problem
stringlengths 18
4.46k
| answer
stringlengths 1
942
| pass_at_n
float64 0.08
0.92
|
|---|---|---|
Calculate the simplified value of the expression:
\[
\frac{4^2 \times 7}{8 \times 9^2} \times \frac{8 \times 9 \times 11^2}{4 \times 7 \times 11}
\] | \frac{44}{9} | 0.583333 |
Given that an ATM password is composed of four digits from $0$ to $9$, and the fourth digit cannot be the same as the third digit, calculate the number of valid passwords possible. | 9000 | 0.5 |
Given a rectangular floor that is $11$ feet wide and $19$ feet long, and tiled with $209$ one-foot square tiles, calculate the number of tiles that a bug visits when it walks from one corner to the opposite corner in a straight line, including the first and the last tile. | 29 | 0.416667 |
A cylindrical tank with radius 6 feet and height 7 feet is lying on its side. The tank is filled with water to a depth of 3 feet. Find the volume of water in the tank, in cubic feet. | 84\pi - 63\sqrt{3} | 0.666667 |
Given $g(x) = 3x^2 + 4x + 5$, calculate $g(x + h) - g(x)$. | h(6x + 3h + 4) | 0.5 |
Given that a 3-digit whole number's digit-sum is $27$, how many of these numbers are even? | 0 | 0.833333 |
Given that Connie divided a number by $3$ and got $27$, determine the correct number that she should have multiplied by $3$ to get the correct answer. | 243 | 0.666667 |
Michael jogs daily around a track consisting of long straight lengths connected by a full circle at each end. The track has a width of 4 meters, and the length of one straight portion is 100 meters. The inner radius of each circle is 20 meters. It takes Michael 48 seconds longer to jog around the outer edge of the track than around the inner edge. Calculate Michael's speed in meters per second. | \frac{\pi}{3} | 0.083333 |
Determine $r_2$ if $q_1(x)$ and $r_1$ are the quotient and remainder, respectively, when the polynomial $x^6$ is divided by $x - \tfrac{1}{3}$, and $q_2(x)$ and $r_2$ are the quotient and remainder, respectively, when $q_1(x)$ is divided by $x - \tfrac{1}{3}$. | \frac{2}{81} | 0.166667 |
If $\frac{x}{3} = y^2$ and $\frac{x}{5} = 5y$, find the value of $x$. | \frac{625}{3} | 0.916667 |
In a class test, $15\%$ of the students scored $60$ points, $20\%$ scored $75$ points, another $40\%$ scored $85$ points, $20\%$ scored $95$ points, and the remainder scored $100$ points, calculate the difference between the mean and the median score. | 3 | 0.333333 |
The numbers from -5 to 10 are arranged in a 4-by-4 square such that the sum of the numbers in each row, each column, and each of the two main diagonals are all the same, find the value of this common sum. | 10 | 0.916667 |
If 300 students voted on two proposals with 230 students voting in favor of the first proposal and 190 students voting in favor of the second proposal, and 40 students voted against both proposals, determine the number of students who voted in favor of both proposals. | 160 | 0.916667 |
Given a real and positive variable $x$ approaching infinity, evaluate the limit of the expression $\log_4{(8x-3)} - \log_4{(3x+4)$. | \log_4 \left(\frac{8}{3}\right) | 0.25 |
Given $G=\frac{9x^2+21x+4m}{9}$ be the square of an expression linear in $x$, and valid when $x = p - 1$ where $p \neq 0$, determine the values of $m$ that satisfy these conditions. | \frac{49}{16} | 0.833333 |
Compute $T_n = 2S_n$, where $S_n = 1-2+3-4+\cdots +(-1)^{n-1}n$, for $n=10, 20, 31$. Calculate the sum of the values of $T_{10}$, $T_{20}$, and $T_{31}$. | 2 | 0.5 |
There are births in West Northland every 6 hours, deaths every 2 days, and a net immigration every 3 days. Calculate the approximate annual increase in population. | 1400 | 0.75 |
Given the expression $(30! - 25!)$, calculate the thousands digit of the result. | 0 | 0.75 |
The point $P(a,b)$ in the $xy$-plane is first rotated counterclockwise by $90^\circ$ around the point $(2,6)$ and then reflected about the line $y = x$. The image of $P$ after these two transformations is at $(-7,4)$. Determine the value of $b - a$. | 15 | 0.916667 |
The numeral 36 in base a represents the same number as 63 in base b. Assuming that both bases are positive integers, find the least possible value of a+b. | 20 | 0.083333 |
Let $a_1, a_2, \ldots$ and $b_1, b_2, \ldots$ be arithmetic progressions with $a_1 = 30, b_1 = 70$. If $a_{50} + b_{50} = 200$, calculate the sum of the first fifty terms of the progression $a_1 + b_1, a_2 + b_2, \ldots$. | 7500 | 0.75 |
The function $g(x)$ satisfies $g(3+x) = g(3-x)$ for all real numbers $x$. If $g(x) = 0$ has exactly three distinct real roots, find the sum of these roots. | 9 | 0.916667 |
Given that points $A$ and $B$ are 12 units apart in a plane, determine how many points $C$ are there such that the perimeter of $\triangle ABC$ is 48 units and the area of $\triangle ABC$ is 72 square units. | 4 | 0.416667 |
What is the minimum number of digits to the right of the decimal point needed to express the fraction $\frac{987654321}{2^{30} \cdot 5^6}$ as a decimal? | 30 | 0.416667 |
Given the equations $x^2+kx+10=0$ and $x^2-kx+10=0$. If, when the roots of the equations are suitably listed, each root of the second equation is $3$ more than the corresponding root of the first equation, then determine the value of $k$. | 3 | 0.916667 |
At noon on a certain day, Denver is \( N \) degrees warmer than Boulder. By 6:00 PM, the temperature in Denver falls by 8 degrees while the temperature in Boulder rises by 2 degrees. At that time, the temperature difference between the two cities is 3 degrees. Find the product of all possible values of \( N \). | 91 | 0.916667 |
Solve the equation $x + \sqrt{3x-2} = 6$. | \frac{15 - \sqrt{73}}{2} | 0.666667 |
Let $x = -2500$. Evaluate the expression $\bigg| \big| |x|-x \big| - |x| \bigg| + x$. | 0 | 0.916667 |
Given a bag of popping corn contains $\frac{3}{4}$ white kernels and $\frac{1}{4}$ neon yellow kernels, where only $\frac{2}{3}$ of the white kernels and $\frac{3}{4}$ of the neon yellow ones will pop, and given that $\frac{1}{4}$ of the popped kernels will also fizz automatically regardless of color, determine the probability that a kernel selected, which pops and then fizzes, was white. | \frac{8}{11} | 0.916667 |
Given that the fraction \(\frac{3}{5}\) is modified by subtracting \(2x\) from the numerator and adding \(x\) to the denominator, and that the resulting fraction is equal to \(\frac{1}{2}\), determine the value of \(x\). | \frac{1}{5} | 0.916667 |
What is the greatest number of consecutive integers whose sum is $36$? | 72 | 0.25 |
A two-digit number is reduced by the sum of its digits and the units digit of the result is $7$. How many two-digit numbers satisfy this condition? | 10 | 0.916667 |
If \( 4^{x+2} = 68 + 4^x \), solve for the value of \( x \). | \log_4\left(\frac{68}{15}\right) | 0.333333 |
Given rectangle $ABCD$ with $E$ the midpoint of diagonal $BD$. Point $E$ is connected to a point $F$ on $DA$ so that $DF=\frac{1}{4}DA$. Find the ratio of the area of $\triangle DFE$ to the area of quadrilateral $ABEF$. | \frac{1}{7} | 0.166667 |
If $x$ varies as the inverse square root of $y$, and $y$ varies as the fourth power of $z$, then determine the value of $n$ for which $x$ varies as the nth power of $z$. | -2 | 0.916667 |
Given the four-digit number 3003, calculate the total number of different numbers that can be formed by rearranging the four digits. | 3 | 0.416667 |
What is the value of $12345 + 23451 + 34512 + 45123 + 51234$? | 166665 | 0.666667 |
Given that five circles are tangent to each other and to two parallel lines, and the radius of the largest circle is $20$ cm while that of the smallest circle is $6$ cm, with the radii forming an arithmetic sequence, find the radius of the middle circle. | 13 | 0.75 |
If Ella completed 9, 13, 8, 14, 12, 10, and 11 hours in the first 7 weeks, calculate how many hours she needs to work in the eighth week to average 12 hours per week over an 8-week period. | 19 | 0.5 |
The sum of $n$ terms of an arithmetic progression with a common difference of $3$ is $180$, and the first term is the square of an integer. Given $n>1$, find the number of possible values for $n$. | 0 | 0.166667 |
Samantha took three exams with a varying number of questions. On the first exam, which had 30 questions, she scored 75%. On the second exam with 50 questions, she secured 80%. The third exam had 20 questions, with each question being double weighted and she answered 65% correctly. What percentage of the overall weighted questions did Samantha answer correctly? | 73.75 | 0.583333 |
The sums of three whole numbers taken in pairs are 20, 25, and 29. Find the middle number. | 12 | 0.75 |
Given a large 15 by 21 rectangular region, calculate the fraction of the region that is shaded if the shaded region constitutes one-fourth of one-half of the rectangle. | \frac{1}{8} | 0.916667 |
How many digits are in the product $3^8 \cdot 6^4$? | 7 | 0.916667 |
Let $n=x-y^{x-y}$. If $x > y$, also add $xy$ to $n$. Find $n$ when $x=3$ and $y=1$. | 5 | 0.916667 |
A poster 4 feet wide is centrally placed on a wall that is 25 feet wide. Calculate the distance from the end of the wall to the nearest edge of the poster. | 10.5 | 0.833333 |
Given that a spaceship orbits around a fictional planet with a radius of 3500 miles at a speed of 550 miles per hour relative to the planet, determine the number of hours required for one complete orbit. | 40 | 0.583333 |
Three fair coins are tossed once. For each head that results, two fair dice are rolled. What is the probability that the total sum of all dice rolls is odd? | \frac{7}{16} | 0.833333 |
Given a large semicircle with radius R and N congruent smaller semicircles inside, each lying on the diameter of the large semicircle and having their diameters completely covering this diameter without gaps or overlaps, find the value of N if the ratio of the total area of the smaller semicircles to the area of the region inside the large semicircle but outside all the smaller semicircles is 1:36. | 37 | 0.833333 |
A circle of radius 6 is inscribed in a rectangle. The ratio of the length of the rectangle to its width is 3:1. Find the area of the rectangle. | 432 | 0.916667 |
Given that $f(x+5) + f(x-5) = f(x)$ for all real $x$, determine the least common positive period $p$ of all such functions $f$. | 30 | 0.5 |
Given Professor Lewis has twelve different language books lined up on a bookshelf: three Arabic, four German, three Spanish, and two French, calculate the number of ways to arrange the twelve books on the shelf keeping the Arabic books together, the Spanish books together, and the French books together. | 362,880 | 0.083333 |
Determine the value of $n$, the number of integer values of $x$ for which $Q = x^4 + 4x^3 + 9x^2 + 2x + 17$ is a prime number. | 4 | 0.166667 |
Anna and her brother observed a freight train as it started crossing a tunnel. Anna counted 8 cars in the first 15 seconds. It took the train 3 minutes to completely pass through the tunnel at a constant speed. Determine the most likely number of cars in the train. | 96 | 0.75 |
Given a pair of standard $6$-sided dice is rolled once, the product of the numbers rolled determines the diameter of a circle. Find the probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference. | \frac{5}{36} | 0.5 |
Two externally tangent circles have radii of lengths 7 and 3, respectively, and their centers at points $A$ and $B$. A line externally tangent to both circles intersects ray $AB$ at point $C$. Calculate the length of $BC$. | 7.5 | 0.083333 |
Given $3^{7} + 1$ and $3^{15} + 1$ inclusive, how many perfect cubes lie between these two values? | 231 | 0.416667 |
Calculate $(2.1)(50.5 + 0.15)$ after increasing $50.5$ by $5\%$. What is the product closest to? | 112 | 0.333333 |
Given that $5y$ varies inversely as the square of $x$ and $3z$ varies directly as $x$, find the value of $y$ when $x = 4$ and $z = 6$ given that when $x = 2$ and $y = 25$, the relationship holds true. | 6.25 | 0.083333 |
Given Jones traveled 100 miles on his first trip and 500 miles on a subsequent trip at a speed four times as fast, compare his new time to the old time. | 1.25 | 0.5 |
Given Mindy made four purchases for $2.96, 6.57, 8.49, and 12.38. Each amount needs to be rounded up to the nearest dollar except the amount closest to a whole number, which should be rounded down. Calculate the total rounded amount. | 31 | 0.75 |
A regular 17-gon has L lines of symmetry, and the smallest positive angle for which it has rotational symmetry is R degrees. Calculate the value of L + R. | \frac{649}{17} | 0.416667 |
Given soda is sold in packs of 8, 14, and 28 cans, determine the minimum number of packs needed to buy exactly 100 cans of soda. | 5 | 0.666667 |
A box contains $23$ red balls, $24$ green balls, $12$ white balls, and $21$ blue balls. Determine the minimum number of balls that must be drawn from the box without replacement to ensure that at least $20$ balls of a single color are drawn. | 70 | 0.416667 |
Given that Elmer's new van has 40% better fuel efficiency than his old truck, and the fuel for the new van costs 30% more per liter than the fuel for the old truck, determine the percent by which Elmer will save or spend more if he uses his new van instead of his old truck for a trip covering 300 kilometers. | 7.14\% | 0.083333 |
What is the largest quotient that can be formed using two numbers chosen from the set $\{-30, -6, 0, 3, 5, 15\}$? | 5 | 0.833333 |
A cell phone plan costs $25$ each month, plus $8$ cents per text message sent, plus $15$ cents for each minute used over $50$ hours. Determine the total cost of the plan for John for the month, given that he sent $250$ text messages and talked for $50.75$ hours. | 51.75 | 0.916667 |
Determine the value of $N$ such that when $(a+b+c+d+e+1)^N$ is expanded and like terms are combined, the expression contains exactly $252$ terms that include all five variables $a, b, c, d,$ and $e$, each to some positive power. | 10 | 0.333333 |
Given integers between 2000 and 6999 have four distinct digits, calculate the number of such integers. | 2520 | 0.416667 |
Two poles are 30 feet and 90 feet high and are 150 feet apart. Find the height at which the lines joining the top of each pole to the foot of the opposite pole intersect. | 22.5 | 0.333333 |
In $\triangle PQR$, $S$ is the midpoint of side $QR$ and $T$ is on side $PR$. If the length of $PR$ is $12$ units and $\angle QPR = 45^\circ, \angle PQR = 90^\circ, \angle PRQ = 45^\circ$ and $\angle RTS = 45^\circ$, find the area of $\triangle PQR$ plus twice the area of $\triangle RST$. | 54 | 0.166667 |
A list of $2023$ positive integers has a unique mode, which occurs exactly $11$ times. What is the least number of distinct values that can occur in the list? | 203 | 0.5 |
Given that vertices P, Q, R, and S of a quadrilateral have coordinates (a, a), (a, -a), (-a, -a), and (-a, a), and the area of the quadrilateral PQRS is 36, calculate the value of a + b. | 6 | 0.166667 |
Given that Liam has written down one integer three times and another integer four times, and the sum of the seven numbers is 161, and one of the numbers is 17, find the value of the other number. | 31 | 0.916667 |
Circle C has a radius of $120$. Circle D, with an integer radius $s$, rolls externally around circle C and returns to its original position after one revolution while remaining externally tangent throughout. Determine the number of possible values for $s$ that are less than $120$. | 15 | 0.75 |
Cagney can frost a cupcake every 15 seconds, while Lacey can frost every 40 seconds. They take a 10-second break after every 10 cupcakes. Calculate the number of cupcakes that they can frost together in 10 minutes. | 50 | 0.083333 |
Suppose $a$, $b$, and $c$ are positive integers such that $a+b+c=2010$ and $b = 2a$. If $a!b!c! = m \cdot 10^n$ where $m$ and $n$ are integers and $m$ is not divisible by 10, what is the smallest possible value of $n$?
A) 586
B) 589
C) 592
D) 595
E) 598 | 589 | 0.5 |
Consider the graphs of $y=3\log{x}$ and $y=\log{(x+4)}$. Determine the number of intersection points of the two graphs. | 1 | 0.916667 |
A large container can hold 420 grams of flour, whereas a smaller container can hold 28 grams of flour. A baker wants to transfer flour from a sack into the smaller containers and then use exactly 3 of these filled smaller containers each to fill medium containers. Calculate the number of medium containers the baker can fill using flour from one large container. | 5 | 0.833333 |
Let's consider the sum of five consecutive odd numbers starting from 997 up to 1005. If the sum $997 + 999 + 1001 + 1003 + 1005 = 5100 - M$, find $M$. | 95 | 0.666667 |
Let \( t_n = \frac{n(n+1)}{2} \) be the \( n \)th triangular number. Find the sum of the reciprocals of the first 2003 triangular numbers, from \( t_1 \) to \( t_{2003} \). | \frac{2003}{1002} | 0.916667 |
Calculate the sum of the digits when $10^{95}-95$ is expressed as a single whole number. | 842 | 0.333333 |
A woman purchases a property for $12,000 and decides to rent it. She saves $15\%$ of each month's rent for maintenance; pays $400 a year in taxes, and targets a $6\%$ return on her investment. Calculate the monthly rent. | 109.80 | 0.5 |
A solid sphere is enclosed within a right circular cylinder. The volume of the cylinder is three times that of the sphere. If the height of the cylinder equals the diameter of the sphere, find the ratio of the height of the cylinder to the radius of its base. | \sqrt{2} | 0.583333 |
Consider all 4-digit palindromes that can be written as $\overline{abba}$, where $a$ is non-zero and $b$ ranges from 1 to 9. Calculate the sum of the digits of the sum of all such palindromes. | 36 | 0.083333 |
Calculate the value of $\left(9^{-1} - 5^{-1}\right)^{-1}$. | -\frac{45}{4} | 0.916667 |
Let $s$ be the result of tripling both the base and exponent of $c^d$, where $d > 0$. Express the result of this operation in terms of $c$ and $d$, and set it equal to the product of $c^d$ by $y^d$. Solve for $y$. | 27c^2 | 0.916667 |
A circle is inscribed in a triangle with side lengths $10, 15$, and $19$. Let the segments of the side of length $10$, made by point of tangency, be $r$ and $s$, with $r < s$. Determine the ratio $r:s$. | 3:7 | 0.083333 |
Given that Anne cycles 3 miles on Monday, 4 miles on Wednesday, and 5 miles on Friday, and her cycling speeds on these days are 6 miles per hour, 4 miles per hour, and 5 miles per hour respectively, calculate the time difference in minutes if she cycled every day at 5 miles per hour. | 6 | 0.5 |
How many primes less than $200$ have $3$ as the ones digit? | 12 | 0.75 |
Given a line $x=m$ intersects the graph of $y=\log_2 x$ and the graph of $y=\log_2 (x + 6)$, find the value of $c+d$ where $m = c + \sqrt{d}$ and $c$ and $d$ are integers. | 6 | 0.583333 |
Given an unfair coin with a $3/4$ probability of turning up heads, find the probability that the total number of heads is odd when the coin is tossed $60$ times. | \frac{1}{2}(1 - \frac{1}{2^{60}}) | 0.083333 |
Given that Professor Lee has ten different language books lined up on a bookshelf: three German, four Spanish, and three French, find the number of ways to arrange the ten books on the shelf keeping both the Spanish and French books together as separate units. | 17280 | 0.25 |
Determine the sum of all values of $z$ for which $f(4z) = 9$, given that $f\left(\dfrac{x}{4}\right) = x^2 - 3x + 2$. | \frac{3}{16} | 0.666667 |
A circle has a diameter of $D$ divided into $2n$ equal parts. On each of these parts, a quarter-circle is constructed standing vertically from each point. Calculate the limiting value of the sum of the lengths of the arcs of these quarter-circles as $n$ becomes very large. | \frac{\pi D}{4} | 0.083333 |
Liam has written down one integer three times and another integer twice. The sum of these five numbers is 130, and one of the numbers is 35. What is the other number? | 20 | 0.833333 |
The least common multiple of $a$ and $b$ is $20$, and the least common multiple of $b$ and $c$ is $21$. Find the least possible value of the least common multiple of $a$ and $c$. | 420 | 0.333333 |
A wooden cube, $n$ units on each side, is painted red on four of its six faces (the top and bottom faces are left unpainted), and then cut into $n^3$ smaller unit cubes. Exactly one-third of the total number of faces of these unit cubes are red. What is the value of $n$, and what is the volume of the original cube in cubic units? | 8 | 0.333333 |
Given \(5^a + 5^b = 2^c + 2^d + 17\), determine the number of integers \(a, b, c, d\) which can possibly be negative. | 0 | 0.916667 |
In a round-robin tournament with 8 teams, each team plays one game against each other team, and each game results in one team winning and one team losing. Find the maximum number of teams that could be tied for the most wins at the end of the tournament. | 7 | 0.166667 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.