instruction_seed stringlengths 40 1.39k | answer stringlengths 1 95 | source stringclasses 1
value | domain listlengths 1 3 | llama8b_solve_rate float64 0 0.98 | _source stringclasses 1
value |
|---|---|---|---|---|---|
Given that $F$ is a focus of the hyperbola $C$: $y^{2}-mx^{2}=3m (m > 0)$, calculate the distance from point $F$ to one asymptote of $C$. Express your answer as a single value. | \sqrt{3} | big_math | [
"Mathematics -> Geometry -> Non-Euclidean Geometry -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of $\lg 5 + \lg 2$. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Logarithmic Functions"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $1 < m < 4$, $F\_1$ and $F\_2$ are the left and right foci of the curve $C: \frac{x^2}{4} + \frac{y^2}{4-m} = 1$, and point $P$ is the intersection point of curve $C$ and curve $E: x^2 - \frac{y^2}{m-1} = 1$ in the first quadrant. Line $l$ is the tangent line to curve $C$ at point $P$. If the incenter of tri... | 3 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a > 0, b > 0)$, where $F$ is the right focus and the distance from point $F$ to the asymptote is equal to $2a$, calculate the eccentricity $(e)$ of the hyperbola. Express your answer as a single value, including the square root symbol if necessary. | \sqrt{5} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given the values $a=-0.3^{2}$, $b=-3^{-2}$, $c={(-\frac{1}{3})^{-2}}$, and $d={(-\frac{1}{5})^0}$, determine their size relationship by ordering them from smallest to largest. Express your answer as a sequence of the variables $a$, $b$, $c$, and $d$, separated by less-than symbols (<). | b < a < d < c | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
A student observed the weather for \( x \) days during the summer holidays. The observations are as follows:
1. There were 7 mornings with sunny weather;
2. There were 5 afternoons with sunny weather;
3. It rained 8 times in total;
4. If it rained in the afternoon, the morning was sunny.
Calculate the total number of ... | 10 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Meteorological stations A and B make weather forecasts simultaneously. If their probabilities of making accurate forecasts are 0.8 and 0.7, respectively, and the accuracy of their forecasts is independent of each other, then find the probability that both meteorological stations make inaccurate forecasts in one forecas... | 0.06 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
The following twelve integers are written in ascending order:
$$
1, x, x, x, y, y, y, y, y, 8, 9, 11
$$
The mean of these twelve integers is 7. Calculate the median of these integers. Express your answer as a single number. | 8 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the real numbers $x$ and $y$ that satisfy the equation $2^{x}+4^{y}=1$, find the maximum value of $x+2y$. Express your answer as a single number. | -2 | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Calculus -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a$ and $b$ are unit vectors, and $a \cdot b = -\frac{1}{2}$. If the plane vector $p$ satisfies $p \cdot a = p \cdot b = \frac{1}{2}$, then find the magnitude of $p$, $|p|$. Express your answer as a single numerical value. | 1 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Using the digits 1, 2, and 3, form a four-digit number with the rule that all three digits must be used, but the same digit cannot be adjacent. What is the total number of such four-digit numbers? | 18 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
A worker spends time in the ratio $1: 2: 3$ to make 1 part $\mathrm{A}$, 1 part $\mathrm{B}$, and 1 part $\mathrm{C}$ respectively. If the worker uses 10 hours to produce 2 parts $\mathrm{A}$, 3 parts $\mathrm{B}$, and 4 parts $\mathrm{C}$, how many hours are needed to produce 14 parts $\mathrm{A}$, 10 parts $\mathrm{B... | 20 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Find the coefficient of $x^5$ in the expansion of $(1-x^3)(1+x)^{10}$. Express your answer as an integer. | 207 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $y=\sin x+\cos x$, the horizontal coordinates of each point on its graph are shortened to $\frac{1}{2}$ of their original length, resulting in the graph of $y=f(x)$. Calculate the smallest positive period of $y=f(x)$ in radians. | \pi | big_math | [
"Mathematics -> Precalculus -> Functions"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Beatrix has fixed a $2 \times 2$ square sheet of steel to a wall. She has two $1 \times 2$ magnetic tiles, both of which she attaches to the steel sheet, in any orientation, so that none of the sheet is visible and the line separating the two tiles cannot be seen. One tile has one black cell and one grey cell; the othe... | 12 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, the line segment with endpoints \( P(-4,0) \) and \( Q(16,0) \) is the diameter of a semi-circle. If the point \( R(0, t) \) is on the circle with \( t > 0 \), find the value of \( t \). Express your answer as a single integer. | 8 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
The first term of an arithmetic sequence is positive, and the sum of the first 3 terms is equal to the sum of the first 11 terms. At which term is the sum of the first few terms of this sequence maximized? Express your answer as a term number (e.g., 7). | 7 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence {a_n}, if a_3 = 9 and a_6 = 15, find the value of a_12. Express your answer as a single numerical value. | 27 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Six students from an environmental group in a class recorded the number of plastic bags their families threw away in one week. The results were as follows (unit: pieces): 33, 25, 28, 26, 25, 31. If there are 45 students in the class, estimate the total number of plastic bags discarded by all the students' families for ... | 1260 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
A hospital has 3 doctors and 6 nurses to be assigned to 3 schools for student physical examinations, with each school being allocated 1 doctor and 2 nurses. Calculate the total number of different allocation methods. Express your answer as a single integer. | 540 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $x^2 + 3x + 5 = 7$, calculate the value of the algebraic expression $3x^2 + 9x - 2$. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Polynomial Operations"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x) = \log_{a}(x-1) + 1$, find the fixed point (x, y) that the graph of this function always passes through, regardless of the value of 'a'. Express your answer as an ordered pair (x, y). | (2,1) | big_math | [
"Mathematics -> Precalculus -> Functions"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Select 4 students to participate in a relay race from a group of 8 students, which includes 6 boys and 2 girls, using stratified sampling by gender. If girls are not to be the last runner, find the number of different arrangements. Express your answer as a numerical value. | 720 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = $$\begin{cases} 2^{x}+1,x\geq 0 \\ |x|,x<0 \end{cases}$$, find all real number values of x such that f(x) = 3. Express your answer as a set of values or list all values explicitly. | \{-3, 1\} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
To understand the vision of 600 students, system sampling was used to extract a sample of 20. Calculate the number of groups the students should be divided into to perform the sampling. Express your answer as a whole number. | 30 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
A class is planning to go camping in the suburbs and has ordered tents from a store. The probability of rain is equal to the probability of no rain, and the probability of receiving the tents on time is also equal. As long as the tents arrive on time, they will not get wet. Calculate the probability that the tents will... | \frac{1}{4} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
The radius of the base of a cone is 3, and the slant height is 5. Find the volume of this cone. Express your answer as a product of pi and an integer. | 12\pi | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Volume"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Given a geometric sequence $\{a_n\}$ with all positive terms and $a_1=1$, where $a_1+a_3+a_5=21$, find the sum of $a_2+a_4+a_6$. Express your answer as a single integer. | 42 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\cos(\pi +\alpha) = \frac{2}{5}$, calculate the value of $\cos 2\alpha$. Express your answer as a fraction. | -\frac{17}{25} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Let f(x) be an even function with the domain of R, and f(2+x) = f(-x), f(1/2) = 1/2. Calculate the value of f(2023/2). Express your answer as a single fraction. | \frac{1}{2} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Find the smallest integer $k$ such that the quadratic equation in $x$, $2x(kx-4)-x^2+6=0$, has no real roots. Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
The warning water level of the second reservoir of Fen River is 905.7 meters. If exceeding the warning water level by 10 meters is denoted as "+10 meters", then how would you denote being below the warning water level by 5 meters? Express your answer as a notation (e.g., '+X meters' or '-X meters'). | -5 meters | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
If \( f(x) \) is an odd function periodic with a period of 5 on \( \mathbf{R} \), and \( f(7) = 9 \), calculate the exact value of \( f(2020) - f(2018) \). Express your answer as a single integer. | 9 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(1,1,0)$ and $\overrightarrow{b}=(-1,0,-2)$, find the value of $k$ such that $k\overrightarrow{a}+\overrightarrow{b}$ is parallel to $2\overrightarrow{a}-\overrightarrow{b}$. Express your answer as an integer. | -2 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Given a sequence $\{a_n\}$ defined by $a_{n+1}=2a_{n}$, and $a_{1}=1$, calculate the sum of the first five terms of the sequence. Express your answer as a single integer. | 31 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $m$ and $n$ are two consecutive integers, and $m < \sqrt{11} < n$, determine the value of $(m+n)$. Express your answer as a single integer. | 7 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Let the sum of the first \(n\) terms of an arithmetic sequence \(\{a_n\}\) be \(S_n\). If \(2a_3=3+a_1\), find the value of \(S_9\). Express your answer as a single number. | 27 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Given three real numbers \( x, y, z \) that satisfy the system of equations
\[
\begin{cases}
\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 2, \\
\frac{1}{x^2} + \frac{1}{y^2} + \frac{1}{z^2} = 1,
\end{cases}
\]
calculate the value of \( \frac{1}{xy} + \frac{1}{yz} + \frac{1}{zx} \). Express your answer as a single numerica... | \frac{3}{2} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Evaluate the expression: $\log_{0.5}0.125+\log_{2}[\log_{3}(\log_{4}64)]$. Provide your answer as a single integer. | 3 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Logarithmic Functions"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Given $\log_{3}{2}=a$ and $\log_{2}{7}=b$, calculate the value of $\log_{3}{7}$ in terms of $a$ and $b$. Express your answer as a product of $a$ and $b$. | ab | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Logarithmic Functions"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $\frac{1}{1-\sqrt[4]{3}}+\frac{1}{1+\sqrt[4]{3}}+\frac{2}{1+\sqrt{3}}$. Express your answer as a single number. | -2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Suppose $p$: $\exists x_0 \in \mathbb{R}$, $mx_0^2 + 1 \leqslant 0$, $q$: $x \in \mathbb{R}$, $x^2 + mx + 1 > 0$. If $p \vee q$ is a true statement, then find the range of the real number $m$. Express your answer in interval notation. | (-\infty, 2) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Quadratic Functions"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $$\sqrt[3]{(2-\pi )^{3}} + \sqrt {(3-\pi)^{2}}$$ and express your answer as a single numerical value. | -1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Find the derivative of the function y = x cos x - sin x. Express your answer as a function of x. | -x sin x | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Let the complex number z satisfy the equation (z - 2i)(2 - i) = 5. Find the value of z. Express your answer in the form a + bi, where a and b are the real and imaginary parts of z. | 2 + 3i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
A portable USB flash drive with a capacity of $2GB (1GB=2^{10}MB)$ is used to store digital photos. If the file size of each digital photo is $16MB$, calculate the number of photos that can be stored on the USB flash drive. Express your answer as a power of 2. | 2^{7} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Given positive numbers a and b satisfying a+2b=6, find the minimum value of 1/(a+2) + 2/(b+1). Express your answer as a fraction in simplest form. | \frac{9}{10} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Find the domain of the function y = 1 / (x + 3). Express your answer as an inequality or equation. | x \neq -3 | big_math | [
"Mathematics -> Precalculus -> Functions"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Given that letters A, B, C, and D represent four different digits selected from 0, 1, 2,..., 9, and (A+B)/(C+D) is an integer that is as large as possible, calculate the value of A+B. | 17 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
In an isosceles triangle, the base angle is $50^{\circ}$. Calculate the vertex angle of this isosceles triangle. Express your answer in degrees as a whole number. | 80 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Three segments are drawn from a point inside an equilateral triangle, each parallel to one of the triangle's sides, with lengths 2 m, 3 m, and 6 m. Find the perimeter of the equilateral triangle. Express your answer in meters as a whole number. | 33 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
A tangent line is drawn to the curve y = 1/x passing through point P(0,2). Calculate the coordinates of the point of tangency. Express your answer as an ordered pair (x, y). | (1,1) | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $3x^2 - 4x + k = 0$ with real roots, find the value of $k$ for which the product of the roots of the equation is a maximum. Express your answer as a single fraction or decimal. | \frac{4}{3} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
The temperature is $-3^{\circ}\mathrm{C}$ and then decreases by $6^{\circ}\mathrm{C}$. Calculate the resulting temperature. Express your answer in degrees Celsius (°C). | -9^{\circ}\mathrm{C} | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given a dihedral angle α-PQ-β, points A and B are on half-plane α, and point C is on half-plane β (A, B, and C are not on PQ, and the distances from A and B to PQ are not equal). Construct a trapezoid with A, B, and C as three of its vertices, such that the fourth vertex also lies on the same half-plane as the dihedral... | 2 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Convert $3\times 10^{10}$ nanometers to meters, given that one nanometer equals one millionth of a millimeter and 1000 millimeters equal 1 meter. Express your answer in meters, rounded to the nearest whole number if necessary. | 30 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
During the Tree Planting Festival, 20 students in a class planted trees along one side of a straight road, with each student planting one tree. The trees planted by adjacent students were 10 meters apart. Initially, the saplings were to be centrally placed next to one of the tree pits. To minimize the total distance wa... | 10 and 11 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given $$(1+x)^{10}=a_{0}+a_{1}(1-x)+a_{2}(1-x)^{2}+\ldots+a_{10}(1-x)^{10}$$, find the value of $a_{9}$. Express your answer as a single integer. | -20 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Among the natural numbers 1, 2, 3,... 100, how many numbers can be divided by 2, 3, and 4? Express your answer as a whole number. | 8 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
A sample data with a capacity of 10 is grouped, and the group intervals and frequencies are as follows: [1,2), 1; [2,3), 1; [3,4), 2; [4,5), 3; [5,6), 1; [6,7), 2. Calculate the frequency of the sample in the interval [1,5). Express your answer as a decimal value rounded to two decimal places. | 0.70 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Given that line $l_1: (k-3)x + (k+4)y + 1 = 0$ is perpendicular to line $l_2: (k+1)x + 2(k-3)y + 3 = 0$, find the value(s) of the real number $k$. Express your answer as a numerical value(s). | 3 \text{ or } -3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
A test consists of 6 multiple-choice questions. The grading rule is as follows: each person initially scores 6 points, each correct answer adds 4 points, each incorrect answer deducts 1 point, and no answer gets 0 points. If 51 students take the test, what is the minimum number of students who must have the same score?... | 3 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Given the polynomial $(1-2x)^7 = a + a_1 x + a_2 x^2 + \ldots + a_7 x^7$, find the value of the sum $a_1 + a_2 + a_3 + a_4 + a_5 + a_6 + a_7$. Provide your answer as a single integer. | -2 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Jonas builds a large \( n \times n \times n \) cube using \( 1 \times 1 \times 1 \) cubes each having the net shown. What is the smallest value of \( n \) for which the sum of the exterior faces of the \( n \times n \times n \) cube can be greater than 1500? Express your answer as a whole number. | 13 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Surface Area"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given set A = {x ∈ R | 0 < 3 - x ≤ 2} and set B = {x ∈ R | 0 ≤ x ≤ 2}, find A ∪ B. Express your answer in interval notation, including any relevant brackets or parentheses. | [0, 3) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the positive terms of a geometric sequence {a_n} satisfy a_3 = 1 and the arithmetic mean between a_5 and 3/2*a_4 is 1/2, find the value of a_1. Express your answer as a single number. | 4 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
A coin is tossed consecutively 3 times. Find the probability that exactly one time it lands on heads. Express your answer as a simplified fraction. | \frac{3}{8} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Combinations"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the reciprocal of $a$ is $-1$, find the value of $a^{2023}$. Express your answer as a single integer. | -1 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Let $f(x)$ be a differentiable function such that $\lim_{\Delta x\to 0} \frac{f(x_0+3\Delta x)-f(x_0)}{\Delta x}=-3$. Using the definition of a derivative, calculate the value of $f'(x_0)$. Express your answer as a single numerical value. | -1 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Given that {a_n} is an arithmetic sequence, with a_2 = -1 and a_8 = 5, find the sum of the first 9 terms of the sequence {a_n}, denoted as S_9. Express your answer as a single integer. | 18 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
In the sequence $\{a_n\}$, $a_1=1$, $a_2=2$, and $a_{n+2}-a_{n}=1+(-1)^{n}$. Calculate the sum of the first 100 terms, $S_{100}$. Express your answer as a single integer. | 2600 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given that b is the geometric mean of a and c, and abc = 27, calculate the value of b. | 3 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation (2+mi)(n-2i) = -4 - 3i, where m and n are real numbers, calculate the value of m/n. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $\sin ^{2}\alpha +2\sin \alpha \cos \alpha -2\sin \alpha -4\cos \alpha =0$, calculate the value of $\cos ^{2}\alpha -\sin \alpha \cos \alpha$. Express your answer as a fraction. | \frac{3}{5} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
In the number "20220420", the frequency of the digit "2" is to be calculated. Find the frequency by dividing the total number of occurrences of the digit "2" by the total number of digits in the number. Express your answer as a simplified fraction. | \frac{1}{2} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
A track and field team consists of 98 athletes in total, with 56 male athletes. By employing stratified sampling proportional to gender, a sample of size 28 should be drawn from all athletes. Calculate the number of female athletes that need to be selected. Express your answer as a whole number. | 12 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
In a class, students have three attitudes towards photography: "like", "dislike", and "neutral". The number of students with a "neutral" attitude is 12 more than those with a "dislike" attitude. A sample of 9 students included 5 students with a "like" attitude, 1 student with a "dislike" attitude, and 3 students with a... | 30 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the average of the data set x₁, x₂,..., x₂₀₁₉ is 100, what is the average of the new data set 2x₁+1, 2x₂+1,..., 2x₂₀₁₉+1? Express your answer as a single numerical value. | 201 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Let complex numbers $z_1$ and $z_2$ correspond to points in the complex plane that are symmetric about the origin, and $z_1 = 2-i$. Calculate the product $z_1 \cdot z_2$ and express your answer as a complex number in the form $a+bi$. | -3+4i | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
The function $y=(2m-1)x+b$ is decreasing on $\mathbb{R}$. Find the condition on $m$ that satisfies this property. Express your answer as an inequality involving the variable $m$. | m < \dfrac{1}{2} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Other"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Given a random variable $X$ that follows the normal distribution $N(3, \frac{1}{4})$, and $P(X > \frac{7}{2}) = 0.1587$, find the probability $P(\frac{5}{2} \leq X \leq \frac{7}{2})$. Express your answer as a decimal value between 0 and 1, rounded to 4 decimal places. | 0.6826 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence $\left\{a_{n}\right\}$ which satisfies: $a_{1}=0, a_{n+1}=\frac{\sqrt{2} a_{n}+\sqrt{6}}{a_{n}-\sqrt{2}}$ for $n \in \mathbf{N}^{*}$, calculate the value of $a_{2007}$. | 0 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Suppose $x$ and $y$ satisfy $x + y = 40$ and both $x$ and $y$ are positive numbers. Find the maximum value of $xy$. Express your answer as a single numerical value. | 400 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(-3,m)$ and $\overrightarrow{b}=(2,-2)$ are perpendicular to each other, calculate the magnitude of $\overrightarrow{a}$. Express your answer as a simplified radical expression. | 3\sqrt{2} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the sum of the first n terms of an arithmetic sequence {a_n} is S_n, and the equation S_4/12 - S_3/9 = 1 holds, find the common difference d. Express your answer as a single numerical value. | 6 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of solutions to the system of equations:
\[
\begin{cases}
x + 3y = 3 \\
||x| - |y|| = 1
\end{cases}
\]
Express your answer as a single integer. | 3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
The graph of the function y = sin(x + π/3) (x ∈ R) is translated to the left by m (m > 0) units. The resulting graph is symmetric about the y-axis. Determine the smallest possible value of m. Express your answer in terms of π. | \frac{\pi}{6} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
The diagram shows the ellipse whose equation is \(x^{2}+y^{2}-xy+x-4y=12\). The curve cuts the \(y\)-axis at points \(A\) and \(C\) and cuts the \(x\)-axis at points \(B\) and \(D\). Calculate the area of the inscribed quadrilateral \(ABCD\). Express your answer as a whole number. | 28 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation \( \sin^{x} \alpha + \cos^{x} \alpha = 1 \) with the constraint \( 0 < \alpha < \frac{\pi}{2} \), find the number of solutions for \( x \). Express your answer as a whole number. | 1 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
How many ordered integer pairs (x, y) are there satisfying (x^2-18)^2 ≡ y^2 (mod 31), where 0 ≤ x, y < 31? Express your answer as a single integer. | 60 | big_math | [
"Mathematics -> Number Theory -> Congruences"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given five numbers -9, $b_1$, $b_2$, $b_3$, -1 form a geometric sequence, and four numbers -9, $a_1$, $a_2$, -3 form an arithmetic sequence, find the value of $b_2(a_2-a_1)$. Express your answer as a single integer. | -6 | big_math | [
"Mathematics -> Algebra -> Sequences and Series",
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Four cards with the same texture, size, and back design have front sides drawn with a circle, a rectangle, an equilateral triangle, and a regular pentagon, respectively. If these cards are randomly placed face down on the table and one card is drawn at random, what is the probability that the drawn card has a front des... | \frac{1}{2} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Given the piecewise function $f(x)=\left\{{\begin{array}{l}{\log_2 x, x>0}\\{|x-2|+2, x\leq 0}\end{array}}\right.$, calculate $f(f(0))$. | 2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
What is the negation of the proposition "$\forall x > 1$, $x^{2}+1\geqslant 0$"? Express your answer as a logical statement. | \exists x > 1, x^{2}+1 < 0 | big_math | [
"Mathematics -> Discrete Mathematics -> Logic"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Given the quadratic function y=x^2+5x+7 and the domain -4 ≤ x ≤ 1, find the range of y. Express your answer as an inequality, including the minimum and maximum values of y. | \frac{3}{4} ≤ y ≤ 13 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Quadratic Functions"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
A pentagon has 5 vertices. From any given vertex, lines can be drawn to the other vertices. However, not all of these lines are diagonals. Specifically, lines drawn to the two adjacent vertices do not count as diagonals because they are sides of the pentagon. Calculate the number of diagonals that can pass through a ve... | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Polygons"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Given a real number \(a\), the set \(M\) is defined as \(M = \left\{x \mid x^{2}-3x-a^{2}+2=0, x \in \mathbf{R}\right\}\). Calculate the number of subsets of the set \(M\). Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Find the inclination angle of the line $x=\tan 45°$. Express your answer in degrees. | 90 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence {a_n} with a common difference d!= 0, and a_1, a_3, a_7 form a geometric sequence, find the exact value of (a_1 + a_3) / (a_2 + a_4). Express your answer as a simplified fraction. | \frac{3}{4} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
In a computer network with $n$ terminals, the probability of each terminal being used on any given day is $p$. Find the average number of terminals used in this network on any given day. Express your answer as an algebraic expression involving $n$ and $p$. | np | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.