problem
stringlengths 18
4.46k
| answer
stringlengths 1
942
| pass_at_n
float64 0.08
0.92
|
|---|---|---|
If $y$ cows can produce $y+2$ cans of milk in $y+1$ days, determine the number of days it will take $y+4$ cows to produce $y+6$ cans of milk. | \frac{y(y+1)(y+6)}{(y+4)(y+2)} | 0.75 |
Given the operations \( a@b = ab - b^2 + b^3 \) and \( a\#b = a + b - ab^2 + ab^3 \), calculate the value of \( \frac{7@3}{7\#3} \). | \frac{39}{136} | 0.166667 |
Given a lawn with a rectangular shape of $120$ feet by $200$ feet, a mower with a $32$-inch swath width, and a $6$-inch overlap between each cut, and a walking speed of $4000$ feet per hour, calculate the time it will take John to mow the entire lawn. | 2.8 | 0.416667 |
Given that the connection from $P$ leads to $R$ or $S$, with $R$ offering paths to $T$ or $U$, and with $T$ and $U$ leading directly to $Q$, $S$ offering paths to $T$, $U$, and directly to $Q$, and $P$ offering a direct path to $Q$, determine the total number of different routes from $P$ to $Q$. | 6 | 0.583333 |
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, calculate the length of $\overline{BC}$ given the area of the triangle is $128$. | 8 | 0.75 |
Calculate the value of the expression: $[x + (y-z)] - [(x+z) - y]$. | 2y - 2z | 0.583333 |
A square board consists of an alternating pattern of light and dark squares similar to a chessboard. Determine the difference in the number of dark squares and the number of light squares in a $9 \times 9$ grid. | 1 | 0.916667 |
Two pitchers, one with a capacity of 800 mL and the other with 500 mL, contain orange juice. The first pitcher is 1/4 full of orange juice and the second pitcher is 3/5 full. Water is added until both pitchers are full. Both pitchers are then poured into one large container. Calculate the fraction of the mixture in the large container that is orange juice. | \frac{5}{13} | 0.916667 |
How many positive integers $b$ have the property that $\log_{b} 512$ is a positive integer. | 3 | 0.833333 |
Given that points $E$ and $F$ are on the same side of diameter $\overline{GH}$ in circle $P$, $\angle GPE = 60^\circ$, and $\angle FPH = 90^\circ$, find the ratio of the area of the smaller sector $PEF$ to the area of the circle. | \frac{1}{12} | 0.416667 |
In a right triangle, the lengths of the legs a and b maintain a ratio of 3 : 4, and the hypotenuse is c. A perpendicular is dropped from the vertex of the right angle to the hypotenuse, dividing it into two segments: r adjacent to a and s adjacent to b. Find the ratio of r to s. | \frac{9}{16} | 0.75 |
Given $\triangle ABC$, $AB = 75$, and $AC=100$. A circle with center $A$ and radius $AB$ intersects $\overline{BC}$ at points $B$ and $X$. Moreover $\overline{BX}$ and $\overline{CX}$ have integer lengths. Calculate the length of $BC$. | 125 | 0.333333 |
Given that there are 18 players on each team, each player receives a minimum salary of $12,000, and the total salary cap for each team is $620,000, calculate the maximum possible salary for the highest-paid player in a team. | 416,000 | 0.5 |
Let $f(x) = \frac{x(x-1)}{2}$. Evaluate $f(x+3)$. | \frac{x^2 + 5x + 6}{2} | 0.916667 |
Given the repeating decimal $0.\overline{57}$, write it in its simplest fractional form and find the sum of the numerator and denominator. | 52 | 0.75 |
Anh traveled 75 miles on the interstate and 15 miles on a mountain pass. The speed on the interstate was four times the speed on the mountain pass. If Anh spent 45 minutes driving on the mountain pass, determine the total time of his journey in minutes. | 101.25 | 0.916667 |
Max's house has 4 bedrooms, each with dimensions 15 feet by 12 feet by 9 feet. Each bedroom has walls that occupy 80 square feet in area, excluding doorways and windows. Calculate the total square feet of walls to be painted. | 1624 | 0.416667 |
Thirty percent less than 90 is one-half more than what number? | 42 | 0.666667 |
Given the numbers from $000$ to $999$, calculate how many have three digits in non-decreasing or non-increasing order, including cases where digits can repeat. | 430 | 0.166667 |
Calculate the expansion for the expression $(1+x^2)(1-x^4)$. | 1 + x^2 - x^4 - x^6 | 0.666667 |
Pablo had four times as many coins as Sofia, who in turn had three times as many coins as Mia. Each of them ends up with an equal number of coins after Pablo distributes some of his coins. Determine the fraction of his coins that Pablo distributes to Sofia and Mia together. | \frac{5}{9} | 0.5 |
Find the times between $8$ and $9$ o'clock, correct to the nearest minute, when the hands of a clock will form an angle of $120^{\circ}$. | 8:22 | 0.666667 |
The diameter of a circle is divided into $n$ equal parts. On each part a semicircle is constructed. Find the value to which the sum of the areas of these semicircles approaches as $n$ becomes very large. | 0 | 0.083333 |
Find the arithmetic mean of the first 35 successive positive integers, starting from 5 and with a common difference of 2. | 39 | 0.916667 |
Given that a set $T$ of points in the $xy$-plane is symmetric about the origin, both coordinate axes, and the line $y = x$, and that the point $(1,4)$ is in $T$, find the smallest number of points in $T$. | 8 | 0.916667 |
If a stationery shop owner purchased 2000 pencils at $0.20 each and sells them for $0.40 each, find how many he must sell to make a profit of exactly $160.00. | 1400 | 0.583333 |
Let's find the smallest positive integers $x$ and $y$ for which the product of $180$ and $x$ is a fourth power and the product of $180$ and $y$ is a sixth power. Find the sum of $x$ and $y$. | 4500 + 4050000 = 4054500 | 0.666667 |
What is the least possible value of $(xy-2)^2 + (x-1+y)^2$ for real numbers $x$ and $y$? | 2 | 0.416667 |
Let $a$ and $b$ be distinct real numbers for which $\frac{a}{b} + \frac{a+5b}{b+5a} = 2$. Find $\frac{a}{b}$. | \frac{3}{5} | 0.916667 |
Given that the even number 138 can be expressed as the sum of two different prime numbers, calculate the largest possible difference between the two primes. | 124 | 0.75 |
Find the area of a polygon with vertices at $(2, 1)$, $(4, 3)$, $(7, 1)$, and $(4, 6)$. | 7.5 | 0.666667 |
Given that Alice is 1.6 meters tall and can reach 50 centimeters above the top of her head, and the ceiling is 3 meters above the floor, find the height of the stool in centimeters if standing on it allows her to just reach a light bulb 15 centimeters below the ceiling while stretching her arm around a hanging decorative item 20 centimeters below the ceiling. | 75 | 0.833333 |
Given whole numbers between $200$ and $500$ contain the digit $3$, determine the number of such numbers. | 138 | 0.5 |
Given that $\log_a b = c^3$ and $a + b + c = 100$, find the number of ordered triples of integers $(a,b,c)$ with $a \ge 2$, $b\ge 1$, and $c \ge 0$. | 1 | 0.916667 |
How many primes less than $200$ have $3$ as the ones digit? | 12 | 0.75 |
Given $0\le x_0<1$, let
\[x_n=\left\{ \begin{array}{ll} 2x_{n-1} &\text{ if }2x_{n-1}<1 \\ 2x_{n-1}-1 &\text{ if }2x_{n-1}\ge 1 \end{array}\right.\]
for all integers $n>0$. Determine the number of $x_0$ values for which $x_0=x_6$. | 63 | 0.5 |
Determine the difference between the number of dark squares and the number of light squares in a $9 \times 9$ grid with alternating colors, where each row starts with the opposite color of the last square from the preceding row. | 1 | 0.583333 |
Cagney can frost a cupcake every 25 seconds and Lacey can frost a cupcake every 35 seconds. They work together for 7 minutes, but there is a 1-minute period where only Cagney is frosting because Lacey takes a break. What is the number of cupcakes they can frost in these 7 minutes? | 26 | 0.166667 |
Determine how many students are enrolled only in the math class, not in science, given that there are 120 students in the eighth grade, 85 students taking a math class, and 65 students taking a science class. | 55 | 0.916667 |
The product of the two $101$-digit numbers $707,070,707,...,070,707$ and $909,090,909,...,090,909$ has tens digit $C$ and units digit $D$. Find the sum of $C$ and $D$. | 9 | 0.333333 |
Each of two boxes contains three chips numbered $1$, $4$, $5$. Calculate the probability that the product of the numbers on the two chips drawn is a multiple of $3$. | 0 | 0.5 |
$\frac{900^2}{306^2-294^2}$ can be simplified to a positive value. | 112.5 | 0.5 |
Given the sticker price of a gadget, calculate the difference in cost of purchasing the gadget at store A, where a 20% discount is applied and two rebates of $60 and $20 are applied subsequently, and store B, where a 35% discount is applied, and if the difference in cost is $20, determine the sticker price of the gadget. | 400 | 0.416667 |
If \( a, b > 0 \) and the triangle in the first quadrant bounded by the coordinate axes and the graph of \( ax + by = 12 \) has an area of 9, find the value of \( ab \). | 8 | 0.416667 |
Given an isosceles triangle $ABC$ with a perimeter of $24$ cm and equal sides $a$, find the maximum area of triangle $ABC$. | 16\sqrt{3} \text{ cm}^2 | 0.583333 |
If a child has $3$ red cubes, $3$ blue cubes, and $4$ green cubes, and wants to build towers measuring $9$ cubes in height, calculate the number of different towers she can make. | 4,200 | 0.083333 |
Given the expression $\frac{x + 2}{x - 2}$, each $x$ is replaced by $\frac{x + 2}{x - 2}$. Calculate the resulting expression for $x = 2$. | 1 | 0.833333 |
Given a quadratic equation \( Dx^2 + Ex + F = 0 \) with roots \( \alpha \) and \( \beta \), find the value of \( p \) such that the roots of the equation \( x^2 + px + q = 0 \) are \( \alpha^2 + 1 \) and \( \beta^2 + 1 \). | \frac{2DF - E^2 - 2D^2}{D^2} | 0.25 |
Given that $\left(64\right)^{0.375}\left(64\right)^{0.125}$, calculate the value of this expression. | 8 | 0.916667 |
Given two circles with radii 5 units and 7 units, determine the maximum number of different values for the number of lines that can be tangent to both circles simultaneously. | 5 | 0.583333 |
Determine the total number of different selections possible for five donuts when choosing from four types of donuts (glazed, chocolate, powdered, and jelly), with the additional constraint of purchasing at least one jelly donut. | 35 | 0.333333 |
Given that Esha can cycle 18 miles in 2 hours and 15 minutes, and she can now stroll 12 miles in 5 hours, calculate the difference in minutes between her current strolling speed and her previous cycling speed. | 17.5 | 0.083333 |
Given the set $\{1,2,3,4,\dots,44,45\}$, two distinct numbers are selected so that the sum of the remaining $43$ numbers is the product of these two numbers. Find the difference between these two numbers. | 9 | 0.916667 |
A merchant purchases products at a discount of $30\%$ from the list price and wants to set a new marked price. They plan to offer a $20\%$ discount on this marked price but still aim to make a $30\%$ profit on the selling price. Determine the percentage of the list price at which the goods must be marked. | 125\% | 0.583333 |
Determine the value(s) of $k$ such that the points $(1,4)$, $(3,-2)$, and $(6, k/3)$ lie on the same straight line. | -33 | 0.916667 |
A triangular array of $3003$ coins has $1$ coin in the first row, $2$ coins in the second row, $3$ coins in the third row, and so on up to $N$ coins in the $N$th row. Calculate the sum of the digits of $N$. | 14 | 0.916667 |
Choose four different numbers from the set $\{3, 5, 7, 9, 11, 13\}$, add the first two and the last two of these numbers together, and then multiply the two sums. What is the smallest result that can be obtained from this process? | 128 | 0.25 |
A standard 12-sided die is rolled once. The number rolled is the diameter of a circle in units. Calculate the probability that the numerical value of the circumference of the circle is less than ten times the numerical value of the area of the circle. | 1 | 0.833333 |
The product of the two $101$-digit numbers $707,070,707,...,070,707$ and $909,090,909,...,090,909$ has tens digit $C$ and units digit $D$. Find the sum of $C$ and $D$. | 9 | 0.333333 |
Triangle DEF has vertices D = (4,0), E = (0,4), and F, where F is on the line x + y = 10. Determine the area of triangle DEF. | 12 | 0.916667 |
A 9 × 9 board is composed of alternating dark and light squares, with the upper-left square being dark. How many more dark squares are there than light squares? | 1 | 0.416667 |
Two cyclists, X and Y, start at the same time to ride from Huntington to Montauk, a distance of 80 miles. Cyclist X travels 6 miles an hour slower than cyclist Y. Cyclist Y reaches Montauk and immediately turns back, meeting X 16 miles from Montauk. Determine the rate of cyclist X. | 12 | 0.666667 |
Given that there are 65 students wearing blue shirts and 85 students wearing yellow shirts, and they are paired into 75 pairs, with 30 pairs consisting of two blue shirts, find the number of pairs consisting of two yellow shirts. | 40 | 0.5 |
Given a structure formed by connecting eight unit cubes, calculate the ratio of the volume in cubic units to the surface area in square units. | \frac{1}{3} | 0.666667 |
A store owner bought $2000$ pencils at $0.08$ each. If he sells them for $0.20$ each, determine the number of pencils he must sell to make a profit of exactly $160.00$. | 1600 | 0.916667 |
If \(5(3y + 7\pi) = Q\), calculate the value of \(10(6y + 14\pi)\). | 4Q | 0.75 |
Given positive real numbers \( x \) and \( y \), define the operation \( x*y = \frac{x \cdot y}{x+y} \). If \( z = x + 2y \), calculate the value of the operation \( x*z \). | \frac{x^2 + 2xy}{2(x + y)} | 0.166667 |
John drove the first quarter of his journey on a highway, then 30 miles through a city, and the remaining one-sixth on country roads. Calculate the total length of John's journey in miles. | \frac{360}{7} | 0.75 |
Jo and Blair take turns counting starting from $2$, with each number said being double the previous number said. What is the $53^{\text{rd}}$ number said? | 2^{53} | 0.416667 |
Given that a rectangular room is 15 feet long and 108 inches wide, calculate the area of the new extended room after adding a 3 feet wide walkway along the entire length of one side, in square yards, where 1 yard equals 3 feet and 1 foot equals 12 inches. | 20 | 0.416667 |
The perimeter of an equilateral triangle exceeds the perimeter of a square by 2016 cm. The length of each side of the triangle exceeds the length of each side of the square by d cm. The square has a perimeter greater than 0. Find the number of positive integers that are NOT possible values for d. | 672 | 0.833333 |
Estimate the value of $\sqrt{90} - \sqrt{88}$. | 0.11 | 0.25 |
A large equilateral triangle with a side length of $20$ units is intended to be filled with non-overlapping equilateral triangles whose side length is $2$ units each, but $10$ of these small triangles are missing from the filling due to damage. Calculate the number of small undamaged triangles needed to fill the remainder of the large triangle. | 90 | 0.75 |
Joe had walked three-fourths of the way from home to school in 3 minutes, and ran the remaining one-fourth of the way to school at a speed 4 times his walking speed. | 3.25 | 0.833333 |
Solve for the number of roots satisfying the equation $\sqrt{9 - x} = 2x\sqrt{9 - x}$. | 2 | 0.833333 |
A circle of radius 3 is centered at $A$. An equilateral triangle with side 6 has a vertex at $A$. Find the difference between the area of the region that lies inside the circle but outside the triangle and the area of the region that lies inside the triangle but outside the circle. | 9(\pi - \sqrt{3}) | 0.166667 |
Alice has \(8p + 2\) quarters and Bob has \(3p + 6\) quarters. Calculate the difference in their money in terms of half-dollars. | 2.5p - 2 | 0.75 |
Given that Tom, Dick, and Harry are playing a game with a biased coin with probabilities of flipping heads as \(\frac{1}{3}\) and tails as \(\frac{2}{3}\), find the probability that all three will have flipped their coins the same number of times before getting their first head. | \frac{1}{19} | 0.916667 |
Suppose the radius of Earth at the equator is about 3950 miles. A jet flies around the Earth along the equator at a speed of 550 miles per hour. Assuming a negligible height above the equator, estimate the number of hours the flight would take. | 45 | 0.583333 |
Determine how many integers n result in $(n+i)^6$ being an integer. | 1 | 0.75 |
The number of revolutions of a wheel, with fixed center and with an outside diameter of $10$ feet, required to cause a point on the rim to go two miles. | \frac{1056}{\pi} | 0.666667 |
Given that Alice and Bob's real numbers are chosen uniformly at random from the intervals [0, 1] and $\left[\frac{1}{3}, \frac{2}{3}\right]$ respectively, determine the number that Carol should choose to maximize her chance of winning. | \frac{1}{2} | 0.5 |
In a cartesian coordinate system, a region termed "rhino's horn segment" is studied. This area is bounded by the quarter-circle in the first quadrant centered at origin $(0,0)$ with radius $4$, the half-circle in the first quadrant centered at $(0,2)$ with radius $2$, and the line segment connecting $(0,0)$ to $(4,0)$. Calculate the area of the "rhino's horn segment". | 2\pi | 0.583333 |
Given that soda is sold in packs of 8, 15, and 32 cans, determine the minimum number of packs needed to buy exactly 120 cans of soda. | 6 | 0.416667 |
A laser is placed at the point $(2,3)$. The laser beam travels in a straight line. The objective is for the beam to hit and bounce off the $y$-axis, then hit and bounce off the $x$-axis, and finally hit the point $(8,3)$. Calculate the total distance the beam will travel along this path. | 2\sqrt{34} | 0.75 |
A regular polygon of n sides is inscribed in a circle of radius \( \sqrt{2} \). The area of the polygon is \( 6 \). Determine the value of n. | 12 | 0.916667 |
Two joggers, 8 miles apart, begin at the same point but head in different directions. They would meet each other in 2 hours if they jogged towards each other, but they would take 4 hours to be 8 miles apart again if they jogged away from each other in the same direction at the beginning. What is the ratio of the speed of the faster jogger to the slower jogger? | 3 | 0.916667 |
Given that Mindy made four purchases for $\textdollar 2.47$, $\textdollar 7.51$, $\textdollar 11.56$, and $\textdollar 4.98$, find her total, to the nearest dollar. | 27 | 0.916667 |
Consider a large equilateral triangle with a side length of 4 units. There's a smaller equilateral triangle whose area is one-third the area of the larger triangle. Find the median of the trapezoid formed by these triangles. | 2 + \frac{2\sqrt{3}}{3} | 0.416667 |
Igor has received test scores of $88, 92, 75, 83,$ and $90$. He hopes to reach an average score of at least $87$ with his next test. What is the minimum score he would need on the upcoming test to meet his goal. | 94 | 0.833333 |
Determine the fraction of the area of the regular octagon $ABCDEFGH$ that is shaded given that the shaded region includes the entire triangles $\triangle BCO$ and $\triangle CDO$, and two-thirds of triangle $\triangle ABO$. | \frac{1}{3} | 0.75 |
Consider a sequence where each term is the sum of the two previous terms. The sequence starts with two numbers. If the last three terms of an 8-term sequence are 21, 34, and 55, determine the first term of the sequence. | 2 | 0.916667 |
Points A and B are on a circle with a radius of 5 and AB = 6. Given that C is the midpoint of the major arc AB, calculate the length of line segment AC. | 3\sqrt{10} | 0.5 |
The average of the first 300 terms of the sequence $2, -4, 6, -8, 10, -12, \ldots,$ can be found by calculating the sum of the first 300 terms and dividing by 300, where the nth term of the sequence is given by $(-1)^{n+1}\cdot 2n$. | -1 | 0.833333 |
Let $M$ be the greatest five-digit number whose digits have a product of $90$. Find the sum of the digits of $M$. | 18 | 0.833333 |
Consider a $3 \times 3$ array where each row and each column is an arithmetic sequence with three terms. The first term of the first row is $3$, and the last term of the first row is $15$. Similarly, the first term of the last row is $9$, and the last term of the last row is $33$. Determine the value of the center square, labeled $Y$. | 15 | 0.666667 |
How many 4-digit positive integers, where each digit is odd, are divisible by 3? | 208 | 0.583333 |
If the sum of all angles except one of a convex polygon is $2970^{\circ}$, calculate the number of sides of the polygon. | 19 | 0.916667 |
Given a basketball player made $7$ baskets during a game, each either a $2$-point or a $3$-point shot, calculate the number of different total scores the player could have achieved. | 8 | 0.916667 |
Given a rectangular storage with length 20 feet, width 15 feet, and height 10 feet, and with the floor and each of the four walls being two feet thick, calculate the total number of one-foot cubical blocks needed for the construction. | 1592 | 0.25 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.