problem stringlengths 13 13.6k | generated_solution stringlengths 190 5.74k | expected_answer stringlengths 0 628 | problem_source stringclasses 3
values | conversations listlengths 2 2 |
|---|---|---|---|---|
Solve for $y$:
$$\frac{y^2 - 3y + 2}{y - 2} = y + 1$$ | Start by multiplying both sides by $y - 2$ to eliminate the denominator:
\[ (y^2 - 3y + 2) = (y + 1)(y - 2) \]
Expand both sides:
\[ y^2 - 3y + 2 = y^2 - y - 2 \]
Subtract $y^2$ from both sides to get:
\[ -3y + 2 = -y - 2 \]
Add $3y$ to both sides:
\[ 2 = 2y - 2 \]
Add $2$ to both sides:
\[ 4 = 2y \]
Divide by $2$... | 2 | augmented_math | [
{
"content": "Solve for $y$:\n\n$$\\frac{y^2 - 3y + 2}{y - 2} = y + 1$$",
"role": "user"
},
{
"content": "Start by multiplying both sides by $y - 2$ to eliminate the denominator:\n\\[ (y^2 - 3y + 2) = (y + 1)(y - 2) \\]\n\nExpand both sides:\n\\[ y^2 - 3y + 2 = y^2 - y - 2 \\]\n\nSubtract $y^2$ from... |
Given a circle centered at the origin, a point $A$ is translated along the circle to a new position $A'$ by rotating the circle about its center by $\frac{\pi}{4}$ radians. If $A$ has coordinates $(3,4)$, find the coordinates of $A'$.
Express your answer as $(a,b)$ with $a$ and $b$ integers. | The rotation matrix for rotating a point $(x, y)$ by an angle $\theta$ counterclockwise is
\[ \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} x \cos \theta - y \sin \theta \\ x \sin \theta + y \cos \theta \end{pmatrix} \]
In t... | (-\frac{\sqrt{2}}{2}, \frac{7\sqrt{2}}{2}) | augmented_math | [
{
"content": "Given a circle centered at the origin, a point $A$ is translated along the circle to a new position $A'$ by rotating the circle about its center by $\\frac{\\pi}{4}$ radians. If $A$ has coordinates $(3,4)$, find the coordinates of $A'$.\n\nExpress your answer as $(a,b)$ with $a$ and $b$ integers."... |
The repeating decimal $0.\overline{009}$ can be expressed as a fraction $\frac{a}{b}$ in lowest terms. Determine the sum of the absolute values of the coefficients of the quadratic equation $ax^2+bx+c=0$ whose roots are $a$ and $b$. | To express $0.\overline{009}$ as a fraction, let $x = 0.\overline{009}$. Multiply both sides by $1000$ to get $1000x = 9.\overline{009}$. Subtracting the first equation from the second equation gives us $999x = 9$, so $x = \frac{9}{999} = \frac{1}{111}$.
Therefore, $a = 1$ and $b = 111$.
The quadratic equation with r... | 224 | augmented_math | [
{
"content": "The repeating decimal $0.\\overline{009}$ can be expressed as a fraction $\\frac{a}{b}$ in lowest terms. Determine the sum of the absolute values of the coefficients of the quadratic equation $ax^2+bx+c=0$ whose roots are $a$ and $b$.",
"role": "user"
},
{
"content": "To express $0.\\o... |
Let $m$ and $n$ be two positive integers such that their greatest common divisor is $18$ and the least common multiple is $1500$. Determine the number of possible values of $m+n$. | ## Step 1: Recall the relationship between GCD and LCM of two numbers
For any two positive integers $m$ and $n$, their product $m \cdot n$ is equal to the product of their greatest common divisor (GCD) and least common multiple (LCM). Therefore, we have $m \cdot n = GCD(m, n) \cdot LCM(m, n)$.
## Step 2: Use the given... | 4 | augmented_math | [
{
"content": "Let $m$ and $n$ be two positive integers such that their greatest common divisor is $18$ and the least common multiple is $1500$. Determine the number of possible values of $m+n$.",
"role": "user"
},
{
"content": "## Step 1: Recall the relationship between GCD and LCM of two numbers\nF... |
Let $(x, y)$ be a solution to the system of equations:
\[\begin{aligned} x^2 + \lfloor y \rfloor &= 25, \\ \lfloor x \rfloor + y^2 &= 33. \end{aligned} \]
Compute $\lfloor x + y \rfloor.$ | We have two equations:
\[ x^2 + \lfloor y \rfloor = 25 \quad \text{(1)} \]
\[ \lfloor x \rfloor + y^2 = 33 \quad \text{(2)} \]
From equation (1), we know that $x^2$ is an integer because the sum of $x^2$ and $\lfloor y \rfloor$ is 25, which is an integer.
Since $x^2$ is an integer, $x$ must also be an integer. Simila... | 9 | augmented_math | [
{
"content": "Let $(x, y)$ be a solution to the system of equations:\n\n\\[\\begin{aligned} x^2 + \\lfloor y \\rfloor &= 25, \\\\ \\lfloor x \\rfloor + y^2 &= 33. \\end{aligned} \\]\n\nCompute $\\lfloor x + y \\rfloor.$",
"role": "user"
},
{
"content": "We have two equations:\n\\[ x^2 + \\lfloor y \... |
Emma visited a local farm and saw 120 baby chicks in the coop. She asked the farmer if she could take some home to raise, and the farmer allowed her to take one-third of them. While collecting the chicks, Emma gathered 2/3 of the total number she was allowed to take. How many more chicks does Emma need to collect to re... | Let's break down the problem step by step.
The farmer allowed Emma to take one-third of the 120 baby chicks, which is $120 \times \frac{1}{3} = 40$ chicks.
Emma gathered 2/3 of the total number she was allowed to take, which is $40 \times \frac{2}{3} = 26.67$ chicks. However, since we can't have a fraction of a chick... | 14 | augmented_gsm8k | [
{
"content": "Emma visited a local farm and saw 120 baby chicks in the coop. She asked the farmer if she could take some home to raise, and the farmer allowed her to take one-third of them. While collecting the chicks, Emma gathered 2/3 of the total number she was allowed to take. How many more chicks does Emma... |
In a group of 10 friends, each person has a different favorite sports team. At a party, each person shakes hands with everyone who supports a team from a different league (there are two leagues, with 5 teams in each league). However, some people are shy and only shake hands with those who support a team that has won a ... | We have 10 friends, with 5 supporting teams from one league and 5 from another.
First, let's find the total number of handshakes if everyone shook hands with everyone from the other league. Since there are 5 people from each league, the total number of handshakes would be $5 \cdot 5 = 25$.
However, we need to conside... | 21 | augmented_math | [
{
"content": "In a group of 10 friends, each person has a different favorite sports team. At a party, each person shakes hands with everyone who supports a team from a different league (there are two leagues, with 5 teams in each league). However, some people are shy and only shake hands with those who support ... |
A group of friends want to share some pizzas that come in three different sizes: small, medium, and large. Each small pizza can feed 2 people, each medium pizza can feed 3 people, and each large pizza can feed 5 people. If there are 12 people in the group and they want to have 1/3 of a pizza left over, how many of each... | The group has 12 people and they want to have 1/3 of a pizza left over, so they need to order pizzas that can feed 12 + 1/3 = 12 1/3 people.
Since each small pizza can feed 2 people, each medium pizza can feed 3 people, and each large pizza can feed 5 people, let's try to find a combination of pizzas that can feed 12 ... | 3 | augmented_math | [
{
"content": "A group of friends want to share some pizzas that come in three different sizes: small, medium, and large. Each small pizza can feed 2 people, each medium pizza can feed 3 people, and each large pizza can feed 5 people. If there are 12 people in the group and they want to have 1/3 of a pizza left ... |
The difference between the cube of a number and twice the number is equal to the square of the number minus three times the number. What is the number? | Let $x$ be the number. According to the problem, we have the equation:
\[ x^3 - 2x = x^2 - 3x \]
To solve for $x$, first move all terms to one side of the equation:
\[ x^3 - x^2 + x = 0 \]
Then factor out the common term $x$:
\[ x(x^2 - x + 1) = 0 \]
Since $x^2 - x + 1$ cannot be factored into real numbers, we have ... | 0 | augmented_math | [
{
"content": "The difference between the cube of a number and twice the number is equal to the square of the number minus three times the number. What is the number?",
"role": "user"
},
{
"content": "Let $x$ be the number. According to the problem, we have the equation:\n\\[ x^3 - 2x = x^2 - 3x \\]\... |
Eva is planning a 240-mile road trip. She decides to drive 20 miles for 10 days. How many miles will she need to drive on the 11th day to reach her destination? | Eva needs to drive a total of 240 miles.
She drives 20 miles a day for 10 days which amounts to $20 * 10 = 200$ miles.
This leaves her with $240 - 200 = 40$ miles to drive on the 11th day to reach her destination.
Thus, Eva needs to drive \boxed{40} miles on the 11th day. | 40 | augmented_gsm8k | [
{
"content": "Eva is planning a 240-mile road trip. She decides to drive 20 miles for 10 days. How many miles will she need to drive on the 11th day to reach her destination?",
"role": "user"
},
{
"content": "Eva needs to drive a total of 240 miles.\nShe drives 20 miles a day for 10 days which amoun... |
A right triangle with one side of length 10 cm and a hypotenuse of length 20 cm is rotated about its hypotenuse. What is the volume of the double cone created by this rotation? | To find the volume of the double cone, we need to first find the radius and height of each cone.
Let's denote the lengths of the triangle's sides as follows: the side of length 10 cm is one of the legs (let's call it $a$), the other leg is $b$ (which we don't know yet), and the hypotenuse is $c = 20$ cm.
Since it's a... | \frac{4000\pi}{3} | augmented_math | [
{
"content": "A right triangle with one side of length 10 cm and a hypotenuse of length 20 cm is rotated about its hypotenuse. What is the volume of the double cone created by this rotation?",
"role": "user"
},
{
"content": "To find the volume of the double cone, we need to first find the radius and... |
Find the smallest positive integer that is divisible by both 5 and 7, and has exactly three distinct prime factors. | To find the smallest positive integer that is divisible by both 5 and 7, and has exactly three distinct prime factors, we first need to identify the two given prime factors. The numbers 5 and 7 are already prime numbers, so we have two prime factors.
The next step is to choose the third prime factor. To keep the integ... | 70 | augmented_math | [
{
"content": "Find the smallest positive integer that is divisible by both 5 and 7, and has exactly three distinct prime factors.",
"role": "user"
},
{
"content": "To find the smallest positive integer that is divisible by both 5 and 7, and has exactly three distinct prime factors, we first need to ... |
Consider two lines: line $p$ parametrized as
\begin{align*}
x &= 3 + 2t,\\
y &= -2 + 5t
\end{align*} and the line $q$ parametrized as
\begin{align*}
x &= 7 + 2s\\
y &= 1 + 5s.
\end{align*}Let $C$ be a point on line $p$, $D$ be a point on line $q$, and let $R$ be the foot of the perpendicular from $C$ to line $q$.
Then... | ## Step 1: Identify the direction vector of line q
The direction vector of line q is given by the coefficients of s in its parametric equations, which is \begin{pmatrix} 2 \\ 5 \end{pmatrix}.
## Step 2: Find a vector perpendicular to line q
A vector perpendicular to line q is given by the vector \begin{pmatrix} -5 \\ ... | \begin{pmatrix} 15/7 \\ -6/7 \end{pmatrix} | augmented_math | [
{
"content": "Consider two lines: line $p$ parametrized as\n\\begin{align*}\nx &= 3 + 2t,\\\\\ny &= -2 + 5t\n\\end{align*} and the line $q$ parametrized as\n\\begin{align*}\nx &= 7 + 2s\\\\\ny &= 1 + 5s.\n\\end{align*}Let $C$ be a point on line $p$, $D$ be a point on line $q$, and let $R$ be the foot of the per... |
Find the surface area of the region in space defined by
\[|x^2 + y^2 + z^2| + |x^2 + y^2 - z^2| \le 16\]and $x,$ $y,$ $z \ge 0.$ | We have
\[|x^2 + y^2 + z^2| + |x^2 + y^2 - z^2| \le 16\]
Since $x,$ $y,$ $z \ge 0,$ we have $x^2 + y^2 + z^2 \ge 0$ and $x^2 + y^2 - z^2 \ge 0.$
Therefore, the equation becomes
\[x^2 + y^2 + z^2 + x^2 + y^2 - z^2 \le 16\]
\[2x^2 + 2y^2 \le 16\]
\[x^2 + y^2 \le 8\]
This is the equation of a circle in the $xy$-plane wi... | 8\pi | augmented_math | [
{
"content": "Find the surface area of the region in space defined by\n\\[|x^2 + y^2 + z^2| + |x^2 + y^2 - z^2| \\le 16\\]and $x,$ $y,$ $z \\ge 0.$",
"role": "user"
},
{
"content": "We have\n\\[|x^2 + y^2 + z^2| + |x^2 + y^2 - z^2| \\le 16\\]\nSince $x,$ $y,$ $z \\ge 0,$ we have $x^2 + y^2 + z^2 \\g... |
What is the minimum value of $3y^2-18y+2412$? | To find the minimum value of $3y^2 - 18y + 2412$, we need to complete the square or use the formula for the vertex of a parabola.
The given expression can be rewritten as:
\[ 3y^2 - 18y + 2412 = 3(y^2 - 6y) + 2412 \]
To complete the square, we take the coefficient of $y$, which is $-6$, divide it by $2$, and square t... | 2385 | augmented_math | [
{
"content": "What is the minimum value of $3y^2-18y+2412$?",
"role": "user"
},
{
"content": "To find the minimum value of $3y^2 - 18y + 2412$, we need to complete the square or use the formula for the vertex of a parabola.\n\nThe given expression can be rewritten as:\n\\[ 3y^2 - 18y + 2412 = 3(y^2 ... |
The graph of the parabola $x = 3y^2 + 4y - 2$ has an $x$-intercept $(d,0)$ and two $y$-intercepts $(0,e)$ and $(0,f)$. Find $d + e + f$. | To find the $x$-intercept $(d,0)$, we substitute $y = 0$ into the equation $x = 3y^2 + 4y - 2$.
\[ x = 3(0)^2 + 4(0) - 2 = -2 \]
So, the $x$-intercept is $(d,0) = (-2,0)$, and $d = -2$.
To find the $y$-intercepts, we set $x = 0$ and solve for $y$.
\[ 0 = 3y^2 + 4y - 2 \]
To solve this quadratic equation, we can us... | -\frac{10}{3} | augmented_math | [
{
"content": "The graph of the parabola $x = 3y^2 + 4y - 2$ has an $x$-intercept $(d,0)$ and two $y$-intercepts $(0,e)$ and $(0,f)$. Find $d + e + f$.",
"role": "user"
},
{
"content": "To find the $x$-intercept $(d,0)$, we substitute $y = 0$ into the equation $x = 3y^2 + 4y - 2$.\n\n\\[ x = 3(0)^2 ... |
In a tetrahedron $ABCD$, the coordinates of $A$, $B$, and $C$ are $(1,2,3)$, $(5,3,1)$, and $(3,4,5)$ respectively. The centroid of the tetrahedron is $(2,3,4)$. Find the coordinates of vertex $D$. | The centroid of a tetrahedron is given by the average of the coordinates of its vertices.
Let the coordinates of $D$ be $(x, y, z)$.
The centroid is $(2, 3, 4)$, so we can set up the equation:
\[ \frac{(1, 2, 3) + (5, 3, 1) + (3, 4, 5) + (x, y, z)}{4} = (2, 3, 4) \]
\[ \frac{(1 + 5 + 3 + x, 2 + 3 + 4 + y, 3 + 1 + 5 ... | (-1, 3, 7) | augmented_math | [
{
"content": "In a tetrahedron $ABCD$, the coordinates of $A$, $B$, and $C$ are $(1,2,3)$, $(5,3,1)$, and $(3,4,5)$ respectively. The centroid of the tetrahedron is $(2,3,4)$. Find the coordinates of vertex $D$.",
"role": "user"
},
{
"content": "The centroid of a tetrahedron is given by the average ... |
The equation $x^3 + 3x^2 - 10x = 0$ can be factored as $x(x^2 + 3x - 10) = 0$, which gives the solutions $x = 0, x = -5$, and $x = 2$. To find the mean of these solutions, we can add them up and divide by 3:
$$\frac{0 + (-5) + 2}{3} = \frac{-3}{3} = -1$$
So, the mean of all solutions for $x$ is $-1$.
Now, let's crea... | We have to solve the equation $\sqrt{x} + \sqrt{x+1} = \sqrt{x+2}$.
Isolate one of the square root expressions:
\[ \sqrt{x} = \sqrt{x+2} - \sqrt{x+1} \]
Square both sides to get rid of the square roots:
\[ x = (x+2) - 2\sqrt{(x+2)(x+1)} + (x+1) \]
Simplify and rearrange:
\[ 2\sqrt{(x+2)(x+1)} = 3 \]
Square both sid... | -\frac{3}{2} | augmented_math | [
{
"content": "The equation $x^3 + 3x^2 - 10x = 0$ can be factored as $x(x^2 + 3x - 10) = 0$, which gives the solutions $x = 0, x = -5$, and $x = 2$. To find the mean of these solutions, we can add them up and divide by 3:\n\n$$\\frac{0 + (-5) + 2}{3} = \\frac{-3}{3} = -1$$\n\nSo, the mean of all solutions for $... |
Consider the sequence defined recursively by $v_1 = b > 0$ and
\[v_{n + 1} = \frac{2}{v_n - 1}\]for $n \ge 1.$ Express $v_{20}$ in terms of $b.$ | The sequence is defined by
\[ v_1 = b \]
\[ v_{n + 1} = \frac{2}{v_n - 1} \text{ for } n \geq 1 \]
Let's find the first few terms of the sequence to see if there is a pattern.
1. $v_1 = b$
2. $v_2 = \frac{2}{v_1 - 1} = \frac{2}{b - 1}$
3. $v_3 = \frac{2}{v_2 - 1} = \frac{2}{\frac{2}{b - 1} - 1} = \frac{2}{\frac{2 - b... | \frac{2}{b - 1} | augmented_math | [
{
"content": "Consider the sequence defined recursively by $v_1 = b > 0$ and\n\\[v_{n + 1} = \\frac{2}{v_n - 1}\\]for $n \\ge 1.$ Express $v_{20}$ in terms of $b.$",
"role": "user"
},
{
"content": "The sequence is defined by\n\\[ v_1 = b \\]\n\\[ v_{n + 1} = \\frac{2}{v_n - 1} \\text{ for } n \\geq... |
Alex and Ben are participating in a 5-day cycling challenge. They aim to cover a total distance of 50 miles. Over the first two days, Alex averages 1.5 miles per day, while on the third day, he cycles 12 miles. Ben maintains an average of 2 miles per day for the first three days. What is the average distance they need ... | To calculate the average distance Alex and Ben need to cover per day over the last two days, let's first compute the distance they have already covered in the first three days and then find the remaining distance they need to cover to meet their goal of 50 miles.
The distance Alex covers in the first three days is giv... | 14.5 | augmented_gsm8k | [
{
"content": "Alex and Ben are participating in a 5-day cycling challenge. They aim to cover a total distance of 50 miles. Over the first two days, Alex averages 1.5 miles per day, while on the third day, he cycles 12 miles. Ben maintains an average of 2 miles per day for the first three days. What is the avera... |
Five distinct points, $E$, $F$, $G$, $H$, and $I$, are to be selected from $2500$ points evenly spaced around a circle. All quintuples are equally likely to be chosen. What is the probability that the chord $\overline{EF}$ intersects both the chord $\overline{GH}$ and the chord $\overline{HI}$? | There are $\binom{2500}{5}$ ways to choose $5$ points from $2500$, so that will be the denominator of our probability.
If $\overline{EF}$ intersects $\overline{GH}$ and $\overline{HI}$, then $\overline{GH}$ and $\overline{HI}$ cannot be diameters. There are $2500$ ways to choose the first point $E$, and then $2499$ wa... | \frac{1}{3} | augmented_math | [
{
"content": "Five distinct points, $E$, $F$, $G$, $H$, and $I$, are to be selected from $2500$ points evenly spaced around a circle. All quintuples are equally likely to be chosen. What is the probability that the chord $\\overline{EF}$ intersects both the chord $\\overline{GH}$ and the chord $\\overline{HI}$?... |
At 8:45 p.m., what is the measure of the smaller angle formed by the minute hand and the hour hand of the clock? | At 8:45 p.m., the hour hand is between the 8 and the 9 on the clock, and the minute hand is on the 9.
1. Find the position of the hour hand:
- The hour hand moves 360 degrees in 12 hours, which means it moves 30 degrees per hour.
- In 45 minutes, the hour hand moves 45/60 * 30 = 0.75 * 30 = 22.5 degrees.
- At... | 7.5 | augmented_math | [
{
"content": "At 8:45 p.m., what is the measure of the smaller angle formed by the minute hand and the hour hand of the clock?",
"role": "user"
},
{
"content": "At 8:45 p.m., the hour hand is between the 8 and the 9 on the clock, and the minute hand is on the 9.\n\n1. Find the position of the hour h... |
A new restaurant offers a free dessert to every customer on Fridays. Last Friday, the desserts came in trays of 24. If they had to prepare 9 trays, and 8 desserts were left uneaten at the end of the night, how many customers received a free dessert if the offer was limited to one per customer? | The restaurant prepared 9 trays of 24 desserts each.
So the total number of desserts is $9 * 24 = 216$.
Out of these, 8 were left uneaten at the end of the night.
So the number of customers who received a free dessert is $216 - 8 = 208$.
Thus \boxed{208} customers received a free dessert. | 208 | augmented_gsm8k | [
{
"content": "A new restaurant offers a free dessert to every customer on Fridays. Last Friday, the desserts came in trays of 24. If they had to prepare 9 trays, and 8 desserts were left uneaten at the end of the night, how many customers received a free dessert if the offer was limited to one per customer?",
... |
Quadrilateral $ABCD$ is formed by four tangents to circle $O$, with $AB$ parallel to $CD$. Find $\angle AOC$.
[asy]
import graph;
unitsize(1.5 cm);
pair A, B, O, C, D, R, S, T, U;
R = dir(115);
S = dir(230);
T = dir(270);
U = dir(65);
A = extension(R, R + rotate(90)*(R), T, T + rotate(90)*(T));
B = extension(S, S +... | Since $AB$ is parallel to $CD$, we have $\angle AOB = \angle COD$ and $\angle AOC = \angle BOD$.
Also, $\angle AOB + \angle BOD = \angle AOC + \angle COD = 180^\circ$, since they are pairs of opposite angles in a quadrilateral.
We also have $\angle AOC + \angle BOD = 180^\circ$, since $\angle AOC$ and $\angle BOD$ ar... | 90^\circ | augmented_math | [
{
"content": "Quadrilateral $ABCD$ is formed by four tangents to circle $O$, with $AB$ parallel to $CD$. Find $\\angle AOC$.\n\n[asy]\nimport graph;\n\nunitsize(1.5 cm);\n\npair A, B, O, C, D, R, S, T, U;\n\nR = dir(115);\nS = dir(230);\nT = dir(270);\nU = dir(65);\nA = extension(R, R + rotate(90)*(R), T, T + r... |
Let $a$ be a real number randomly selected from the interval $[1, 5]$. Then, $m$ and $n$ are two relatively prime positive integers such that $m/n$ is the probability that the equation $x^3 + ax^2 + 3a^2 = (2a^2 - 3a)x$ has at least one real root in the interval $[-2, 2]$. Find the value of $m + n$. | Rearrange the equation to get $x^3 + (a - 2a^2 + 3a)x^2 + 3a^2 = 0$.
Notice that $x = 0$ is a root of the equation.
We are interested in finding when the equation has at least one real root in the interval $[-2, 2]$.
Divide the equation by $x$ (since $x = 0$ is a root, this is valid) to get:
\[ x^2 + (a - 2a^2 + 3a)... | 2 | augmented_math | [
{
"content": "Let $a$ be a real number randomly selected from the interval $[1, 5]$. Then, $m$ and $n$ are two relatively prime positive integers such that $m/n$ is the probability that the equation $x^3 + ax^2 + 3a^2 = (2a^2 - 3a)x$ has at least one real root in the interval $[-2, 2]$. Find the value of $m + n... |
If $2x+y=8$ and $x^2+xy+y^2=21$, then what is $x^2-y^2$? | We have two equations:
\begin{align*}
2x + y = 8 \quad &(1) \\
x^2 + xy + y^2 = 21 \quad &(2)
\end{align*}
From (1), we can express $y$ in terms of $x$:
\[ y = 8 - 2x \]
Substitute $y = 8 - 2x$ into (2):
\[ x^2 + x(8 - 2x) + (8 - 2x)^2 = 21 \Rightarrow x^2 + 8x - 2x^2 + 64 - 32x + 4x^2 = 21 \Rightarrow 3x... | 11 | augmented_math | [
{
"content": "If $2x+y=8$ and $x^2+xy+y^2=21$, then what is $x^2-y^2$?",
"role": "user"
},
{
"content": "We have two equations:\n \\begin{align*}\n 2x + y = 8 \\quad &(1) \\\\\n x^2 + xy + y^2 = 21 \\quad &(2)\n \\end{align*}\n\nFrom (1), we can express $y$ in terms of $x$:\n\\[ y = 8 - 2x \... |
Maria has 3 rabbits, and each rabbit eats 5 carrots per day. How many carrots will Maria's rabbits eat in 6 days? | Maria has 3 rabbits, and each rabbit eats 5 carrots per day.
So in total, the rabbits eat 3 * 5 = 15 carrots per day.
To find out how many carrots they will eat in 6 days, we multiply 15 carrots by 6 days, which equals 15 * 6 = 90.
Thus, the rabbits will eat \boxed{90} carrots in 6 days. | 90 | augmented_gsm8k | [
{
"content": "Maria has 3 rabbits, and each rabbit eats 5 carrots per day. How many carrots will Maria's rabbits eat in 6 days?",
"role": "user"
},
{
"content": "Maria has 3 rabbits, and each rabbit eats 5 carrots per day.\nSo in total, the rabbits eat 3 * 5 = 15 carrots per day.\nTo find out how ma... |
Samantha, Sarah, and Sophia are collecting sand for a large sandbox in the park. Each has a bucket they fill with sand before emptying it into the sandbox. If Samantha's bucket holds three times as much sand as Sarah's, and Sarah's holds 1.5 gallons more than Sophia's, and Sophia's is 3 gallons, how much sand do the th... | Let's break this down step by step:
* Sophia's bucket holds 3 gallons.
* Sarah's bucket holds 1.5 gallons more than Sophia's, so it holds 3 + 1.5 = 4.5 gallons.
* Samantha's bucket holds three times as much sand as Sarah's, so it holds 3 * 4.5 = 13.5 gallons.
Now, let's calculate the total amount of sand collected by... | 84 | augmented_gsm8k | [
{
"content": "Samantha, Sarah, and Sophia are collecting sand for a large sandbox in the park. Each has a bucket they fill with sand before emptying it into the sandbox. If Samantha's bucket holds three times as much sand as Sarah's, and Sarah's holds 1.5 gallons more than Sophia's, and Sophia's is 3 gallons, h... |
A sequence of integers is called "zigzag" if the difference between consecutive terms alternates between positive and negative. How many zigzag sequences of length 7 can be formed using the integers 1 through 10, with no integer repeated? | To form a zigzag sequence of length 7 using the integers 1 through 10, we need to select 7 integers without repetition. The sequence must start with either an increase or a decrease from the first term to the second term.
Let's consider the two cases separately:
Case 1: The sequence starts with an increase.
1. Choos... | 240 | augmented_math | [
{
"content": "A sequence of integers is called \"zigzag\" if the difference between consecutive terms alternates between positive and negative. How many zigzag sequences of length 7 can be formed using the integers 1 through 10, with no integer repeated?",
"role": "user"
},
{
"content": "To form a z... |
Find the maximum value of $x^2+y^2$, where $x$ and $y$ are real numbers such that $\sin^2 x + \sin^2 y = \sin^2 (x+y)$. | Using the given equation $\sin^2 x + \sin^2 y = \sin^2 (x+y)$, we will derive an expression for $x$ and $y$ in terms of trigonometric functions.
Expand $\sin^2 (x+y)$ using the angle addition formula:
\[ \sin^2 (x+y) = (\sin x \cos y + \cos x \sin y)^2 = \sin^2 x \cos^2 y + 2 \sin x \cos x \sin y \cos y + \cos^2 x \si... | \frac{\pi^2}{8} | augmented_math | [
{
"content": "Find the maximum value of $x^2+y^2$, where $x$ and $y$ are real numbers such that $\\sin^2 x + \\sin^2 y = \\sin^2 (x+y)$.",
"role": "user"
},
{
"content": "Using the given equation $\\sin^2 x + \\sin^2 y = \\sin^2 (x+y)$, we will derive an expression for $x$ and $y$ in terms of trigon... |
Express $\frac{2}{5}$ as a percentage. | To convert the fraction $\frac{2}{5}$ to a percentage, we first convert it to a decimal by dividing the numerator by the denominator:
\[ \frac{2}{5} = 2 : 5 = 0.4 \]
Then, to convert the decimal to a percentage, we multiply by 100:
\[ 0.4 \times 100 = 40\% \]
So, $\frac{2}{5}$ as a percentage is $\boxed{40\%}$ | 40\% | augmented_math | [
{
"content": "Express $\\frac{2}{5}$ as a percentage.",
"role": "user"
},
{
"content": "To convert the fraction $\\frac{2}{5}$ to a percentage, we first convert it to a decimal by dividing the numerator by the denominator:\n\\[ \\frac{2}{5} = 2 : 5 = 0.4 \\]\n\nThen, to convert the decimal to a perc... |
Tom is on a road trip and wants to know how far he can travel in a certain amount of time. He knows that his car's fuel efficiency is 30 miles per gallon, and he has 15 gallons of gas in the tank. If he drives at an average speed of 45 miles per hour, how many hours can he drive before running out of gas? | To find out how many hours Tom can drive before running out of gas, we need to calculate the total distance he can travel with the gas he has and then divide that by his average speed.
1. Calculate the total distance Tom can travel with 15 gallons of gas:
\[ \text{Total Distance} = \text{Fuel Efficiency} \times \text{... | 10 | augmented_math | [
{
"content": "Tom is on a road trip and wants to know how far he can travel in a certain amount of time. He knows that his car's fuel efficiency is 30 miles per gallon, and he has 15 gallons of gas in the tank. If he drives at an average speed of 45 miles per hour, how many hours can he drive before running out... |
Simplify $\dfrac{1+10^{-1}+10^{-2}}{1+10^{-1}+10^{-2}+10^{-3}} \cdot 0.3$ | \[ \frac{1+10^{-1}+10^{-2}}{1+10^{-1}+10^{-2}+10^{-3}} \cdot 0.3 = \frac{1+\frac{1}{10}+\frac{1}{100}}{1+\frac{1}{10}+\frac{1}{100}+\frac{1}{1000}} \cdot 0.3 = \frac{\frac{100}{100}+\frac{10}{100}+\frac{1}{100}}{\frac{1000}{1000}+\frac{100}{1000}+\frac{10}{1000}+\frac{1}{1000}} \cdot 0.3 = \frac{\frac{111}{100}}{\frac{... | 0.3 | augmented_math | [
{
"content": "Simplify $\\dfrac{1+10^{-1}+10^{-2}}{1+10^{-1}+10^{-2}+10^{-3}} \\cdot 0.3$",
"role": "user"
},
{
"content": "\\[ \\frac{1+10^{-1}+10^{-2}}{1+10^{-1}+10^{-2}+10^{-3}} \\cdot 0.3 = \\frac{1+\\frac{1}{10}+\\frac{1}{100}}{1+\\frac{1}{10}+\\frac{1}{100}+\\frac{1}{1000}} \\cdot 0.3 = \\frac... |
A lighthouse is built on a small rock near a straight coastline. The lighthouse emits light to a spot located 150 meters from its base. The angle formed by the light beam and the vertical line from the top of the lighthouse is 30 degrees. Assuming the coastline is straight and can be considered a horizontal line, what ... | To find the height of the lighthouse above the water level, we can use trigonometric principles. Given that the angle formed by the light beam and the vertical line from the top of the lighthouse is 30 degrees, and the beam hits the coastline 150 meters from the base of the lighthouse, we can form a right-angled triang... | 50\sqrt{3} | augmented_math | [
{
"content": "A lighthouse is built on a small rock near a straight coastline. The lighthouse emits light to a spot located 150 meters from its base. The angle formed by the light beam and the vertical line from the top of the lighthouse is 30 degrees. Assuming the coastline is straight and can be considered a ... |
In a triangle $ABC$, the angle $A$ is $150^\circ$, and the sides $AB$ and $AC$ are equal. Find the ratio of the length of side $BC$ to the length of the altitude from vertex $A$ to side $BC$. | Given that angle $A$ is $150^\circ$ and sides $AB$ and $AC$ are equal, we know that triangle $ABC$ is isosceles.
Let $D$ be the foot of the altitude from vertex $A$ to side $BC$. Since triangle $ABC$ is isosceles, $BD = DC$.
In triangle $ABD$, angle $A$ is $150^\circ$, angle $D$ is $90^\circ$, and $BD$ is half of $BC... | 4 + 2\sqrt{3} | augmented_math | [
{
"content": "In a triangle $ABC$, the angle $A$ is $150^\\circ$, and the sides $AB$ and $AC$ are equal. Find the ratio of the length of side $BC$ to the length of the altitude from vertex $A$ to side $BC$.",
"role": "user"
},
{
"content": "Given that angle $A$ is $150^\\circ$ and sides $AB$ and $AC... |
Let $x$ and $y$ be real numbers such that $x^3 + y^3 = 2(x + y).$ Find the maximum value of $x^2 - y^2.$ | Let $x^3 + y^3 = 2(x + y)$. We know that
\[ x^3 + y^3 = (x + y)(x^2 - xy + y^2) \]
So we can rewrite the equation as:
\[ (x + y)(x^2 - xy + y^2) = 2(x + y) \]
Dividing both sides by $(x + y)$ gives us:
\[ x^2 - xy + y^2 = 2 \]
We want to find the maximum value of $x^2 - y^2$.
Notice that:
\[ x^2 - y^2 = (x^2 - xy +... | 2 | augmented_math | [
{
"content": "Let $x$ and $y$ be real numbers such that $x^3 + y^3 = 2(x + y).$ Find the maximum value of $x^2 - y^2.$",
"role": "user"
},
{
"content": "Let $x^3 + y^3 = 2(x + y)$. We know that\n\\[ x^3 + y^3 = (x + y)(x^2 - xy + y^2) \\]\n\nSo we can rewrite the equation as:\n\\[ (x + y)(x^2 - xy ... |
In a survey of 100 students, it was found that 60 students like pizza, 40 students like sushi, and 20 students like both pizza and sushi. If the number of students who like pizza but not sushi is 10 more than the number of students who like sushi but not pizza, how many students like only pizza? | Let's denote the number of students who like pizza but not sushi as $P$ and the number of students who like sushi but not pizza as $S$.
We know that:
1. The total number of students who like pizza is $P + 20 = 60$ (since 20 students like both pizza and sushi).
2. The total number of students who like sushi is $S + 20 ... | 40 | augmented_math | [
{
"content": "In a survey of 100 students, it was found that 60 students like pizza, 40 students like sushi, and 20 students like both pizza and sushi. If the number of students who like pizza but not sushi is 10 more than the number of students who like sushi but not pizza, how many students like only pizza?",... |
Ava's living room is 240 sq ft and her dining room is 120 sq ft. If she wants to build a new game room that is three times as large as the combined space of her living and dining rooms, how many sq ft will the new game room have? | The combined space of Ava's living and dining rooms is $240 + 120 = 360$ sq ft.
The new game room will be three times as large as this combined space.
Thus the area of the new game room will be $360 * 3 = 1080$ sq ft.
So the new game room will have \boxed{1080} sq ft. | 1080 | augmented_gsm8k | [
{
"content": "Ava's living room is 240 sq ft and her dining room is 120 sq ft. If she wants to build a new game room that is three times as large as the combined space of her living and dining rooms, how many sq ft will the new game room have?",
"role": "user"
},
{
"content": "The combined space of ... |
In a quadrilateral, the angles are $80^{\circ}$, $60^{\circ}$, $80^{\circ}$, and $x$. If the sum of the measures of the interior angles is equal to the product of the number of sides and the average of the measures of the angles, find the value of $x$. | The sum of the interior angles in a quadrilateral is given by the formula $(n - 2) \times 180$, where $n$ is the number of sides. For a quadrilateral, this sum is $(4 - 2) \times 180 = 360$ degrees.
The average of the measures of the angles is the sum of the angles divided by the number of sides.
We are given that th... | 140 | augmented_math | [
{
"content": "In a quadrilateral, the angles are $80^{\\circ}$, $60^{\\circ}$, $80^{\\circ}$, and $x$. If the sum of the measures of the interior angles is equal to the product of the number of sides and the average of the measures of the angles, find the value of $x$.",
"role": "user"
},
{
"content... |
Let $a_1, a_2, \ldots, a_{96}$ be a sequence of real numbers, not necessarily distinct, such that $a_1 + a_2 + \ldots + a_{96} = 0$. Find the smallest possible value of $\sum_{1 \le i < j \le 96} a_i a_j^2$. | We can write the given sum as
\[ S = \sum_{1 \le i < j \le 96} a_i a_j^2 = \sum_{i = 1}^{96} a_i \sum_{j = i + 1}^{96} a_j^2 = \sum_{i = 1}^{96} a_i \left( \sum_{j = 1}^{96} a_j^2 - \sum_{j = 1}^i a_j^2 \right) \]
Note that $\sum_{j = 1}^{96} a_j^2$ is a constant, so we can denote it by $C$. Then we have
\[ S = \sum_{... | 0 | augmented_math | [
{
"content": "Let $a_1, a_2, \\ldots, a_{96}$ be a sequence of real numbers, not necessarily distinct, such that $a_1 + a_2 + \\ldots + a_{96} = 0$. Find the smallest possible value of $\\sum_{1 \\le i < j \\le 96} a_i a_j^2$.",
"role": "user"
},
{
"content": "We can write the given sum as\n\\[ S = ... |
Let $f(n)$ represent the base prime representation of a natural number $n$. What is the largest natural number $n$ such that the sequence $f(1), f(2), f(3), \ldots, f(n)$ forms an arithmetic progression? | Let's examine the function $f(n)$ and its values for the first few natural numbers.
$f(1) = 2$ since 2 is the smallest prime number.
$f(2) = 3$ since 3 is the next prime number.
$f(3) = 5$ since 5 is the next prime number.
$f(4) = 7$ since 7 is the next prime number.
$f(5) = 11$ since 11 is the next prime number.
... | 5 | augmented_math | [
{
"content": "Let $f(n)$ represent the base prime representation of a natural number $n$. What is the largest natural number $n$ such that the sequence $f(1), f(2), f(3), \\ldots, f(n)$ forms an arithmetic progression?",
"role": "user"
},
{
"content": "Let's examine the function $f(n)$ and its value... |
Emily purchased 4 cookies on Monday, 2 cookies on Wednesday, and 6 cookies on Saturday. If she has 5 cookies remaining, how many cookies did Emily eat? | Let's solve the new question.
To find out how many cookies Emily ate, we need to subtract the number of cookies remaining from the total number of cookies she purchased.
The total number of cookies purchased is $4 + 2 + 6 = 6 + 6 = 12$.
Since Emily has 5 cookies remaining, she must have eaten $12 - 5 = 7$ cookies.
Thu... | 7 | augmented_gsm8k | [
{
"content": "Emily purchased 4 cookies on Monday, 2 cookies on Wednesday, and 6 cookies on Saturday. If she has 5 cookies remaining, how many cookies did Emily eat?",
"role": "user"
},
{
"content": "Let's solve the new question.\n\nTo find out how many cookies Emily ate, we need to subtract the num... |
A spacecraft is approaching a planet along a parabolic trajectory, with the planet at the focus. At the vertex of the parabola, the spacecraft is 2500 km away from the planet. When it crosses the orbit of the planet's largest moon, the spacecraft is traveling at a speed of 20 km/s relative to the moon. The moon's orbit... | ## Step 1: Determine the equation of the parabolic trajectory of the spacecraft.
Given that the planet is at the focus of the parabola and the vertex is 2500 km away from the planet, we can set up the equation of the parabola. The general form of a parabola with its vertex at the origin and opening to the right is \(y^... | 0 | augmented_math | [
{
"content": "A spacecraft is approaching a planet along a parabolic trajectory, with the planet at the focus. At the vertex of the parabola, the spacecraft is 2500 km away from the planet. When it crosses the orbit of the planet's largest moon, the spacecraft is traveling at a speed of 20 km/s relative to the ... |
A group of 5 musicians, consisting of a lead vocalist, a drummer, a guitarist, a bassist, and a keyboardist, are to be seated on a stage in a specific arrangement. The lead vocalist must be seated in the center, and the drummer must be seated on one of the two ends. The guitarist and bassist must be seated on either si... | To solve this problem, we can break it down into several steps:
1. The lead vocalist must be seated in the center. Since there is only one center position, there is only one way to seat the lead vocalist.
2. The drummer must be seated on one of the two ends. There are two ends, so there are 2 ways to seat the drummer... | 4 | augmented_math | [
{
"content": "A group of 5 musicians, consisting of a lead vocalist, a drummer, a guitarist, a bassist, and a keyboardist, are to be seated on a stage in a specific arrangement. The lead vocalist must be seated in the center, and the drummer must be seated on one of the two ends. The guitarist and bassist must ... |
How many four-digit numbers have the property that their square is a palindrome? | Let the four-digit number be $x$. We need to find the values of $x$ such that $x^2$ is a palindrome.
The square of a four-digit number can have 7 or 8 digits. A 7-digit palindrome has the form $abccba$, and an 8-digit palindrome has the form $abbaabba$.
To find the possible values of $x$, we can start by looking at t... | 3 | augmented_math | [
{
"content": "How many four-digit numbers have the property that their square is a palindrome?",
"role": "user"
},
{
"content": "Let the four-digit number be $x$. We need to find the values of $x$ such that $x^2$ is a palindrome.\n\nThe square of a four-digit number can have 7 or 8 digits. A 7-digit... |
In a game, a player can win either $2 or $5 on each turn, with probabilities of 0.4 and 0.6, respectively. If the player's expected winnings per turn is $3.80, what is the number of turns the player needs to play to have at least a 90% chance of winning a total of $15 or more? | Let's solve this problem step-by-step.
Step 1: Find the expected value of the player's winnings per turn.
\[ \text{Expected value} = (0.4)(2) + (0.6)(5) = 0.8 + 3 = 3.8 \]
Step 2: Find the variance of the player's winnings per turn.
\[ \text{Variance} = (0.4)(2 - 3.8)^2 + (0.6)(5 - 3.8)^2 = (0.4)(-1.8)^2 + (0.6)(1.2)... | 6 | augmented_math | [
{
"content": "In a game, a player can win either $2 or $5 on each turn, with probabilities of 0.4 and 0.6, respectively. If the player's expected winnings per turn is $3.80, what is the number of turns the player needs to play to have at least a 90% chance of winning a total of $15 or more?",
"role": "user"... |
A certain bacteria population triples every 5 days. After 10 days, the population reaches 25,000. Find the initial population. | We know that the population triples every 5 days. This means that the population grows by a factor of 3 every 5 days.
Let's denote the initial population as $P_0$.
After 5 days, the population will be $3P_0$.
After another 5 days (total of 10 days), the population will triple again, so it will be $3(3P_0) = 9P_0$.
... | 2777.78 | augmented_math | [
{
"content": "A certain bacteria population triples every 5 days. After 10 days, the population reaches 25,000. Find the initial population.",
"role": "user"
},
{
"content": "We know that the population triples every 5 days. This means that the population grows by a factor of 3 every 5 days.\n\nLet'... |
If $y = \displaystyle\frac{x^2}{2x-1}$, what is the value of $x$ when $y = 2$? | We are given $y = \displaystyle\frac{x^2}{2x-1}$ and we need to find the value of $x$ when $y = 2$.
Substitute $y = 2$ into the equation:
\[ 2 = \frac{x^2}{2x-1} \]
To solve for $x$, multiply both sides by $2x-1$:
\[ 2(2x-1) = x^2 \Rightarrow 4x - 2 = x^2 \Rightarrow x^2 - 4x + 2 = 0 \]
This is a quadratic equation.... | 2 + \sqrt{2} | augmented_math | [
{
"content": "If $y = \\displaystyle\\frac{x^2}{2x-1}$, what is the value of $x$ when $y = 2$?",
"role": "user"
},
{
"content": "We are given $y = \\displaystyle\\frac{x^2}{2x-1}$ and we need to find the value of $x$ when $y = 2$.\n\nSubstitute $y = 2$ into the equation:\n\\[ 2 = \\frac{x^2}{2x-1} \... |
A bakery is famous for its delicious cookies, which are made with a special blend of ingredients. The recipe calls for a ratio of 3 parts flour to 2 parts sugar. If the bakery wants to make a batch of cookies that uses 120 grams of flour, how many grams of sugar should they use?
Note: I've created a problem that requi... | Let's start by understanding the ratio of flour to sugar in the recipe: 3 parts flour to 2 parts sugar.
Since the bakery wants to use 120 grams of flour, we need to find out how many grams of sugar they should use.
We can set up a proportion to relate the ratio of flour to sugar:
$\frac{3}{2} = \frac{120}{x}$
where... | 80 | augmented_math | [
{
"content": "A bakery is famous for its delicious cookies, which are made with a special blend of ingredients. The recipe calls for a ratio of 3 parts flour to 2 parts sugar. If the bakery wants to make a batch of cookies that uses 120 grams of flour, how many grams of sugar should they use?\n\nNote: I've crea... |
Emily wants to buy 12 new art supplies. Each supply costs $3. If Emily already has $20, how much more money does she need to save up? | To find out how much more money Emily needs, we need to calculate the total cost of the art supplies she wants to buy.
The cost of 1 art supply is $3.
The cost of 12 art supplies is $3 * 12 = 36$.
Emily already has $20, so we subtract the amount she has from the total cost to find out how much more she needs:
$36 - ... | 16 | augmented_gsm8k | [
{
"content": "Emily wants to buy 12 new art supplies. Each supply costs $3. If Emily already has $20, how much more money does she need to save up?",
"role": "user"
},
{
"content": "To find out how much more money Emily needs, we need to calculate the total cost of the art supplies she wants to buy.... |
Let $P$ be a point on the ellipse $\frac{x^2}{4} + y^2 = 1,$ and let $Q$ be a point on the hyperbola $x^2 - 4y^2 = 1.$ Find the smallest possible distance $PQ.$ | The ellipse is given by the equation $\frac{x^2}{4} + y^2 = 1$, and the hyperbola is given by $x^2 - 4y^2 = 1$. We want to find the smallest distance between any point on the ellipse and any point on the hyperbola.
Let $P = (x_1, y_1)$ be a point on the ellipse and $Q = (x_2, y_2)$ be a point on the hyperbola.
The di... | 1 | augmented_math | [
{
"content": "Let $P$ be a point on the ellipse $\\frac{x^2}{4} + y^2 = 1,$ and let $Q$ be a point on the hyperbola $x^2 - 4y^2 = 1.$ Find the smallest possible distance $PQ.$",
"role": "user"
},
{
"content": "The ellipse is given by the equation $\\frac{x^2}{4} + y^2 = 1$, and the hyperbola is give... |
On the Cartesian plane, the point $A(a,b)$ is the center of a circle with radius $r$, and the point $B(c,d)$ is the center of a circle with radius $R$. If $A$ is moved vertically upwards 8 units and horizontally to the right 2 units, and $B$ is moved vertically downwards 2 units and horizontally to the left 10 units, t... | To find the maximum distance between the two points of intersection, we need to consider the position of the circles after moving $A$ and $B$.
Initially:
- Circle centered at $A$ has equation $(x - a)^2 + (y - b)^2 = r^2$.
- Circle centered at $B$ has equation $(x - c)^2 + (y - d)^2 = R^2$.
After moving $A$ and $B$:
... | 2\sqrt{61} | augmented_math | [
{
"content": "On the Cartesian plane, the point $A(a,b)$ is the center of a circle with radius $r$, and the point $B(c,d)$ is the center of a circle with radius $R$. If $A$ is moved vertically upwards 8 units and horizontally to the right 2 units, and $B$ is moved vertically downwards 2 units and horizontally t... |
Given that $g(x^2) = x - 1$ for all $x \geq 0$, find the value of $g^{-1}(g(4) + g(16))$. | To find $g^{-1}(g(4) + g(16))$, we need to follow these steps:
1. Find $g(4)$ and $g(16)$ using the given function.
2. Add the results of $g(4)$ and $g(16)$.
3. Find the inverse of the result from step 2.
Given $g(x^2) = x - 1$, we can substitute $x^2 = 4$ and $x^2 = 16$ into the function.
1. For $x^2 = 4$, we have ... | 25 | augmented_math | [
{
"content": "Given that $g(x^2) = x - 1$ for all $x \\geq 0$, find the value of $g^{-1}(g(4) + g(16))$.",
"role": "user"
},
{
"content": "To find $g^{-1}(g(4) + g(16))$, we need to follow these steps:\n\n1. Find $g(4)$ and $g(16)$ using the given function.\n2. Add the results of $g(4)$ and $g(16)$.... |
Consider the function $g(x) = \frac{x^2}{e^x - e^{-x}}$. Determine whether this function is even, odd, or neither. | To determine whether the function $g(x)$ is even, odd, or neither, we need to examine $g(-x)$ and compare it with $g(x)$.
For an even function: $g(-x) = g(x)$
For an odd function: $g(-x) = -g(x)$
Let's find $g(-x)$:
\[ g(-x) = \frac{(-x)^2}{e^{-x} - e^{x}} \]
Simplify the expression:
\[ g(-x) = \frac{x^2}{e^{-x} - e... | \text{odd} | augmented_math | [
{
"content": "Consider the function $g(x) = \\frac{x^2}{e^x - e^{-x}}$. Determine whether this function is even, odd, or neither.",
"role": "user"
},
{
"content": "To determine whether the function $g(x)$ is even, odd, or neither, we need to examine $g(-x)$ and compare it with $g(x)$.\n\nFor an even... |
Consider the function $f(x) = \sin x + \sin 2x + \sin 3x$. Determine the smallest positive value of $x$ for which $f(x) = 0$. | To find the smallest positive value of $x$ for which $f(x) = 0$, we need to solve the equation:
\[ \sin x + \sin 2x + \sin 3x = 0 \]
Use the sum-to-product identity for sine:
\[ \sin A + \sin B = 2 \sin \left( \frac{A + B}{2} \right) \cos \left( \frac{A - B}{2} \right) \]
Applying this identity to the equation, we ge... | \frac{\pi}{2} | augmented_math | [
{
"content": "Consider the function $f(x) = \\sin x + \\sin 2x + \\sin 3x$. Determine the smallest positive value of $x$ for which $f(x) = 0$.",
"role": "user"
},
{
"content": "To find the smallest positive value of $x$ for which $f(x) = 0$, we need to solve the equation:\n\\[ \\sin x + \\sin 2x + \... |
The diagram shows the number of fish in a tank as a function of time. The green fish are a species that is constantly being introduced into the tank, and the blue fish are a species that is being introduced at the same rate, but only after a 4-month delay.
[asy]
size(200);
draw((0,0)--(8,0)--(8,50)--(0,50)--cycle);
dr... | The green fish population increases at a rate of 20 fish per month for the first 3 months, reaching a maximum of 50 fish. It then decreases at a rate of 10 fish per month.
The blue fish population increases at a rate of 10 fish per month, starting from the fourth month.
To find when the number of blue fish exceeds th... | \text{June} | augmented_math | [
{
"content": "The diagram shows the number of fish in a tank as a function of time. The green fish are a species that is constantly being introduced into the tank, and the blue fish are a species that is being introduced at the same rate, but only after a 4-month delay.\n\n[asy]\nsize(200);\ndraw((0,0)--(8,0)--... |
Solve the inequality
$$\frac{x^2-2x-3}{x^2-4} \geq 0.$$ | To solve the inequality, we first need to factor the numerator and denominator:
\[ \frac{x^2-2x-3}{x^2-4} = \frac{(x-3)(x+1)}{(x-2)(x+2)} \]
Next, we find the critical points by setting each factor equal to zero:
\[ x - 3 = 0 \Rightarrow x = 3 \]
\[ x + 1 = 0 \Rightarrow x = -1 \]
\[ x - 2 = 0 \Rightarrow x = 2 \]
\[ ... | (-\infty, -2) \cup [-1, 2) \cup [3, \infty) | augmented_math | [
{
"content": "Solve the inequality\n\n$$\\frac{x^2-2x-3}{x^2-4} \\geq 0.$$",
"role": "user"
},
{
"content": "To solve the inequality, we first need to factor the numerator and denominator:\n\\[ \\frac{x^2-2x-3}{x^2-4} = \\frac{(x-3)(x+1)}{(x-2)(x+2)} \\]\n\nNext, we find the critical points by setti... |
A five-digit palindrome is a number that reads the same forward and backward, such as 12321. If a five-digit palindrome is chosen at random from the set of all five-digit palindromes, what is the probability that it is divisible by 3? | ## Step 1: Understand what makes a five-digit number a palindrome
A five-digit palindrome reads the same forward and backward, e.g., 12321.
## Step 2: Determine the total number of five-digit palindromes
The first digit can be any number from 1 to 9 (since a five-digit number cannot start with 0), the second digit can... | \frac{1}{3} | augmented_math | [
{
"content": "A five-digit palindrome is a number that reads the same forward and backward, such as 12321. If a five-digit palindrome is chosen at random from the set of all five-digit palindromes, what is the probability that it is divisible by 3?",
"role": "user"
},
{
"content": "## Step 1: Unders... |
Find the sum of the nonreal roots of $x^4-4x^3+6x^2-4x=2005$, given that one of the real roots is a rational number. | Observe that the left side of the equation is almost the expansion of $(x-1)^4$. Let's add $1$ to both sides to see how close we get to $(x-1)^4$:
\begin{align*}
x^4-4x^3+6x^2-4x+1 &= 2005 + 1 \\
(x-1)^4 &= 2006
\end{align*}
Taking the fourth root of both sides:
\[ x-1 = \sqrt[4]{2006}, i\sqrt[4]{2006}, -\sqrt[4]{2006... | 2 | augmented_math | [
{
"content": "Find the sum of the nonreal roots of $x^4-4x^3+6x^2-4x=2005$, given that one of the real roots is a rational number.",
"role": "user"
},
{
"content": "Observe that the left side of the equation is almost the expansion of $(x-1)^4$. Let's add $1$ to both sides to see how close we get to... |
On a tough week, Haji's mother sells goods worth $800, which is half the amount she sells on a good week. What's the total amount of money she makes if she has 5 good weeks and 3 tough weeks? | The amount Haji's mother sells on a good week is $800 * 2 = 1600$.
The amount she sells on 5 good weeks is $1600 * 5 = 8000$.
The amount she sells on 3 tough weeks is $800 * 3 = 2400$.
So the total amount of money she makes is $8000 + 2400 = 10400$.
The total amount of money she makes is \boxed{10400} dollars. | 10400 | gsm8k | [
{
"content": "On a tough week, Haji's mother sells goods worth $800, which is half the amount she sells on a good week. What's the total amount of money she makes if she has 5 good weeks and 3 tough weeks?",
"role": "user"
},
{
"content": "The amount Haji's mother sells on a good week is $800 * 2 = ... |
Let $f(3x+2) = 5x - 3$. For what value of $x$ is $f(x) = 2x$? | We start by substituting $(3x + 2)$ for $x$ in the given equation $f(3x+2) = 5x - 3$ to express $f$ in terms of its argument.
Given that $f(3x+2) = 5x - 3$, we want to find the expression for $f(x)$.
First, solve $3x + 2 = y$ for $x$:
\[ 3x + 2 = y \Rightarrow 3x = y - 2 \Rightarrow x = \frac{y - 2}{3} \]
Now substi... | -19 | augmented_math | [
{
"content": "Let $f(3x+2) = 5x - 3$. For what value of $x$ is $f(x) = 2x$?",
"role": "user"
},
{
"content": "We start by substituting $(3x + 2)$ for $x$ in the given equation $f(3x+2) = 5x - 3$ to express $f$ in terms of its argument.\n\nGiven that $f(3x+2) = 5x - 3$, we want to find the expression... |
Let $f(x) = x^3 + ax^2 + bx + c$ be a polynomial with integer coefficients, where $a$, $b$, and $c$ are distinct. How many different remainders can $f(x)$ leave upon division by $120$, as $x$ ranges over all positive integers that are not divisible by $2$, $3$, or $5$? | Since $x$ is not divisible by $2$, $3$, or $5$, we can write $x = 30k + r$, where $k$ is a positive integer and $r \in \{1, 7, 11, 13, 17, 19, 23, 29\}$.
We need to find the remainder of $f(x)$ when divided by $120$.
First, consider $x^3$ modulo $120$:
\[ x^3 \equiv (30k + r)^3 \equiv 30^3k^3 + 3 \cdot 30^2k^2r + 3 \... | 8 | augmented_math | [
{
"content": "Let $f(x) = x^3 + ax^2 + bx + c$ be a polynomial with integer coefficients, where $a$, $b$, and $c$ are distinct. How many different remainders can $f(x)$ leave upon division by $120$, as $x$ ranges over all positive integers that are not divisible by $2$, $3$, or $5$?",
"role": "user"
},
... |
A round pizza is cut into $16$ equal-sized slices. Some of the slices are topped with pepperoni, while the rest are topped with mushrooms. If $3/4$ of the slices with pepperoni are adjacent to a slice with mushrooms, and $1/2$ of the slices with mushrooms are adjacent to a slice with pepperoni, how many slices of each ... | Assume that there are $p$ slices with pepperoni and $m$ slices with mushrooms. Since the pizza is cut into $16$ equal-sized slices, we have:
\[ p + m = 16 \]
The statement that $3/4$ of the slices with pepperoni are adjacent to a slice with mushrooms means that $3/4$ of the $p$ slices with pepperoni are adjacent to a ... | 8 | augmented_math | [
{
"content": "A round pizza is cut into $16$ equal-sized slices. Some of the slices are topped with pepperoni, while the rest are topped with mushrooms. If $3/4$ of the slices with pepperoni are adjacent to a slice with mushrooms, and $1/2$ of the slices with mushrooms are adjacent to a slice with pepperoni, ho... |
Solve the equation
$\left(\frac{x+5}{3}\right)^2+2=5.$ | \begin{align*}
\left(\frac{x+5}{3}\right)^2+2=5 \quad &\Rightarrow \quad \left(\frac{x+5}{3}\right)^2=3\\
&\Rightarrow \quad \frac{x+5}{3}=\pm\sqrt{3}\\
&\Rightarrow \quad x+5=3(\pm\sqrt{3})\\
&\Rightarrow \quad x+5=\pm3\sqrt{3}\\
&\Rightarrow \quad x=-5\pm3\sqrt{3}\\
&\Rightarrow \quad x=-5-3\sqrt{3} \text{ or } x=-5+... | -5\pm3\sqrt{3} | augmented_math | [
{
"content": "Solve the equation \n$\\left(\\frac{x+5}{3}\\right)^2+2=5.$",
"role": "user"
},
{
"content": "\\begin{align*}\n\\left(\\frac{x+5}{3}\\right)^2+2=5 \\quad &\\Rightarrow \\quad \\left(\\frac{x+5}{3}\\right)^2=3\\\\\n&\\Rightarrow \\quad \\frac{x+5}{3}=\\pm\\sqrt{3}\\\\\n&\\Rightarrow \\q... |
Samantha owns a farm that produces strawberries and blueberries. She can harvest her strawberries every 4 months and sell the harvest for $80. She can harvest her blueberries every 6 months and sell the harvest for $40. How much money can Samantha earn every year? | Let's calculate the total amount of money Samantha can earn from strawberries and blueberries in a year.
For strawberries, she can harvest every 4 months, which means she can harvest 3 times a year (12 months / 4 months = 3). Each harvest earns her $80, so she can earn a total of $80 x 3 = $240 per year from strawberr... | 320 | augmented_gsm8k | [
{
"content": "Samantha owns a farm that produces strawberries and blueberries. She can harvest her strawberries every 4 months and sell the harvest for $80. She can harvest her blueberries every 6 months and sell the harvest for $40. How much money can Samantha earn every year?",
"role": "user"
},
{
... |
A pyramid with a square base is filled with sand to 2/3 of its height. What percent of the pyramid's volume is occupied by the sand? Express your answer as a decimal to the nearest thousandth. | The volume of the pyramid is given by the formula:
\[ V = \frac{1}{3}Bh \]
where $B$ is the area of the base and $h$ is the height.
Let's denote the area of the base as $B$ and the height as $h$. Then the volume of the pyramid is $V = \frac{1}{3}Bh$.
The volume of the sand-filled portion is also a pyramid with the sa... | 66.667 | augmented_math | [
{
"content": "A pyramid with a square base is filled with sand to 2/3 of its height. What percent of the pyramid's volume is occupied by the sand? Express your answer as a decimal to the nearest thousandth.",
"role": "user"
},
{
"content": "The volume of the pyramid is given by the formula:\n\\[ V =... |
Problem:
Let $p$ be a prime number such that $p^2 - 2$ is a perfect square. Find the number of positive divisors of $p^3 + 1$.
(No solution provided) | ## Step 1: Understand the given problem and identify key information.
We need to find the number of positive divisors of $p^3 + 1$, given that $p$ is a prime number and $p^2 - 2$ is a perfect square.
## Step 2: Recognize the relationship between $p^2 - 2$ being a perfect square and possible values of $p$.
Since $p^2 -... | 6 | augmented_math | [
{
"content": "Problem:\n\nLet $p$ be a prime number such that $p^2 - 2$ is a perfect square. Find the number of positive divisors of $p^3 + 1$.\n\n(No solution provided)",
"role": "user"
},
{
"content": "## Step 1: Understand the given problem and identify key information.\nWe need to find the numbe... |
A palindrome is a sequence that reads the same forward and backward. For example, the sequence HTTH is a palindrome, but THTH is not. How many distinct palindromes are possible if a coin is flipped eight times and the sequence of heads and tails is recorded? | Let's analyze the problem step by step:
1. The length of the palindrome is 8.
2. A palindrome reads the same forward and backward. This means that the first half of the sequence is the same as the second half in reverse order.
3. Since the length is 8, the first half will have 4 characters, and the second half will ... | 16 | augmented_math | [
{
"content": "A palindrome is a sequence that reads the same forward and backward. For example, the sequence HTTH is a palindrome, but THTH is not. How many distinct palindromes are possible if a coin is flipped eight times and the sequence of heads and tails is recorded?",
"role": "user"
},
{
"cont... |
Find the value of $x$ that satisfies the equation $\lfloor x \rfloor + \lceil x \rceil = 17$. | The equation is
\[ \lfloor x \rfloor + \lceil x \rceil = 17 \]
Since $\lfloor x \rfloor$ and $\lceil x \rceil$ are the floor and ceiling functions, they will be integers.
Let's consider two cases:
Case 1: $x$ is an integer.
In this case, $\lfloor x \rfloor = \lceil x \rceil = x$. The equation becomes:
\[ 2x = 17 \R... | 8.5 | augmented_math | [
{
"content": "Find the value of $x$ that satisfies the equation $\\lfloor x \\rfloor + \\lceil x \\rceil = 17$.",
"role": "user"
},
{
"content": "The equation is\n\\[ \\lfloor x \\rfloor + \\lceil x \\rceil = 17 \\]\n\nSince $\\lfloor x \\rfloor$ and $\\lceil x \\rceil$ are the floor and ceiling fun... |
In the convex hexagon $ABCDEF$, all interior angles are equal and the lengths of the sides are consecutive positive integers. If the length of side $AB$ is 5, what is the length of side $CD$? | We know that the sum of the interior angles of a hexagon is $(6 - 2) \times 180 = 720$ degrees.
Since all interior angles of the hexagon $ABCDEF$ are equal, we can find each interior angle by dividing the sum of the interior angles by the number of angles:
\[ \text{Each interior angle} = 720 : 6 = 120 \text{ degrees} ... | 7 | augmented_math | [
{
"content": "In the convex hexagon $ABCDEF$, all interior angles are equal and the lengths of the sides are consecutive positive integers. If the length of side $AB$ is 5, what is the length of side $CD$?",
"role": "user"
},
{
"content": "We know that the sum of the interior angles of a hexagon is... |
Emily's laptop has 80 gigabytes of storage, with 45 gigabytes already occupied. She plans to uninstall a program that takes up 8 gigabytes of space and then install a new game that requires 15 gigabytes. If she decides to transfer all her files to a new laptop with a 120-gigabyte hard drive, how many free gigabytes wil... | The laptop originally had 80 GB of space with 45 GB occupied.
So the amount of free space on the original laptop was $80 - 45 = 35$ GB.
Uninstalling the program would free up 8 GB. So the amount of free space now becomes $35 + 8 = 43$ GB.
Installing the new game will take up 15 GB. So the amount of free space now becom... | 68 | augmented_gsm8k | [
{
"content": "Emily's laptop has 80 gigabytes of storage, with 45 gigabytes already occupied. She plans to uninstall a program that takes up 8 gigabytes of space and then install a new game that requires 15 gigabytes. If she decides to transfer all her files to a new laptop with a 120-gigabyte hard drive, how m... |
In the figure, circle $O$ has radius 6 units. Chord $CD$ has length 8 units and is parallel to segment $KB$. If $KA$ = 12 units and points $K$, $A$, $O$, and $B$ are collinear, what is the length of the line segment $KD$? Express your answer in simplest radical form. | To find the length of $KD$, we can use the properties of the circle and the given information.
Since $CD$ is parallel to $KB$, we can draw a line from $O$ that is perpendicular to both $CD$ and $KB$. This line intersects $CD$ at its midpoint $M$. The line also intersects $KB$ at a point $N$.
Now we have right triangl... | 4\sqrt{19} | augmented_math | [
{
"content": "In the figure, circle $O$ has radius 6 units. Chord $CD$ has length 8 units and is parallel to segment $KB$. If $KA$ = 12 units and points $K$, $A$, $O$, and $B$ are collinear, what is the length of the line segment $KD$? Express your answer in simplest radical form.",
"role": "user"
},
{
... |
In a game of darts, Alex throws a dart at a circular target with a diameter of 20 cm. The target has a concentric inner circle with a diameter of 10 cm. If Alex's probability of hitting the target is 0.8, what is the probability that he will hit the inner circle on his next throw, assuming that his throws are independe... | The probability that Alex hits the inner circle is the ratio of the area of the inner circle to the area of the whole target.
Let's find the areas of the circles:
\[ \text{Area of the whole target} = \pi r^2 = \pi (10)^2 = 100\pi \]
\[ \text{Area of the inner circle} = \pi r^2 = \pi (5)^2 = 25\pi \]
The probability o... | 0.2 | augmented_math | [
{
"content": "In a game of darts, Alex throws a dart at a circular target with a diameter of 20 cm. The target has a concentric inner circle with a diameter of 10 cm. If Alex's probability of hitting the target is 0.8, what is the probability that he will hit the inner circle on his next throw, assuming that hi... |
Find the number of positive three-digit integers that are multiples of both 4 and 9 and have a sum of digits equal to 18. | To solve this problem, we need to consider the following conditions:
1. The number is a three-digit positive integer.
2. The number is a multiple of both 4 and 9.
3. The sum of the digits of the number is 18.
First, let's consider the second condition. Since the number is a multiple of both 4 and 9, it must be a multi... | 10 | augmented_math | [
{
"content": "Find the number of positive three-digit integers that are multiples of both 4 and 9 and have a sum of digits equal to 18.",
"role": "user"
},
{
"content": "To solve this problem, we need to consider the following conditions:\n1. The number is a three-digit positive integer.\n2. The num... |
Emily travels to school by walking two days a week, taking the train four days a week, and getting a ride from her mom one day a week. Walking takes her 45 minutes. The train takes 15 minutes longer than walking because it stops at multiple stations. Her mom's ride is the fastest, reducing her walking time by half. How... | Let's compute the time spent in each of the three modes of commuting to school.
Time spent walking to school is 45 minutes and she walks two days a week. So she spends 2 * 45 = 90 minutes walking to school.
Time spent on the train is 15 minutes more than walking. So it is 45 + 15 = 60 minutes and she takes the train fo... | 352.5 | augmented_gsm8k | [
{
"content": "Emily travels to school by walking two days a week, taking the train four days a week, and getting a ride from her mom one day a week. Walking takes her 45 minutes. The train takes 15 minutes longer than walking because it stops at multiple stations. Her mom's ride is the fastest, reducing her wal... |
An equilateral triangle is inscribed in a circle and its probability of landing on a point inside this triangle when randomly picked is $\frac{1}{2}$. What is the ratio of the side length of the equilateral triangle to the radius of the circle? Express your answer in simplest form. | ## Step 1: Understand the problem and the given conditions
We are given an equilateral triangle inscribed in a circle. The probability of randomly picking a point inside the triangle is $\frac{1}{2}$. This implies that the area of the triangle is half the area of the circle.
## Step 2: Recall the formula for the area ... | \sqrt{\frac{2\pi}{\sqrt{3}}} | augmented_math | [
{
"content": "An equilateral triangle is inscribed in a circle and its probability of landing on a point inside this triangle when randomly picked is $\\frac{1}{2}$. What is the ratio of the side length of the equilateral triangle to the radius of the circle? Express your answer in simplest form.",
"role": ... |
Let $a$, $b$, and $c$ be positive integers such that $a$ and $b$ are relatively prime. Find the smallest possible value of $c$ such that $a+c$ and $b+c$ share a common factor greater than 1. | We are looking for the smallest $c$ such that $a+c$ and $b+c$ share a common factor greater than 1. Since $a$ and $b$ are relatively prime, they have no common factors greater than 1.
Consider the smallest prime number, which is 2. If $a$ and $b$ are both odd, then adding 1 to each would make them both even, thus shar... | 1 | augmented_math | [
{
"content": "Let $a$, $b$, and $c$ be positive integers such that $a$ and $b$ are relatively prime. Find the smallest possible value of $c$ such that $a+c$ and $b+c$ share a common factor greater than 1.",
"role": "user"
},
{
"content": "We are looking for the smallest $c$ such that $a+c$ and $b+c$... |
Find the least positive integer $n$ such that $5n$ is divisible by 8, $3n$ is divisible by 7, and $11n$ is divisible by 5. | First, find the conditions for $n$ to satisfy each divisibility requirement:
1. $5n$ is divisible by 8:
Since 5 and 8 are coprime (their greatest common divisor is 1), $n$ must be divisible by 8.
2. $3n$ is divisible by 7:
Since 3 and 7 are coprime, $n$ must be divisible by 7.
3. $11n$ is divisible by 5:
Si... | 280 | augmented_math | [
{
"content": "Find the least positive integer $n$ such that $5n$ is divisible by 8, $3n$ is divisible by 7, and $11n$ is divisible by 5.",
"role": "user"
},
{
"content": "First, find the conditions for $n$ to satisfy each divisibility requirement:\n\n1. $5n$ is divisible by 8:\n Since 5 and 8 are ... |
Let $f(x) = x^5 + x^4 + x^3 + x^2 + x + 1,$ and let $g(x)$ be the remainder when $f(x)$ is divided by $x^3 - 2x^2 - x + 1$. Find the remainder when $|g(-1)|$ is divided by 17. | We need to perform polynomial division to find $g(x)$. Divide $f(x) = x^5 + x^4 + x^3 + x^2 + x + 1$ by $x^3 - 2x^2 - x + 1$.
\[ \begin{array}{c} x^2 + 3x + 4 \\ x^3 - 2x^2 - x + 1 \enclose{longdiv}{x^5 + x^4 + x^3 + x^2 + x + 1} \\ \underline{x^5 - 2x^4 - x^3 + x^2} \\ 3x^4 + 2x^3 + x \\ \underline{3x^4 - 6x^3 - 3x^2... | 6 | augmented_math | [
{
"content": "Let $f(x) = x^5 + x^4 + x^3 + x^2 + x + 1,$ and let $g(x)$ be the remainder when $f(x)$ is divided by $x^3 - 2x^2 - x + 1$. Find the remainder when $|g(-1)|$ is divided by 17.",
"role": "user"
},
{
"content": "We need to perform polynomial division to find $g(x)$. Divide $f(x) = x^5 + ... |
In a certain coding system, each letter of the alphabet is assigned a unique three-digit code, with the first digit representing the letter's position in the alphabet (A=1, B=2,..., Z=26). How many more occurrences of the digit 2 are there in the codes for the letters A to Z than occurrences of the digit 9? | We know that the digit 2 occurs in the codes for the letters B, K, T, and the digit 9 occurs in the codes for the letters I, R.
The digit 2 appears in the codes for the letters B, K, T, which are the 2nd, 11th, and 20th letters of the alphabet, respectively.
The digit 9 appears in the codes for the letters I, R, whic... | 1 | augmented_math | [
{
"content": "In a certain coding system, each letter of the alphabet is assigned a unique three-digit code, with the first digit representing the letter's position in the alphabet (A=1, B=2,..., Z=26). How many more occurrences of the digit 2 are there in the codes for the letters A to Z than occurrences of th... |
Given a regular hexagon $ABCDEF$ with sides of length $4$, let $G$, $H$, $I$, $J$, $K$, and $L$ be the midpoints of sides $AB$, $BC$, $CD$, $DE$, $EF$, and $FA$, respectively. Additionally, let $M$ be the intersection of diagonals $AC$ and $DF$, and let $N$ be the intersection of diagonals $BE$ and $FG$. If the area of... | ## Step 1: First, we need to determine the areas of the pentagons ACGJM and BHINL.
The area of a regular hexagon can be found using the formula: Area = (3 * sqrt(3)) / 2 * side^2, where the side is the length of one side of the hexagon.
## Step 2: To find the areas of the pentagons ACGJM and BHINL, we need to divide t... | 1 | augmented_math | [
{
"content": "Given a regular hexagon $ABCDEF$ with sides of length $4$, let $G$, $H$, $I$, $J$, $K$, and $L$ be the midpoints of sides $AB$, $BC$, $CD$, $DE$, $EF$, and $FA$, respectively. Additionally, let $M$ be the intersection of diagonals $AC$ and $DF$, and let $N$ be the intersection of diagonals $BE$ an... |
The bookstore can buy a novel for $5 and sell it for $12. The owner wants to use the profits from novel sales to attend a conference, which costs $250. After two weeks, she is 30% short of the registration fee. How many novels did she sell during those two weeks? | Let's break it down step by step:
The owner wants to attend a conference that costs $250.
After two weeks, she is 30% short of the registration fee, which means she has 70% of the fee.
So, the amount of money she has is $250 * 70 / 100 = 17500 / 100 = 175$.
This money comes from the profit of selling novels.
The prof... | 25 | augmented_gsm8k | [
{
"content": "The bookstore can buy a novel for $5 and sell it for $12. The owner wants to use the profits from novel sales to attend a conference, which costs $250. After two weeks, she is 30% short of the registration fee. How many novels did she sell during those two weeks?",
"role": "user"
},
{
... |
Emily's dad asks her to buy cooking oil. He gives her $15 and instructs her to spend the entire amount on the best deal for cooking oil. She finds a 12 oz bottle for $2, an 18 oz bottle for $3, a 32 oz bottle for $5, a 64 oz bottle for $8, and a 128 oz bottle for $12. How many bottles of cooking oil will Emily buy? | Since Emily wants to spend the entire $15, she will look for the combination of bottles that will allow her to spend the most money without going over $15.
First, she will buy the 128 oz bottle for $12. This leaves her with $15 - $12 = $3.
Next, she will buy the 18 oz bottle for $3. This leaves her with $3 - $3 = $0.... | 2 | augmented_gsm8k | [
{
"content": "Emily's dad asks her to buy cooking oil. He gives her $15 and instructs her to spend the entire amount on the best deal for cooking oil. She finds a 12 oz bottle for $2, an 18 oz bottle for $3, a 32 oz bottle for $5, a 64 oz bottle for $8, and a 128 oz bottle for $12. How many bottles of cooking o... |
Find the value of $N$ if $30\%$ of $N$ is equal to the difference between $12\%$ of 2010 and $8\%$ of 2008. | Let's solve this step by step.
First, calculate $12\%$ of 2010:
\[ 12\% \text{ of } 2010 = 0.12 \times 2010 = 241.2 \]
Next, calculate $8\%$ of 2008:
\[ 8\% \text{ of } 2008 = 0.08 \times 2008 = 160.64 \]
Now find the difference between the two:
\[ 241.2 - 160.64 = 80.56 \]
This difference is equal to $30\%$ of $N$... | 268.53 | augmented_math | [
{
"content": "Find the value of $N$ if $30\\%$ of $N$ is equal to the difference between $12\\%$ of 2010 and $8\\%$ of 2008.",
"role": "user"
},
{
"content": "Let's solve this step by step.\n\nFirst, calculate $12\\%$ of 2010:\n\\[ 12\\% \\text{ of } 2010 = 0.12 \\times 2010 = 241.2 \\]\n\nNext, cal... |
Let
\[\mathbf{A} = \begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix}.\]
Compute $\mathbf{A}^{100}.$
Another problem inspired by this one:
Find the smallest positive integer $n$ such that
\[\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} + \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 &... | Note that $\mathbf{A}^3 = \mathbf{I}$, the $3 \times 3$ identity matrix, since
\[\mathbf{A}^3 = \begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix} \begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix} \begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix} = \begin{pmatrix} 1 & 0 & ... | \begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix} | augmented_math | [
{
"content": "Let\n\\[\\mathbf{A} = \\begin{pmatrix} 0 & 0 & 1 \\\\ 1 & 0 & 0 \\\\ 0 & 1 & 0 \\end{pmatrix}.\\]\nCompute $\\mathbf{A}^{100}.$\n\nAnother problem inspired by this one:\n\nFind the smallest positive integer $n$ such that\n\\[\\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1 \\end{pmatrix} +... |
Mira had just been given some marbles by her friend. On the way home, she added exactly twice as many marbles to the ones she already had, and then lost exactly one-third of all the marbles. What fraction of the marbles that she received from her friend were lost after Mira added and lost some marbles? Express your ans... | Let $x$ be the initial number of marbles Mira had.
After adding twice as many marbles, she has $x + 2x = 3x$ marbles.
Then she loses one-third of all the marbles, which is $\frac{1}{3}(3x) = x$ marbles.
So, the number of marbles lost is $x$, which is the same as the initial number of marbles she received from her fr... | \frac{1}{3} | augmented_math | [
{
"content": "Mira had just been given some marbles by her friend. On the way home, she added exactly twice as many marbles to the ones she already had, and then lost exactly one-third of all the marbles. What fraction of the marbles that she received from her friend were lost after Mira added and lost some mar... |
Find all three-digit palindromic numbers that are multiples of 7 and have exactly one digit equal to 0. What is the sum of all such numbers? | A three-digit palindromic number with exactly one digit equal to 0 must be of the form $aba$, where $a$ and $b$ are digits and $a \neq 0$. For the number to be a multiple of 7, the value of $aba$ must be divisible by 7.
Since $aba = 100a + 10b + a = 101a + 10b$, we can write:
\[ 101a + 10b \equiv 0 \pmod{7} \]
Notice... | 707 | augmented_math | [
{
"content": "Find all three-digit palindromic numbers that are multiples of 7 and have exactly one digit equal to 0. What is the sum of all such numbers?",
"role": "user"
},
{
"content": "A three-digit palindromic number with exactly one digit equal to 0 must be of the form $aba$, where $a$ and $b$... |
Let $a$, $b$, $c$, and $d$ be the roots of $x^4-x^3+2x^2-x+1=0$.
Find the smallest possible value of $\sin\left(\tan^{-1} a+\tan^{-1} b+\tan^{-1} c+\tan^{-1} d\right)$. | The given equation is $x^4-x^3+2x^2-x+1=0$. Let's start by factorizing the polynomial.
Divide the equation by $x^2$:
\[ x^2 - x + 2 - \frac{1}{x} + \frac{1}{x^2} = 0 \]
Rearrange to group terms symmetrically:
\[ \left(x^2 + \frac{1}{x^2}\right) - \left(x - \frac{1}{x}\right) + 2 = 0 \]
Observe that this can be rewri... | 0 | augmented_math | [
{
"content": "Let $a$, $b$, $c$, and $d$ be the roots of $x^4-x^3+2x^2-x+1=0$.\nFind the smallest possible value of $\\sin\\left(\\tan^{-1} a+\\tan^{-1} b+\\tan^{-1} c+\\tan^{-1} d\\right)$.",
"role": "user"
},
{
"content": "The given equation is $x^4-x^3+2x^2-x+1=0$. Let's start by factorizing the ... |
A certain three-digit integer $ABC$ can be expressed in base 10 and also in base 8. If it is known that $A+B+C=7$ and that the base 8 representation is obtained from the base 10 representation by permuting the digits, find the smallest possible value of $ABC$ in base 10. | We have a three-digit integer $ABC$ that can be expressed in both base 10 and base 8, where $A+B+C=7$. Since the base 8 representation is obtained from the base 10 representation by permuting the digits, we can write the base 8 representation as $A'B'C'$.
In base 10, the value of $ABC$ is $100A + 10B + C$.
In base 8,... | 124 | augmented_math | [
{
"content": "A certain three-digit integer $ABC$ can be expressed in base 10 and also in base 8. If it is known that $A+B+C=7$ and that the base 8 representation is obtained from the base 10 representation by permuting the digits, find the smallest possible value of $ABC$ in base 10.",
"role": "user"
},
... |
Given the functions $h(x) = x^2 + 1$ and $j(x) = \sqrt{x - 1},$ and their inverses $h^{-1}$ and $j^{-1},$ calculate \[j(h^{-1}(j^{-1}(h(j(7))))).\] | First, calculate $j(7)$:
\[ j(7) = \sqrt{7 - 1} = \sqrt{6} \]
Now, calculate $h(j(7))$:
\[ h(j(7)) = h(\sqrt{6}) = (\sqrt{6})^2 + 1 = 7 \]
Next, calculate $j^{-1}(h(j(7)))$:
\[ j^{-1}(h(j(7))) = j^{-1}(7) \]
To find $j^{-1}(7)$, we need to find the value of $x$ such that $j(x) = 7$:
\[ j(x) = \sqrt{x - 1} = 7 \Right... | \sqrt{6} | augmented_math | [
{
"content": "Given the functions $h(x) = x^2 + 1$ and $j(x) = \\sqrt{x - 1},$ and their inverses $h^{-1}$ and $j^{-1},$ calculate \\[j(h^{-1}(j^{-1}(h(j(7))))).\\]",
"role": "user"
},
{
"content": "First, calculate $j(7)$:\n\\[ j(7) = \\sqrt{7 - 1} = \\sqrt{6} \\]\n\nNow, calculate $h(j(7))$:\n\\[ ... |
At the beach, Sophia saw surfers riding kayaks and paddleboats. Kayaks have one paddle and paddleboats have two paddles. 8 teenagers were riding kayaks and 12 adults were riding paddleboats. How many paddles did Sophia see at the beach? | Since each kayak has 1 paddle and there were 8 teenagers riding kayaks, the total number of paddles from kayaks is 8 * 1 = 8.
Since each paddleboat has 2 paddles and there were 12 adults riding paddleboats, the total number of paddles from paddleboats is 12 * 2 = 24.
Thus, the total number of paddles Sophia saw is 8 + ... | 32 | augmented_gsm8k | [
{
"content": "At the beach, Sophia saw surfers riding kayaks and paddleboats. Kayaks have one paddle and paddleboats have two paddles. 8 teenagers were riding kayaks and 12 adults were riding paddleboats. How many paddles did Sophia see at the beach?",
"role": "user"
},
{
"content": "Since each kaya... |
Given that $\triangle ABC$ is a right triangle whose legs measure $x$ and $y$ and whose hypotenuse measures $z$, the volume of the rectangular solid with edges $x,y,$ and $z$ is 162 cubic centimeters. Find the value of $x^2+y^2$. | ## Step 1: Recall the Pythagorean Theorem for a right triangle
The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. In the given triangle $\triangle ABC$, this means $z^2 = x^2 + y^2$.
## Step 2: Ex... | 145 | augmented_math | [
{
"content": "Given that $\\triangle ABC$ is a right triangle whose legs measure $x$ and $y$ and whose hypotenuse measures $z$, the volume of the rectangular solid with edges $x,y,$ and $z$ is 162 cubic centimeters. Find the value of $x^2+y^2$.",
"role": "user"
},
{
"content": "## Step 1: Recall the... |
What is the range of the function $y = \frac{x^3 - 2x^2 - 5x + 6}{x^2 + 1}$? (Express your answer using interval notation.) | To find the range of the function $y = \frac{x^3 - 2x^2 - 5x + 6}{x^2 + 1}$, we need to consider the behavior of the function as $x$ varies.
Since the denominator is always positive ($x^2 + 1 \geq 1$ for all $x$), the sign of the function is determined by the numerator.
Let's rewrite the function in a more manageable... | (-\infty, \infty) | augmented_math | [
{
"content": "What is the range of the function $y = \\frac{x^3 - 2x^2 - 5x + 6}{x^2 + 1}$? (Express your answer using interval notation.)",
"role": "user"
},
{
"content": "To find the range of the function $y = \\frac{x^3 - 2x^2 - 5x + 6}{x^2 + 1}$, we need to consider the behavior of the function... |
A boat travels 30 miles upstream and then returns to its starting point. If the speed of the boat in still water is 10 miles per hour and the speed of the current is 2 miles per hour, how long did the round trip take? | To find the time it took for the round trip, we need to calculate the time spent going upstream and the time spent going downstream.
Speed of the boat in still water = 10 miles per hour
Speed of the current = 2 miles per hour
Upstream speed = boat speed - current speed = 10 - 2 = 8 miles per hour
Downstream speed = b... | 6.25 | augmented_math | [
{
"content": "A boat travels 30 miles upstream and then returns to its starting point. If the speed of the boat in still water is 10 miles per hour and the speed of the current is 2 miles per hour, how long did the round trip take?",
"role": "user"
},
{
"content": "To find the time it took for the r... |
What is the ratio of the area of the region bounded by the graph of the equation $y = x^2$ and the $x$-axis for $0 \leq x \leq 2$ to the area of the region bounded by the graph of the equation $y = x^2$ and the $x$-axis for $0 \leq x \leq 3$? | To find the ratio of the areas, we first need to calculate the area of each region.
The area of the region bounded by the graph of $y = x^2$ and the $x$-axis for $0 \leq x \leq 2$ is
\[ \int_0^2 x^2 \, dx = \left[ \frac{x^3}{3} \right]_0^2 = \frac{2^3}{3} - \frac{0^3}{3} = \frac{8}{3} \]
The area of the region bounde... | \frac{8}{27} | augmented_math | [
{
"content": "What is the ratio of the area of the region bounded by the graph of the equation $y = x^2$ and the $x$-axis for $0 \\leq x \\leq 2$ to the area of the region bounded by the graph of the equation $y = x^2$ and the $x$-axis for $0 \\leq x \\leq 3$?",
"role": "user"
},
{
"content": "To fi... |
At a family reunion, five adults and four children want to take a group photo. Each adult wants to be standing next to at least one child, and each child wants to be standing next to at least two adults. In how many different arrangements can they stand for the photo? | To solve this problem, let's break it down into manageable steps. We need to consider the restrictions:
1. Each adult wants to be standing next to at least one child.
2. Each child wants to be standing next to at least two adults.
Given these conditions, let's analyze the possible arrangements:
- Since each child ne... | 2880 | augmented_math | [
{
"content": "At a family reunion, five adults and four children want to take a group photo. Each adult wants to be standing next to at least one child, and each child wants to be standing next to at least two adults. In how many different arrangements can they stand for the photo?",
"role": "user"
},
{... |
Find the sum of the roots of all polynomials of degree 3 with integer coefficients that have exactly one root equal to 2 and the other two roots distinct and not equal to 2, 3 or 4. | Let the polynomial be $p(x) = a(x - 2)(x - r)(x - s)$, where $r$ and $s$ are the other two roots.
Since the polynomial has integer coefficients, by the Rational Root Theorem, $r$ and $s$ must be rational numbers.
The sum of the roots is $2 + r + s$.
To find $r + s$, we can use Vieta's formulas:
\[ r + s = -\frac{b}{... | 8 | augmented_math | [
{
"content": "Find the sum of the roots of all polynomials of degree 3 with integer coefficients that have exactly one root equal to 2 and the other two roots distinct and not equal to 2, 3 or 4.",
"role": "user"
},
{
"content": "Let the polynomial be $p(x) = a(x - 2)(x - r)(x - s)$, where $r$ and $... |
If $0.8= \frac{y}{186+x}$, find the smallest value of $x$, if $x$ and $y$ are positive integers.
Determine the largest possible value of $n$ for which the inequality $\frac{a}{b} < \frac{a+1}{b+n}$ holds true, where $a$ and $b$ are positive integers and $n$ is a non-negative integer. | We can rewrite the inequality as:
\[ \frac{a}{b} < \frac{a+1}{b+n} \]
Cross-multiply:
\[ a(b+n) < b(a+1) \]
Expand:
\[ ab + an < ab + b \]
Subtract $ab$ from both sides:
\[ an < b \]
Divide by $a$:
\[ n < \frac{b}{a} \]
Since $n$ is an integer, the largest possible value of $n$ is the integer part of $\frac{b}{a}$... | 4 | augmented_math | [
{
"content": "If $0.8= \\frac{y}{186+x}$, find the smallest value of $x$, if $x$ and $y$ are positive integers.\n\nDetermine the largest possible value of $n$ for which the inequality $\\frac{a}{b} < \\frac{a+1}{b+n}$ holds true, where $a$ and $b$ are positive integers and $n$ is a non-negative integer.",
"... |
Lily's aloe vera plant produces 3 offsets 3 times a year. After 5 years, how many offsets will the mother plant have produced? | The mother plant produces 3 offsets, 3 times a year.
So, every year it produces 3 * 3 = 9 offsets.
In 5 years, the mother plant will produce 9 * 5 = 45 offsets.
Thus, after 5 years, the mother plant will have produced \boxed{45} offsets. | 45 | augmented_gsm8k | [
{
"content": "Lily's aloe vera plant produces 3 offsets 3 times a year. After 5 years, how many offsets will the mother plant have produced?",
"role": "user"
},
{
"content": "The mother plant produces 3 offsets, 3 times a year.\nSo, every year it produces 3 * 3 = 9 offsets.\nIn 5 years, the mother p... |
The operation $\diamond$ is defined as $a \diamond b = a^3 \div b$. For how many integer values of $x$ will the value of $5 \diamond x$ be a positive integer? | ## Step 1: Understand the operation $\diamond$ and the condition for $5 \diamond x$ to be a positive integer.
The operation $\diamond$ is defined as $a \diamond b = a^3 \div b$. For $5 \diamond x$ to be a positive integer, $5^3 \div x$ must be a positive integer.
## Step 2: Determine the factors of $5^3$ since $x$ mus... | 4 | augmented_math | [
{
"content": "The operation $\\diamond$ is defined as $a \\diamond b = a^3 \\div b$. For how many integer values of $x$ will the value of $5 \\diamond x$ be a positive integer?",
"role": "user"
},
{
"content": "## Step 1: Understand the operation $\\diamond$ and the condition for $5 \\diamond x$ to ... |
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