problem
stringlengths 18
4.46k
| answer
stringlengths 1
942
| pass_at_n
float64 0.08
0.92
|
|---|---|---|
How many ways can a student schedule $4$ science courses – physics, chemistry, biology, and earth science – in an $8$-period day if no two science courses can be taken in consecutive periods? | 120 | 0.75 |
A ferry boat shuttles tourists to an island every hour starting at 10 AM until its last trip, which starts at 2 PM. On the 10 AM trip, there were 85 tourists on the ferry boat, and on each successive trip, the number of tourists was 3 more than on the previous trip. Calculate the total number of tourists the ferry took to the island that day. | 455 | 0.916667 |
Find the number of distinct points in the $xy$-plane where the graphs of $(x+2y-8)(3x-y+6)=0$ and $(2x-3y+2)(4x+y-16)=0$ intersect. | 4 | 0.916667 |
Given the product of the digits of a 3-digit positive integer equals 36, determine the number of such integers. | 21 | 0.333333 |
Find the fifth term in the expansion of \((\frac{a}{x} + \frac{x}{a^3})^8\). | \frac{70}{a^8} | 0.916667 |
A modified sign pyramid with five levels, where a cell gets a "+" if the two cells below it have the same sign, and it gets a "-" if the two cells below it have different signs. If a "-" is to be at the top of the pyramid, calculate the number of possible ways to fill the five cells in the bottom row. | 16 | 0.666667 |
Calculate the value of $(3(3(3(3(3(3+2)+2)+2)+2)+2)+2)$. | 1457 | 0.5 |
An iterative process is used to find an average of the numbers -1, 0, 5, 10, and 15. Arrange the five numbers in a certain sequence. Find the average of the first two numbers, then the average of the result with the third number, and so on until the fifth number is included. What is the difference between the largest and smallest possible final results of this iterative average process? | 8.875 | 0.416667 |
Given the equation $m+n = mn - 1$, determine the number of pairs $(m,n)$ of integers that satisfy this equation. | 4 | 0.833333 |
Ike and Mike enter a bakery with a total of $50.00 to spend. Sandwiches cost $6.00 each and pastries cost $1.50 each. They decide to buy as many sandwiches as they can with their money, then spend the remainder on pastries. Calculate the total number of items (sandwiches and pastries) they will buy. | 9 | 0.916667 |
Given a box with dimensions $4\text{-in} \times 3\text{-in} \times 6\text{-in}$ and blocks with dimensions $3\text{-in} \times 1\text{-in} \times 2\text{-in}$, calculate the maximum number of blocks that can fit in the box. | 12 | 0.5 |
Given Professor Chang has ten different language books, with two Arabic, four German, and four Spanish, calculate the number of arrangements of the books on the shelf while keeping the Arabic books together and the Spanish books together. | 34560 | 0.166667 |
What is the least possible value of the expression (x+1)(x+2)(x+3)(x+4) + 2021 where x is a real number? | 2020 | 0.75 |
If the product $\dfrac{4}{3}\cdot \dfrac{5}{4}\cdot \dfrac{6}{5}\cdot \ldots\cdot \dfrac{a}{b} = 16$, calculate the sum of $a$ and $b$. | 95 | 0.833333 |
Given that A can finish a task in 12 days and B is 75% more efficient than A, determine the number of days it takes B to complete the same task. | \frac{48}{7} | 0.833333 |
A rug is made with three differently colored regions, and their areas form an arithmetic progression. The inner rectangle is two feet wide, and each of the two shaded regions is $2$ feet wide on all four sides. Determine the length in feet of the inner rectangle. | 4 | 0.25 |
The Red Robin High School chess team consists of three boys and four girls. A photographer wants to take a picture of the team for a school magazine. She decides to have them sit in a row with a boy at each end and the remaining team members in the middle. Determine the number of arrangements possible. | 720 | 0.75 |
A symposium began at 3:00 p.m. and lasted for 1500 minutes. What time did the symposium end? | 4:00 \text{ p.m.} | 0.916667 |
$\dfrac{13! - 12!}{10!}$ | 1584 | 0.666667 |
Given that in a 60-question multiple choice math contest, students receive 5 points for a correct answer, 0 points for an answer left blank, and -2 points for an incorrect answer, and Evelyn's total score on the contest was 150, determine the maximum number of questions that Evelyn could have answered correctly. | 38 | 0.833333 |
Consider the graphs of $y=3\log{x}$ and $y=\log{3x}$. Determine the value(s) of $x$ where they intersect. | \sqrt{3} | 0.916667 |
Given the sequence where $x_{k+1} = x_k + \frac{1}{3}$ for $k = 1, 2, \dots, n-1$ and $x_1 = 2$, find the sum $x_1 + x_2 + \dots + x_n$. | \frac{n(n+11)}{6} | 0.583333 |
Calculate the sum: $$\dfrac{3}{15}+\dfrac{6}{15}+\dfrac{9}{15}+\dfrac{12}{15}+\dfrac{15}{15}+\dfrac{18}{15}+\dfrac{21}{15}+\dfrac{24}{15}+\dfrac{27}{15}+\dfrac{75}{15}.$$. | 14 | 0.916667 |
Queen Middle School has 1500 students. Each student takes 6 classes a day. Each teacher teaches 5 classes. Each class has 25 students and 1 teacher. Find the number of teachers at Queen Middle School. | 72 | 0.833333 |
A bakery prepares a giant cookie sheet measuring 30 inches by 24 inches. The cookies are intended to be 3 inches by 4 inches each. Calculate the number of full cookies that can be made from the sheet. | 60 | 0.916667 |
Given the ratio of $w$ to $x$ is $4:3$, the ratio of $y$ to $z$ is $5:3$, and the ratio of $z$ to $x$ is $1:6$, determine the ratio of $w$ to $y$. | \frac{24}{5} | 0.916667 |
Let P be the total number of pages in the novel. On the first day, Paul read \(\frac{1}{6}\) of the pages plus 10 more. On the second day, he read \(\frac{1}{5}\) of the remaining pages plus 14 pages, and on the third day, he read \(\frac{1}{4}\) of the remaining pages plus 16 pages. After these readings, exactly 48 pages were left. Determine the total number of pages in the novel. | 161 | 0.5 |
Given the polynomial expression \( x^4 - 61x^2 + 60 \), for how many integers \( x \) is the expression negative. | 12 | 0.583333 |
Calculate the sum of the numbers in the four corners of a 9x9 checkerboard extended with numbers 10 to 90. | 200 | 0.25 |
A right rectangular prism whose edge lengths are $\log_{5}x, \log_{6}x,$ and $\log_{7}x$ and whose surface area is twice its volume. Find the value of $x$. | 210 | 0.25 |
Given a rectangle divided into a 2x4 grid of equally spaced points, calculate the total number of distinct triangles that can be formed using three of these points as vertices. | 48 | 0.416667 |
A month has 30 days in which the number of Mondays and Fridays are the same. How many of the seven days of the week could be the first day of this month? | 3 | 0.083333 |
A set of 36 square blocks is arranged into a 6 × 6 square. Calculate the number of different combinations of 4 blocks that can be selected from that set so that no two are in the same row or column. | 5400 | 0.916667 |
Given Zara has 5 marbles: an Aggie, a Bumblebee, a Steelie, a Tiger, and a Cat's Eye, find the number of ways to arrange them in a row on her shelf such that the Steelie and the Tiger are not next to each other and the Aggie and the Bumblebee are not adjacent. | 48 | 0.916667 |
The circle having (0,0) and (10,10) as the endpoints of a diameter intersects the x-axis at a point. Find the x-coordinate of this point. | 10 | 0.75 |
Given the numbers $1357, 2468, 3579, 4680, 5791$, calculate their sum. | 17875 | 0.333333 |
Given points A and B are 12 units apart in a plane, find the number of points C such that the perimeter of triangle ABC is 60 units and the area of triangle ABC is 150 square units. | 0 | 0.25 |
Find the value of $x$ and $y$ that satisfies the equation: $5^{-6} = \frac{5^{56/x}}{5^{32/y} \cdot 25^{22/x}}$. | x = 3, y = 4 | 0.333333 |
A store owner buys a piece of equipment at $50$ dollars less 20%. He then wishes to sell the equipment at a gain of 25% of his cost after allowing a 15% discount on his marked price. Determine the price at which the equipment should be marked. | 58.82 | 0.166667 |
Six points on a circle are numbered 1, 2, 3, 4, 5, and 6 in clockwise order. A bug jumps in a clockwise direction from one point to another around the circle; if it is on an odd-numbered point, it moves two points, and if it is on an even-numbered point, it moves one point. If the bug begins on point 6, after 2023 jumps calculate the point where the bug will be. | 1 | 0.166667 |
A box contains 21 red balls, 17 green balls, 24 yellow balls, 10 blue balls, 14 white balls, and 14 black balls. Find the minimum number of balls that must be drawn without replacement to guarantee that at least 16 balls of a single color will be drawn. | 84 | 0.166667 |
Two cyclists, $C$ and $D$, start at the same time to ride from town X to town Y and town Z respectively, where town Y is $90$ miles away from X and town Z is $120$ miles away. Cyclist $C$ travels $6$ mph slower than cyclist $D$, who reaches town Y and then travels $30$ miles further before meeting cyclist $C$, who is still en route to town Y. Determine the speed of cyclist $C$. | 6 | 0.083333 |
Given that all three vertices of $\bigtriangleup ABC$ lie on the parabola defined by $y = 2x^2$, with $A$ at the origin and $\overline{BC}$ parallel to the $x$-axis, find the length of $BC$ given that the area of the triangle is $128$. | 8 | 0.5 |
Six different awards are to be given to four students, with each student receiving at least one award, find the total number of different ways the awards can be distributed. | 1560 | 0.833333 |
What is the value of $\dfrac{13! - 12!}{10!}$? | 1584 | 0.75 |
Given a wooden cube $n$ units on a side is painted blue on all six faces and then cut into $n^3$ unit cubes, find the value of $n$ if exactly one-third of the total number of faces of the unit cubes are blue. | 3 | 0.75 |
Two pictures, each 2 feet across, are hung in the center of a wall that is 25 feet wide with 1 foot of space between them. Calculate the distance from the end of the wall to the nearest edge of the first picture. | 10 | 0.166667 |
Three wheels are spun, and each wheel selects one number. Calculate the probability that the sum of the three selected numbers is even, given the first wheel has a probability $\frac{3}{4}$ of landing on an even number, the second wheel has a probability $\frac{1}{2}$, and the third wheel has a probability $\frac{1}{4}$ of landing on an even number. | \frac{1}{2} | 0.916667 |
Given the quadratic equation $y = x^2 + px + r$, where $r = q - \frac{p^2}{4}$, determine the value of $q$ for which the minimum value of $y$ is zero. | \frac{p^2}{2} | 0.916667 |
Given that Queen Middle School has 1500 students, each student takes 5 classes a day, each teacher teaches 5 classes, and each class has 25 students and 1 teacher, determine the total number of teachers at Queen Middle School. | 60 | 0.416667 |
Find the minimum value of $\sqrt{x^2+y^2}$ if $8x+15y=120$ and $x \geq 0$. | \frac{120}{17} | 0.833333 |
Given the numbers $1, 2, 3, 4, 5, 6, 7, 8, 9$ written in a $3\times3$ array of squares, with each consecutive pair sharing an edge, and the numbers in the four corners adding up to $20$ and the numbers in the middle row adding up to $15$, calculate the number in the center. | 5 | 0.916667 |
If a box of marbles contains $\frac{2}{3}$ green marbles, while the rest are yellow, and the number of yellow marbles is tripled with the number of green marbles staying the same, what fraction of the marbles will be yellow? | \frac{3}{5} | 0.833333 |
Given $x = 1+3^p$ and $y = 1+3^{-p}$, determine the expression for $y$ in terms of $x$. | \frac{x}{x-1} | 0.916667 |
Solve the equation where \(\log(3x) - 4 \log 9 = 3\). | 2187000 | 0.916667 |
Given that Moe needs to mow his larger rectangular lawn measuring $120$ feet by $200$ feet, the swath of the mower remains $28$ inches, but this time he overlaps each pass by $6$ inches to ensure no grass is missed, and Moe walks at the rate of $4000$ feet per hour with the mower, calculate the number of hours it will take Moe to mow the entire lawn. | 3.3 | 0.583333 |
If five fish are traded for three loaves of bread and two loaves of bread are traded for seven bags of rice, determine the value of one fish in terms of bags of rice. | 2.1 | 0.25 |
Anthony's family ordered a 16-slice pizza for dinner. Anthony ate one slice, shared another slice equally with his brother Ben, and shared a third slice equally with his cousin Chris. What fraction of the pizza did Anthony eat? | \frac{1}{8} | 0.916667 |
Find $n$ which represents the number of integer values of $x$ such that $Q = x^4 + 4x^3 + 10x^2 + 4x + 29$ is the square of an integer. Determine $n$. | 0 | 0.75 |
Given that the two children from the youngest generation receive a $60\%$ discount, the two members of the middle generation pay full price, and the two members of the oldest generation receive a $20\%$ discount as senior citizens, and the senior ticket costs $\$7.50$, calculate the total amount Grandfather must pay for the concert and a $5 handling fee. | 46.25 | 0.083333 |
When simplified, $\log_{16}{32} \div \log_{16}{\frac{1}{2}}$, calculate the result. | -5 | 0.916667 |
A positive integer $N$ with three digits in its base ten representation is chosen at random, with each three-digit number having an equal chance of being chosen. Calculate the probability that $\log_3 N$ is an integer. | \frac{1}{450} | 0.916667 |
Two integers have a sum of $24$. When two more integers are added to the first two, the sum becomes $39$. Finally, when two more integers are added to the sum of the previous $4$ integers, the sum becomes $58$. What is the minimum number of even integers among the $6$ integers? | 2 | 0.5 |
Given the graphs of $y = -2|x-a| + b$ and $y = 2|x-c| + d$, where they intersect at points $(5,10)$ and $(11,6)$, find the value of $a+c$. | 16 | 0.666667 |
The $y$-intercepts, $R$ and $S$, of two perpendicular lines intersecting at the point $B(4,10)$ have a sum of zero. Find the area of $\triangle BRS$. | 8\sqrt{29} | 0.75 |
Carlos is constructing a rectangular storage unit using one-foot cubical blocks. The storage unit is 15 feet long, 12 feet wide, and 8 feet high. The walls and the floor are 1.5 feet thick. Calculate the total number of blocks contained in the storage unit. | 738 | 0.166667 |
Given that $a$, $b$, and $c$ are real numbers, solve the equation $4x - 7 + a = 2bx + c$ for a unique solution $x$, under the condition that $c \neq 0$. | x = \frac{c + 7 - a}{4 - 2b} | 0.25 |
Find the integer value nearest to $(\sqrt{5}+\sqrt{3})^4$. | 248 | 0.833333 |
Julia drives to a mountain retreat, covering 120 miles on the interstate, 40 miles on a mountain road, and 5 miles on a dirt track. She drives twice as fast on the interstate as on the mountain road and four times as fast on the interstate as on the dirt track. If Julia spent 60 minutes driving on the interstate, calculate the total time of her entire trip. | 110 | 0.75 |
What is the sum of the exponents of the prime factors of the square root of the largest perfect square that divides $15!$? | 10 | 0.75 |
When a certain unfair die is rolled, an odd number is $4$ times as likely to appear as an even number. The die is rolled twice. Calculate the probability that the sum of the numbers rolled is odd. | \frac{8}{25} | 0.25 |
A rectangular floor measures $a$ by $b$ feet, where $a$ and $b$ are positive integers and $b > a$. A person paints a rectangle on the floor with the sides of the rectangle parallel to the floor, but now leaving a border of width $2$ feet all around. The unpainted border part occupies half of the total area of the floor. Determine the number of possibilities for the ordered pair $(a, b)$. | 3 | 0.916667 |
In triangle $ABC$, point $D$ is on side $BC$ such that $BD:DC = 1:2$. A line through $A$ and $D$ intersects $BC$ at $E$. If the area of triangle $ABE$ is $30$, find the total area of triangle $ABC$. | 90 | 0.583333 |
Each disc can now hold a maximum of 75 minutes, and the narration takes 6 hours and 45 minutes to complete. How many minutes of narration will each disc contain? | 67.5 | 0.666667 |
Given a list of the lengths of 24 students' names from a classroom, with 9 names of 4 letters each, 6 names of 5 letters, 2 names of 6 letters, and 7 names of 7 letters, determine the median length of these names. | 5 | 0.833333 |
Let $x=-1008$. Evaluate the expression $\Bigg\vert\Big\vert |x| - 2x\Big\vert - |x|\Bigg\vert - x$. | 3024 | 0.75 |
Given that Max visits Sylvia every 5 days, Nora every 6 days, and Olivia every 7 days, and all three visited her yesterday, calculate the number of days in the next 365-day period when exactly two friends visit her. | 27 | 0.666667 |
Given the sequence of even counting numbers starting from $0$, find the sum of the first $1500$ terms. Then, given the sequence of odd counting numbers, find the sum of the first $1500$ terms, and calculate their difference. | 1500 | 0.916667 |
What is the smallest positive integer $n$ such that $\sqrt{n} - \sqrt{n-1} < 0.02$? | 626 | 0.75 |
On a digital watch that displays time in a 24-hour format, with the time shown as hours and minutes, calculate the largest possible sum of the digits in the display. | 24 | 0.083333 |
Given Marissa has 2035 coins, consisting of pennies and nickels, with at least three pennies and at least two nickels, calculate the difference in cents between the greatest and least amounts of money she can have. | 8120 | 0.416667 |
What is the value of $4321 + 3214 + 2143 - 1432$? | 8246. | 0.916667 |
Given that Isabella's fort has dimensions $15$ feet in length, $12$ feet in width, and $6$ feet in height, with one-foot thick floor and walls, determine the number of one-foot cubical blocks required to construct this fort. | 430 | 0.416667 |
Doug can paint a room in 4 hours, and Dave can paint the same room in 6 hours. They start painting together but must stop for a 30-minute meeting after which they continue to finish the room. Let $t$ be the total time, in hours, required for them to complete the job working together, including the meeting. Which equation correctly represents this situation? | \left(\frac{5}{12}\right)(t-0.5) = 1 | 0.666667 |
Given right triangle $\triangle ABC$ has a hypotenuse $AC$ of length 10, where $A$ is at the right angle. Point $M$ is the midpoint of hypotenuse $\overline{AC}$, and $D$ is a point on leg $\overline{AB}$ such that $AD:DB = 1:2$. Determine the area of $\triangle ADM$. | 4 | 0.25 |
A 20-quart radiator contains 18 quarts of water and 2 quarts of oil. Five quarts of this mixture are removed and replaced with pure antifreeze. This replacement process is done two more times. Calculate the fractional part of the final mixture that is water. | \frac{243}{640} | 0.416667 |
Determine the value of $m$ such that the polynomial $6x^3 - 12x^2 + mx - 24$ is divisible by $x - 4$. | -42 | 0.916667 |
Given that in a bag of marbles, $\frac{1}{3}$ of the marbles are blue, $\frac{1}{3}$ are red, and the rest are yellow, if the number of red marbles is doubled and the number of blue marbles stays the same, determine the fraction of the marbles that will be red. | \frac{1}{2} | 0.166667 |
Given that points $A$ and $B$ are $8$ units apart, find the number of lines in a given plane containing $A$ and $B$ that are $3$ units from $A$ and $4$ units from $B$, and also intersect the line $y = x$ at a $45^\circ$ angle. | 0 | 0.25 |
December 31, 2020, was a leap day. What time will it be 2023 minutes after midnight? | 9:43 \text{ AM} | 0.583333 |
Calculate the sum of the numbers $2143 + 3412 + 4213 + 1324$. | 11092 | 0.75 |
Given the distance light travels in one year is approximately $9,460,800,000,000$ kilometers, calculate the distance light travels in $70$ years. | 6.62256 \times 10^{14} | 0.333333 |
Given a boat with a speed of 20 mph in still water and a stream current of 4 mph, calculate the ratio of the average speed for a round trip downstream and then upstream to the speed in still water. | \frac{24}{25} | 0.5 |
Evaluate the expression $3 + \sqrt{3} + \frac{1}{3 + \sqrt{3}} + \frac{1}{\sqrt{3} - 3}$. | 3 + \frac{2\sqrt{3}}{3} | 0.916667 |
An artist begins filling in a large mosaic at 9:00 AM. By 12:45 PM she has completed one-fourth of the mosaic. Find the time when she will finish the entire mosaic. | 12:00 \text{ AM next day} | 0.666667 |
How many whole numbers are between $\sqrt[3]{50}$ and $\sqrt{200}$? | 11 | 0.75 |
What is the value of $\dfrac{12!-10!}{8!}$? | 11790 | 0.916667 |
A list of $2023$ positive integers has a unique mode, which occurs exactly $11$ times. Determine the least number of distinct values that can occur in the list. | 203 | 0.333333 |
In Marco's first $6$ soccer games, he scored $2, 5, 1, 4, 6,$ and $3$ goals. In his seventh game, he scored fewer than $10$ goals, and his goals-per-game average for these $7$ games was an integer. In his eighth game, he also scored fewer than $10$ goals and the goals-per-game average for the $8$ games became an integer too. Find the product of the number of goals he scored in the seventh and eighth games. | 28 | 0.416667 |
Twelve points are spaced around a $3 \times 3$ square at intervals of one unit. Two of the 12 points are chosen at random. Find the probability that the two points are one unit apart. | \frac{2}{11} | 0.333333 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.