problem
stringlengths 18
4.46k
| answer
stringlengths 1
942
| pass_at_n
float64 0.08
0.92
|
|---|---|---|
The sum of 81 consecutive integers is $9^5$. What is their median?
|
729
| 0.666667 |
Marty wants to paint a box. He can choose to use either blue, green, yellow, black, or red paint. However, if he chooses to use a sponge, he can only do so with green or yellow paints. Marty can also style the paint by painting with a brush, a roller, or a sponge. How many different combinations of color and painting method can Marty choose?
|
12
| 0.666667 |
In a Biology class, Jake wants to prepare a nutrient solution by mixing 0.05 liters of nutrient concentrate with 0.03 liters of water to obtain 0.08 liters of solution. He needs to make a total of 0.64 liters of this solution for an experiment. How many liters of water will he need to use?
|
0.24
| 0.916667 |
A flagpole is supported by a wire which extends from the top of the pole to a point on the ground 5 meters from its base. When Ben walks 3 meters from the base of the pole toward the point where the wire is attached to the ground, his head just touches the wire. Ben is 1.8 meters tall. How many meters tall is the flagpole?
|
4.5 \text{ meters}
| 0.583333 |
We know the following to be true:
- $Z$ and $K$ are integers where $1000 < Z < 2000$ and $K > 0$.
- $Z = K \times K^3$.
What is the value of $K$ for which $Z$ is a perfect fourth power?
|
6
| 0.916667 |
In MODIFIED SHORT BINGO, a $5\times5$ card is filled by marking the middle square as WILD. Modify the assignment of numbers such that the first column consists of 5 distinct numbers from the set $10-25$. The rest of the squares are filled with numbers as follows: 5 distinct numbers from $26-35$ in the second column, 4 distinct numbers $36-45$ in the third column (skipping the WILD square), 5 distinct numbers from $46-55$ in the fourth column, and 5 distinct numbers from $56-65$ in the fifth column.
How many distinct possibilities are there for the values in the first column of a MODIFIED SHORT BINGO card, where the order of the numbers matters?
|
524160
| 0.916667 |
What is the length of the diagonal of a square with side length $100\sqrt{3}$ cm? Express your answer in simplest form.
|
100\sqrt{6} \text{ cm}
| 0.916667 |
Define the operation $X \diamond Y$ such that $X \diamond Y = 4X - 3Y + 2$. Calculate $X$ if $X \diamond 6 = 35$.
|
\frac{51}{4}
| 0.916667 |
In the prime factorization of $30!$, what is the exponent of $2$ and the exponent of $5$?
|
7
| 0.416667 |
Below is the graph of $y = a \tan bx$ for some positive constants $a$ and $b.$ The graph has a period of $\frac{2\pi}{5}$ and passes through the point $\left(\frac{\pi}{10}, 1\right)$. Find $ab.$
|
\frac{5}{2}
| 0.833333 |
What is the sum and the count of all positive integer divisors of 91?
|
4
| 0.083333 |
The Great Eighteen Soccer League has three divisions, with six teams in each division. Each team plays each of the other teams in its own division three times and every team in the other divisions twice. How many league games are scheduled?
|
351
| 0.416667 |
Suppose the graph of a function $y = g(x)$ has the property that if it is shifted $30$ units to the right, then the resulting graph is identical to the original graph of $y = g(x)$.
What is the smallest positive $b$ such that if the graph of $y = g\left(\frac{x}{4}\right)$ is shifted $b$ units to the right, then the resulting graph is identical to the original graph of $y = g\left(\frac{x}{4}\right)$?
|
120
| 0.916667 |
Calculate the sum of the coefficients in the polynomial $4(2x^{6} + 9x^3 - 6) + 8(x^4 - 6x^2 + 3)$ when it is fully simplified.
|
4
| 0.166667 |
If $4$ lunks can be traded for $2$ kunks, and $2$ kunks will buy $6$ apples, how many lunks are needed to purchase two dozen apples?
|
16
| 0.916667 |
I have 7 marbles numbered 1 through 7 in a bag. Suppose I take out three different marbles at random. What is the expected value of the sum of the numbers on the marbles?
|
12
| 0.916667 |
John now has six children. What is the probability that exactly three of them are girls? (Assume a boy is equally likely to be born as a girl.)
|
\frac{5}{16}
| 0.833333 |
In triangle $ABC$, altitudes $AD$, $BE$, and $CF$ intersect at the orthocenter $H$. If $\angle ABC = 58^\circ$ and $\angle ACB = 23^\circ$, then find the measure of $\angle BHC$, in degrees.
|
81^\circ
| 0.916667 |
What is the smallest number with three different prime factors, none of which can be less than 10?
|
2431
| 0.416667 |
What is \[3 - 5x - 7x^2 + 9 + 11x - 13x^2 - 15 + 17x + 19x^2 + 6x^3\] in terms of $x$?
|
6x^3 - x^2 + 23x - 3
| 0.916667 |
Two fair, six-sided dice are rolled. What is the probability that the sum of the two numbers showing is less than 12?
|
\frac{35}{36}
| 0.916667 |
Simplify $(1)(2a)(3b)(4a^2b)(5a^3b^2)$.
|
120a^6b^4
| 0.75 |
Given $\begin{vmatrix} x & y & z \\ a & b & c \\ p & q & r \end{vmatrix} = 2$, find $\begin{vmatrix} 3x & 3y & 3z \\ 3a & 3b & 3c \\ 3p & 3q & 3r \end{vmatrix}.$
|
54
| 0.916667 |
What is the measure of the smaller angle between the hands of a 12-hour clock at 3:45 pm, in degrees? Express your answer as a decimal to the nearest tenth.
|
157.5^\circ
| 0.916667 |
Determine the value of the positive number $b$ such that the terms $25, b, \frac{1}{4}$ are the first, second, and third terms, respectively, of a geometric sequence.
|
\frac{5}{2}
| 0.833333 |
Factor the following expression: $75x^2 + 50x$.
|
25x(3x + 2)
| 0.916667 |
A circular plaza includes a three-layer arrangement: a central water feature, surrounded by a decorative tile area, then bordered by a circular walkway. The water feature has a diameter of 16 feet. The decorative tile area is 12 feet wide uniformly around the water feature, and the walkway is 10 feet wide. What is the diameter of the circle that forms the outer boundary of the walkway?
|
60 \text{ feet}
| 0.916667 |
I have 6 books, three of which are identical copies of the same math book, and the rest are different. In how many ways can I arrange them on a shelf?
|
120
| 0.833333 |
Find the least positive integer $x$ that satisfies $x + 3567 \equiv 1543 \pmod{14}$.
|
6
| 0.916667 |
In a configuration, two concentric circles form two regions: an outer gray region and an inner white region. The area of the gray region within the larger circle is equal to four times the area of the white circular region. What is the ratio of the radius of the small circle to the radius of the large circle?
|
\frac{1}{\sqrt{5}}
| 0.25 |
$\textbf{Mia's Classic Postcard Collection}$
Mia organizes her postcard collection by country and by the decade in which they were printed. The prices she paid for them at an antique shop were: Italy and Germany, $8$ cents each, Canada $5$ cents each, and Japan $7$ cents each. (Italy and Germany are European countries, Canada and Japan are in North America and Asia respectively.)
In her collection:
- 1950s: Italy 5, Germany 9, Canada 3, Japan 6
- 1960s: Italy 12, Germany 5, Canada 7, Japan 8
- 1970s: Italy 11, Germany 13, Canada 6, Japan 9
- 1980s: Italy 10, Germany 15, Canada 10, Japan 5
- 1990s: Italy 6, Germany 7, Canada 11, Japan 9
In dollars and cents, how much did her Asian and North American postcards issued in the '60s, '70s, and '80s cost her?
|
\$2.69
| 0.416667 |
How many possible distinct arrangements are there of the letters in the word "BALLOON"?
|
1260
| 0.916667 |
Evaluate $\log_{\sqrt{10}} (1000\sqrt{10})$.
|
7
| 0.916667 |
Three years ago, there were 30 trailer homes on Oak Street with an average age of 15 years. In that year, several brand new trailer homes were added to Oak Street. Today, the average age of all the trailer homes on Oak Street is 12 years. How many new trailer homes were added three years ago?
|
20
| 0.666667 |
A sphere-shaped balloon with a radius of 2 feet is filled with helium. This helium is then evenly transferred into 64 smaller sphere-shaped balloons. What is the radius of each of the smaller balloons, in feet?
|
\frac{1}{2}
| 0.916667 |
Fully simplify the following expression: $[(1+2+3+4+5+6)\div 3] + [(3\cdot5 + 12) \div 4]$.
|
13.75
| 0.916667 |
Rebecca purchases hot dogs that come in packages of seven, and hot dog buns that come in packages of nine. What is the smallest number of hot dog packages she must buy to have an equal number of hot dogs and hot dog buns?
|
9
| 0.666667 |
Find \( n \) such that \( 2^7 \cdot 3^4 \cdot n = 9! \).
|
35
| 0.75 |
How many one-quarters are there in one-eighth?
|
\frac{1}{2}
| 0.916667 |
Nora has three stamp albums. Each page she fills contains an equal number of stamps. If she fills the first album with 945 stamps, the second with 1260 stamps, and the third with 1575 stamps, what is the maximum number of stamps she could be putting on each page?
|
315
| 0.916667 |
Simplify this expression to a common fraction: \(\frac{1}{\frac{1}{(\frac{1}{3})^1}+\frac{1}{(\frac{1}{3})^2}+\frac{1}{(\frac{1}{3})^3}}\)
|
\frac{1}{39}
| 0.916667 |
The hypotenuse of a 30-60-90 triangle is $6\sqrt{2}$ units. How many square units are in the area of this triangle?
|
9\sqrt{3}
| 0.833333 |
Let $a\star b = a^2b + 2b - a$. If $7\star x = 85$, find the value of $x$.
|
\frac{92}{51}
| 0.916667 |
Four concentric circles are drawn with radii of 2, 4, 6, and 8. The innermost circle is painted black, the ring around it is white, the next ring beyond it is black, and the outermost ring is white. What is the ratio of the black area to the white area?
|
\frac{3}{5}
| 0.916667 |
Allie and Betty play a game where they alternately roll a standard die. The scoring function $g(n)$ awards points as per the following criteria:
\[g(n) = \left\{
\begin{array}{cl}
6 & \text{if } n \text{ is a multiple of 2 and 3}, \\
3 & \text{if } n \text{ is a multiple of 3 but not 2}, \\
2 & \text{if } n \text{ is a multiple of 2 but not 3}, \\
0 & \text{otherwise}.
\end{array}
\right.\]
Allie rolls the die four times, receiving the outcomes 6, 3, 2, and 4. Betty rolls the die four times as well, getting 5, 2, 3, and 6. Calculate the product of Allie's total points and Betty's total points.
|
143
| 0.916667 |
Find the monic quadratic polynomial, in $x,$ with real coefficients, which has $-3 - i \sqrt{8}$ as a root.
|
x^2 + 6x + 17
| 0.916667 |
Andy and Alexa baked a total of 30 cookies. This time, Alexa ends up eating three times the number of cookies eaten by Andy. Determine the maximum number of cookies Andy could have eaten.
|
7
| 0.833333 |
In right triangle $XYZ$, $\angle X = \angle Z$ and $XY = 10$. What is the area of $\triangle XYZ$?
|
50
| 0.25 |
Determine the ratio of $a$ to $b$ if: $\frac{9a-4b}{12a-3b} = \frac{4}{7}$.
|
\frac{16}{15}
| 0.916667 |
What is the smallest odd number with five different prime factors?
|
15015
| 0.833333 |
Eight people can paint two identical houses in four hours. If their efficiency decreases by 20% when they are tired, how long would it take five people to paint the same two houses, assuming they are tired?
|
8 \text{ hours}
| 0.916667 |
In the land of Eldoria, a valid license plate consists of two letters followed by three digits. However, the first letter of these license plates must be a vowel (A, E, I, O, U). How many valid license plates are possible in Eldoria?
|
130,000
| 0.916667 |
In a right triangle $ABC$, $\tan A = \frac{3}{4}$, $AC = 6$, and $\angle C = 90^\circ$. Find the length of side $AB$.
|
7.5
| 0.833333 |
What is the smallest integer \( k \) such that \( k > 1 \) and \( k \) has remainder \( 1 \) when divided by any of \( 23, \) \( 7, \) \( and \) \( 3? \)
|
484
| 0.833333 |
Find the smallest composite number that has no prime factors less than 20.
|
529
| 0.833333 |
Coach Grunt is preparing the 6-person starting lineup for his basketball team, the Grunters. There are 15 players on the team. Two of them, Ace and Zeppo, are league All-Stars and will definitely be in the starting lineup. However, one player, Xander, is recovering from an injury and cannot be in the lineup. How many different starting lineups are possible?
|
495
| 0.75 |
A domino set is modified such that each integer from 0 to 12 is paired with every other integer from 0 to 12 exactly once. A $\textit{double}$ is defined as a domino with the same number on both squares. Calculate the probability that a randomly selected domino from this set will be a $\textit{double}$. Express the answer as a common fraction.
|
\frac{1}{7}
| 0.583333 |
The arithmetic mean of 15 scores is 85. When the highest and lowest scores are removed, the new mean becomes 90. If the highest of the 15 scores is 105, what is the lowest score?
|
0
| 0.5 |
A square is divided into four congruent rectangles, each rectangle is as shown. If the perimeter of each of these four rectangles is 40 inches, what is the perimeter of the square, in inches?
[asy]
draw((0,0)--(0,4)--(4,4)--(4,0)--cycle);
draw((1,0)--(1,4));
draw((2,0)--(2,4));
draw((3,0)--(3,4));
[/asy]
|
64 \text{ inches}
| 0.75 |
April has five different basil plants and four different tomato plants. In how many ways can she arrange the plants in a row if she puts all the tomato plants next to each other?
|
17280
| 0.916667 |
How many distinct sequences of four letters can be made from the letters in EQUALS if each sequence must begin with L and must include the letter 'S' somewhere in the sequence, while no letter can appear in the sequence more than once?
|
36
| 0.75 |
A valid license plate in New Xanadu consists of two letters followed by three digits and then one letter again. How many valid license plates are possible?
|
17,576,000
| 0.916667 |
Find the units digit of the following within the indicated number base: $65_8 + 75_8$.
|
2
| 0.75 |
An infinite geometric series has a first term of $512$ and a sum of $3072$. What is its common ratio?
|
\frac{5}{6}
| 0.916667 |
Allie and Betty play a game where they take turns rolling a standard die. If a player rolls $n$, she is awarded $f(n)$ points, where \[f(n) = \left\{
\begin{array}{cl}
9 & \text{ if } n \text{ is a multiple of 3}, \\
3 & \text{ if } n \text{ is only a multiple of 2}, \\
1 & \text{ otherwise.}
\end{array}
\right.\] This time, Allie rolls the die three times, getting a 6, 3, and 4. Betty rolls the die four times, getting 1, 2, 5, and 6. Calculate the product of Allie's total points and Betty's total points.
|
294
| 0.75 |
You recently purchased over 150 eggs. The eggs are arranged in containers, each ideally holding 12 eggs. This time, three containers each contain only 11 eggs, while all other containers are full with 12 eggs each. What is the smallest number of eggs you could have now?
|
153
| 0.916667 |
Contractor Maria agreed to complete a job in 40 days. After 10 days, she found that the 10 people assigned to the work had already completed $\frac{2}{5}$ of the job. Assuming all workers maintain the same rate of productivity, what is the minimum number of workers she must retain to ensure the job is completed on time?
|
5
| 0.666667 |
Yan is somewhere between his home and the school. He can either walk directly to school or walk back home and then ride his bicycle to school. He rides 5 times as fast as he walks, and both choices require the same amount of time. What is the ratio of Yan's distance from his home to his distance from the school?
|
\frac{2}{3}
| 0.75 |
If $x$ is a positive number such that \[\sqrt{12x}\cdot\sqrt{18x}\cdot\sqrt{6x}\cdot\sqrt{9x}=27,\] find all possible values for $x$.
|
\frac{1}{2}
| 0.75 |
As $n$ ranges over the positive integers, what is the maximum possible value for the greatest common divisor of $13n + 4$ and $7n + 3$?
|
11
| 0.916667 |
The hypotenuse of a right triangle measures 12 inches and one angle is $30^{\circ}$. What is the area of the triangle in square inches?
|
18\sqrt{3}
| 0.666667 |
The noon temperatures for ten consecutive days were recorded as $75^\circ$, $74^\circ$, $76^\circ$, $77^\circ$, $80^\circ$, $81^\circ$, $83^\circ$, $85^\circ$, $83^\circ$, and $85^\circ$ Fahrenheit. Calculate the mean noon temperature, in degrees Fahrenheit, for these ten days.
|
79.9
| 0.833333 |
Define $X\star Y$ as $X\star Y = \frac{(X+Y)}{4}$. What is the value of $(3\star 11) \star 7$?
|
2.625
| 0.833333 |
What is the area, in square units, of a triangle with vertices at $A(2, 3), B(9, 3), C(4, 12)$?
|
31.5 \text{ square units}
| 0.75 |
How many positive integers less than $201$ are multiples of either $5$ or $11$, but not both at once?
|
52
| 0.916667 |
Express $3x^2 + 9x + 20$ in the form $a(x - h)^2 + k$. What is the value of $h$?
|
-\frac{3}{2}
| 0.916667 |
How many positive three-digit integers are there in which each of the first two digits is prime and the last digit is a non-prime odd number?
|
32
| 0.833333 |
Compute: $\left(\frac{3}{5}\right)^4 \cdot \left(\frac{2}{9}\right)^{1/2}$.
|
\frac{81\sqrt{2}}{1875}
| 0.583333 |
I have six apples and eight oranges. If a fruit basket must contain at least one piece of fruit, how many different kinds of fruit baskets can I create? (The apples are identical, and the oranges are identical. A fruit basket consists of some number of pieces of fruit, and the arrangement of the fruit in the basket does not matter.)
|
62
| 0.833333 |
Compute $4 \begin{pmatrix} 3 \\ -9 \end{pmatrix} - 3 \begin{pmatrix} 2 \\ -8 \end{pmatrix} + 2 \begin{pmatrix} 1 \\ -6 \end{pmatrix}$.
|
\begin{pmatrix} 8 \\ -24 \end{pmatrix}
| 0.916667 |
The diameter, in inches, of a sphere with three times the volume of a sphere of radius 6 inches can be expressed in the form $a\sqrt[3]{b}$ where $a$ and $b$ are positive integers and $b$ contains no perfect cube factors. Compute $a+b$.
|
15
| 0.916667 |
What is the sum of all of the positive even factors of $720$?
|
2340
| 0.75 |
The mean of the set of numbers $\{90,88,81,84,87,x\}$ is 85. Calculate the median of this set of six numbers.
|
85.5
| 0.333333 |
Multiply the base-10 numbers 312 and 57, then write the product in base-7. What is the units digit of the base-7 representation?
|
4
| 0.833333 |
Simplify $\sin (2x - y) \cos (3y) + \cos (2x - y) \sin (3y)$.
|
\sin (2x + 2y)
| 0.583333 |
For how many different digits $n$ is the three-digit number $23n$ divisible by $n$?
Note: $23n$ refers to a three-digit number with the unit digit of $n,$ not the product of $23$ and $n.$
|
3
| 0.916667 |
A relatively prime date is a date where the numbers of the month and the day are relatively prime. For example, February 17 is a relatively prime date because the greatest common factor of 2 and 17 is 1. Determine how many relatively prime dates are in February during a leap year.
|
15
| 0.916667 |
How many even natural-number factors divisible by 5 does $n = 2^3 \cdot 5^2 \cdot 11^1$ have?
|
12
| 0.583333 |
Rationalize the denominator of $\frac{4}{3\sqrt[4]{7}}$. The answer can be written in the form of $\frac{A\sqrt[4]{B}}{C}$, where $A$, $B$, and $C$ are integers, $C$ is positive, and $B$ is not divisible by the fourth power of any prime. Find $A+B+C$.
|
368
| 0.916667 |
Consider a generalized domino set where each domino contains two squares, and each square can contain an integer from 0 to 11. Each integer from this range is paired with every integer (including itself) exactly once to form a complete set of dominos. What is the probability that a domino randomly selected from this set will be a $\textit{double}$ (containing the same integer on both squares)?
|
\frac{2}{13}
| 0.583333 |
Compute $\cos 225^\circ$ and $\sin 225^\circ$.
|
-\frac{\sqrt{2}}{2}
| 0.083333 |
Evaluate $\log_{\sqrt{12}} (1728\sqrt{12})$.
|
7
| 0.916667 |
Calculate the residue of $-963 + 100 \pmod{35}$. The answer should be an integer in the range $0,1,2,\ldots,33,34$.
|
12
| 0.916667 |
A 6-foot by 12-foot floor is tiled with square tiles of size 1 foot by 1 foot. Each tile has a pattern consisting of four white quarter circles of radius 1/3 foot centered at each corner of the tile. The remaining portion of the tile is shaded. Calculate the total square feet of the floor that are shaded.
|
72 - 8\pi \text{ square feet}
| 0.916667 |
An infinite geometric series has a first term of $400$ and sums up to $2500$. What is the common ratio?
|
\frac{21}{25}
| 0.916667 |
If \( a, b, c, \) and \( d \) are real numbers satisfying:
\begin{align*}
a+b+c &= 5, \\
a+b+d &= 3, \\
a+c+d &= 8, \text{ and} \\
b+c+d &= 17,
\end{align*}
find \( ab + cd \).
|
30
| 0.916667 |
A square is divided into four congruent rectangles, each rectangle having the longer side parallel to the sides of the square. If the perimeter of each rectangle is 40 inches, what is the perimeter of the square?
|
64 \text{ inches}
| 0.75 |
How many license plates consist of 3 letters followed by 3 digits, where the sequence is letter-digit-letter-digit-letter-digit, and each digit alternates between odd and even starting with an odd number?
|
2,\!197,\!000
| 0.75 |
Find the sum of the coefficients in the polynomial $3(x^8 - 2x^5 + x^3 - 6) - 5(2x^4 + 3x^2) + 2(x^6 - 5)$.
|
-51
| 0.833333 |
What is the constant term of the expansion of \( \left(5x + \frac{2}{5x}\right)^8 \)?
|
1120
| 0.833333 |
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