problem
stringlengths 18
4.46k
| answer
stringlengths 1
942
| pass_at_n
float64 0.08
0.92
|
|---|---|---|
In rectangle $JKLM$, $JL=2$, and $LM=4$. Points $N$, $O$, and $P$ are midpoints of $\overline{LM}$, $\overline{MJ}$, and $\overline{JK}$, respectively. Point $Q$ is the midpoint of $\overline{NP}$. Find the area of the triangle formed by points $Q$, $O$, and $P$.
|
1
| 0.166667 |
Given that the sequence starts with 2 and alternates by adding 2 between consecutive terms, find the 30th term of this arithmetic sequence.
|
60
| 0.75 |
Given a set $\{4,6,8,12,14,18\}$, select three different numbers, add two of these numbers, multiply their sum by the third number, and finally subtract the smallest number you initially selected. Find the smallest result that can be obtained from this process.
|
52
| 0.333333 |
The diameter of a circle is divided into n equal parts. On each part a quarter-circle is constructed. As n becomes very large, calculate the sum of the lengths of the arcs of the quarter-circles.
|
\frac{\pi D}{2}
| 0.083333 |
Given a three-digit number with digits that sum exactly to $14$ and have the first digit equal to the last digit, find the total count of such numbers that are not divisible by $5$.
|
4
| 0.416667 |
Given that $(xy-2)^2 + (x+y-1)^2$ represents a sum of squares of real numbers, calculate the least possible value.
|
2
| 0.25 |
Star lists the whole numbers $1$ through $50$ once. Emilio copies Star's numbers, but he replaces each occurrence of the digit $2$ by the digit $1$ and each occurrence of the digit $3$ by the digit $2$. Calculate the difference between Star's sum and Emilio's sum.
|
210
| 0.083333 |
In a quadrilateral ABCD, if $\angle A = 70^\circ$, $\angle B = 60^\circ$, and $\angle C = 40^\circ$, calculate the measurement of $\angle D$.
|
190^{\circ}
| 0.5 |
In ∆ABC, points D and E are interior points such that lines AD, BE, and CE intersect at point F. Let x, y, z, w, v represent the angles at F in degrees. Solve for x in terms of y, z, w, v.
|
360^\circ - y - z - w - v
| 0.333333 |
Queen High School has 1500 students, each taking 6 classes a day, and there are 25 students in each class. Each teacher teaches 5 classes per day. Calculate the number of teachers needed at Queen High School.
|
72
| 0.833333 |
A circle with a radius of 6 is inscribed in a rectangle. The ratio of the length of the rectangle to its width is 3:1. Calculate the area of the rectangle.
|
432
| 0.916667 |
Eight spheres, each with a radius of 2, are placed one in each octant. Each sphere is tangent to the coordinate planes. Calculate the radius of the smallest sphere, centered at the origin, that can enclose all eight of these spheres.
|
2\sqrt{3} + 2
| 0.916667 |
Ten friends ate at a restaurant and agreed to share the bill equally. However, because Sam forgot his money, each of his nine friends paid an extra $3 to cover his share of the total bill. What is the total bill?
|
270
| 0.916667 |
Determine the number of integer solutions $(x, y)$ to the equation $x^3 + 4x^2 - 11x + 30 = 8y^3 + 24y^2 + 18y + 7$.
|
0
| 0.666667 |
A fair 8-sided die is rolled twice. Find the probability that the sum of the numbers that come up is a prime number.
|
\frac{23}{64}
| 0.916667 |
A circle is inscribed in a triangle with side lengths 12, 16, and 20. Let the segments of the side of length 12, made by a point of tangency, be $r$ and $s$, with $r<s$. Determine the ratio $r:s$.
|
\frac{1}{2}
| 0.333333 |
Andrew and Bella are jogging on a circular track at different speeds, with Andrew completing a lap every 75 seconds and Bella finishing a lap every 120 seconds. Both start from the same line simultaneously. A photographer, standing at a point one-third of the way from the starting line around the track, will take a photo between 9 and 10 minutes after they start, capturing one-fourth of the track around his standing point. Determine the probability that both Andrew and Bella will be captured in this photo.
|
0
| 0.25 |
How many ordered pairs (m, n) of positive integers, with m ≥ n, have the property that their squares differ by 150?
|
0
| 0.583333 |
Determine the median of the list of numbers given by:
\[1, 2, 3, \ldots, 3030, 1^2, 2^2, 3^2, \ldots, 3030^2, 1^3, 2^3, 3^3, \ldots, 3030^3.\]
A) $2297339.5$
B) $2297340.5$
C) $2297341.5$
D) $2297342.5$
|
2297340.5
| 0.166667 |
A square is drawn inside a rectangle. The ratio of the width of the rectangle to a side of the square is $2:1$. The ratio of the rectangle's length to its width is $3:1$. Calculate the percent of the rectangle's area that is contained within the square.
|
8.33\%
| 0.25 |
Given $x$ such that $0 \leq x < 4$, calculate the minimum value of the expression $\frac{x^2 + 2x + 6}{2x + 2}.$
|
\sqrt{5}
| 0.75 |
Given the set $B$ of all positive integers exclusively having prime factors from the set $\{2, 3, 5, 7\}$, Calculate the infinite sum:
\[
\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8} + \frac{1}{9} + \frac{1}{10} + \frac{1}{12} + \frac{1}{14} + \frac{1}{15} + \frac{1}{16} + \frac{1}{18} + \frac{1}{20} + \frac{1}{21} + \dots
\]
and express this sum, $\Sigma$, as a fraction in simplest form $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers.
|
\frac{35}{8}
| 0.916667 |
Given a list of the first 12 positive integers such that for each $2\le i\le 12$, either $a_i + 1$ or $a_i-1$ or both appear somewhere before $a_i$ in the list, calculate the number of such lists.
|
2048
| 0.083333 |
Emily is riding her bicycle at 15 miles per hour and Emerson is jogging at 9 miles per hour in the same direction. If Emily sees Emerson 1 mile ahead and then sees him 1 mile behind her after passing, how long can she see him in minutes.
|
20
| 0.916667 |
Let $\overline{AB}$ be a diameter in a circle of radius $7$. Let $\overline{CD}$ be a chord in the circle that intersects $\overline{AB}$ at a point $E$ such that $BE=3$ and $\angle AEC = 45^{\circ}.$ Calculate $CE^2+DE^2$.
|
98
| 0.083333 |
In a square $EFGH$ with side length 2 units, a circle $\Omega$ is inscribed. The circle intersects the line segment $\overline{HG}$ at $N$. The line $\overline{EN}$ intersects $\Omega$ again at a point $Q$ (different from $N$). Calculate the length of $EQ$.
|
\frac{\sqrt{5}}{5}
| 0.416667 |
Given the product $2! \cdot 4! \cdot 6! \cdot 8! \cdot 10!$, find the number of perfect squares that are divisors of this product.
|
360
| 0.083333 |
Given that Maria scored $5, 2, 4, 3, 6, 2, 7, 4, 1,$ and $3$ goals in her first $10$ soccer matches, her total score after $11$ matches is divisible by $11$, and her total score after $12$ matches is divisible by $12$. Find the product of the number of goals she scored in the eleventh and twelfth matches.
|
7 \times 4 = 28
| 0.166667 |
Let $M = 39 \cdot 48 \cdot 77 \cdot 150$. Calculate the ratio of the sum of the odd divisors of $M$ to the sum of the even divisors of $M$.
|
\frac{1}{62}
| 0.5 |
Given that x and y are distinct nonzero real numbers such that x + 3/x = y + 3/y, find the value of xy.
|
3
| 0.916667 |
The polynomial $(x+y)^{11}$ is expanded in decreasing powers of $x$. Evaluate the third and fourth terms of the expansion when $x=p$ and $y=q$, where $p$ and $q$ are positive numbers whose sum is two. Determine the value of $p$.
|
\frac{3}{2}
| 0.083333 |
A cell phone plan costs $25 each month, plus $8 cents per text message sent, plus $12 cents for each minute used over 40 hours. Calculate the total cost for Jenny in February, given that she sent 150 text messages and talked for 41 hours.
|
44.20
| 0.833333 |
If \(10^{3x} = 1000\), find the value of \(10^{-x}\).
|
\frac{1}{10}
| 0.833333 |
Rectangle $ABCD$ is inscribed in a semicircle with diameter $\overline{FE}$. Let $DA=20,$ and let $FD=AE=7.$ Calculate the area of $ABCD$.
|
60\sqrt{21}
| 0.333333 |
Determine the minimum number of small droppers required to fill a medicine container completely with a total volume of $265$ milliliters, using only small droppers that hold $19$ milliliters.
|
14
| 0.833333 |
Given that Jenny has 30 coins consisting of nickels and dimes, and if her nickels were dimes and her dimes were nickels, she would have $\$1.20 more, determine the total value of her coins.
|
1.65
| 0.666667 |
Given Elmer upgraded to a new car with $60\%$ better fuel efficiency in kilometers per liter and uses a type of diesel that is $30\%$ more expensive per liter, calculate the percentage that Elmer saves on fuel costs using his new car over the same distance compared to his previous car.
|
18.75\%
| 0.916667 |
Compute the sum $g \left(\frac{1}{2021} \right) - g \left(\frac{2}{2021} \right) + g \left(\frac{3}{2021} \right) - g \left(\frac{4}{2021} \right) + \cdots - g \left(\frac{2020}{2021} \right)$ where $g(x) = x^2(1-x)^2$.
|
0
| 0.916667 |
The sum of two numbers is \( S \). Suppose \( x \) is added to each number and then each of the resulting numbers is multiplied by \( k \). What is the sum of the final two numbers?
|
kS + 2kx
| 0.333333 |
The parabolas $y = ax^2 - 4$ and $y = 6 - bx^2$ intersect the coordinate axes in exactly four points, and these four points are the vertices of a kite of area $24$. Determine the value of $a + b$.
|
\frac{125}{72}
| 0.416667 |
Given the number $15!$, calculate the probability that a randomly chosen positive integer divisor of this number is odd.
|
\frac{1}{12}
| 0.083333 |
Given a cube with 4-inch edges is to be constructed from 64 smaller cubes with 1-inch edges, where 48 are colored red and 16 are colored white, determine the fraction of the surface area that is white when constructed to have the smallest possible white surface area showing.
|
\frac{1}{12}
| 0.25 |
Three primes $p, q$, and $r$ satisfy $p + q + r = s$ where $s$ is the smallest square number greater than 15. If $1 < p < q < r$, find $p$.
|
2
| 0.916667 |
Given that the parabola $y=2x^2$ is defined and all three vertices of $\bigtriangleup ABC$ lie on the parabola, point $A$ is at the origin, and $\overline{BC}$ is parallel to the $x$-axis, determine the length of $BC$ if the area of the triangle is $128$.
|
8
| 0.833333 |
Given quadrilateral $EFGH$, with side lengths $EF = 6$, $FG = 19$, $GH = 6$, and $HE = 10$, and where $EG$ is an integer, calculate the value of $EG$.
|
15
| 0.25 |
Given a quadrilateral $Q$ with vertices $A$, $B$, $C$, and $D$, determine how many circles in the plane of $Q$ can be drawn such that a diameter of each circle is defined by a pair of vertices from $Q$.
|
6
| 0.833333 |
Given the operation $x \otimes y = x^3 + 3 - y$, calculate the value of $k \otimes (k \otimes (k\otimes k))$.
|
k^3 + 3 - k
| 0.833333 |
Two cyclists start from two points that are $50$ miles apart. If they travel towards each other, they meet in $2$ hours. However, if one cyclist starts from a point $10$ miles behind his original position and the other remains at his starting position, and they would be $70$ miles apart in $3$ hours. Determine the ratio of the speed of the faster cyclist to the speed of the slower cyclist.
|
\frac{17}{13}
| 0.083333 |
Given a 4-inch cube constructed from 64 smaller 1-inch cubes, with 50 red and 14 white cubes, arrange these cubes such that the white surface area exposed on the larger cube is minimized, and calculate the fraction of the total surface area of the 4-inch cube that is white.
|
\frac{1}{16}
| 0.25 |
Determine how many pairs of positive integers (a, b) with $a+b \leq 90$ satisfy the equation $\frac{a+b^{-1}}{a^{-1}+b} = 17.$
|
5
| 0.5 |
In a chess tournament, there are $n$ junior players and $3n$ senior players, with each participant playing exactly one game against every other player. If there are no draws and the ratio of the number of games won by junior players to the number of games won by senior players is $3/7$, find the value of $n$.
|
4
| 0.25 |
When $10^{100} - 94$ is expressed as a single whole number, calculate the sum of its digits.
|
888
| 0.333333 |
A rectangular solid has its side, front, and bottom faces with areas $15\text{ in}^2$, $20\text{ in}^2$, and $12\text{ in}^2$ respectively. Determine the volume of this solid.
|
60
| 0.916667 |
Using a calculator with only two keys [+1] and [x2], starting with the display "1," calculate the fewest number of keystrokes needed to reach "256".
|
8
| 0.75 |
Three primes $p, q$, and $s$ satisfy $p + q = s + 4$ and $1 < p < q$. Solve for $p$.
|
2
| 0.75 |
In a triangle, if one angle remains the same, one of its enclosing sides is tripled, and the other side is doubled, calculate the factor by which the area of the triangle is multiplied.
|
6
| 0.833333 |
Given that in a kingdom $\frac{1}{3}$ of the knights are green, and $\frac{1}{5}$ of the knights are magical, and the fraction of green knights who are magical is $3$ times the fraction of yellow knights who are magical, find the fraction of green knights who are magical.
|
\frac{9}{25}
| 0.416667 |
Ahn chooses a two-digit integer, subtracts it from 300, and triples the result. Find the smallest number Ahn can get.
|
603
| 0.833333 |
Given Alice has 26 apples and each person must receive at least 3 apples, calculate the total number of ways of distributing the apples among Alice, Becky, and Chris.
|
171
| 0.916667 |
Given the function $g$ defined by the table and given that $v_0=3$ and $v_{n+1} = g(v_n + 1)$ for $n \ge 0$, find $v_{2004}$.
|
1
| 0.166667 |
Determine the number of distinct triangles that can be formed using three of the dots below as vertices, where the dots form a $3 \times 3$ grid.
|
76
| 0.916667 |
If \((4x - 2)^4! = b_{24}x^{24} + b_{23}x^{23} + \cdots + b_0\), determine the sum \(b_{24} + b_{23} + \cdots + b_0\).
|
16777216
| 0.75 |
Given that 30% of the objects in the urn are beads, and 50% of the coins in the urn are silver, calculate the percentage of objects in the urn that are gold coins.
|
35\%
| 0.916667 |
Kiana has two older twin siblings, and their ages, together with her own, multiply together to equal 162. Calculate the sum of their ages.
|
20
| 0.25 |
Tom's Hat Shoppe increased all original prices by $30\%$. Then, the shoppe is having a sale where all prices are $10\%$ off these increased prices. What is the sale price of an item relative to its original price?
|
1.17
| 0.083333 |
Orvin went to the store with enough money to buy 40 balloons at full price. He discovered a revised store promotion: after buying 4 balloons at full price, the next balloon could be purchased at half off. Determine the greatest number of balloons Orvin could buy if he takes full advantage of this offer.
|
44
| 0.083333 |
Given that Fluffy the rabbit ate twice as many carrots each day from April 1st to April 3rd, and a total of 84 carrots over these three days, calculate the number of carrots Fluffy ate on April 3rd.
|
48
| 0.5 |
Peter's family ordered a 16-slice pizza for dinner. Peter ate two slices and shared another slice equally with his two brothers, Paul and John. What fraction of the pizza did Peter eat.
|
\frac{7}{48}
| 0.416667 |
Determine the maximum number of quarters that could be in a coin box containing 120 coins, consisting of only nickels, dimes, and quarters, and totaling $10.00.
|
20
| 0.916667 |
Given that each licence candy costs $24$ cents, calculate the smallest possible value of $n$ when Casper has exactly enough money to buy $10$ pieces of jelly candies, $16$ pieces of chocolate candies, or $18$ pieces of gummies.
|
30
| 0.166667 |
Given two angles form a linear pair and are both prime numbers, and the larger angle $a^{\circ}$ is more than the smaller angle $b^{\circ}$, find the least possible value of $b$.
|
7
| 0.916667 |
Two lines with slopes $-\frac{1}{3}$ and $3$ intersect at $(3,3)$. Find the area of the triangle enclosed by these two lines and the line $x+y=12$.
|
22.5
| 0.083333 |
Given a test with 30 questions, a scoring system of +4 points for each correct answer, -1 point for each incorrect answer, and 0 points for each unanswered question, and Mary scores 70 points, calculate the maximum number of questions she could have answered correctly.
|
20
| 0.833333 |
Evaluate the expression $(3(3(3(3(3(3-2)-2)-2)-2)-2)-2)$.
|
1
| 0.416667 |
Given that $P$ is a point inside rectangle $ABCD$, the distances from $P$ to the vertices of the rectangle are $PA = 5$ inches, $PD = 12$ inches, and $PC = 13$ inches. Find $PB$, which is $x$ inches.
|
5\sqrt{2}
| 0.583333 |
Given that Penelope takes 50 equal waddles to walk between consecutive telephone poles, and Hector takes 15 equal jumps to cover the same distance, and the 51st pole is 6336 feet from the first pole, calculate how much longer Hector's jump is than Penelope's waddle.
|
5.9136
| 0.833333 |
A positive integer \( n \) has 48 divisors and 5n has 72 divisors. Find the greatest integer k such that 5^k divides n.
|
1
| 0.75 |
Two circles with radii $r$ and $s$ (where $r > s$) are placed such that the distance between their centers is $d$. Determine the possible number of different values of $k$ representing the number of lines that are simultaneously tangent to both circles.
|
5
| 0.5 |
The expression $\frac{2022}{2021} - \frac{2021}{2022}$ is equal to the fraction $\frac{p}{q}$ in which $p$ and $q$ are positive integers whose greatest common divisor is $1$. Find the value of $p$.
|
4043
| 0.916667 |
Calculate the sum of the digits of the number \(10^{100} - 57\).
|
889
| 0.25 |
A car moves such that its speed for the third and subsequent kilometers varies inversely as the square of the integral number of kilometers already traveled. If the third kilometer is traversed in $3$ hours, determine the time needed, in hours, to traverse the $n$th kilometer.
|
\frac{3(n-1)^2}{4}
| 0.416667 |
Given Emma's telephone number has the form $555-ab-cde-fgh$, where $a$, $b$, $c$, $d$, $e$, $f$, $g$, and $h$ are distinct digits in increasing order, and none is either $0$, $1$, $4$, or $5$, determine the total number of different telephone numbers Emma could have.
|
0
| 0.666667 |
Given that $x$ is a positive real number, find the equivalent expression for $\sqrt[4]{x^3\sqrt{x}}$.
|
x^{7/8}
| 0.916667 |
The number $24!$ has many positive integer divisors. What is the probability that a divisor randomly chosen from these is odd?
|
\frac{1}{23}
| 0.416667 |
Three primes p, q, and r satisfy p + q = r + 2 and 1 < p < q. Find the value of p.
|
2
| 0.416667 |
If the product $\dfrac{4}{3}\cdot \dfrac{5}{4}\cdot \dfrac{6}{5}\cdot \ldots\cdot \dfrac{a}{b} = 12$, find the sum of $a$ and $b$.
|
71
| 0.666667 |
Carl encounters three types of rocks in a different section of the cave: $6$-pound rocks worth $\$$20 each, $3$-pound rocks worth $\$$9 each, and $2$-pound rocks worth $\$$4 each. Given that Carl can carry up to $24$ pounds and there are at least $10$ of each type available, find the maximum value, in dollars, of the rocks he can carry out of the cave.
|
80
| 0.833333 |
Find the number of ordered triples of integers (x, y, z) that satisfy |x+y| + z = 23 and xy + |z| = 119.
|
4
| 0.25 |
Suppose the estimated cost to build an interstellar telescope is $30$ billion dollars and it is shared equally by the $300$ million people in the country. Calculate the amount each person needs to pay.
|
100
| 0.916667 |
Evaluate the expression $\dfrac{13!-12!}{10!}$.
|
1584
| 0.833333 |
The maximum number of the five integers that can be less than -5.
|
4
| 0.083333 |
Given a rectangle with dimensions $a$ and $b$ ($a < b$), determine the number of rectangles with dimensions $x$ and $y$ such that $x < \frac{a}{2}$ and $y < \frac{a}{2}$, and the perimeter of the new rectangle is half the perimeter of the original rectangle and its area is half the area of the original rectangle.
|
0
| 0.166667 |
Given that there are eleven books, including three Italian, three German, and five French books, determine the number of ways to arrange the books on the shelf, keeping the Italian books together, the German books together, and the French books together.
|
25920
| 0.916667 |
A list of $2520$ positive integers has a unique mode, which occurs exactly $12$ times, calculate the least number of distinct values that can occur in the list.
|
229
| 0.75 |
Determine how many solution-pairs in the positive integers exist for the equation $4x + 7y = 548$.
|
19
| 0.916667 |
Determine how many numbers between 3000 and 4000 have the property that their units digit equals the product of the other three digits modulo 10.
|
100
| 0.25 |
Given that a semipro basketball league has teams with 15 players each and each player must receive a minimum salary of $20,000, and the total of all players' salaries for each team must not exceed $500,000, determine the maximum possible salary for the highest-paid player on the team.
|
220,000
| 0.583333 |
Given that spinner P has numbers 2, 4, 5, spinner Q has numbers 1, 3, 5, and spinner R has numbers 2, 5, 7, determine the probability that the sum of numbers obtained from rotating spinners P, Q, and R is an odd number.
|
\frac{4}{9}
| 0.25 |
Given the mean score of the students in the morning class is 90, the mean score of the students in the afternoon class is 75, and the ratio of the number of students in the morning class to the afternoon class is $\frac{5}{6}$, calculate the mean score of all the students in both classes combined.
|
\frac{900}{11}
| 0.916667 |
Brenda's rate of building bricks per hour is 1/8, Brandon's rate of building bricks per hour is 1/12, but when they work together their combined rate is reduced by 15 bricks per hour, and they complete the chimney in 6 hours, calculate the number of bricks in the chimney.
|
360
| 0.25 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.