problem stringlengths 11 2.6k | answer stringlengths 1 359 | source stringclasses 10
values | domain listlengths 1 3 | llama8b_solve_rate float64 0.11 0.89 |
|---|---|---|---|---|
The equation of the tangent line to the circle $x^2+y^2-4x+2y=0$ at the point $P(1,1)$ is: ______. | x-2y+1=0 | cn_k12 | [
"Mathematics -> Geometry -> Plane Geometry -> Circles",
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.734375 |
There are four numbers, and their sum is 45. If you add 2 to the first number, subtract 2 from the second number, multiply the third number by 2, and divide the fourth number by 2, the results are all the same. Find the exact values of the original four numbers in order, separated by commas. | 8, 12, 5, 20 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.578125 |
James injured his ankle and decides to slowly start working back up to his previous running goals and then surpass them. Before the injury, he was able to run 100 miles per week. He wants to get up to 20% more than that total in 280 days and each week he will increase miles walked in the week by the same amount. How... | 3 | gsm8k | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.109375 |
A sells a bicycle to B at a profit of 20%. B sells it to C at a profit of 25%. C pays Rs. 225 for it. What is the cost price of the bicycle for A? | 150 | orca_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.640625 |
A courtyard is 18 meters long and some meters wide and is to be paved with bricks of dimensions 12 cm by 6 cm. The total number of bricks required is 30,000. How wide is the courtyard? | 12 | orca_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.828125 |
Find the solution set of the inequality $5-x^2 > 4x$. | solution set of the inequality 5 - x^2 > 4x is (-5, 1) | cn_k12 | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.359375 |
Given the function f(x) = (a-1)x + 1 is a decreasing function on R, determine the condition on 'a' that satisfies this behavior. Express your answer as an inequality in the format a [relation] [value], where [relation] is <, >, ≤, or ≥. | a < 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.890625 |
Given that the real number $m$ is the middle term of an arithmetic sequence with first term $1$ and last term $5$, find the value of $m$. Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.6875 |
A car leaves Station A and travels 10 kilometers, then continues at a constant speed of 60 kilometers per hour for t hours. Derive the mathematical expression that represents the relationship between the distance s (in kilometers) the car has traveled from Station A and the time t (in hours). Express your answer as an ... | s = 10 + 60t | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.484375 |
Given a geometric sequence {a_n} with a_1 = a_8 = 3, find the formula for the sum of the first n terms, S_n. Express your answer as a formula in terms of n. | S_n = 3n | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.375 |
If x gets 25% more than y and y gets 20% more than z, the share of z out of a certain amount is Rs. 200. What is the total amount? | 740 | orca_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.78125 |
Two snow plows are working to clear the snow. The first one can clear the street in 1 hour, while the second one can do it in 75% of that time. Starting simultaneously, both machines worked together for 20 minutes, after which the first machine stopped working. How much more time does the second machine need to finish ... | 10 \text{ minutes} | olympiads | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.484375 |
How many integer pairs (m, n) satisfy the equation m^3 + 6m^2 + 5m = 27n^3 + 9n^2 + 9n + 1? Express your answer as a single integer. | 0 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.109375 |
Given that a complex number $z$ satisfies $(1+2i) \overline{z} = 4+3i$, find $z$. | 2 + i | cn_k12 | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.5 |
Given that the circle $x^2 + y^2 - 6x + 6y + 14 = 0$ is symmetrical about the line $l: ax + 4y - 6 = 0$, find the slope of the line $l$. Express your answer as a fraction or integer. | -\frac{3}{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.578125 |
Find the smallest positive period of the function $f(x)=\frac{\cos x+\sin x}{\cos x-\sin x}$. Express your answer as a single numerical value. | \pi | big_math | [
"Mathematics -> Precalculus -> Functions"
] | 0.5 |
Express 121 trillion yuan in scientific notation. Provide your answer in the form $a \times 10^{n}$ where $1 \leq a < 10$ and $n$ is an integer. | 1.21 \times 10^{14} | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.703125 |
Given a geometric sequence {a_n} with a positive common ratio, and a_3 * a_9 = 2a_5^2, and a_2 = 1, find the value of a_1. Express your answer as a numerical value. | \frac{\sqrt{2}}{2} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.671875 |
Given the circle $x^2+y^2+2x-4y+4=0$ is symmetric about the line $y=2x+b$, then $b=\boxed{4}$. | 4 | cn_k12 | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.78125 |
Let $S=\{0, 1, 2, 3, \ldots, n\}$. The sum of the absolute differences between any two elements in set $S$ is ______. | \frac{1}{6}n^3+ \frac{1}{2}n^2+ \frac{1}{3}n | cn_k12 | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.171875 |
49 ã — 49 = x ^ 4. What is the value of x? | 1 | orca_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.125 |
The partial fraction decomposition of
\[\frac{x^2 - 19}{x^3 - 2x^2 - 5x + 6}\]is
\[\frac{A}{x - 1} + \frac{B}{x + 2} + \frac{C}{x - 3}.\]Find the product $ABC.$ | 3 | math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.421875 |
I am trying to make a three-digit number by picking three different numbers. Given the numbers 1, 2, 6, 7, and 8, what is the difference between the largest number and the smallest number? | 750 | orca_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.1875 |
Let \(D\) be a point on side \(BC\) of \(\triangle ABC\). Points \(E\) and \(F\) are the centroids of \(\triangle ABD\) and \(\triangle ACD\), respectively. The line segment \(EF\) intersects \(AD\) at point \(G\). Find the value of \(\frac{DG}{GA}\). | \frac{1}{2} | olympiads | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.234375 |
Given the system of linear equations in $x$ and $y$,
$$\begin{cases} \lambda x - 12y = 2 \\ 5x + 6y = -1 \end{cases}$$
If the system has infinitely many solutions, then $\lambda = \_\_\_\_\_\_.$ | \lambda = -10 | cn_k12 | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.671875 |
In the diagram, $\triangle P Q R$ is equilateral and has a side length of $6 \mathrm{~cm}$. Each of the shaded triangles is equilateral and has a side length of $2 \mathrm{~cm}$. Calculate the fraction of the area of $\triangle P Q R$ that is shaded. Express your answer as a simplified fraction. | \frac{1}{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.125 |
Evaluate the expression: 11 + sqrt ( - 4 + 6 × some number ÷ 3 ) = 13. What is that number? | 4 | orca_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.796875 |
Two real numbers a and b are randomly chosen from the interval (0,1). Find the probability that the equation x^2 - sqrt(a) x + b = 0 has real roots. Express your answer as a numerical value between 0 and 1. | \frac{1}{8} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.46875 |
A can do a piece of work in 4 days. B can do it in 8 days. With the assistance of C, they completed the work in 2 days. In how many days can C alone do the work? | 8 | orca_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.859375 |
A standard fair coin is tossed three times. What is the probability that the three outcomes are all the same? Express your answer as a simplified fraction. | \frac{1}{4} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.828125 |
Given that the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is $60^{\circ}$, ($|\overrightarrow{a}| = 2, |\overrightarrow{b}| = 1$), determine the projection of $\overrightarrow{a}$ on $\overrightarrow{b}$. | 1 | cn_k12 | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.78125 |
Simplify: $(\frac{3}{a+1}-1)÷\frac{a-2}{a^2+2a+1}$. | -a-1 | cn_k12 | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.296875 |
Given that $f(x)=x^{2}+3ax+4$, and $b-3 \leqslant x \leqslant 2b$ is an even function, find the value of $a-b$. | -1 | cn_k12 | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.234375 |
The function f(x) is an odd function defined on the real numbers. When x < 0, f(x) = (x + 1)^3e^(x+1) - e. How many extremum points does the function f(x) have? Provide your answer as a whole number. | 2 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.328125 |
John won $155250 in the lottery and decided to give one-thousandth of the winnings to each of the top students in his college. The total amount received by these students is $15525. How many students received the money from John? | 100 | orca_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.84375 |
Given the sets $A={0,1,2}$ and $B={a+2,a^2+3}$, if $A \cap B = {1}$, then the value of the real number $a$ is $\_\_\_\_\_\_\_\_$. | -1 | cn_k12 | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.890625 |
Camila has only gone hiking 7 times in her life. Amanda has gone on 8 times as many hikes as Camila, and Steven has gone on 15 more hikes than Amanda. If Camila wants to say that she has hiked as many times as Steven and plans to go on some hikes a week, it would take Camila 16 weeks to achieve her goal. How many hikes... | 4 | orca_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.609375 |
In $\triangle ABC$, the three sides $a$, $b$, and $c$ satisfy the equation $b^{2}+a^{2}-c^{2}= \sqrt {3}ab$. Find the measure of $\angle C$. | C=\frac{\pi}{6} | cn_k12 | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.109375 |
In the right triangle \(ABC\), the leg \(AB\) is 21, and the leg \(BC\) is 28. A circle with center \(O\) on the hypotenuse \(AC\) is tangent to both legs.
Find the radius of the circle. | 12 | olympiads | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.109375 |
Given two vectors perpendicular to the edge of a dihedral angle on its two faces, which are (0,-1,3) and (2,2,4), respectively, calculate the cosine value of this dihedral angle. Express your answer as a fraction with a square root if necessary, and include the ± symbol if the cosine value can be positive or negative. | \pm \dfrac{\sqrt{15}}{6} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.21875 |
Given the equation (1+2i)z = 1-i, calculate the value of the complex number z. Express your answer in the form a + bi, where a and b are real numbers. | -\frac{1}{5} - \frac{3}{5}i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.8125 |
Determine the range of the function $f(x) = 4^x - 2^{x+1} + 3$. | [2, +\infty) | cn_k12 | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.484375 |
The Razorback t-shirt shop sells various types of merchandise including t-shirts, hats, and scarves. Each t-shirt is sold for $98, each hat for $45, and each scarf for $60. During the Arkansas game and the Texas tech game, they sold a total of 163 items. During the Arkansas game, they sold 89 items including 42 t-shirt... | \$6456 | orca_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.8125 |
Given the sets $M = \{x \mid -1 \leq x < 3\}$, and $N = \{x \mid 2 < x \leq 5\}$, find the union $M \cup N$. | \{x \mid -1 \leq x \leq 5\} | cn_k12 | [
"Mathematics -> Algebra -> Other"
] | 0.359375 |
In the city, there is a block of flats. This block has some floors. Half of the floors have 6 apartments and the other half have 5 apartments. One apartment can accommodate a maximum of 4 residents. The maximum number of residents that can live in this block of flats is 264. How many floors does this block have? | 12 | orca_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.8125 |
You and a friend go to a Mexican restaurant. You order 2 tacos and 3 enchiladas, and your friend orders 3 tacos and 5 enchiladas. Your bill is some amount plus tax, and your friend's bill is 12.70 dollars plus tax. How much was your bill before tax if each enchilada costs $2? | $7.80 | orca_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.484375 |
A train is moving at a speed of 132 km/h. The length of the train is 110 meters. At the same time, another train, 165 meters long, is moving in the opposite direction with a speed of 96 km/h. How long will it take for the two trains to completely cross each other? | 4.34 | orca_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.46875 |
If $x \cdot \log_{3}2 = 1$, then $2^{x} =$ \_\_\_\_\_\_. | 3 | cn_k12 | [
"Mathematics -> Algebra -> Intermediate Algebra -> Logarithmic Functions"
] | 0.890625 |
Xiaohong and Xiaoming both brought money to buy the book "Fun Math". At the bookstore, Xiaohong found that she was short 2 yuan and 2 jiao, while Xiaoming was short 1 yuan and 8 jiao. When they combined their money, they had just enough to buy one copy. What is the price of each "Fun Math" book in yuan? | 4 | olympiads | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.265625 |
Given $$f(x)= \begin{cases} \left.\begin{matrix}f(x-4)，x>0 \\ \int_{ x }^{ \frac {\pi}{4} } \cos t \, dt，x\leq0\end{matrix}\right.\end{cases},$$ find $f(2012)$. | \frac{\sqrt{2}}{2} | cn_k12 | [
"Mathematics -> Calculus -> Other"
] | 0.78125 |
Eugene built a complex model house out of toothpicks and a deck of playing cards. For every card, he used 64 toothpicks for the walls and an additional 12 toothpicks for each of the windows and doors. The deck of playing cards had 52 cards, and Eugene used all but 23 of the cards to build his house. There are three win... | 7 | orca_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.671875 |
Calculate the limit of the function:
$$\lim _{x \rightarrow 2} \frac{\ln \left(9-2 x^{2}\right)}{\sin 2 \pi x}$$ | -\frac{4}{\pi} | olympiads | [
"Mathematics -> Calculus -> Limits -> Other"
] | 0.359375 |
Calculate the limit of the function:
\[
\lim_{x \rightarrow 0} \frac{e^{5x} - e^{3x}}{\sin 2x - \sin x}
\] | 2 | olympiads | [
"Mathematics -> Calculus -> Differential Calculus -> Limits"
] | 0.203125 |
At the end of the first semester of the first year in a certain school, a survey on students' interests and hobbies was conducted. It was found that 68% of the students like physics, and 72% of the students like chemistry. What is the minimum percentage of students who like both physics and chemistry? | 40\% | cn_k12 | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.765625 |
Given the hyperbola $x^{2}- \frac {y^{2}}{3}=1$, find the angle between its two asymptotes. Express your answer in degrees. | 60^{\circ} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.71875 |
In a school located at the heart of the city, there are only 2 wheelers and 4 wheelers parked. If there are 11 four-wheelers parked there, what is the total number of wheels in the parking area? | 44 | orca_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.359375 |
Given the left and right foci of ellipse $E$ as $F_1$ and $F_2$ respectively, the maximum distance from a point $P$ on ellipse $E$ to focus $F_1$ is 7 and the minimum distance is 1. Calculate the square root of the eccentricity of ellipse $E$, expressing your answer as a simplified fraction or radical. | \frac{\sqrt{3}}{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.625 |
Find the number of integer solutions to the inequality $\log_{3}|x-2| < 2$. | 17 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Inequalities"
] | 0.53125 |
There are 200 more red apples than green apples in a grocery store. A truck arrives and delivers some green apples. There were originally 32 green apples. Now, there are 140 more green apples than red apples in the store. How many green apples did the truck deliver? | 340 | orca_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.4375 |
Given that $a$ and $b$ are real numbers, if for any $x\in R$, $\left(\ln a+b\right)e^{x}-a^{2}ex\geqslant 0$ always holds, then the minimum value of $\frac{b}{a}$ is ____. | 1 | cn_k12 | [
"Mathematics -> Algebra -> Other"
] | 0.171875 |
If the graph of the function $y=\sin(2x+\varphi)$ ($0<\varphi<\pi$) is symmetric about the line $x= \frac{\pi}{3}$, then the value of $\varphi$ is \_\_\_\_\_\_. | \frac{5\pi}{6} | cn_k12 | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.109375 |
Let the set \( A = \{a_{1}, a_{2}, a_{3}, a_{4}\} \). If the set of sums of elements from all possible three-element subsets of \( A \) is \( B = \{-1, 3, 5, 8\} \), then find the set \( A \). | \{-3, 0, 2, 6\} | olympiads | [
"Mathematics -> Discrete Mathematics -> Combinatorics -> Other"
] | 0.125 |
Find all positive integers $x$ and $y$ such that
$$
\frac{1}{x}+\frac{1}{y}=\frac{1}{4}
$$ | (5, 20), (6, 12), (8, 8), (12, 6), (20, 5) | olympiads | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.453125 |
On a certain road, 20% of the motorists exceed the posted speed limit and receive speeding tickets, but some percent of the motorists who exceed the posted speed limit do not receive speeding tickets. If 25% of the motorists on that road exceed the posted speed limit, what percent of the motorists who exceed the speed ... | 20\% | orca_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Other"
] | 0.234375 |
A circular truncated cone has a top radius of 2, a bottom radius of 4, and a slant height of 2√10. Calculate the volume of the truncated cone. Express your answer in terms of π. | 56\pi | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Volume"
] | 0.296875 |
If $ rac{x-y}{z-y}=-10$, what is the value of $ rac{x-z}{y-z}$? | 11 | omnimath | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.28125 |
Josephine receives a bill from the hospital. 50 percent of the bill is for medication. 25 percent of the remaining bill is for overnight stays, and 175$ is for food. The rest of the bill, which is 1700$, is for the ambulance ride. What is the total amount of the bill? | $5000 | orca_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.6875 |
Given two boxes each containing three chips numbered 1, 2, 3, a chip is randomly drawn from each box, and the numbers on these two chips are multiplied. What is the probability that the resulting product is an even number? Express your answer as a fraction. | \frac{5}{9} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.625 |
James took a bike ride for some hours. In the second hour, he traveled 12 miles, which was 20 percent farther than he traveled the first hour. If he traveled 25 percent farther in the third hour than he did in the second hour, James traveled 37 miles during the entire ride. How many hours did James ride his bike? | 3 | orca_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.734375 |
The solution set of the inequality ${{2}^{{{x}^{2}}-2x-2}} < 2$ is. | (-1,3) | cn_k12 | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.796875 |
Points P, Q, and R are on a number line. Q is halfway between P and R. Given that P is at -6 and Q is at -1, find the position of R. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 0.859375 |
Find three consecutive natural numbers such that the sum of their squares is 2030, and the cube of the middle number minus the square of the middle number is divisible by 7. What is the middle number? | 26 | orca_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.53125 |
Two people, A and B, are participating in a shooting competition. The probability of hitting the target for A and B are $\frac{1}{2}$ and $\frac{2}{3}$, respectively (the two people hitting the target are independent events). If both of them shoot twice, what is the probability that they hit the target the same number ... | \frac{13}{36} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Other"
] | 0.125 |
In a workshop, there are 90 workers. Each worker can produce either 15 bolts or 24 nuts per day on average. If one bolt needs to be matched with two nuts, how should the workers be allocated to ensure that the bolts and nuts are exactly matched? | x = 40, y = 50 | cn_k12 | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.21875 |
On the ellipse $\frac{x^2}{25} + y^2 = 1$, the distance from a point $P$ to one of the foci is $2$. Calculate the distance from point $P$ to the other focus. Express your answer as a single numerical value. | 8 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.328125 |
He found out that they harvest 25 sacks of unripe oranges and some sacks of ripe oranges per day. After 73 days of harvest, they will have 365 sacks of ripe oranges. How many sacks of ripe oranges do they harvest per day? | 5 | orca_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.71875 |
A perfect power is an integer $n$ that can be represented as $a^{k}$ for some positive integers $a \geq 1$ and $k \geq 2$. Find the sum of all prime numbers $0<p<50$ such that $p$ is 1 less than a perfect power. | 41 | omnimath | [
"Mathematics -> Number Theory -> Other"
] | 0.421875 |
Given an arithmetic sequence {a_n} that satisfies a_2 + a_4 = 4 and a_3 + a_5 = 10, find the sum of its first 10 terms. Express your answer as a single integer. | 95 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.84375 |
Given the function $f(x) =
\begin{cases}
3x - \frac{1}{2}, & \text{if } x < 1\\
2^{x}, & \text{if } x \geq 1
\end{cases}$,
find the value of $f[f\left( \frac{1}{2} \right)]$. | 2 | cn_k12 | [
"Mathematics -> Algebra -> Functions"
] | 0.8125 |
Given $\frac{a}{b}=\frac{3}{4}$, then $\frac{{b-a}}{b}=$____. | \frac{1}{4} | cn_k12 | [
"Mathematics -> Algebra -> Prealgebra -> Fractions"
] | 0.890625 |
A high school with three grades has a total of 2800 students. A stratified sampling method is required to select a sample of size 40. It is known that there are 910 students in the first grade; the number of students sampled from the second grade accounts for 3/10 of the total sample size. Calculate the number of stude... | 15 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.125 |
James and Lisa ordered 2 small pizzas. Each pizza has a certain number of slices. James ate 2/3 of all the slices and ate 8 slices of pizza. How many slices does each pizza have? | 6 | orca_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.78125 |
If the equation \\(kx^{2}-2x+1=0\\) has real roots, then the range of values for \\(k\\) is \_\_\_\_\_\_\_\_. | k\leqslant 1 | cn_k12 | [
"Mathematics -> Algebra -> Equations and Inequalities -> Quadratic Functions"
] | 0.78125 |
If $f(x)$ is an odd function defined on $\mathbb{R}$, and when $x > 0$, $f(x) = \sqrt[3]{x+1}$, then find the expression for $f(x)$ when $x < 0$. Express your answer as a function of $x$. | -\sqrt[3]{-x+1} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.359375 |
Find the distance from the center of the circle $x^{2}+y^{2}-2x+2y=0$ to the line $y=x+1$. | \frac{3\sqrt{2}}{2} | cn_k12 | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.421875 |
Given the inequality system $\left\{\begin{array}{l}x＞1\\ x≤a\end{array}\right.$ has a solution set with a total of $3$ integer solutions, find the range of values for $a$. Express your answer as an inequality (e.g., 4 ≤ a < 5). | 4 ≤ a < 5 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.140625 |
Blanche, Rose, and Dorothy liked to collect sea glass when they went to the beach. Blanche found 12 pieces of green and 3 pieces of red sea glass. Rose found 9 pieces of red and 11 pieces of blue sea glass. Dorothy found twice as many pieces of red glass as Blanche and Rose and a certain multiple of blue sea glass piec... | 3:1 | orca_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.140625 |
In the space rectangular coordinate system Oxyz, given point D(2,1,0), vector m=(4,1,2), and m is perpendicular to plane DEF. Calculate the distance from point O to plane DEF. Express your answer as a simplified fraction, including radicals if necessary. | \frac{3\sqrt{21}}{7} | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.5625 |
Find the sum of the lengths of the edges of a regular tetrahedron with a side length of 3 centimeters (cm) plus 3. | 21 | orca_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.53125 |
Eugene has 127 pencils. He gets 14 more from Joyce. Then, he decides to give 7 pencils to each of his friends. If he gives pencils to all his friends, how many pencils is Eugene left with? | 1 | orca_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.59375 |
Bobby stands 10 feet from Sam, and Eric stands 8 feet from Bobby. What is the least number of feet that Eric could be from Sam? | 2 | math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.203125 |
A, B, and C are partners. A receives 2/3 of the profits, while B and C divide the remainder equally. A's income is increased by Rs. 300 when the rate of profit rises from one percentage to another. A's capital is Rs. 15,000. What is the change in the rate of profit that causes A's income to increase by Rs. 300? | 3\% | orca_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.34375 |
If the product of 18 integers is negative and all integers are distinct and greater than 1, at most how many of the integers can be negative? | 17 | orca_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.3125 |
Find the domain of the function $f(x) = \tan(\arccos(x^2)).$ | [-1,0) \cup (0,1] | math | [
"Mathematics -> Precalculus -> Functions"
] | 0.734375 |
A train is moving at 6/7 of its usual speed. The train is late by some minutes. The usual time for the train to complete the journey is 1.4999999999999993 hours. How many minutes is the train late? | 15 | orca_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.328125 |
Rs. 1440 are divided among A, B, and C so that A receives 1/3 as much as B. B receives a certain fraction as much as C. The amount B received is Rs. 270. What is the ratio of the amount B received to the amount C received? | 1:4 | orca_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.828125 |
Given vectors \(\overrightarrow{a}+\overrightarrow{b}=(0,\sqrt{2},0)\) and \(\overrightarrow{a}-\overrightarrow{b}=(2,\sqrt{2},-2\sqrt{3})\), calculate \(cos\langle\overrightarrow{a},\overrightarrow{b}\rangle\). Express your answer as a single value. | -\frac{\sqrt{6}}{3} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.296875 |
Given the arithmetic sequence -3, -7, -11, …, find the general term formula in terms of n. Express your answer as a formula using standard algebraic notation. | -4n+1 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.109375 |
John's marks were wrongly entered as 82 instead of some number. Due to that, the average marks for the class got increased by half (1/2). The number of students in the class is 80. What were John's actual marks? | 42 | orca_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.296875 |
Taequan is playing a dice game where he rolls three, fair, 6-sided dice twice. If he gets the exact same numbers on his second try as his first, in any order, he wins. If he rolled a 2, 3, and 4 on his first try, what is the probability that he will win with his second roll? | 1/36 | orca_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Combinations"
] | 0.703125 |
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