problem
stringlengths 18
4.46k
| answer
stringlengths 1
942
| pass_at_n
float64 0.08
0.92
|
|---|---|---|
Given that Mary is $25\%$ older than Sally, and Sally is $45\%$ younger than Danielle, the sum of their ages is $25.5$ years. Express Mary's age on her next birthday as an integer. | 8 | 0.083333 |
Charlie sets out to read a book in a week for her summer project. She reads an average of 40 pages per day for the first two days and an average of 45 pages per day for the next four days. She completes the book by reading 20 pages on the seventh day. Determine the total number of pages in the book. | 280 | 0.916667 |
Given the values $1256, 2561, 5612, 6125$, calculate the sum. | 15554 | 0.833333 |
If the product $\dfrac{4}{3}\cdot \dfrac{5}{4}\cdot \dfrac{6}{5}\cdot \ldots\cdot \dfrac{c}{d} = 16$, determine the sum of $c$ and $d$. | 95 | 0.416667 |
Evaluate the expression $(16)^{-2^{-3}}$. | \frac{1}{\sqrt{2}} | 0.083333 |
How many whole numbers are between $\sqrt{50}$ and $\sqrt{200}$? | 7 | 0.916667 |
Keiko walks around an athletic track that has a similar structure as mentioned before, with straight sides and semicircular ends. The width of the track is $8$ meters. It takes her $48$ seconds longer to walk around the outside edge of the track than around the inside edge. Assuming Keiko walks at a constant speed throughout, determine her speed in meters per second. | \frac{\pi}{3} | 0.5 |
Given three positive consecutive integers whose sum is $c = 3a + 3$, compute the average of the next three consecutive integers starting from $c$. | 3a + 4 | 0.25 |
A fair coin is tossed 4 times. Calculate the probability of getting at least two consecutive heads. | \frac{1}{2} | 0.333333 |
Let $U$ be the set of the $1500$ smallest positive multiples of $5$, and let $V$ be the set of the $1500$ smallest positive multiples of $8$. Calculate the number of elements common to $U$ and $V$. | 187 | 0.833333 |
The real numbers \(a, b, c\) form a geometric sequence and satisfy \(a \leq b \leq c \leq 1\). The quadratic \(bx^2 + cx + a\) has exactly one root. Find this root. | -\frac{\sqrt[3]{4}}{2} | 0.083333 |
The total area of all the faces of a rectangular solid is $26\text{cm}^2$, and the total length of all its edges is $28\text{cm}$. Determine the length in cm of any one of its interior diagonals. | \sqrt{23} | 0.916667 |
A circle has (2,2) and (10,8) as the endpoints of its diameter. It intersects the x-axis at a second point. Find the x-coordinate of this point. | 6 | 0.083333 |
In trapezoid $ABCD$, $\overline{AD}$ is perpendicular to $\overline{DC}$, $AD = AB = 5$, and $DC = 10$. Additionally, $E$ is on $\overline{DC}$ such that $\overline{BE}$ is parallel to $\overline{AD}$. Find the area of the parallelogram formed by $\overline{BE}$. | 25 | 0.916667 |
An inverted cone filled with water has a base radius of 15 cm and a height of 15 cm. The water is then transferred into a cylindrical container with a base radius of 30 cm. Determine the height of the water level in the cylinder. | 1.25 | 0.916667 |
Given the function $f$ that satisfies $f(x+5)+f(x-5) = f(x)$ for all real $x$, determine the smallest positive period $p$ for these functions. | 30 | 0.416667 |
A bug starts crawling on a number line from position -3. It first moves to -7, then turns around and stops briefly at 0 before continuing on to 8. Calculate the total number of units the bug crawls. | 19 | 0.833333 |
Given the set $\{-7, -5, -3, 0, 2, 4, 6\}$, calculate the maximum possible sum of three different numbers where at least one of the numbers must be negative. | 7 | 0.916667 |
How many primes less than $150$ have $3$ as the ones digit? | 9 | 0.75 |
Given trapezoid $ABCD$, $\overline{AD}$ is perpendicular to $\overline{DC}$, $AD = AB = 4$, and $DC = 8$. Additionally, $E$ is on $\overline{DC}$, and $\overline{BE}$ is parallel to $\overline{AD}$. Furthermore, $E$ is exactly midway between $D$ and $C$. Find the area of $\triangle BEC$. | 8 | 0.916667 |
If rose bushes are spaced about 2 feet apart, calculate the number of bushes needed to surround a circular patio whose radius is 15 feet. | 47 | 0.5 |
Given the scale model of the Empire State Building has a ratio of $1:50$, calculate the height in feet of the scale model of the building, rounded to the nearest whole number. | 29 | 0.833333 |
Given a square is originally painted blue, each time the square is changed, the middle ninth of each blue square turns red. After four changes, determine the fractional part of the original area of the blue square that remains blue. | \frac{4096}{6561} | 0.916667 |
Given that set $A$ has $30$ elements, set $B$ has $20$ elements, and set $C$ has $10$ elements, calculate the smallest possible number of elements in $A \cup B \cup C$. | 30 | 0.916667 |
A store sells 9 pencils and 10 notebooks for $\mathdollar 5.35$ and 6 pencils and 4 notebooks for $\mathdollar 2.50$. If a special offer allows buying pencils in packs of 4 for a 10% discount on the per-pencil price, calculate the cost of 24 pencils and 15 notebooks. | 9.24 | 0.916667 |
Given that a triangular corner with side lengths DB=EB=1.5 is cut from an equilateral triangle ABC of side length 4.5, determine the perimeter of the remaining quadrilateral. | 12 | 0.416667 |
Given \(\frac{\log{a}}{p}=\frac{\log{b}}{q} =\frac{\log{c}}{r} =\log{x}\), where all logarithms are to the same base and \(x \not= 1\). If \(\frac{a^2}{bc}=x^y\), find \(y\). | 2p - q - r | 0.916667 |
A box contains 35 red balls, 25 green balls, 22 yellow balls, 15 blue balls, 14 white balls, and 12 black balls. Calculate the minimum number of balls that must be drawn from the box without replacement to guarantee that at least 18 balls of a single color will be drawn. | 93 | 0.666667 |
Alice and Bob play a game on a circle divided into 18 equally-spaced points, labeled from 1 to 18. Both start at point 18. Alice moves 7 points clockwise each turn while Bob moves 13 points counterclockwise each turn. Find the minimum number of turns required for Alice and Bob to meet at the same point. | 9 | 0.916667 |
Given the equations of the parabola \( y = x^2 + 2x - 3 \) and the horizontal line \( y = m \), where \( -8<m<5 \), determine the value of \( s \) as \( m \) approaches 0, where \( s = \frac{M(-2m) - M(m)}{2m} \), and \( M(m) \) is the \( x \)-coordinate of the left-most intersection point of the parabola and the line. | \frac{3}{8} | 0.75 |
Lisa and her father observed a train passing by a crossing, with 10 train cars passing in 15 seconds at a constant speed. The entire train passed the crossing in 3 minutes and 30 seconds. Determine the total number of train cars that passed the crossing. | 140 | 0.916667 |
Two cyclists, C and D, start from Albany towards New York, a distance of 100 miles. Cyclist C travels 5 mph slower than cyclist D. After reaching New York, D turns back immediately and meets C 20 miles from New York. Find the speed of cyclist C. | 10 | 0.5 |
Moe has a new rectangular lawn measuring 100 feet by 180 feet. He uses a mower with a swath that cuts 30 inches wide, but he overlaps each cut by 6 inches to ensure complete coverage. Moe walks at a speed of 4500 feet per hour while mowing. Calculate the number of hours it will take Moe to mow his entire lawn. | 2 | 0.916667 |
Elmer's new car gives $100\%$ better fuel efficiency, measured in kilometers per liter, than his old car. His new car uses diesel fuel, which is $25\%$ more expensive per liter than the gasoline his old car used. Determine the percent by which Elmer will save money if he uses his new car instead of his old car for a long trip. | 37.5\% | 0.833333 |
Let $Q$ be the product of two numbers: 7,123,456,789 and 23,567,891,234. Determine the number of digits in $Q$. | 21 | 0.083333 |
Three fair coins are tossed once. For each head that results, two fair dice are to be rolled. Calculate the probability that the sum of all die rolls is odd. | \frac{7}{16} | 0.75 |
Mr. Thompson has three nephews who visit him regularly. One visits him every six days, another every eight days, and the last one every ten days. All three visited him on January 1, 2020. Calculate the number of days during the year 2020 when Mr. Thompson did not receive a visit from any of his nephews. | 257 | 0.25 |
Three runners start running simultaneously from the same point on a 400-meter circular track, running clockwise. They maintain constant speeds of 5.0, 5.5, and 6.0 meters per second. Determine the time at which all the runners will be together again. | 800 | 0.583333 |
For how many integers x does a triangle with side lengths 12, 30, and x have all its angles acute? | 5 | 0.583333 |
Consider a grid with dimensions $9 \times 9$, consisting of alternating dark and light squares. Calculate the difference in the number of dark squares compared to light squares. | 1 | 0.583333 |
A collector offers to buy state quarters for 500% of their face value. If Bryden has four state quarters, each with a face value of $0.50, calculate the total amount he will receive from the collector. | 10 \text{ dollars} | 0.916667 |
What is the sum of the digits of the greatest prime number that is a divisor of $32,767$? | 7 | 0.75 |
Given $(b_1, b_2, ... b_7)$ be a list of the first 7 even positive integers such that for each $2 \le i \le 7$, either $b_i + 2$ or $b_i - 2$ or both appear somewhere before $b_i$ in the list, determine the number of such lists. | 64 | 0.083333 |
Given that $x$ and $y$ are distinct nonzero real numbers such that $x + \frac{3}{x} = y + \frac{3}{y}$, find the value of $xy$. | 3 | 0.916667 |
Given a set $T$ of 5 integers taken from $\{2,3, \dots, 15\}$ such that if $c$ and $d$ are elements of $T$ with $c<d$, then $d$ is not a divisor of $c$, determine the greatest possible value of an element in $T$. | 15 | 0.833333 |
Find the result when the Greatest Common Divisor (GCD) of $7260$ and $540$ is diminished by $12$, then add $5$ to the result. | 53 | 0.333333 |
Given the first container is $\frac{4}{5}$ full of water and the second container is initially empty, with all the water being poured into the second container filling it to $\frac{2}{3}$ of its capacity, determine the ratio of the volume of the first container to the volume of the second container. | \frac{5}{6} | 0.916667 |
The taxi fare in Metropolis City is $3.00 for the first $\frac{3}{4}$ mile and additional mileage charged at the rate $0.30 for each additional 0.1 mile. You plan to give the driver a $3 tip. Calculate the number of miles you can ride for $15. | 3.75 | 0.416667 |
First, $a$ is chosen at random from the set $\{1, 2, 3, \ldots, 99, 100\}$, and then $b$ is chosen at random from the same set. The probability that the integer $2^a + 5^b$ has a units digit of $8$ is?
A) $\frac{1}{16}$
B) $\frac{1}{8}$
C) $\frac{3}{16}$
D) $\frac{1}{5}$
E) $0$ | 0 | 0.666667 |
Calculate the sum $\sum^{50}_{i=1} \sum^{150}_{j=1} (2i + 3j)$. | 2081250 | 0.666667 |
Joe had walked one-third of the way from home to school when he realized he was very late. He ran the rest of the way to school. He ran 4 times as fast as he walked. Joe took 9 minutes to walk the one-third distance to school. Calculate the total time it took Joe to get from home to school. | 13.5 | 0.666667 |
Samantha leaves her house at 7:15 a.m. to catch the school bus, starts her classes at 8:00 a.m., and has 8 classes that last 45 minutes each, a 40-minute lunch break, and spends an additional 90 minutes in extracurricular activities. If she takes the bus home and arrives back at 5:15 p.m., calculate the total time spent on the bus. | 110 | 0.166667 |
Three distinct vertices of a tetrahedron are chosen at random. Find the probability that the plane determined by these three vertices does not pass through the interior of the tetrahedron. | 1 | 0.916667 |
Find the value of $y^{-1/3}$ if $\log_4 (\log_{5} (\log_{3} (y^2))) = 0$. | 3^{-5/6} | 0.666667 |
Calculate the product of the 10 factors $\Big(1 - \frac{1}{2}\Big)\Big(1 - \frac{1}{3}\Big)\Big(1 - \frac{1}{4}\Big)\cdots\Big(1 - \frac{1}{11}\Big)$. | \frac{1}{11} | 0.916667 |
A circle with center $O$ is divided into $18$ equal arcs. Points on the circumference are labeled sequentially from $A$ to $R$. Calculate the sum of the degrees in angles $x$ and $y$, where $x$ is the angle at circumference subtended by three arcs, and $y$ is subtended by five arcs. | 80 | 0.916667 |
Suppose the euro is now worth 1.5 dollars. If Marco has 600 dollars and Juliette has 350 euros, find the percentage by which the value of Juliette's money is greater than or less than the value of Marco's money. | 12.5\% | 0.833333 |
Given 18 parking spaces in a row, 14 cars arrive and occupy spaces at random, followed by Auntie Em, who requires 2 adjacent spaces, determine the probability that the remaining spaces are sufficient for her to park. | \frac{113}{204} | 0.166667 |
Let \( m \) be the smallest positive integer such that \( m \) is divisible by 30, \( m^2 \) is a perfect cube, and \( m^3 \) is a perfect square. Determine the number of digits of \( m \). | 9 | 0.833333 |
Given that $\log_8 (y-4) = 1.5$, find the value of $y$. | 16\sqrt{2} + 4 | 0.583333 |
Given a $4 \times 4$ square grid, where each unit square is painted white or black with equal probability and then rotated $180\,^{\circ}$ clockwise, calculate the probability that the grid becomes entirely black after this operation. | \frac{1}{65536} | 0.416667 |
Moe uses a mower to cut his rectangular 120-foot by 100-foot lawn. The mower cuts a swath 30 inches wide, but he overlaps each cut by 6 inches to ensure no grass is missed. He walks at the rate of 4000 feet per hour while pushing the mower. Determine the number of hours it will approximately take Moe to mow the entire lawn. | 1.5 | 0.833333 |
What is the arithmetic mean of the first n positive even integers? | n+1 | 0.916667 |
If $|2x - \log z| = 2x + \log z$ where $2x$ and $\log z$ are real numbers, determine $x$ and $z$.
**A)** $x = 0$
**B)** $z = 1$
**C)** $x = 0 \text{ and } z = 1$
**D)** $x(z-1) = 0$
**E)** None of these | x(z-1) = 0 | 0.583333 |
Evaluate \((x^x)^{(x^x)}\) at \(x = 3\). | 27^{27} | 0.833333 |
A large rectangle has area 168 square units and is divided into four rectangles by two segments, with one segment parallel to the width and the other not necessarily parallel to the length. Given three of these resulting rectangles have areas 33, 45, and 20, find the area of the fourth rectangle. | 70 | 0.666667 |
What is the sum of the digits of the greatest prime number that is a divisor of $32,767$? | 7 | 0.5 |
What is the greatest number of consecutive integers whose sum is $36$? | 72 | 0.25 |
The state income tax where Linda lives is levied at the rate of p% of the first $35000 of annual income plus (p + 3)% of any amount above $35000. Linda noticed that the state income tax she paid amounted to (p + 0.5)% of her annual income. What is her annual income? | 42000 | 0.916667 |
Given a regular tetrahedron with edges of length 2, calculate the length of the shortest path from the midpoint of one edge to the midpoint of the opposite edge, subject to a wind multiplier of 1.5. | 3 | 0.166667 |
Given a regular decagon, the probability that exactly one of the sides of the triangle formed by connecting three randomly chosen vertices of the decagon is also a side of the decagon. | \frac{1}{2} | 0.083333 |
Define a sequence \( \{u_n\} \) such that \( u_1=7 \) and the relationship \( u_{n+1}-u_n = 5n + 2(n-1) \) for \( n=1,2,3,\ldots \). Express \( u_n \) as a polynomial in \( n \) and determine the algebraic sum of its coefficients. | 7 | 0.75 |
Given a cell phone plan costing $30$ dollars each month, plus $0.10$ dollars for each text message sent, plus $0.12$ dollars for each minute used over $25$ hours, and a $5$ dollar discount for total talking time exceeding $35$ hours, calculate the total cost for a customer who sent $150$ text messages and talked for $36$ hours in a month. | 119.20 | 0.166667 |
How many subsets containing three different numbers can be selected from the set $\{ 12, 18, 25, 33, 47, 52 \}$ so that the sum of the three numbers is divisible by 3? | 7 | 0.666667 |
Let $s_1$ be the sum of the first $n$ terms of the arithmetic sequence $3, 10, \cdots$ and let $s_2$ be the sum of the first $n$ terms of the arithmetic sequence $5, 8, \cdots$. Determine the number of values of $n$ for which the equation $s_1 = s_2$ is true. | 1 | 0.75 |
The area of $\triangle ABC$ is $120$. Given that the base of $\triangle ABC$ is $30$ and the lengths of sides $BC$ and $AC$ are $17$ and $25$, calculate the length of the altitude to the base $AB$. | 8 | 0.583333 |
Given the equation $x^{6} + y^2 = 6y$, find the number of ordered pairs of integers $(x, y)$ that satisfy this equation. | 2 | 0.5 |
Given Aaron needs to read an average of 15 pages per day to complete his book club's challenge over the next 6 days, and that in the first 5 days he reads 18, 12, 23, 10, and 17 pages each day, calculate the number of pages he needs to read on the sixth day to meet the challenge. | 10 | 0.833333 |
Claire holds 2 green and 2 red jelly beans in her hand. Daniel holds 2 green, 3 yellow, and 4 red jelly beans in his hand. Calculate the probability that the colors of the jelly beans picked by Claire and Daniel match. | \frac{1}{3} | 0.833333 |
Let \( M = 57^4 + 4 \cdot 57^3 + 6 \cdot 57^2 + 4 \cdot 57 + 1 \). Calculate the number of positive integers that are factors of \( M \). | 25 | 0.75 |
What is the largest product that can be formed using two numbers chosen from the set $\{-20, -4, -1, 3, 5, 9\}$? | 80 | 0.916667 |
When the polynomial $x^3 + 3x^2 - 4$ is divided by the polynomial $x^2 + x - 2$, calculate the remainder. | 0 | 0.916667 |
John begins work on four equally time-consuming tasks at 2:00 PM. He finishes the third task at 5:00 PM. Determine the time at which he finishes the fourth and final task. | 6:00 \text{ PM} | 0.916667 |
What is the hundreds digit of $(30! - 25!)$? | 0 | 0.833333 |
Points A and B lie on a plane with AB = 10. Calculate the number of locations for point C on this plane such that the triangle with vertices A, B, and C forms a right triangle with an area of 20 square units. | 8 | 0.583333 |
Determine the number of quadratic polynomials $ax^2+bx+c$ with real coefficients for which the set of roots matches the set of coefficients, and additionally, one of the coefficients is the arithmetic mean of the other two. | 0 | 0.083333 |
Points A and B lie on a coordinate plane with AB = 10. Determine how many locations for point C exist on this plane such that the triangle with vertices A, B, and C forms a right triangle with an area of 15 square units. | 8 | 0.583333 |
Consider equations of the form \( x^2 + bx + c = 0 \). How many such equations have real roots if coefficients \( b \) and \( c \) are selected from the sets \( B = \{2, 4, 6\} \) and \( C = \{1, 3, 5\} \), respectively? | 6 | 0.666667 |
Consider the sequence $1, -4, 9, -16, 25, -36, \ldots,$ where the $n$th term is $(-1)^{n+1} \cdot n^2$. Calculate the average of the first 100 terms of this sequence. | -50.5 | 0.583333 |
Given $10^{3x} = 1000$, calculate $10^{-x}$. | \frac{1}{10} | 0.583333 |
Josanna's test scores are $92, 78, 84, 76,$ and $88$. She aims to raise her test average by at least $5$ points with her next test. What is the minimum test score she would need on the next test to reach this goal? | 114 | 0.416667 |
Find the number of distinct terms in the expansion of $[(2a+5b)^3(2a-5b)^3]^3$ when simplified. | 10 | 0.583333 |
Given a rectangular racing track consisting of two parallel 100 meters long straight sections and two semicircular ends, where the inner semicircle has a radius of 50 meters and the outer semicircle has a radius of 52 meters, find the runner's average speed in meters per second, given that it takes a runner 40 seconds longer to complete a lap around the outer edge than the inner edge. | \frac{\pi}{10} | 0.416667 |
Given Eliot's test scores are \(88, 92, 75, 85,\) and \(80\), find the minimum score he would need on his next test to raise his test average by 5 points. | 114 | 0.916667 |
The length of the rectangle PQRS is 8 inches and its width is 6 inches. Diagonal PR is divided into four equal segments by points M, N, and O. Find the area of triangle QMN. | 6 | 0.583333 |
Given the equation $a \cdot b + 45 = 10 \cdot \text{lcm}(a, b) + 18 \cdot \text{gcd}(a, b)$, where $\text{gcd}(a, b)$ denotes the greatest common divisor of $a$ and $b$, and $\text{lcm}(a, b)$ denotes their least common multiple, calculate the number of ordered pairs $(a, b)$ of positive integers that satisfy this equation. | 4 | 0.333333 |
The number of arrangements of the Blue Bird High School chess team with a girl at each end and the three boys in the middle can be found by determining the number of possible permutations. | 12 | 0.833333 |
From a regular decagon, a triangle is formed by connecting three randomly chosen vertices of the decagon. Determine the probability that at least one of the sides of the triangle is also a side of the decagon. | \frac{7}{12} | 0.083333 |
Consider the infinite series $1 - \frac{1}{3} - \frac{1}{9} + \frac{1}{27} - \frac{1}{81} - \frac{1}{243} + \frac{1}{729} - \cdots$. Let \( T \) be the sum of this series. Find \( T \). | \frac{15}{26} | 0.333333 |
A fair 8-sided die is rolled twice. What is the probability that the first number that comes up is greater than or equal to the second number, and the sum of the two rolls is even? | \frac{5}{16} | 0.75 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.