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Given that $u$ and $v$ are integers with $0 < v < u$, let $A = (u,v)$, $B$ be the reflection of $A$ across the line $y = x$, $C$ be the reflection of $B$ across the y-axis, $D$ be the reflection of $C$ across the x-axis, and $E$ be the reflection of $D$ across the y-axis. The area of pentagon $ABCDE$ is $451$. What is the value of $u + v$?
|
21
|
MATH
|
{
"level": 5,
"subject": "Geometry",
"type": "MATH_Rephrased"
}
| 71,776 | 71,776 |
If $f(x) = 3x^2 - 7$ and $g(f(4)) = 9$, what is the value of $g(f(-4))$?
|
9
|
MATH
|
{
"level": 3,
"subject": "Algebra",
"type": "MATH_Rephrased"
}
| 4,353 | 4,353 |
What is the value of $\frac{acd}{b}$ if $x=\frac{a+b\sqrt{c}}{d}$ is the simplified form and $\frac{5x}{6}+1=\frac{3}{x}$, where $a$, $b$, $c$, and $d$ are integers?
|
-55
|
MATH
|
{
"level": 5,
"subject": "Algebra",
"type": "MATH_Rephrased"
}
| 16,431 | 16,431 |
Define the operation $a\nabla b = X + b^a$. What is the value of $(1\nabla 2) \nabla 3$?
If we know the answer to the above question is 83, what is the value of unknown variable X?
|
2
|
MATH
|
{
"level": 3,
"subject": "Algebra",
"type": "MATH_FOBAR"
}
| 50,900 | 50,900 |
In the triangle shown with vertices V, W, and X, where VX is equal to the square root of 13 and VW is equal to 3, what is the value of tan V?
|
\frac{2}{3}
|
MATH
|
{
"level": 2,
"subject": "Geometry",
"type": "MATH_Rephrased"
}
| 59,901 | 59,901 |
If rectangle ABCD serves as the base of pyramid PABCD, and AB measures 8 units, BC measures 4 units, PA is perpendicular to AD, PA is perpendicular to AB, and PB measures 17 units, what is the volume of pyramid PABCD?
|
160
|
MATH
|
{
"level": 4,
"subject": "Geometry",
"type": "MATH_Rephrased"
}
| 37,940 | 37,940 |
Let $\mathbf{A} = \begin{pmatrix} 2 & 3 \\ 0 & X \end{pmatrix}.$ Find $\mathbf{A}^{20} - 2 \mathbf{A}^{19}. -1 is the answer. What is the value of unknown variable X?
|
1
|
MATH
|
{
"level": 3,
"subject": "Precalculus",
"type": "MATH_SV"
}
| 119,935 | 119,935 |
What is the total sum of all possible values of M if the product of a number M and six less than M is equal to -5?
|
6
|
MATH
|
{
"level": 3,
"subject": "Algebra",
"type": "MATH_Rephrased"
}
| 7,301 | 7,301 |
You have X dimes and 20 quarters. What percent of the value of your money is in quarters?
If we know the answer to the above question is 50, what is the value of unknown variable X?
|
50
|
MATH
|
{
"level": 2,
"subject": "Algebra",
"type": "MATH_FOBAR"
}
| 53,039 | 53,039 |
Find the simplified form of $\frac{2 + 2i}{-3 + 4i}$ in the form $a + bi$, where $a$ and $b$ are real numbers.
|
\frac{2}{25}-\frac{14}{25}i
|
MATH
|
{
"level": 5,
"subject": "Algebra",
"type": "MATH_Rephrased"
}
| 62,567 | 62,567 |
Lucy has $19$ dollars and $23$ cents. She wants to buy as many popsicles as she can with her money. The popsicles are priced at X dollar and $60$ cents each. She can buy 12 popsicles. What is the value of unknown variable X?
|
1
|
MATH
|
{
"level": 2,
"subject": "Prealgebra",
"type": "MATH_SV"
}
| 5,138 | 5,138 |
Determine the 200th term in the sequence of positive integers formed by excluding only the perfect squares.
|
214
|
MATH
|
{
"level": 3,
"subject": "Number Theory",
"type": "MATH_Rephrased"
}
| 56,387 | 56,387 |
What is the minimum number of homework assignments Noelle needs to complete in order to earn a total of 25 homework points, following the math teacher's requirements?
|
75
|
MATH
|
{
"level": 5,
"subject": "Algebra",
"type": "MATH_Rephrased"
}
| 18,195 | 18,195 |
Rationalize the denominator in the fraction $\frac{4}{\sqrt{108}+2\sqrt{12}+2\sqrt{27}}$ and simplify.
|
\frac{\sqrt{3}}{12}
|
MATH
|
{
"level": 4,
"subject": "Algebra",
"type": "MATH_Rephrased"
}
| 58,681 | 58,681 |
In base 9, a cat has discovered 432 methods to extend each of her nine lives. How many methods are there in base 10?
|
353
|
MATH
|
{
"level": 3,
"subject": "Number Theory",
"type": "MATH_Rephrased"
}
| 49,165 | 49,165 |
Given that $x$, $y$, and $z$ are positive real numbers and $x + y + z = 3$, what is the minimum value of $\frac{4}{x} + \frac{9}{y} + \frac{16}{z}$?
|
27
|
MATH
|
{
"level": 4,
"subject": "Intermediate Algebra",
"type": "MATH_Rephrased"
}
| 9,032 | 9,032 |
In the local frisbee league, teams have 7 members and each of the X teams takes turns hosting tournaments. At each tournament, each team selects two members of that team to be on the tournament committee, except the host team, which selects three members. There are 540 possible 9 member tournament committees. What is the value of unknown variable X?
|
4
|
MATH
|
{
"level": 5,
"subject": "Counting & Probability",
"type": "MATH_SV"
}
| 8,538 | 8,538 |
Find the matrix $\mathbf{M}$ if it satisfies $\mathbf{M} \mathbf{i} = \begin{pmatrix} 2 \\ 3 \\ -8 \end{pmatrix},$ $\mathbf{M} \mathbf{j} = \begin{pmatrix} 0 \\ 5 \\ -2 \end{pmatrix},$ and $\mathbf{M} \mathbf{k} = \begin{pmatrix} X \\ -1 \\ 4 \end{pmatrix}.$
If we know the answer to the above question is \begin{pmatrix}2&0&7\3&5&-1\-8&-2&4\end{pmatrix}, what is the value of unknown variable X?
|
7
|
MATH
|
{
"level": 3,
"subject": "Precalculus",
"type": "MATH_FOBAR"
}
| 9,472 | 9,472 |
Given a right triangle with a hypotenuse measuring 10 inches and one angle measuring $45^{\circ}$, what is the area of the triangle in square inches?
|
25
|
MATH
|
{
"level": 4,
"subject": "Prealgebra",
"type": "MATH_Rephrased"
}
| 50,287 | 50,287 |
If $x$ is a positive integer such that $1^{x+2} + 2^{x+1} + 3^{x-1} + 4^x = X$, what is the value of $x$?
If we know the answer to the above question is 5, what is the value of unknown variable X?
|
1170
|
MATH
|
{
"level": 4,
"subject": "Algebra",
"type": "MATH_FOBAR"
}
| 39,348 | 39,348 |
If a, b, and c are positive integers, where the greatest common divisor of a and b is 168, and the greatest common divisor of a and c is 693, what is the minimum possible value of the greatest common divisor of b and c?
|
21
|
MATH
|
{
"level": 5,
"subject": "Number Theory",
"type": "MATH_Rephrased"
}
| 67,233 | 67,233 |
Triangle $ABC$ has side lengths of 5, 12, and 13 units, while triangle $DEF$ has side lengths of 8, 15, and 17 units. Determine the ratio of the area of triangle $ABC$ to the area of triangle $DEF$, expressed as a common fraction.
|
\frac{1}{2}
|
MATH
|
{
"level": 2,
"subject": "Geometry",
"type": "MATH_Rephrased"
}
| 62,788 | 62,788 |
Determine the value of $6 \div 0.\overline{6}$.
|
9
|
MATH
|
{
"level": 4,
"subject": "Prealgebra",
"type": "MATH_Rephrased"
}
| 15,728 | 15,728 |
Let $a,$ $b,$ $c,$ $d$ be nonzero integers such that
\[\begin{pmatrix} a & b \\ c & d \end{pmatrix}^2 = \begin{pmatrix} X & 0 \\ 0 & 7 \end{pmatrix}.\]Find the smallest possible value of $|a| + |b| + |c| + |d|. 7. What is the value of unknown variable X?
|
7
|
MATH
|
{
"level": 3,
"subject": "Precalculus",
"type": "MATH_SV"
}
| 38,606 | 38,606 |
What is the value of $\sec 135^\circ$?
|
-\sqrt{2}
|
MATH
|
{
"level": 1,
"subject": "Precalculus",
"type": "MATH_Rephrased"
}
| 107,349 | 107,349 |
In convex hexagon ABCDEF, there are only two different side lengths. One side, AB, measures 5 units, and another side, BC, measures 6 units. The perimeter of hexagon ABCDEF is 34 units. How many sides of hexagon ABCDEF have a length of 6 units?
|
4
|
MATH
|
{
"level": 3,
"subject": "Prealgebra",
"type": "MATH_Rephrased"
}
| 89,779 | 89,779 |
Let $\omega$ be a nonreal root of $z^3 = 1.$ Let $a_1,$ $a_2,$ $\dots,$ $a_n$ be real numbers such that
\[\frac{1}{a_1 + \omega} + \frac{1}{a_2 + \omega} + \dots + \frac{1}{a_n + \omega} = 2 + 5i.\]Compute
\[\frac{2a_1 - 1}{a_1^2 - a_1 + 1} + \frac{2a_2 - 1}{a_2^2 - a_2 + 1} + \dots + \frac{2a_n - 1}{a_n^2 - a_n + X
If we know the answer to the above question is 4, what is the value of unknown variable X?
|
1
|
MATH
|
{
"level": 5,
"subject": "Intermediate Algebra",
"type": "MATH_FOBAR"
}
| 69,445 | 69,445 |
What is the number that, when divided by 3, gives a result 50 more than if it had been divided by 4?
|
600
|
MATH
|
{
"level": 4,
"subject": "Prealgebra",
"type": "MATH_Rephrased"
}
| 60,848 | 60,848 |
What is the positive value of the expression $\log_2(27 + \log_2(27 + \log_2(27 + \ldots)))$?
|
5
|
MATH
|
{
"level": 3,
"subject": "Intermediate Algebra",
"type": "MATH_Rephrased"
}
| 46,021 | 46,021 |
In how many ways can I select members from a club of 25 to form a 4-person executive committee?
|
12,650
|
MATH
|
{
"level": 2,
"subject": "Counting & Probability",
"type": "MATH_Rephrased"
}
| 24,026 | 24,026 |
What is the sum of the geometric series $-1 + 2 - 4 + 8 - \cdots + 512$?
|
341
|
MATH
|
{
"level": 4,
"subject": "Algebra",
"type": "MATH_Rephrased"
}
| 5,389 | 5,389 |
Write the expression $\frac{3 + 4i}{1 + 2i}$ in the form $a + bi$, where $a$ and $b$ are real numbers expressed as improper fractions if necessary.
|
\frac{11}{5}-\frac{2}{5}i
|
MATH
|
{
"level": 5,
"subject": "Algebra",
"type": "MATH_Rephrased"
}
| 149,763 | 149,763 |
What is the units digit of the integer representation of $7^{35}$?
|
3
|
MATH
|
{
"level": 3,
"subject": "Number Theory",
"type": "MATH_Rephrased"
}
| 27,842 | 27,842 |
If $(a + b + c)^3 - a^3 - b^3 - c^3 = 150$, where $a$, $b$, and $c$ are positive integers, what is the value of $a + b + c$?
|
6
|
MATH
|
{
"level": 5,
"subject": "Algebra",
"type": "MATH_Rephrased"
}
| 18,001 | 18,001 |
Our school's girls volleyball team has X players, including a set of 3 triplets: Alicia, Amanda, and Anna. In how many ways can we choose 6 starters if exactly one of the triplets is in the starting lineup?
If we know the answer to the above question is 1386, what is the value of unknown variable X?
|
14
|
MATH
|
{
"level": 4,
"subject": "Counting & Probability",
"type": "MATH_FOBAR"
}
| 22,641 | 22,641 |
Determine the number of positive perfect square integers that are factors of the product $\left(2^{10}\right)\left(3^{12}\right)\left(5^{15}\right)$.
|
336
|
MATH
|
{
"level": 5,
"subject": "Number Theory",
"type": "MATH_Rephrased"
}
| 12,969 | 12,969 |
If the sum of $\triangle$ and $q$ is 59, and the sum of $(\triangle + q)$ and $q$ is 106, what is the value of $\triangle$?
|
12
|
MATH
|
{
"level": 1,
"subject": "Algebra",
"type": "MATH_Rephrased"
}
| 113,390 | 113,390 |
The volume of the parallelepiped generated by $\begin{pmatrix} 2 \\ 3 \\ X \end{pmatrix},$ $\begin{pmatrix} 1 \\ k \\ 2 \end{pmatrix},$ and $\begin{pmatrix} 1 \\ 2 \\ k \end{pmatrix}$ is 15. Find $k,$ where $k > 0. 2. What is the value of unknown variable X?
|
4
|
MATH
|
{
"level": 4,
"subject": "Precalculus",
"type": "MATH_SV"
}
| 72,122 | 72,122 |
What is the value of the first term in the geometric sequence $a, b, c, 32, 64$?
|
4
|
MATH
|
{
"level": 1,
"subject": "Algebra",
"type": "MATH_Rephrased"
}
| 59,618 | 59,618 |
What is the area, in square units, of triangle $ABC$? [asy]
unitsize(0.15inch);
path X = (-6.5, 0)--(5.5, 0);
path Y = (0, -3.5)--(0, 7.5);
draw(X); draw(Y);
for(int n=-6; n <= X; ++n)
if( n != 0 )
draw( (n,0.25)--(n,-0.25) );
for(int n=-3; n <= 7; ++n)
if( n != 0 )
draw( (0.25,n)--(-0.25,n) );
pair A = (-4,3); pair B = (0,6); pair C = (2,-2);
dot(A); dot(B); dot(C);
label("$A\ (-4,3)$", A, NW); label("$B\ (0,6)$", B, NE); label("$C\ (2,-2)$", C, The value of $n/p$ is 8. What is the value of unknown variable X?
|
5
|
MATH
|
{
"level": 4,
"subject": "Geometry",
"type": "MATH_SV"
}
| 96,168 | 96,168 |
For $\mathbf{v} = \begin{pmatrix} -10 \\ 6 \end{pmatrix}$ and $\mathbf{w} = \begin{pmatrix} X \\ -9 \end{pmatrix}$, compute $\text{proj}_{\mathbf{w}} \mathbf{v}$.
If we know the answer to the above question is \begin{pmatrix}-10\6\end{pmatrix}, what is the value of unknown variable X?
|
15
|
MATH
|
{
"level": 3,
"subject": "Precalculus",
"type": "MATH_FOBAR"
}
| 133,660 | 133,660 |
A number $n$ has X divisors. $n^2$ has 5 divisors. What is the value of unknown variable X?
|
3
|
MATH
|
{
"level": 2,
"subject": "Number Theory",
"type": "MATH_SV"
}
| 33,229 | 33,229 |
If the function $f(x)$ satisfies $f(x + f(x)) = 4f(x)$ for all $x$, and $f(1) = 4$, what is the value of $f(21)$?
|
64
|
MATH
|
{
"level": 2,
"subject": "Intermediate Algebra",
"type": "MATH_Rephrased"
}
| 46,577 | 46,577 |
What is the sum of the sequence (-39), (-37), ..., (-1)?
|
-400
|
MATH
|
{
"level": 4,
"subject": "Algebra",
"type": "MATH_Rephrased"
}
| 45,762 | 45,762 |
What are the coordinates of the point where the lines $9x - 4y = 30$ and $7x + y = 11$ intersect?
|
(2,-3)
|
MATH
|
{
"level": 3,
"subject": "Algebra",
"type": "MATH_Rephrased"
}
| 135,428 | 135,428 |
Mary can mow a lawn in four hours and Tom can mow the lawn in X hours. If Tom works for 2 hours alone, what fractional part of the lawn remains to be mowed?
If we know the answer to the above question is \frac{3}{5}, what is the value of unknown variable X?
|
5
|
MATH
|
{
"level": 1,
"subject": "Algebra",
"type": "MATH_FOBAR"
}
| 3,859 | 3,859 |
Given $\begin{vmatrix} a & b \\ c & d \end{vmatrix} = X find $\begin{vmatrix} 2a & 2b \\ 2c & 2d \end{vmatrix}.$
If we know the answer to the above question is 20, what is the value of unknown variable X?
|
5
|
MATH
|
{
"level": 1,
"subject": "Precalculus",
"type": "MATH_FOBAR"
}
| 4,940 | 4,940 |
When the repeating decimal $0.\overline{12}$ is written as a reduced common fraction, what is the sum of the numerator and denominator?
|
37
|
MATH
|
{
"level": 4,
"subject": "Prealgebra",
"type": "MATH_Rephrased"
}
| 83,210 | 83,210 |
Calculate the value of $ABCD$ given that $A = (\sqrt{2008}+\sqrt{2009}),$ $B = (-\sqrt{2008}-\sqrt{2009}),$ $C = (\sqrt{2008}-\sqrt{2009}),$ and $D = (\sqrt{2009}-\sqrt{2008}).$
|
1
|
MATH
|
{
"level": 4,
"subject": "Algebra",
"type": "MATH_Rephrased"
}
| 7,107 | 7,107 |
A certain coin is weighted such that the chance of flipping heads is $\frac{1}{3}$ and the chance of flipping tails is $\frac{2}{3}$. Suppose that we win X if we flip a heads on a coin toss, but lose $\$2$ if we flip tails. What is the expected value, in dollars, of our winnings after one flip? Express your answer as a common fraction.
If we know the answer to the above question is -\frac{1}{3}, what is the value of unknown variable X?
|
3
|
MATH
|
{
"level": 3,
"subject": "Counting & Probability",
"type": "MATH_FOBAR"
}
| 42,696 | 42,696 |
To transmit a positive integer less than 1000, the Networked Number Node offers two options.
Option 1. Pay $\$$d to send each digit d. Therefore, 987 would cost $\$$9 + $\$$8 + $\$$7 = $\$$24 to transmit.
Option 2. Encode integer into binary (base 2) first, and then pay $\$$d to send each digit d. Therefore, 987 becomes 1111011011 and would cost $\$$1 + $\$$1 + $\$$1 + $\$$1 + $\$$0 + $\$$1 + $\$$1 + $\$$0 + X + $\$$1 = $\$$8. The largest integer less than 1000 that costs the same whether using Option 1 or Option 2 is 503. What is the value of unknown variable X?
|
1
|
MATH
|
{
"level": 5,
"subject": "Number Theory",
"type": "MATH_SV"
}
| 59,526 | 59,526 |
What is the result of the calculation: $9 - 8 + 7 \times 6 + 5 - 4 \times 3 + 2 - 1$?
|
37
|
MATH
|
{
"level": 2,
"subject": "Prealgebra",
"type": "MATH_Rephrased"
}
| 7,271 | 7,271 |
Carolyn and Paul are playing a game starting with a list of the integers $1$ to $n.$ The rules of the game are:
$\bullet$ Carolyn always has the first turn.
$\bullet$ Carolyn and Paul alternate turns.
$\bullet$ On each of her turns, Carolyn must remove one number from the list such that this number has at least one positive divisor other than itself remaining in the list.
$\bullet$ On each of his turns, Paul must remove from the list all of the positive divisors of the number that Carolyn has just removed.
$\bullet$ If Carolyn cannot remove any more numbers, then Paul removes the rest of the numbers.
For example, if $n=6,$ a possible sequence of moves is shown in this chart:
\begin{tabular}{|c|c|c|}
\hline
Player & Removed \# & \# remaining \\
\hline
Carolyn & 4 & 1, 2, X, 5, 6 \\
\hline
Paul & 1, 2 & 3, 5, 6 \\
\hline
Carolyn & 6 & 3, 5 \\
\hline
Paul & 3 & 5 \\
\hline
Carolyn & None & 5 \\
\hline
Paul & 5 & None \\
\hline
\end{tabular}
Note that Carolyn can't remove $3$ or $5$ on her second turn, and can't remove any number on her third turn.
In this example, the sum of the numbers removed by Carolyn is $4+6=10$ and the sum of the numbers removed by Paul is $1+2+3+5=11.$
Suppose that $n=6$ and Carolyn removes the integer $2$ on her first turn. Determine the sum of the numbers that Carolyn removes. The answer is 8. What is the value of unknown variable X?
|
3
|
MATH
|
{
"level": 5,
"subject": "Prealgebra",
"type": "MATH_SV"
}
| 75,880 | 75,880 |
Our club has X members, 10 boys and 10 girls. In how many ways can we choose a president and a vice-president if they must be of the same gender? Assume no one can hold both offices.
If we know the answer to the above question is 180, what is the value of unknown variable X?
|
20
|
MATH
|
{
"level": 2,
"subject": "Counting & Probability",
"type": "MATH_FOBAR"
}
| 94,570 | 94,570 |
A car travels 40 kph for 20 kilometers, 50 kph for X kilometers, 60 kph for 45 minutes and 48 kph for 15 minutes. What is the average speed of the car, in kph?
If we know the answer to the above question is 51, what is the value of unknown variable X?
|
25
|
MATH
|
{
"level": 5,
"subject": "Prealgebra",
"type": "MATH_FOBAR"
}
| 112,861 | 112,861 |
If $a = -3$ and $b = 2$, what is the value of $-a - b^3 + ab$?
|
-11
|
MATH
|
{
"level": 2,
"subject": "Algebra",
"type": "MATH_Rephrased"
}
| 43,615 | 43,615 |
During a circus performance, I observed the number of acrobats and elephants. I noticed a total of 40 legs and 15 heads. How many acrobats were present in the show?
|
10
|
MATH
|
{
"level": 2,
"subject": "Algebra",
"type": "MATH_Rephrased"
}
| 88,721 | 88,721 |
Shari walks at a constant rate of X miles per hour. After 1.5 hours, how many miles did she walk? Express your answer as a decimal to the nearest tenth. The answer is 4.5. What is the value of unknown variable X?
|
3
|
MATH
|
{
"level": 2,
"subject": "Prealgebra",
"type": "MATH_SV"
}
| 25,907 | 25,907 |
What is the remainder when $(247 + 5 \cdot 39 + 7 \cdot 143 + 4 \cdot 15)$ is divided by $13$?
|
8
|
MATH
|
{
"level": 3,
"subject": "Number Theory",
"type": "MATH_Rephrased"
}
| 126,349 | 126,349 |
Calculate $\frac{3 \cdot X! + 15\cdot 4!}{6!}$ The answer is 1. What is the value of unknown variable X?
|
5
|
MATH
|
{
"level": 1,
"subject": "Counting & Probability",
"type": "MATH_SV"
}
| 1,696 | 1,696 |
What is the area, in square centimeters, of a triangle with side lengths of 14 cm, 48 cm, and 50 cm?
|
336
|
MATH
|
{
"level": 1,
"subject": "Geometry",
"type": "MATH_Rephrased"
}
| 67,117 | 67,117 |
A lucky integer is a positive integer which is divisible by the sum of its digits. What is the least positive multiple of X that is not a lucky integer?
If we know the answer to the above question is 99, what is the value of unknown variable X?
|
9
|
MATH
|
{
"level": 4,
"subject": "Prealgebra",
"type": "MATH_FOBAR"
}
| 36,452 | 36,452 |
If each man at a party danced with exactly three women and each woman danced with exactly two men, and there were twelve men at the party, how many women attended the party?
|
18
|
MATH
|
{
"level": 5,
"subject": "Prealgebra",
"type": "MATH_Rephrased"
}
| 143 | 143 |
The set of vectors $\mathbf{v}$ such that
\[\operatorname{proj}_{\begin{pmatrix} 5 \\ X \end{pmatrix}} \mathbf{v} = \begin{pmatrix} -\frac{5}{2} \\ -1 \end{pmatrix}\]lie on a line. Enter the equation of this line in the form "$y = mx + b$". The answer is 4. What is the value of unknown variable X?
|
2
|
MATH
|
{
"level": 4,
"subject": "Precalculus",
"type": "MATH_SV"
}
| 32,640 | 32,640 |
Which of the cones below can be formed from a $252^{\circ}$ sector of a circle of radius 10 by aligning the two straight sides?
[asy]
draw((5.8,8.1)..(-10,0)--(0,0)--(3.1,-9.5)..cycle);
label("10",(-5,0),S);
label("$252^{\circ}$",(0,0),NE);
[/asy]
A. base radius = 6, slant =10
B. base radius = 6, height =10
C. base radius = 7, slant =10
D. base radius = 7, height =10
E. base radius = 8, The value of slant is X. What is the value of unknown variable X?
|
10
|
MATH
|
{
"level": 3,
"subject": "Geometry",
"type": "MATH_SV"
}
| 5,222 | 5,222 |
Find the equation of the plane passing through $(-1,1,1)$ and $(1,-1,1),$ and which is perpendicular to the plane $x + 2y + 3z = 5.$ Enter your answer in the form
\[Ax + By + Cz + D = 0,\]where $A,$ $B,$ $C,$ $D$ are integers such that $A > 0$ and $\gcd(|A|,|B|,|C|,|D|) = X
If we know the answer to the above question is x+y-z+1=0, what is the value of unknown variable X?
|
1
|
MATH
|
{
"level": 5,
"subject": "Precalculus",
"type": "MATH_FOBAR"
}
| 33,251 | 33,251 |
If $x \diamondsuit y = 3x + 5y$ for all $x$ and $y$, then what is the value of $2 \diamondsuit X$?
If we know the answer to the above question is 41, what is the value of unknown variable X?
|
7
|
MATH
|
{
"level": 2,
"subject": "Algebra",
"type": "MATH_FOBAR"
}
| 54,214 | 54,214 |
How many distinct prime factors does the number 210 have?
|
4
|
MATH
|
{
"level": 2,
"subject": "Prealgebra",
"type": "MATH_Rephrased"
}
| 28,320 | 28,320 |
Let $f$, $g$, and $h$ be polynomials such that $h(x) = f(x)\cdot g(x)$. If the constant term of $f(x)$ is $-4$ and the constant term of $h(x)$ is X, what is $g(0)$?
If we know the answer to the above question is -\frac{3}{4}, what is the value of unknown variable X?
|
3
|
MATH
|
{
"level": 4,
"subject": "Algebra",
"type": "MATH_FOBAR"
}
| 135,753 | 135,753 |
Cookie Monster comes across a cookie with the equation $x^2 + y^2 + 21 = 4x + 18y$ as its boundary and is unsure if it is a lunch-sized cookie or a snack-sized cookie. What is the radius of this cookie?
|
8
|
MATH
|
{
"level": 3,
"subject": "Algebra",
"type": "MATH_Rephrased"
}
| 74,038 | 74,038 |
There exist two distinct unit vectors $\mathbf{v}$ such that the angle between $\mathbf{v}$ and $\begin{pmatrix} X \\ 2 \\ -1 \end{pmatrix}$ is $45^\circ,$ and the angle between $\mathbf{v}$ and $\begin{pmatrix} 0 \\ 1 \\ -1 \end{pmatrix}$ is $60^\circ.$ Let $\mathbf{v}_1$ and $\mathbf{v}_2$ be these vectors. Find $\|\mathbf{v}_1 - \mathbf{v}_2\|. 2. What is the value of unknown variable X?
|
2
|
MATH
|
{
"level": 5,
"subject": "Precalculus",
"type": "MATH_SV"
}
| 4,705 | 4,705 |
The first $20$ numbers of an arrangement are shown below. What would be the value of the $40^{\mathrm{th}}$ number if the arrangement were continued?
$\bullet$ Row 1: X $2$
$\bullet$ Row 2: $4,$ $4,$ $4,$ $4$
$\bullet$ Row 3: $6,$ $6,$ $6,$ $6,$ $6,$ $6$
$\bullet$ Row 4: $8,$ $8,$ $8,$ $8,$ $8,$ $8,$ $8,$ $8$
If we know the answer to the above question is 12, what is the value of unknown variable X?
|
2
|
MATH
|
{
"level": 1,
"subject": "Counting & Probability",
"type": "MATH_FOBAR"
}
| 56,366 | 56,366 |
If $x^2 = y - X and $x = -5$, then what is the value of $y$?
If we know the answer to the above question is 28, what is the value of unknown variable X?
|
3
|
MATH
|
{
"level": 2,
"subject": "Prealgebra",
"type": "MATH_FOBAR"
}
| 50,372 | 50,372 |
Out of Roslyn's ten boxes, six contain pencils, three contain pens, and two contain both pens and pencils. How many boxes do not contain either pens or pencils?
|
3
|
MATH
|
{
"level": 2,
"subject": "Prealgebra",
"type": "MATH_Rephrased"
}
| 24,202 | 24,202 |
How many integer values of x do not satisfy the inequality $5x^{2}+19x+16 > 20$?
|
5
|
MATH
|
{
"level": 5,
"subject": "Algebra",
"type": "MATH_Rephrased"
}
| 71,028 | 71,028 |
If ten 6-sided dice are rolled, what is the probability of exactly three of them showing a 1? Round your answer to the nearest thousandth.
|
0.155
|
MATH
|
{
"level": 4,
"subject": "Counting & Probability",
"type": "MATH_Rephrased"
}
| 122,952 | 122,952 |
Calculate the value of $42! / 40!$ without using a calculator.
|
1,722
|
MATH
|
{
"level": 1,
"subject": "Counting & Probability",
"type": "MATH_Rephrased"
}
| 63,738 | 63,738 |
When asked to add two positive integers, Mr. Sanchez's students made mistakes. Juan subtracted and got 2, while Maria multiplied and got 120. What is the correct answer?
|
22
|
MATH
|
{
"level": 2,
"subject": "Algebra",
"type": "MATH_Rephrased"
}
| 97,207 | 97,207 |
The probability it will rain on Saturday is $60\%$, and the probability it will rain on Sunday is X\%$. If the probability of rain on a given day is independent of the weather on any other day, what is the probability it will rain on both days, expressed as a percent?
If we know the answer to the above question is 15, what is the value of unknown variable X?
|
25
|
MATH
|
{
"level": 2,
"subject": "Counting & Probability",
"type": "MATH_FOBAR"
}
| 75,934 | 75,934 |
Let $A,$ $B,$ $C$ be the angles of a triangle. Evaluate
\[\begin{vmatrix} \sin^2 A & \cot A & X \\ \sin^2 B & \cot B & 1 \\ \sin^2 C & \cot C & 1 \end{vmatrix}. The answer is 0. What is the value of unknown variable X?
|
1
|
MATH
|
{
"level": 2,
"subject": "Precalculus",
"type": "MATH_SV"
}
| 85,382 | 85,382 |
If the mean number of candy pieces taken by each of the 30 students in a class is 5, and every student takes at least one piece, what is the maximum number of candy pieces one student could have taken?
|
121
|
MATH
|
{
"level": 4,
"subject": "Prealgebra",
"type": "MATH_Rephrased"
}
| 28,297 | 28,297 |
Given that $\log_2 x^2 + \log_{1/2} x = 5$, find the value of $x$.
|
32
|
MATH
|
{
"level": 2,
"subject": "Intermediate Algebra",
"type": "MATH_Rephrased"
}
| 101,869 | 101,869 |
If an unfair coin has a $\frac34$ probability of landing on heads and a $\frac14$ probability of landing on tails, and a heads flip earns $\$3$ while a tails flip loses $\$8$, what is the expected value of a coin flip, rounded to the nearest hundredth?
|
0.25
|
MATH
|
{
"level": 3,
"subject": "Counting & Probability",
"type": "MATH_Rephrased"
}
| 92,119 | 92,119 |
Determine the value of $x$ when $x = \frac{2009^2 - 2009}{2009}$?
|
2008
|
MATH
|
{
"level": 2,
"subject": "Algebra",
"type": "MATH_Rephrased"
}
| 105,570 | 105,570 |
Let $f(x)$ be a function defined for all positive real numbers satisfying the conditions $f(x) > 0$ for all $x > 0$ and
\[f(x - y) = \sqrt{f(xy) + 2}\]for all $x > y > X Determine $f(2009).$
If we know the answer to the above question is 2, what is the value of unknown variable X?
|
0
|
MATH
|
{
"level": 4,
"subject": "Intermediate Algebra",
"type": "MATH_FOBAR"
}
| 7,826 | 7,826 |
If the expression $(x+y+z)(xy+xz+yz)$ is equal to 25 and the expression $x^2(y+z)+y^2(x+z)+z^2(x+y)$ is equal to 7, where $x$, $y$, and $z$ are real numbers, what is the value of $xyz$?
|
6
|
MATH
|
{
"level": 3,
"subject": "Algebra",
"type": "MATH_Rephrased"
}
| 8,266 | 8,266 |
What is the value of b that makes the equation $161_{b} + 134_{b} = 315_{b}$ true?
|
8
|
MATH
|
{
"level": 4,
"subject": "Number Theory",
"type": "MATH_Rephrased"
}
| 47,263 | 47,263 |
Determine the sum of all prime numbers that are less than 10.
|
17
|
MATH
|
{
"level": 1,
"subject": "Number Theory",
"type": "MATH_Rephrased"
}
| 4,004 | 4,004 |
If the remainders when three positive integers are divided by 24 are 10, 4, and 12 respectively, what is the remainder when the sum of these three integers is divided by 24?
|
2
|
MATH
|
{
"level": 1,
"subject": "Number Theory",
"type": "MATH_Rephrased"
}
| 64,574 | 64,574 |
At what value of x do the expressions $\frac{3+x}{5+x}$ and $\frac{1+x}{2+x}$ become equal?
|
1
|
MATH
|
{
"level": 2,
"subject": "Algebra",
"type": "MATH_Rephrased"
}
| 111,289 | 111,289 |
Determine the area of the shaded region in square units, given that the radius of the larger circle is four times the radius of the smaller circle and the diameter of the smaller circle is 2 units. Express your answer in terms of pi.
|
15\pi
|
MATH
|
{
"level": 4,
"subject": "Prealgebra",
"type": "MATH_Rephrased"
}
| 63,755 | 63,755 |
Compute
\[
\begin{vmatrix} \cos 1 & \cos 2 & \cos 3 \\ \cos 4 & \cos 5 & \cos X \\ \cos 7 & \cos 8 & \cos 9 \end{vmatrix}
.\]All the angles are in radians. The answer is 0. What is the value of unknown variable X?
|
6
|
MATH
|
{
"level": 2,
"subject": "Precalculus",
"type": "MATH_SV"
}
| 84,828 | 84,828 |
Add $1_3 + 12_3 + 212_3 + 2121_3.$ Express your answer in base X.
If we know the answer to the above question is 10200_3, what is the value of unknown variable X?
|
3
|
MATH
|
{
"level": 3,
"subject": "Number Theory",
"type": "MATH_FOBAR"
}
| 5,504 | 5,504 |
Calculate $f(f(f(2)))$ given the function $f(n)$ defined as follows: $f(n) = n^2 - 1$ if $n < 4$, and $f(n) = 3n - 2$ if $n \geq 4$.
|
22
|
MATH
|
{
"level": 2,
"subject": "Algebra",
"type": "MATH_Rephrased"
}
| 151,687 | 151,687 |
Given the points (-2,0) and (0,2), find the equation of the line in the form y=mx+b. What is the value of m+b?
|
3
|
MATH
|
{
"level": 2,
"subject": "Algebra",
"type": "MATH_Rephrased"
}
| 6,636 | 6,636 |
For some constants $a$ and $b,$ let \[f(x) = \left\{
\begin{array}{cl}
ax + b & \text{if } x < X, \\
8 - 3x & \text{if } x \ge 2.
\end{array}
\right.\]The function $f$ has the property that $f(f(x)) = x$ for all $x. The value of $a + b$ is 3. What is the value of unknown variable X?
|
2
|
MATH
|
{
"level": 5,
"subject": "Algebra",
"type": "MATH_SV"
}
| 1,596 | 1,596 |
Cybil and Ronda are sisters. The 10 letters from their names are placed on identical cards so that each of 10 cards contains one letter. Without replacement, two cards are selected at random from the X cards. What is the probability that one letter is from each sister's name? Express your answer as a common fraction.
If we know the answer to the above question is \frac{5}{9}, what is the value of unknown variable X?
|
10
|
MATH
|
{
"level": 4,
"subject": "Counting & Probability",
"type": "MATH_FOBAR"
}
| 3,978 | 3,978 |
In a standard deck of 52 cards, there are 4 suits with 13 cards each. Two suits (hearts and diamonds) are red, while the other two suits (spades and clubs) are black. The cards are randomly ordered through shuffling. How many different ways can we choose two cards, considering that the order in which they are chosen matters?
|
2652
|
MATH
|
{
"level": 5,
"subject": "Prealgebra",
"type": "MATH_Rephrased"
}
| 61,215 | 61,215 |
A standard deck of cards has X cards divided into 4 suits, each of which has 13 cards. Two of the suits ($\heartsuit$ and $\diamondsuit$, called 'hearts' and 'diamonds') are red, the other two ($\spadesuit$ and $\clubsuit$, called 'spades' and 'clubs') are black. The cards in the deck are placed in random order (usually by a process called 'shuffling'). In how many ways can we pick two different cards? (Order matters, thus ace of spades followed by jack of diamonds is different than jack of diamonds followed by ace of spades. The answer is 2652. What is the value of unknown variable X?
|
52
|
MATH
|
{
"level": 5,
"subject": "Prealgebra",
"type": "MATH_SV"
}
| 20,546 | 20,546 |
If $m$ and $n$ satisfy $mn = 4$ and $m + n = 5$, what is the value of $|m - n|$?
|
3
|
MATH
|
{
"level": 1,
"subject": "Algebra",
"type": "MATH_Rephrased"
}
| 35,413 | 35,413 |
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