problem
stringlengths 18
4.46k
| answer
stringlengths 1
942
| pass_at_n
float64 0.08
0.92
|
|---|---|---|
Given the length of the straight sections of the track is 200 meters, the width between the inside and outside of the track is 8 meters, and it takes Keiko 48 seconds longer to jog around the outside edge of the track than around the inside edge, determine Keiko's jogging speed in meters per second.
|
\frac{\pi}{3}
| 0.5 |
Given the number $1225$, find the number of different four-digit numbers that can be formed by rearranging its digits.
|
12
| 0.75 |
A triangle and a trapezoid share the same altitude and have equal areas. The base of the triangle is 24 inches. If one base of the trapezoid is 10 inches longer than the other base, find the median of the trapezoid.
|
12
| 0.916667 |
Given that any even integer greater than 7 can be expressed as the sum of two different prime numbers, determine the largest possible difference between the two primes for the even number 156.
|
146
| 0.583333 |
Given that a 7-digit palindrome must consist of the digits 1, 1, 4, 4, 4, 6, 6, calculate the total number of such palindromes.
|
6
| 0.083333 |
A point is chosen randomly from within a circular region with radius $r$. A related concentric circle with radius $\sqrt{r}$ contains points that are closer to the center than to the boundary. Calculate the probability that a randomly chosen point lies closer to the center than to the boundary.
|
\frac{1}{4}
| 0.916667 |
Given the expression $(3(3(3(3(3(3+2)+2)+2)+2)+2)+2)$, calculate its value.
|
1457
| 0.833333 |
How many different integers can be expressed as the sum of three distinct members of the set $\{2, 5, 8, 11, 14, 17, 20, 23\}$?
|
16
| 0.166667 |
What number should be removed from the list $1,2,3,4,5,6,7,8,9,10,12$ so that the average of the remaining numbers is $6.5$?
|
2
| 0.166667 |
Given that the monogram of a person's first, middle, and last initials is in alphabetical order with no letter repeated, and the last initial is 'M', calculate the number of possible monograms.
|
66
| 0.166667 |
Given that the euro is worth 1.1 dollars, and Charles has 600 dollars and Fiona has 450 euros, calculate the percentage by which the value of Fiona's money is greater than or less than the value of Charles' money.
|
17.5\%
| 0.583333 |
Given the graphs of $y = -|x-(a+1)| + b$ and $y = |x-(c-1)| + (d-1)$ intersect at points $(3,4)$ and $(7,2)$. Find $a+c$.
|
10
| 0.416667 |
Given that M/8 = 8/N, where M ≤ 64, calculate the number of ordered pairs of positive integers (M,N).
|
7
| 0.833333 |
Given that the base $b$ is a positive integer that satisfies the equation $\log_{b} 1024$ is a positive integer, calculate the number of values of $b$.
|
4
| 0.916667 |
What is the hundreds digit of $(30! - 20! +10!)$?
|
8
| 0.666667 |
Given that John drove at an average speed of 60 km/h, then stopped 30 minutes for lunch, and then drove at an average speed of 90 km/h, covering a total distance of 300 km in 4 hours including his lunch break, determine the equation that could be used to solve for the time t in hours that John drove before his lunch break.
|
60t + 90\left(\frac{7}{2} - t\right) = 300
| 0.083333 |
Let $n$ be the smallest positive integer such that $n$ is divisible by $30$, $n^2$ is a perfect cube, and $n^3$ is a perfect square. What is the number of digits of $n$?
|
9
| 0.666667 |
If a medium jar can hold $50$ milliliters of spices and a larger pantry container can hold $825$ milliliters of spices, and at most one jar might not be transferred successfully, calculate the minimum number of medium jars Harry must prepare to fill a large pantry container.
|
18
| 0.166667 |
Find the equation of the straight line passing through the point (1,3) and perpendicular to the line 2x-6y-8=0.
|
y + 3x - 6 = 0
| 0.083333 |
Given the expressions $k(a+bc)$ and $(a+b)(a+c)$ are equal, where $k$ is a constant, and the condition $a + b + c = 1$ holds, determine the value of $k$.
|
k=1
| 0.916667 |
In a survey with 45 people preferring Apples, 75 people preferring Bananas, 30 people preferring Cherries, and 50 people preferring Dragonfruit, calculate the percentage of surveyed people who preferred Bananas.
|
37.5\%
| 0.916667 |
Given the expression $\log_4{16} \div \log_{4}{\frac{1}{16}} + \log_4{32}$, simplify the expression.
|
1.5
| 0.166667 |
Given that the students in sixth grade run an average of $18$ minutes per day, seventh grade run an average of $20$ minutes per day and eighth grade run an average of $12$ minutes per day, and there are three times as many sixth graders as seventh graders and the number of eighth graders is half the number of seventh graders, and seventh graders spend an extra $5$ minutes on a stretching routine, determine the average number of minutes spent on physical activities per day by these students.
|
\frac{170}{9}
| 0.833333 |
Find the area of the rectangular region defined by the vertical lines $x = 2e$ and $x = -3f$, and the horizontal lines $y = 4g$ and $y = -5h$, where $e, f, g,$ and $h$ are positive numbers.
|
8eg + 10eh + 12fg + 15fh
| 0.166667 |
Given the sum of two numbers is S, if 5 is added to each number and then each of the resulting numbers is tripled, determine the sum of the final two numbers.
|
3S + 30
| 0.916667 |
Consider all triangles ABC with AB = AC, and point D on AC such that BD ⊥ AC. Given conditions are AC and CD are integers, and BD^2 = 68. Find the smallest possible value of AC when CD is a prime number less than 10.
|
18
| 0.166667 |
If $f(3x) = \frac{5}{3+x}$ for all $x>0$, find $3f(x)$.
|
\frac{45}{9 + x}
| 0.916667 |
Let $T_1$ be a triangle with side lengths $20, 21,$ and $29$. For $n \ge 1$, if $T_n = \triangle ABC$ and $D, E,$ and $F$ are the points of tangency of the incircle of $\triangle ABC$ to the sides $AB, BC$, and $AC,$ respectively, then $T_{n+1}$ is a triangle with side lengths $AD, BE,$ and $CF,$ if it exists. What is the perimeter of the last triangle in the sequence $( T_n )$.
|
35
| 0.166667 |
Consider the quadratic equation $2x^2 - 5x + m = 0$. Find the value of $m$ such that the sum of the roots of the equation is maximized while ensuring that the roots are real and rational.
|
\frac{25}{8}
| 0.083333 |
Given medians AD and BE of triangle ABC are perpendicular, AD=6, and BE=9. Compute the area of triangle ABC.
|
36
| 0.5 |
The probability that a randomly chosen divisor of $25!$ is odd.
|
\frac{1}{23}
| 0.666667 |
Find the arithmetic mean of $\frac{x + 2a}{x}$ and $\frac{x - 3a}{x}$, where $x \neq 0$ and $a$ is a constant.
|
1 - \frac{a}{2x}
| 0.666667 |
Given that Sadie exchanged $d$ U.S. dollars with an exchange rate of $8$ Canadian dollars for every $5$ U.S. dollars, and after her shopping spent $80$ Canadian dollars and had exactly $d$ Canadian dollars left, calculate the sum of the digits of $d$.
|
7
| 0.833333 |
In $\triangle ABC$, $D$ is on $AC$ and $E$ is on $BC$. Also, $AB \perp BC$, $AE \perp BC$, and $BD = DC = CE = x$. Find the length of $AC$.
|
2x
| 0.833333 |
Calculate the simplified value of the expression: $2 - \frac{1}{2 + \sqrt{5}} + \frac{1}{2 - \sqrt{5}}.$
|
2 - 2\sqrt{5}
| 0.333333 |
Calculate the number of digits in the number $4^{25}5^{22}$ when written in decimal form.
|
31
| 0.416667 |
A coffee shop offers a promotion stating, "25% off on coffee today: quarter-pound bags are available for $4.50 each." Determine the regular price for a half-pound of coffee, in dollars.
|
12
| 0.916667 |
Given the sprinter's age is $30$ years, calculate the target heart rate, in beats per minute, which is $85\%$ of the theoretical maximum heart rate found by subtracting the sprinter's age from $225$.
|
166
| 0.666667 |
Let $\angle PQR = 40^{\circ}$ and $\angle PQS = 15^{\circ}$. Find the smallest possible degree measure for $\angle SQR$.
|
25^\circ
| 0.916667 |
Given a number twelve times as large as $x$ is increased by five, determine one third of the result in terms of $x$.
|
4x + \frac{5}{3}
| 0.916667 |
The photographer wants to arrange three boys and three girls in a row such that a boy or a girl could be at each end, and the rest alternate in the middle, calculate the total number of possible arrangements.
|
72
| 0.416667 |
What is the smallest positive integer that is neither prime nor square and that has no prime factor less than 60?
|
4087
| 0.833333 |
The point $Q(c,d)$ in the $xy$-plane is first rotated clockwise by $90^\circ$ around the point $(2,3)$ and then reflected about the line $y = x$. The image of $Q$ after these two transformations is at $(4, -1)$. Find the value of $d - c$.
|
-1
| 0.166667 |
To hit at least 90% of his targets in 60 rounds, John successfully hits 54 targets. After 40 rounds, he has 20 targets remaining. Calculate the maximum number of rounds he can afford to miss.
|
0
| 0.25 |
Given that Mina writes down one integer three times and another integer four times, and their sum is $135$, and one of the numbers is $15$, find the other number.
|
25
| 0.916667 |
Two cards are dealt from a deck composed of three red cards labeled $A$, $B$, $C$, three green cards labeled $A$, $B$, $C$, and three blue cards labeled $A$, $B$, $C$. A winning pair consists of two cards of the same color or two cards with the same letter. Calculate the probability of drawing a winning pair.
|
\frac{1}{2}
| 0.916667 |
Mary divides a circle into 15 sectors. The central angles of these sectors, also measured in degrees, are all integers and form an arithmetic sequence. Find the degree measure of the smallest possible sector angle.
|
3
| 0.666667 |
Given that the least common multiple of $a$ and $b$ is $16$, and the least common multiple of $b$ and $c$ is $21$, determine the least possible value of the least common multiple of $a$ and $c$.
|
336
| 0.166667 |
Let the number of $2 pairs of socks be x, the number of $3 pairs of socks be y, and the number of $5 pairs of socks be z. Given that x + y + z = 15 and 2x + 3y + 5z = 35, determine the value of x.
|
12
| 0.083333 |
Given the set $A$ of positive integers that have no prime factors other than $2$ and $7$, express the infinite sum of the reciprocals of the elements of $A$ as a fraction $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, and calculate $m + n$.
|
10
| 0.083333 |
Calculate the sum of $\frac{3}{20} + \frac{5}{200} + \frac{7}{2000} + 5$.
|
5.1785
| 0.333333 |
Determine the number of significant digits in the measurement of the side of a square whose computed area is $2.3406$ square inches to the nearest ten-thousandth of a square inch.
|
5
| 0.416667 |
Given that the area of triangle $XYZ$ is $100$ square units, and the geometric mean between sides $XY$ and $XZ$ is $15$ units, find $\sin X$.
|
\frac{8}{9}
| 0.916667 |
Consider the operation "add the reciprocal of," defined by $a\diamond b=a+\frac{1}{b}$. Evaluate the expression $((3\diamond4)\diamond5)-(3\diamond(4\diamond5))$.
|
\frac{89}{420}
| 0.833333 |
If Menkara has a $3 \times 7$ index card, and she shortens the length of one side by $2$ inches, resulting in an area of $15$ square inches, determine the area of the card if she shortens the length of the other side by $2$ inches.
|
7
| 0.416667 |
Find the sum of the squares of all real numbers satisfying the equation $x^{64} - 64^{16} = 0$.
|
16
| 0.75 |
Eric sets a new goal to complete a modified triathlon in 3 hours. He plans to swim 0.5 miles at an average speed of 1.5 miles per hour, run 4 miles at an average speed of 5 miles per hour, and then ride a bicycle for 20 miles. What must his average speed in miles per hour be for the bicycle ride to achieve his goal?
|
\frac{75}{7}
| 0.916667 |
Five different awards are to be given to four students. Each student will receive at least one award. Calculate the total number of ways the awards can be distributed.
|
240
| 0.916667 |
Given that A can do a piece of work in $12$ days, and B is $66\frac{2}{3}\%$ more efficient than A, calculate the number of days it will take for A and B to complete the same piece of work when working together.
|
4.5
| 0.916667 |
Given the area of a square is $1.2105$ square inches to the nearest ten-thousandth of a square inch, determine the number of significant digits in the measurement of the diagonal of the square.
|
5
| 0.75 |
Given a tetrahedron, calculate the sum of the number of edges, the number of vertices, and the number of faces, assuming Joe counted one vertex twice.
|
15
| 0.666667 |
$\dfrac{13!-12!}{10!}$
|
1584
| 0.833333 |
Triangle $DEF$ has a right angle at $D$. The sides of the triangle serve as diameters for semicircles. The area of the semicircle on $\overline{DE}$ equals $12.5\pi$, and the arc length of the semicircle on $\overline{DF}$ is $7\pi$. Determine the radius of the semicircle on $\overline{EF}$.
|
\sqrt{74}
| 0.916667 |
Each of a group of $60$ girls is either blonde or brunette and has either blue eyes or brown eyes. If $20$ are blue-eyed blondes, $36$ are brunettes, and $25$ are brown-eyed, then calculate the number of brown-eyed brunettes.
|
21
| 0.416667 |
In John's first $6$ basketball games, he scored $10, 5, 8, 6, 11,$ and $4$ points. In his seventh game, he scored fewer than $15$ points and his points-per-game average for the seven games was an integer. For the eighth game, he also scored fewer than $15$ points, and the average for the $8$ games became an integer again. Determine the product of the number of points he scored in the seventh and eighth games.
|
35
| 0.833333 |
A positive integer $n$ not exceeding $120$ is chosen such that if $n\le 60$, then the probability of choosing $n$ is $p$, and if $n > 60$, then the probability of choosing $n$ is $3p$. Determine the probability that a perfect square is chosen.
|
\frac{1}{15}
| 0.416667 |
The bakery owner turns on his muffin machine at $\text{9:00}\ {\small\text{AM}}$ and completes one fourth of the day's job at $\text{12:15}\ {\small\text{PM}}$. Determine the time at which the muffin machine will complete the job.
|
10:00 PM
| 0.916667 |
Given 4 mathematics courses and 8 available periods in a day, find the number of ways a student can schedule the courses so that no two mathematics courses are taken in consecutive periods.
|
120
| 0.583333 |
A digital watch displays time in a 24-hour format showing only hours and minutes. Find the largest possible sum of the digits in the display.
|
24
| 0.083333 |
Given that group A consisting of $a$ cows produces $b$ gallons of milk in $c$ days, and group B, which is $20\%$ more efficient in milk production than group A, has $d$ cows, calculate the number of gallons of milk that group B will produce in $e$ days.
|
\frac{1.2bde}{ac}
| 0.583333 |
Given that Peter won 5 games and lost 4 games, Emma won 4 games and lost 5 games, and Jordan lost 2 games, find the number of games Jordan won.
|
2
| 0.5 |
Given points $Q_1, Q_2, \ldots, Q_9$ on a straight line, in the order stated (not necessarily evenly spaced). Let $Q$ be a point on the line, and let $t$ be the sum of the undirected lengths $QQ_1, QQ_2, \ldots, QQ_9$. Determine the location of point $Q$ such that $t$ is minimized.
|
Q_5
| 0.833333 |
Given that 63 students wear red shirts and 81 students wear green shirts, and the 144 students are divided into 72 pairs, while 27 of these pairs consist of two students wearing red shirts, find the number of pairs with two students wearing green shirts.
|
36
| 0.25 |
Given that $a_n = \frac{(n+7)!}{(n-1)!}$, determine the rightmost digit of $a_k$ when it stops changing for the smallest positive integer $k$ such that the rightmost digit of $a_k$ stops changing after reaching $k+5$.
|
0
| 0.916667 |
Given the track consists of 3 semicircular arcs with radii $R_1 = 120$ inches, $R_2 = 70$ inches, and $R_3 = 90$ inches, and a ball with a diameter of 6 inches rolls from point A to point B, calculate the distance the center of the ball travels.
|
271\pi
| 0.166667 |
Given that the activity lasts for 120 minutes and eight children participate, with only four children playing at any one time, calculate the total time each child plays.
|
60
| 0.666667 |
Stephan has 60% more chocolates than Mary, and Clara has 40% more chocolates than Mary. Calculate the percentage difference in chocolates between Stephan and Clara.
|
14.29\%
| 0.416667 |
Jasmine wishes to fill the decorative container holding 2650 milliliters entirely with craft bottles that hold 150 milliliters each. Calculate the number of craft bottles necessary for this purpose.
|
18
| 0.916667 |
Right triangle DEF has leg lengths DE = 18 and EF = 24. If the foot of the altitude from vertex E to hypotenuse DF is F', then find the number of line segments with integer length that can be drawn from vertex E to a point on hypotenuse DF.
|
10
| 0.083333 |
The base of an isosceles triangle is 20 inches. A line is drawn parallel to the base, which divides the triangle into two regions where the area of the smaller region is $\frac{1}{4}$ the area of the triangle. Determine the length of this line parallel to the base.
|
10
| 0.916667 |
A square is drawn inside a rectangle. The ratio of the width of the rectangle to a side of the square is $3:1$ and the ratio of the rectangle's length to its width is $3:2$. Calculate the percentage of the rectangle’s area that is inside the square.
|
7.41\%
| 0.666667 |
Given that Alex wins $\frac{1}{3}$ of the rounds, Mel's probability of winning is $\frac{p}{3}$ and Chelsea is three times as likely to win as Mel, determine the probability that Alex wins four rounds, Mel wins one round, and Chelsea wins one round.
|
\frac{5}{162}
| 0.916667 |
Peter's family ordered a 16-slice pizza for dinner. Peter ate one slice of pizza alone and then shared another slice with his brother Paul and their sister Mary, each having an equal part of that slice. Calculate the amount of pizza that Peter ate in total.
|
\frac{1}{12}
| 0.75 |
Express the repeating decimal $7.171717\ldots$ as a fraction and compute the sum of its numerator and denominator in their lowest terms.
|
809
| 0.916667 |
Two candles of the same length are made of different materials such that one burns out completely at a uniform rate in $5$ hours and the other in $3$ hours. At what time should the candles be lighted so that, at 5 P.M., one stub is three times the length of the other?
|
2:30 \text{ PM}
| 0.75 |
Consider a set of $n$ numbers where $n > 2$. Two of these numbers are $1 - \frac{1}{n}$ and $1 - \frac{2}{n}$, and the remaining $n-2$ numbers are all $1$. Calculate the arithmetic mean of these $n$ numbers.
|
1 - \frac{3}{n^2}
| 0.916667 |
Given the expression $2 - (-3)^2 - 4 - (-5) - 6^2 - (-7)$, simplify the arithmetic expression.
|
-35
| 0.916667 |
If $7 = k \cdot 3^s$ and $126 = k \cdot 9^s$, determine the value of $s$.
|
2 + \log_3 2
| 0.083333 |
John went to the bookstore and purchased 20 notebooks totaling $62. Some notebooks were priced at $2 each, some at $5 each, and some at $6 each. John bought at least one notebook of each type. Let x be the number of $2 notebooks, y be the number of $5 notebooks, and z be the number of $6 notebooks. Solve for x.
|
14
| 0.333333 |
A $9 \times 9$ grid has a total of 81 squares, and the first square is dark. What is the difference in the number of dark squares and light squares on this grid?
|
1
| 0.75 |
Rectangle $ABCD$ is inscribed in a semicircle with diameter $\overline{FE}$. Let $DA=20$, and let $FD=AE=12$. Calculate the area of rectangle $ABCD$.
|
160\sqrt{6}
| 0.083333 |
In how many ways can $435$ be written as the sum of an increasing sequence of two or more consecutive positive integers?
|
7
| 0.666667 |
Alex sent 200 text messages and talked for 51 hours, and the cell phone plan costs $25$ dollars each month, $5$ cents per text message sent, and $15$ cents for each minute used over $50$ hours. Calculate the total cost for Alex's cell phone plan in February.
|
44
| 0.833333 |
A rectangle has a length of 12 meters and a width of 8 meters. The rabbit runs 8 meters along a diagonal toward the opposite corner, makes a 90-degree right turn, and runs 3 more meters. Calculate the average of the shortest distances from the rabbit to each side of the rectangle.
|
5
| 0.583333 |
A line $x = k$ intersects the graphs of $y = \log_5 x$ and $y = \log_5 (x + 8)$. The distance between the points of intersection is $1$. Express $k$ in the form $a + \sqrt{b}$, where $a$ and $b$ are integers, and find $a + b$.
|
2
| 0.5 |
Given Michael walks at the rate of $4$ feet per second on a long straight path, and the garbage truck travels at $8$ feet per second in the same direction as Michael, calculate the number of times Michael and the truck will meet.
|
1
| 0.083333 |
What is the smallest positive even integer $n$ such that the product $3^{1/6}3^{2/6}\cdots3^{n/6}$ is greater than $500$?
|
8
| 0.833333 |
Find the total number of ordered pairs of integers \((x, y)\) that satisfy the equation \(x^{4} + y^2 = 4y\).
|
2
| 0.25 |
A standard 6-sided die and an 8-sided die are rolled once. The sum of the numbers rolled determines the side length of a square. What is the probability that the numerical value of the area of the square is less than the numerical value of the square's perimeter?
|
\frac{1}{16}
| 0.75 |
Given that a class with \( n > 15 \) students achieved an average quiz score of \( 12 \) and a subset of \( 15 \) students from this class has an average score of \( 20 \), determine the average score of the remaining students in terms of \( n \).
|
\frac{12n-300}{n-15}
| 0.916667 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.