a common fraction. | Four prime numbers are randomly selected without replacement from the first ten prime numbers. What is the probability that the sum of the four selected numbers is odd? Express your answer as a common fraction. |
The sum of two positive integers is 30 and their product is 216. What is the largest possible value of the greatest common divisor of the two integers? | The sum of two positive integers is 30 and their product is 216. What is the largest possible value of the greatest common divisor of the two integers? |
A square garden is enlarged so that the new perimeter is twice the old perimeter. What is the ratio of the original area to the enlarged area? Express your answer as a common fraction. | A square garden is enlarged so that the new perimeter is twice the old perimeter. What is the ratio of the original area to the enlarged area? Express your answer as a common fraction. |
In a triangle, the midpoint of the side joining the vertices (0, 0) and (4, 8) lies on a line passing through the vertex (0, 0) and another point (2, -4). Find the area of the triangle formed by these three vertices. | In a triangle, the midpoint of the side joining the vertices (0, 0) and (4, 8) lies on a line passing through the vertex (0, 0) and another point (2, -4). Find the area of the triangle formed by these three vertices. |
What is the sum of the coefficients in the expansion of $(2x + 3)^4$? | What is the sum of the coefficients in the expansion of $(2x + 3)^4$? |
A parabola is graphed below.
[asy]
import graph;
size(180);
real lsf=0.5;
pen dps=linewidth(0.7)+fontsize(10);
defaultpen(dps); pen ds=black;
real xmin=-3.25,xmax=4.25,ymin=-7.25,ymax=6.25;
Label laxis; laxis.p=fontsize(10);
xaxis("$x$",xmin,xmax,defaultpen+black,Ticks(laxis,Step=1.0,Size=2,NoZero),Arrows(6),abo... | A parabola is graphed below.
[asy]
import graph;
size(180);
real lsf=0.5;
pen dps=linewidth(0.7)+fontsize(10);
defaultpen(dps); pen ds=black;
real xmin=-3.25,xmax=4.25,ymin=-7.25,ymax=6.25;
Label laxis; laxis.p=fontsize(10);
xaxis("$x$",xmin,xmax,defaultpen+black,Ticks(laxis,Step=1.0,Size=2,NoZero),Arrows(6),abo... |
In a trapezoid with parallel sides of length 14 cm and 21 cm, the altitude is 8 cm. Two right triangles are formed by dropping perpendiculars from the vertices of the shorter side to the longer side. Determine the sum of the lengths of the legs of the larger triangle, given that the area of the smaller triangle is 48 s... | In a trapezoid with parallel sides of length 14 cm and 21 cm, the altitude is 8 cm. Two right triangles are formed by dropping perpendiculars from the vertices of the shorter side to the longer side. Determine the sum of the lengths of the legs of the larger triangle, given that the area of the smaller triangle is 48 s... |
For any real number $m,$ the graph of
\[y = 3x^2 + mx + 5m\]
passes through a fixed point $(c,d).$ Find $(c,d).$ | For any real number $m,$ the graph of
\[y = 3x^2 + mx + 5m\]
passes through a fixed point $(c,d).$ Find $(c,d).$ |
The prime factorization of 2007 is $3^{2}\times223$. How many ordered pairs of positive integers $(x,y)$ satisfy the equation $xy=2007$? | The prime factorization of 2007 is $3^{2}\times223$. How many ordered pairs of positive integers $(x,y)$ satisfy the equation $xy=2007$? |
Let $a$ and $b$ be integers. Determine $a$ if $x^2 + 3x + 2$ is a factor of $ax^{15} + bx^{14} + 1$. | Let $a$ and $b$ be integers. Determine $a$ if $x^2 + 3x + 2$ is a factor of $ax^{15} + bx^{14} + 1$. |
The function $f(x)$ satisfies
\[f(x + y) = f(x) f(y)\]for all real numbers $x$ and $y.$ Find all possible values of $f(0).$
Enter all the possible values, separated by commas. | The function $f(x)$ satisfies
\[f(x + y) = f(x) f(y)\]for all real numbers $x$ and $y.$ Find all possible values of $f(0).$
Enter all the possible values, separated by commas. |
Consider a $5\times5$ grid of points, where each point is one unit away from its nearest neighbors.
[asy] unitsize(0.25inch); defaultpen(linewidth(0.7)); int i, j; for(i = 0; i < 5; ++i) for(j = 0; j < 5; ++j) dot(((real)i, (real)j)); [/asy]
A "descending path" is a sequence of distinct points in the grid such that... | Consider a $5\times5$ grid of points, where each point is one unit away from its nearest neighbors.
[asy] unitsize(0.25inch); defaultpen(linewidth(0.7)); int i, j; for(i = 0; i < 5; ++i) for(j = 0; j < 5; ++j) dot(((real)i, (real)j)); [/asy]
A "descending path" is a sequence of distinct points in the grid such that... |
Find all real values of $a$ such that the equation $ax^2 + (a + 3)x + 4 = 0$ has at least one root that is a root of the equation $2x^2 + 3x + 1 = 0$. | Find all real values of $a$ such that the equation $ax^2 + (a + 3)x + 4 = 0$ has at least one root that is a root of the equation $2x^2 + 3x + 1 = 0$. |
Find the smallest value of $b$ such that the quadratic expression $2x^2 + bx + 32$ can be expressed as the difference of two perfect squares with integer coefficients. | Find the smallest value of $b$ such that the quadratic expression $2x^2 + bx + 32$ can be expressed as the difference of two perfect squares with integer coefficients. |
What is the value of $x$ if \begin{align*}x &= y+5,\\
y &= z+10,\\
z &= w+20,\\
\text{and }\qquad w &= 80?
\end{align*} | What is the value of $x$ if \begin{align*}x &= y+5,\\
y &= z+10,\\
z &= w+20,\\
\text{and }\qquad w &= 80?
\end{align*} |
The volume and surface area of a sphere are in the ratio of 3:1. If the sphere is circumscribed about a cube, what is the ratio of the volume of the cube to the surface area of the cube? | The volume and surface area of a sphere are in the ratio of 3:1. If the sphere is circumscribed about a cube, what is the ratio of the volume of the cube to the surface area of the cube? |
Tom and Alex are standing on opposite sides of a rectangular lake, 0.8 miles wide and 2.4 miles long. Tom has a small boat that can travel at a constant rate of 3 miles per hour. If Tom rows directly across the lake to Alex's side, it takes him 16 minutes. How long would it take Tom to row along the length of the lake ... | Tom and Alex are standing on opposite sides of a rectangular lake, 0.8 miles wide and 2.4 miles long. Tom has a small boat that can travel at a constant rate of 3 miles per hour. If Tom rows directly across the lake to Alex's side, it takes him 16 minutes. How long would it take Tom to row along the length of the lake ... |
What is the remainder when the sum of the first 102 counting numbers is divided by 5250?
A new problem:
The product of the first 50 even counting numbers is divided by 25. What is the remainder? | What is the remainder when the sum of the first 102 counting numbers is divided by 5250?
A new problem:
The product of the first 50 even counting numbers is divided by 25. What is the remainder? |
Solve for $x$ in the equation $24-5=x+1$. | Solve for $x$ in the equation $24-5=x+1$. |
Consider a function $h(x)$ defined on the domain $[-6,2]$. If we define a new function $j(x)$ by $j(x) = h(3x)$, then what is the domain of $j(x)$? Express your answer in interval notation. | Consider a function $h(x)$ defined on the domain $[-6,2]$. If we define a new function $j(x)$ by $j(x) = h(3x)$, then what is the domain of $j(x)$? Express your answer in interval notation. |
Find the sum of all angles $x \in [0^\circ, 360^\circ]$ that satisfy
\[\cos^4 x - \sin^4 x = \frac{1}{\sin x} - \frac{1}{\cos x}.\] | Find the sum of all angles $x \in [0^\circ, 360^\circ]$ that satisfy
\[\cos^4 x - \sin^4 x = \frac{1}{\sin x} - \frac{1}{\cos x}.\] |
Solve for $x$ in the equation $x^{3/2} = -8$. | Solve for $x$ in the equation $x^{3/2} = -8$. |
Find the remainder when $$12345 + 12346 + 12347 + 12348 + 12349 + 12350$$ is divided by 5. | Find the remainder when $$12345 + 12346 + 12347 + 12348 + 12349 + 12350$$ is divided by 5. |
Let $g(x)=x^2-3x+1$. What is the value of $g(g(g(g(g(g(2))))))$? | Let $g(x)=x^2-3x+1$. What is the value of $g(g(g(g(g(g(2))))))$? |
Let $S$ be a set of positive integers such that for each element $n \in S$, the base-$n$ representation of the integer $2022$ is a palindrome. Determine the number of elements in $S$. | Let $S$ be a set of positive integers such that for each element $n \in S$, the base-$n$ representation of the integer $2022$ is a palindrome. Determine the number of elements in $S$. |
Let $a$, $b$, and $c$ be positive integers such that $\sqrt{a+b\sqrt{c}}=m+n\sqrt{p}$, where $m$, $n$, and $p$ are positive integers. Find the smallest possible value of $m+n+p$. | Let $a$, $b$, and $c$ be positive integers such that $\sqrt{a+b\sqrt{c}}=m+n\sqrt{p}$, where $m$, $n$, and $p$ are positive integers. Find the smallest possible value of $m+n+p$. |
Problem:
Compute
\[\cos^2 4^\circ + \cos^2 8^\circ + \cos^2 12^\circ + \dots + \cos^2 176^\circ.\]
Note: The approach to solve this problem would be different from the original problem, as it involves cosine instead of sine, and the Pythagorean identity would need to be used to simplify the expression. | Problem:
Compute
\[\cos^2 4^\circ + \cos^2 8^\circ + \cos^2 12^\circ + \dots + \cos^2 176^\circ.\]
Note: The approach to solve this problem would be different from the original problem, as it involves cosine instead of sine, and the Pythagorean identity would need to be used to simplify the expression. |
Let $h(x) = x^2 - 4$ and $j(x) = \frac{x}{2}$. Find the value of $h(j(2))$. | Let $h(x) = x^2 - 4$ and $j(x) = \frac{x}{2}$. Find the value of $h(j(2))$. |
Consider the operation $\odot$ defined as $a \odot b = \frac{a+b}{2}$ if $a$ and $b$ have different parities, and $a \odot b = a + b$ otherwise. What is the value of $(3 \odot 5) \odot 2$? | Consider the operation $\odot$ defined as $a \odot b = \frac{a+b}{2}$ if $a$ and $b$ have different parities, and $a \odot b = a + b$ otherwise. What is the value of $(3 \odot 5) \odot 2$? |
Solve the equation
\[\frac{3y + y^2}{y - 2} \left( y - \frac{3 + y}{y - 2} \right) = 15.\]Enter all the solutions, separated by commas. | Solve the equation
\[\frac{3y + y^2}{y - 2} \left( y - \frac{3 + y}{y - 2} \right) = 15.\]Enter all the solutions, separated by commas. |
How many of the numbers from the set $\{1,\ 2,\ 3,\ldots,\ 50\}$ have a prime factor other than 2 or 3? | How many of the numbers from the set $\{1,\ 2,\ 3,\ldots,\ 50\}$ have a prime factor other than 2 or 3? |
If the angle between the vectors $\mathbf{a}$ and $\mathbf{b}$ is $43^\circ,$ and the angle between the vectors $-\mathbf{a}$ and $\mathbf{b}$ is $137^\circ,$ what is the angle between the vectors $\mathbf{a}$ and $-\mathbf{b}$? | If the angle between the vectors $\mathbf{a}$ and $\mathbf{b}$ is $43^\circ,$ and the angle between the vectors $-\mathbf{a}$ and $\mathbf{b}$ is $137^\circ,$ what is the angle between the vectors $\mathbf{a}$ and $-\mathbf{b}$? |
A circle is inscribed in a triangle with sides $3x$, $4x$, and $5x$. Find the area of the triangle when the radius of the circle is 2. | A circle is inscribed in a triangle with sides $3x$, $4x$, and $5x$. Find the area of the triangle when the radius of the circle is 2. |
Two congruent cylinders each have radius 8 inches and height 3 inches. The radius of one cylinder and the height of the other are both increased by the same nonzero number of inches. The resulting volumes are equal. How many inches is the increase? Express your answer as a common fraction. | Two congruent cylinders each have radius 8 inches and height 3 inches. The radius of one cylinder and the height of the other are both increased by the same nonzero number of inches. The resulting volumes are equal. How many inches is the increase? Express your answer as a common fraction. |
On a Cartesian coordinate plane, points $(1, 2)$ and $(4, 5)$ are opposite vertices of a rectangle. If the sides of the rectangle are parallel to the axes, what is the area of the rectangle? | On a Cartesian coordinate plane, points $(1, 2)$ and $(4, 5)$ are opposite vertices of a rectangle. If the sides of the rectangle are parallel to the axes, what is the area of the rectangle? |
Let $w$ be a complex number such that $|w| = 7.$ Find $w \times \overline{w}.$ | Let $w$ be a complex number such that $|w| = 7.$ Find $w \times \overline{w}.$ |
A bakery sold a total of 250 loaves of bread in a day, with different types of bread being sold at different prices. The prices and quantities of each type of bread are shown in the table below.
| Type of Bread | Price per Loaf | Quantity Sold |
| --- | --- | --- |
| Whole Wheat | $2.50 | 30 |
| White | $2.00 | 50 |
|... | A bakery sold a total of 250 loaves of bread in a day, with different types of bread being sold at different prices. The prices and quantities of each type of bread are shown in the table below.
| Type of Bread | Price per Loaf | Quantity Sold |
| --- | --- | --- |
| Whole Wheat | $2.50 | 30 |
| White | $2.00 | 50 |
|... |
Given that $\csc x - \cot x = \frac{2}{5},$ enter all possible values of $\cos x.$ | Given that $\csc x - \cot x = \frac{2}{5},$ enter all possible values of $\cos x.$ |
What is the smallest positive integer $k$ such that $3^{2k} \equiv 1 \pmod{10}$?
Note: This problem requires a different approach than the original, as it involves finding the smallest positive integer $k$ that satisfies the congruence relation, rather than finding the remainder of a specific power of a number. | What is the smallest positive integer $k$ such that $3^{2k} \equiv 1 \pmod{10}$?
Note: This problem requires a different approach than the original, as it involves finding the smallest positive integer $k$ that satisfies the congruence relation, rather than finding the remainder of a specific power of a number. |
When three positive integers are divided by $12$, the remainders are $7,$ $9,$ and $10,$ respectively.
When the sum of the three integers is divided by $12$, what is the remainder? | When three positive integers are divided by $12$, the remainders are $7,$ $9,$ and $10,$ respectively.
When the sum of the three integers is divided by $12$, what is the remainder? |
Given two points $(x_1, y_1)$ and $(x_2, y_2)$, the operation $\circ$ is defined as:
$$(x_1, y_1)\circ(x_2, y_2) = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right).$$
If $(a, b) \circ (c, d) = \left(\frac{3}{5}, \frac{4}{5}\right)$ and $(a - c)^2 + (b - d)^2 = 2$, determine the value of $a^2 + b^2 + c^2 + d^2$. | Given two points $(x_1, y_1)$ and $(x_2, y_2)$, the operation $\circ$ is defined as:
$$(x_1, y_1)\circ(x_2, y_2) = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right).$$
If $(a, b) \circ (c, d) = \left(\frac{3}{5}, \frac{4}{5}\right)$ and $(a - c)^2 + (b - d)^2 = 2$, determine the value of $a^2 + b^2 + c^2 + d^2$. |
Let $A$ be a geometric sequence and $B$ be an arithmetic sequence. Each sequence stops as soon as one of its terms is less than $-200.$ The sequences are such that the difference between the largest terms of both sequences is $54$. Find the difference between the number of terms in sequence $A$ and sequence $B$.
$\bul... | Let $A$ be a geometric sequence and $B$ be an arithmetic sequence. Each sequence stops as soon as one of its terms is less than $-200.$ The sequences are such that the difference between the largest terms of both sequences is $54$. Find the difference between the number of terms in sequence $A$ and sequence $B$.
$\bul... |
The greatest common divisor of two positive integers $m$ and $n$ is 17. Determine the smallest positive integer $a$ so that $4m + 7n + a$ is divisible by 17. | The greatest common divisor of two positive integers $m$ and $n$ is 17. Determine the smallest positive integer $a$ so that $4m + 7n + a$ is divisible by 17. |
A man invested $\$P$ in stocks. The value of the stocks increased annually and had varying annual increase rates. The first year after the investment the value increased by $10\%$, the second year by $20\%$, and the third year by $30\%$. At the end of the fourth year, after the annual increase of $40\%$ of the third ye... | A man invested $\$P$ in stocks. The value of the stocks increased annually and had varying annual increase rates. The first year after the investment the value increased by $10\%$, the second year by $20\%$, and the third year by $30\%$. At the end of the fourth year, after the annual increase of $40\%$ of the third ye... |
Let $f(x)=\frac{p(x)}{q(x)}$ be a rational function, where $p(x)$ and $q(x)$ are monic polynomials. Suppose that $y=a$ is a horizontal asymptote of the graph of $f(x)$. If $a=2$ and $p(x)$ is divisible by $x^2+1$, what is the smallest possible degree of $q(x)$? | Let $f(x)=\frac{p(x)}{q(x)}$ be a rational function, where $p(x)$ and $q(x)$ are monic polynomials. Suppose that $y=a$ is a horizontal asymptote of the graph of $f(x)$. If $a=2$ and $p(x)$ is divisible by $x^2+1$, what is the smallest possible degree of $q(x)$? |
From a circular piece of paper with radius $BC$, Jeff removes the unshaded sector shown. Using the larger shaded sector, he joins edge $BC$ to edge $BA$ (without overlap) to form a cone of radius 12 centimeters and of volume $432\pi$ cubic centimeters. What is the number of degrees in the measure of angle $ABC$ of th... | From a circular piece of paper with radius $BC$, Jeff removes the unshaded sector shown. Using the larger shaded sector, he joins edge $BC$ to edge $BA$ (without overlap) to form a cone of radius 12 centimeters and of volume $432\pi$ cubic centimeters. What is the number of degrees in the measure of angle $ABC$ of th... |
In how many ways can I choose 2 cooks on a backpacking trip with 8 people if any of the 8 people may be cooks? | In how many ways can I choose 2 cooks on a backpacking trip with 8 people if any of the 8 people may be cooks? |
Find $\sec \frac{5 \pi}{3}.$ | Find $\sec \frac{5 \pi}{3}.$ |
A group of students are surveyed about their preferences for studying math and science. It is found that 15 students prefer studying math, 20 students prefer studying science, and 8 students prefer studying both math and science. However, 5 students who prefer studying math also prefer studying a third subject, English... | A group of students are surveyed about their preferences for studying math and science. It is found that 15 students prefer studying math, 20 students prefer studying science, and 8 students prefer studying both math and science. However, 5 students who prefer studying math also prefer studying a third subject, English... |
The product of positive integers $x$, $y$ and $z$ equals 2004. What is the minimum possible value of the sum $x + y + z$? | The product of positive integers $x$, $y$ and $z$ equals 2004. What is the minimum possible value of the sum $x + y + z$? |
Determine all positive integers $n\geq 2$ such that $n^2+1$ is a perfect square. | Determine all positive integers $n\geq 2$ such that $n^2+1$ is a perfect square. |
Find the value of the expression $$(a+b+c)^2-(a^2+b^2+c^2)$$ when $a+b+c=0$ and $ab+bc+ca=1$. | Find the value of the expression $$(a+b+c)^2-(a^2+b^2+c^2)$$ when $a+b+c=0$ and $ab+bc+ca=1$. |
Find the area of the triangle bounded by the x-axis and the lines 2y - 5x = 1 and 4y + 2x = 20. | Find the area of the triangle bounded by the x-axis and the lines 2y - 5x = 1 and 4y + 2x = 20. |
It is now 12:00:00 midnight, as read on a 12-hour digital clock. In 122 hours, 39 minutes and 44 seconds the time will be $A:B:C$. What is the value of $A + B + C$? | It is now 12:00:00 midnight, as read on a 12-hour digital clock. In 122 hours, 39 minutes and 44 seconds the time will be $A:B:C$. What is the value of $A + B + C$? |
Simplify the following expression: $(9x^9+7x^8+4x^7) + (x^{11}+x^9+2x^7+3x^3+5x+8).$ Express your answer as a polynomial with the degrees of the terms in decreasing order. | Simplify the following expression: $(9x^9+7x^8+4x^7) + (x^{11}+x^9+2x^7+3x^3+5x+8).$ Express your answer as a polynomial with the degrees of the terms in decreasing order. |
Find all positive three-digit integers such that the sum of the squares of their digits is equal to the product of their digits. | Find all positive three-digit integers such that the sum of the squares of their digits is equal to the product of their digits. |
The mean (average) of 4, 7, and 15 is equal to the mean (average) of 9 and x. What is the value of x? | The mean (average) of 4, 7, and 15 is equal to the mean (average) of 9 and x. What is the value of x? |
If $a$, $b$ are nonzero real numbers such that $a^2+b^2=8ab$, find the value of $\left|\frac{a+b}{a-b}\right|$. | If $a$, $b$ are nonzero real numbers such that $a^2+b^2=8ab$, find the value of $\left|\frac{a+b}{a-b}\right|$. |
The figure shows a large rectangle with side lengths of $(x+7)$ and $(x+5)$. In the large rectangle, there is a square hole with side length of $(x-2)$. The area of the large rectangle is 56 square units more than the combined area of the two smaller rectangles formed by the square hole. What is the area of the large r... | The figure shows a large rectangle with side lengths of $(x+7)$ and $(x+5)$. In the large rectangle, there is a square hole with side length of $(x-2)$. The area of the large rectangle is 56 square units more than the combined area of the two smaller rectangles formed by the square hole. What is the area of the large r... |
There are two 3-digit positive integers, where the hundreds digit of one number is the same as the tens digit of the other, and vice versa. When you add these two numbers together, the result is a number that ends in 9. What is the sum of the units digits of these two numbers? | There are two 3-digit positive integers, where the hundreds digit of one number is the same as the tens digit of the other, and vice versa. When you add these two numbers together, the result is a number that ends in 9. What is the sum of the units digits of these two numbers? |
Tom and Alex each bought 10 ounces of tea in a 16-ounce cup. Tom drank 3 ounces of his tea and then added 1 ounce of honey. Alex added 1 ounce of honey, stirred the tea well, and then drank 3 ounces. What is the resulting ratio of the amount of honey in Tom's tea to that in Alex's tea? Express your answer as a fraction... | Tom and Alex each bought 10 ounces of tea in a 16-ounce cup. Tom drank 3 ounces of his tea and then added 1 ounce of honey. Alex added 1 ounce of honey, stirred the tea well, and then drank 3 ounces. What is the resulting ratio of the amount of honey in Tom's tea to that in Alex's tea? Express your answer as a fraction... |
A cylindrical water tank is $\frac{1}{5}$ full. If three liters were added, the tank would be $\frac{1}{4}$ full. How many liters does the tank hold when it is full? | A cylindrical water tank is $\frac{1}{5}$ full. If three liters were added, the tank would be $\frac{1}{4}$ full. How many liters does the tank hold when it is full? |
Let $g(x) = 3x^4-25x^3+42x^2-30x+45$. Find $g(5)$. | Let $g(x) = 3x^4-25x^3+42x^2-30x+45$. Find $g(5)$. |
A group of friends decided to divide a bill evenly among themselves. If 6 friends were originally going to split the bill, but 3 more friends joined in, they found that the amount each friend would pay decreased by $8. If 3 friends left after that, the remaining friends had to pay $15 more than they would have paid if ... | A group of friends decided to divide a bill evenly among themselves. If 6 friends were originally going to split the bill, but 3 more friends joined in, they found that the amount each friend would pay decreased by $8. If 3 friends left after that, the remaining friends had to pay $15 more than they would have paid if ... |
In a sequence of real numbers, the sum of any three consecutive terms forms an arithmetic progression with a common difference of $d$. If the sum of the first three terms is 15 and the sum of the next three terms is 18, determine the value of the 100th term. | In a sequence of real numbers, the sum of any three consecutive terms forms an arithmetic progression with a common difference of $d$. If the sum of the first three terms is 15 and the sum of the next three terms is 18, determine the value of the 100th term. |
Given the graph of function y=f(x) shown below:
Draw the graph of the function
y=|f(x)|+|f(-x)| and find its minimum value.
I hope it meets your request. Let me know if you need another problem. | Given the graph of function y=f(x) shown below:
Draw the graph of the function
y=|f(x)|+|f(-x)| and find its minimum value.
I hope it meets your request. Let me know if you need another problem. |
Find constants $A,$ $B,$ and $C$ so that
\[\frac{x^2 - 7}{(x - 2)(x - 3)(x - 5)} = \frac{A}{x - 2} + \frac{B}{x - 3} + \frac{C}{x - 5}.\]Enter the ordered triple $(A,B,C).$ | Find constants $A,$ $B,$ and $C$ so that
\[\frac{x^2 - 7}{(x - 2)(x - 3)(x - 5)} = \frac{A}{x - 2} + \frac{B}{x - 3} + \frac{C}{x - 5}.\]Enter the ordered triple $(A,B,C).$ |
Given that $f(x) = x^3 - 2x^2 + x$, find the number of solutions to the equation $f(\cos x) = 0$ in the range $0 \le x \le \pi$. | Given that $f(x) = x^3 - 2x^2 + x$, find the number of solutions to the equation $f(\cos x) = 0$ in the range $0 \le x \le \pi$. |
Problem:
In a Cartesian plane, a particle starts at the point $(0,0)$ and moves along a line with a slope of $\frac{3}{4}$ until it intersects a circle centered at the origin with a radius of $5$. The particle then reflects off the circle and moves along a line perpendicular to the original line. Find the distance betw... | Problem:
In a Cartesian plane, a particle starts at the point $(0,0)$ and moves along a line with a slope of $\frac{3}{4}$ until it intersects a circle centered at the origin with a radius of $5$. The particle then reflects off the circle and moves along a line perpendicular to the original line. Find the distance betw... |
The planet Zorvath follows an elliptical orbit with its sun at one focus. At its nearest point (perigee), it is 3 astronomical units (AU) from the sun, while at its furthest point (apogee) it is 9 AU away. When Zorvath is midway along its orbit, how far is it from the sun, in AU? | The planet Zorvath follows an elliptical orbit with its sun at one focus. At its nearest point (perigee), it is 3 astronomical units (AU) from the sun, while at its furthest point (apogee) it is 9 AU away. When Zorvath is midway along its orbit, how far is it from the sun, in AU? |
An octagon is inscribed in a circle, and the eight vertices are connected to form a larger octagon. The smaller octagon has a side length of 6 cm, and the larger octagon has a side length of 12 cm. If the radius of the circle is 15 cm, what is the area of the region between the two octagons?
(Note: The problem is desi... | An octagon is inscribed in a circle, and the eight vertices are connected to form a larger octagon. The smaller octagon has a side length of 6 cm, and the larger octagon has a side length of 12 cm. If the radius of the circle is 15 cm, what is the area of the region between the two octagons?
(Note: The problem is desi... |
Evaluate $\lfloor-2.54\rfloor+\lceil25.4\rceil$. | Evaluate $\lfloor-2.54\rfloor+\lceil25.4\rceil$. |
Expand $(x+8)(3y+12)$. | Expand $(x+8)(3y+12)$. |
Determine if the point $(5, 3)$ lies inside or outside the circle defined by the equation $(x - 2)^2 + (y + 1)^2 = 16$. Express your answer as either "Inside" or "Outside". | Determine if the point $(5, 3)$ lies inside or outside the circle defined by the equation $(x - 2)^2 + (y + 1)^2 = 16$. Express your answer as either "Inside" or "Outside". |
Consider the line parameterized by
\begin{align*}
x&= 3s - 1,\\
y& = 2s-3.
\end{align*}Find a vector $\begin{pmatrix}c \\ d \end{pmatrix}$ pointing from the origin to this line that is parallel to $\begin{pmatrix}1 \\2 \end{pmatrix}$. | Consider the line parameterized by
\begin{align*}
x&= 3s - 1,\\
y& = 2s-3.
\end{align*}Find a vector $\begin{pmatrix}c \\ d \end{pmatrix}$ pointing from the origin to this line that is parallel to $\begin{pmatrix}1 \\2 \end{pmatrix}$. |
If $$x^{\log_{10}3}=4$$ find the value of $$x^{2\log_{10}9}.$$ | If $$x^{\log_{10}3}=4$$ find the value of $$x^{2\log_{10}9}.$$ |
Find $5^{-1} \pmod{241}$, as a residue modulo 241. (Give an answer between 0 and 240, inclusive.) | Find $5^{-1} \pmod{241}$, as a residue modulo 241. (Give an answer between 0 and 240, inclusive.) |
A department store has 8 shelves, and each shelf can hold 12 boxes of cereal. If the store manager randomly assigns 15 boxes of a new cereal to the shelves, what is the probability that no shelf will have more than 2 boxes of the new cereal? | A department store has 8 shelves, and each shelf can hold 12 boxes of cereal. If the store manager randomly assigns 15 boxes of a new cereal to the shelves, what is the probability that no shelf will have more than 2 boxes of the new cereal? |
Is the function $g(x) = \frac{x^3}{x^4-1}$ an even function, odd function, or neither? | Is the function $g(x) = \frac{x^3}{x^4-1}$ an even function, odd function, or neither? |
Consider a sequence of strings of 1's and 0's defined recursively as follows:
S_1 = 1
S_2 = 01
S_n = S_{n-1}S_{n-2} for n > 2
Find the number of 1's in S_{100}. | Consider a sequence of strings of 1's and 0's defined recursively as follows:
S_1 = 1
S_2 = 01
S_n = S_{n-1}S_{n-2} for n > 2
Find the number of 1's in S_{100}. |
Let $\mathbf{P}$ be a $2 \times 2$ matrix such that $\mathbf{P}^3 = \mathbf{I}$, where $\mathbf{I}$ is the $2 \times 2$ identity matrix. Compute the value of $\mathbf{P} \mathbf{P}^T$. | Let $\mathbf{P}$ be a $2 \times 2$ matrix such that $\mathbf{P}^3 = \mathbf{I}$, where $\mathbf{I}$ is the $2 \times 2$ identity matrix. Compute the value of $\mathbf{P} \mathbf{P}^T$. |
What is the greatest common factor of 252 and 96? | What is the greatest common factor of 252 and 96? |
Let $a, b, c, d, z$ be complex numbers such that $|a| = |b| = |c| = |d| > 0$ and
$$az^3 + bz^2 + cz + d = 0.$$
Find the largest possible value of $|z|$ such that $\text{Im}(z) > 0$. | Let $a, b, c, d, z$ be complex numbers such that $|a| = |b| = |c| = |d| > 0$ and
$$az^3 + bz^2 + cz + d = 0.$$
Find the largest possible value of $|z|$ such that $\text{Im}(z) > 0$. |
A water tank can hold 1200 cubic meters of water. If the water is poured into a cylindrical container with a height of 2 meters and a base radius of 0.5 meters, how many containers can be filled completely? | A water tank can hold 1200 cubic meters of water. If the water is poured into a cylindrical container with a height of 2 meters and a base radius of 0.5 meters, how many containers can be filled completely? |
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