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 the derivative of the function $f(x)$ is $f′(x)$, and they satisfy the relationship $f(x)=x^{2}+3xf′(2)$, find the value of $f′(2)$. Express your answer as a single number. | -2 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
A certain school has 160 staff members, including 104 full-time teachers, 32 administrative staff, and 24 logistics support staff. If a stratified sampling method is used to draw a sample of size 20, calculate the number of administrative staff members to be drawn. Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence $1, \sqrt{3}, \sqrt{5}, \sqrt{7}, 3, \sqrt{11}, \ldots, \sqrt{2n-1}$, find the term number of the value 5. Express your answer as a whole number. | 13 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation of the parabola y = -2x^2, find the coordinates of its focus. Express your answer as an ordered pair (x, y). | (0, -\frac{1}{8}) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Three students, A, B, and C, each take a bucket to a communal tap to fill with water at the same time. The time it takes for them to fill their buckets is 1.5 minutes, 0.5 minutes, and 1 minute, respectively. If they can only fill their buckets one at a time, and the students who are not filling their buckets have to w... | 5 \text{ minutes} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Four vertices of a quadrilateral are located at \((7,6), (-5,1), (-2,-3)\), and \((10,2)\). Calculate the area of the quadrilateral in square units. Express your answer as a whole number. | 63 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Evaluate the expression \((999 - 99 + 9) \div 9\) and provide your answer as a whole number. | 101 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) defined as follows:
$$
f(x) =
\begin{cases}
f(x+2), & \text{if } x < 3 \\
(\frac{1}{2})^{x}, & \text{if } x \geq 3
\end{cases}
$$
Find the value of f(-4). Express your answer as a fraction. | \frac{1}{16} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the terminal side of angle α passes through point P(cos(π/7), -sin(π/7)), and 0 < α < 2π, find the value of α in radians. Express your answer as a fraction of π. | \frac{13\pi}{7} | big_math | [
"Mathematics -> Trigonometry -> Trigonometric Functions"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $| \overrightarrow{a}|=5$, $| \overrightarrow{b}|=4$, and $\overrightarrow{a} \cdot \overrightarrow{b}=-10$, find the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ (denoted as $\langle \overrightarrow{a}, \overrightarrow{b} \rangle$). Express your answer in radians as a fraction of pi. | \frac{2\pi}{3} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
In the polar coordinate system, find the distance between point $A(2, \frac{\pi}{6})$ and point $B(2, -\frac{\pi}{6})$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
A bar of soap loses 10% of its volume each time it is used. What is the smallest number of times the soap must be used so that less than 50% of its original volume remains? Express your answer as a whole number. | 7 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Find the largest number among 0, 2, -1, and -2. | 2 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given \(\alpha \in \left(0, \frac{\pi}{2}\right)\), calculate the minimum value of the expression \(\frac{\sin^3 \alpha}{\cos \alpha} + \frac{\cos^3 \alpha}{\sin \alpha}\). Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation of the ellipse $\frac{x^2}{4} + \frac{y^2}{9} = 1$, calculate the length of the major axis. Express your answer as a numerical value. | 6 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
A person contracted the flu, and after two rounds of infection, a total of 196 people were infected. Calculate the average number of people infected by one person in each round of infection. Express your answer as a whole number. | 13 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Determine the base in which the equation 12 × 43 = 1000 holds true. Express your answer as a single integer. | 6 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $x^{2}+2mx+9$ is a perfect square trinomial, find the value of $m$. Express your answer as a numerical value, including the possibility of multiple values (e.g., ±). | \pm 3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Two circles \( C_{1} \) and \( C_{2} \) have their centers at the point \( (3, 4) \) and touch a third circle, \( C_{3} \). The center of \( C_{3} \) is at the point \( (0, 0) \) and its radius is 2. Find the sum of the radii of the two circles \( C_{1} \) and \( C_{2} \). Express your answer as a whole number. | 10 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given a complex number $z$ that satisfies $(1-i)z = 1+i$, find the value of $|z+i|$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Keegan paddled the first 12 km of his 36 km kayak trip before lunch. What fraction of his overall trip remains to be completed after lunch? Express your answer as a simplified fraction. | \frac{2}{3} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given a sequence $\{a_n\}$ with the sum of the first $n$ terms $S_n = n^2 - 8n$, and the $k$-th term $a_k = 5$, calculate the value of $k$. Express your answer as a whole number. | 7 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Given a quadratic equation $x^2 - 3x - 5 = 0$ with roots $x_1$ and $x_2$, calculate the value of $x_1 + x_2 - x_1 \cdot x_2$. Express your answer as a single integer. | 8 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Quadratic Functions"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
In triangle $ riangle ABC$, if $a^{2}+c^{2}=b^{2}+\sqrt{3}ac$, then find the measure of angle $B$ in degrees. | 30 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{OA}=(1,-1,0)$ and $\overrightarrow{OB}=(1+k,3k,k-2)$, and the condition $\overrightarrow{OA}⊥\overrightarrow{OB}$, calculate the value of $k$. Express your answer as a fraction or decimal. | \frac{1}{2} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
In triangle \( \triangle ABC \), \( \angle A = 80^\circ \). Points \( D, E, F \) are on sides \( BC, AB \), and \( AC \) respectively, with \( BD = BE \) and \( CD = CF \). Find the measure of \( \angle EDF \) in degrees. | 50^\circ | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Let $\{a_n\}$ be an arithmetic sequence with a common difference of $-2$. If $a_1+a_4+a_7+\ldots+a_{97}=50$, then find the value of $a_3+a_6+a_9+\ldots+a_{99}$. Express your answer as a single integer. | -82 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Three valves A, B, and C release water into a tank at constant rates. With all three valves open, the tank fills in 1 hour. With only valves A and C open, it takes 1.5 hours to fill the tank. With only valves B and C open, it takes 2 hours to fill the tank. Find the time required to fill the tank with only valves A and... | 1.2 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
If the sum of a complex number and its modulus is $5+\sqrt{3}i$, find the complex number. Express your answer in the form $a + bi$. | \frac{11}{5} + \sqrt{3}i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
A fair coin is tossed. If the first toss shows heads, it is tossed again: if the second toss shows heads, record 0; if the second toss shows tails, record 1. If the first toss shows tails, record any random real number in the closed interval [0, 1]. This process is repeated two independent times, obtaining two real num... | \frac{7}{16} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
School A and school B each have 3 volunteers participating in a public welfare activity. After the activity, they take a group photo with 3 people in each row. If volunteers from the same school cannot stand next to each other in each row, calculate the total number of different ways to arrange them. Express your answe... | 72 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Five different products are arranged in a row on a shelf, among which A and B must be adjacent, while C and D must not be adjacent. Find the total number of different arrangements. Express your answer as a single integer. | 24 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given an infinite geometric series with common ratio \( r \) (where \( |r| < 1 \)), the sum of the series is 15, and the sum of the squares of each term of this series is 45. Calculate the first term of this series. | 5 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of the complex number $\frac{4+3i}{2-i}$. Express your answer in the form a+bi, where a and b are integers. | 1+2i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
A triangle has two sides of length 2 and 4, and the third side is a solution of the equation x^2 - 11x + 30 = 0. What is the perimeter of this triangle? Express your answer as a whole number. | 11 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Quadratic Functions"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
If \( n = 3^{0} + 3^{1} + 3^{2} + \cdots + 3^{2015} + 3^{2016} \), then find the remainder when \( n \) is divided by 4. Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
The diagonal of square $I$ is $a+b$. Square $II$ has twice the area of square $I$. Find the area of square $II$. Express your answer as a formula. | (a+b)^2 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the geometric sequence $(b_{n})$ with common ratio $2$ satisfies $\log_{2}b_{n} \cdot \log_{2}b_{n+1} - 2 = n^2 + 3n$, find the first term of the sequence $(b_{n})$. Express your answer as a single numerical value. | 4 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
If \( 2 \times 10^{3} \) is represented as \( 2 E 3 \) on a certain calculator, how would the product of \( 2 E 3 \) and \( 3 E 2 \) be represented? Express your answer in calculator notation (e.g., 6 E 5). | 6 E 5 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
The sum of two prime numbers is 85. One of these primes is even, and the other is odd. Find the product of these two prime numbers. Express your answer as a whole number. | 166 | big_math | [
"Mathematics -> Number Theory -> Prime Numbers"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Simulate the execution of the following program to find the final value of s:
```basic
i=1
WHILE i < 8
i=i+2
s=2*i+3
WEND
PRINT s
END
```
Express your answer as a single integer. | 17 | big_math | [
"Mathematics -> Discrete Mathematics -> Algorithms"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the line with a slope of $-2$ and a y-intercept of $3$. Express your answer in the standard form of a linear equation (Ax + By + C = 0). | 2x+y-3=0 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Given the general term formula of the sequence $\{a_n\}$, which is $a_n = n^2 - 14n + 65$, determine the term numbers for which $a_n = 20$. Express your answer as a list of whole numbers. | 5, 9 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Quadratic Functions"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Determine the number of equilateral triangles that can be formed using the vertices of a cube. Express your answer as a whole number. | 8 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given sets $A=\left\{{x|y=\sqrt{1-x}}\right\}$ and $B=\{x|x^{2} \lt 3\}$, calculate the intersection $A\cap B$. Express your answer in interval notation, including all necessary details. | (-\sqrt{3},1] | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Find the highest common divisor (HCD) of 6432 and 132, and then subtract 8 from it. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Number Theory -> Greatest Common Divisors (GCD)"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Given the plane vectors $a=(1,0)$ and $b=(-\frac{1}{2},\frac{\sqrt{3}}{2})$, calculate the angle between $a$ and $a+b$. Express your answer in radians using pi notation (e.g. pi/3). | \frac{\pi}{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Let the integer part of \(\sqrt{5}\) be \(m\) and the fractional part be \(n\). Calculate the value of \(m\left(m - \frac{1}{n}\right)^3\). Express your answer as a multiple of \(\sqrt{5}\). | -10\sqrt{5} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number equation $(\sqrt{3} + 3i)z = 3i$, solve for $z$ in the form $a+bi$. Express your answer in the form $\boxed{a+bi}$. | \frac{3}{4} + \frac{\sqrt{3}}{4}i | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Rectangular tiles, which measure 6 by 4, are arranged without overlapping to create a square. Calculate the minimum number of these tiles needed to make a square. Express your answer as a whole number. | 6 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $\frac{a+2i}{i} = b - i$, where $a$ and $b$ are real numbers, find the sum of $a$ and $b$. Express your answer as a single integer. | 3 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Let x be a real number. Find the minimum value of (x^2 - 4x + 3)(x^2 + 4x + 3). Express your answer as a single integer. | -16 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of the complex number $z=| \sqrt {3}-i|+i^{2017}$, where $i$ is the imaginary unit. Express your answer in the form $a + bi$. | 2+i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Given a point $P(-4,3)$ on the terminal side of angle $\alpha$, determine the value of $\sin (\frac{\pi}{2} + \alpha)$ as a fraction. | -\frac{4}{5} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression |2-x| + (√(x-2))^2 - √(4x^2 - 4x + 1) for x ≥ 2. Provide the final simplified form of the expression. | -3 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
Given point C (-2, -2), $\overrightarrow{CA} \cdot \overrightarrow{CB} = 0$, and line CA intersects the x-axis at point A, line CB intersects the y-axis at point B, find the equation of the trajectory of the midpoint M of segment AB in the form Ax + By + C = 0. | x + y + 2 = 0 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
In the complex plane, consider a regular 20-gon inscribed in the unit circle with vertices corresponding to complex numbers \( z_{1}, z_{2}, \cdots, z_{20} \). Determine the number of distinct points corresponding to the complex numbers \( z_{1}^{1995}, z_{2}^{1995}, \cdots, z_{20}^{1995} \). Express your answer as a w... | 4 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a} = (2k-3, -6)$ and $\overrightarrow{c} = (2, 1)$, and $\overrightarrow{a} \parallel \overrightarrow{c}$, find the real number $k$ that satisfies this condition. Express your answer as a single real number. | -\frac{9}{2} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
In a plane Cartesian coordinate system, a lattice point is a point with integer coordinates. Determine the number of lattice points (x, y) that satisfy the inequality (|x|-1)^2 + (|y|-1)^2 < 2. Express your answer as a single integer. | 16 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
The tap water company charges households according to the following standard: if the water usage per household does not exceed 5 cubic meters, the charge is 2.8 yuan per cubic meter; if the water usage per household exceeds 5 cubic meters per month, the excess part is charged at 3 yuan per cubic meter. Given that Xiao ... | 10 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Five doors are randomly either open or closed. Calculate the probability that exactly two of the five doors are open. Express your answer as a simplified fraction. | \frac{5}{16} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Combinations"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given that \(x= -\frac{\pi}{8}\) and \(x_0\) are respectively a symmetry axis and a zero point of the function \(f(x) = A \sin(\omega x + \phi)\) (\(A > 0, \omega > 0, |\phi| < \frac{\pi}{2}\)), and that the minimum value of \(|x_0 + \frac{\pi}{8}|\) is \(\frac{\pi}{4}\), determine the value of \(\phi\). Express your a... | -\frac{\pi}{4} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the result of multiplying the sum of the sequence $81, 83, 85, 87, 89, 91, 93, 95, 97, 99$ by $2$. Express your answer as a single integer. | 1800 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z(1-2i)=2+i$, solve for $z$ in its simplest form. | i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Xiaoming is trying to write the word "egg" but has forgotten the correct order of the letters. He randomly arranges the letters e, g, and g. What is the probability that he writes the word "egg" correctly? Express your answer as a simplified fraction. | \frac{1}{3} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
The arithmetic mean of ten different positive integers is 10. What is the maximum possible value of the largest of these ten numbers? Express your answer as a single integer. | 55 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Find the minimum value of the expression $\frac{2}{x}+\frac{x}{2}$ given that $x > 0$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence $\{a_n\}$ with the sum of the first $n$ terms denoted as $S_n$, if $S_{17}=170$, find the value of $a_7+a_8+a_{12}$. Express your answer as a single integer. | 30 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Determine the domain of the function $f(x)=\frac{\sqrt{-x^{2}-3x+4}}{\log(x+1)}$. Express your answer in interval notation. | (-1,0)\cup(0,1] | big_math | [
"Mathematics -> Algebra -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given $\overrightarrow{a}=(1,0,1)$, $\overrightarrow{b}=(-2,-1,1)$, and $\overrightarrow{c}=(3,1,0)$, calculate the magnitude of the vector $\overrightarrow{a} - \overrightarrow{b} + 2\overrightarrow{c}$. Express your answer as a simplified radical expression if necessary. | 3\sqrt{10} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Given the Monge circle of the ellipse $\Gamma$ as $C: x^{2}+y^{2}=3b^{2}$, calculate the eccentricity of the ellipse $\Gamma$. Express your answer as a fraction with a square root, if necessary. | \frac{\sqrt{2}}{2} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given line l: x - 2y + 1 = 0, find the y-intercept of line m that is perpendicular to line l and passes through point (-1, 0). Express your answer as a single integer. | -2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Linear Equations"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Find the maximum value of the function $f(x)=\cos 2x+6\cos(\frac{\pi}{2}-x)$. Express your answer as a single number. | 5 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, the radius of the larger circle is 3 times the radius of the smaller circle. Calculate the fraction of the area of the larger circle that is not shaded. Express your answer as a simplified fraction. | \frac{8}{9} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = x / sqrt(1 - x), calculate the value of f(-3). Express your answer as a single value. | -3/2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
The sequence $\{a_n\}$ satisfies $a_1=1$, and for any $n \in \mathbb{N}^*$, it holds that $a_{n+1}=a_1+a_n+n$. Calculate the value of $\dfrac{1}{a_1}+ \dfrac{1}{a_2}+\ldots+ \dfrac{1}{a_{2016}}$. Express your answer as a fraction in simplest form. | \dfrac{4032}{2017} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
There are 7 light-emitting diodes (LEDs) in a row, each of which can emit red or green light when lit. If exactly 3 LEDs are lit at a time, and adjacent LEDs cannot be lit simultaneously, how many different messages can be represented by the different positions or colors of the lit LEDs? Express your answer as a single... | 80 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Find the imaginary part of the complex number $\frac{2i}{1+i}$. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
A teacher and four students stand in a row for a photo. The students ask the teacher to stand in the middle. Calculate the total number of different ways the 5 people can stand, considering the restriction on the teacher's position. Express your answer as a single integer. | 24 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Permutations"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
If $\alpha$ is an angle in the fourth quadrant and $\tan \alpha =-\frac{5}{12}$, find the value of $\sin \alpha$. Express your answer as a fraction, including the sign of the sine value. | -\frac{5}{13} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation x^6 = -64, find the product of the roots with positive real part. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given a hyperbola with equation $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > 0, b > 0$), where $F\_1$ and $F\_2$ are the left and right foci, respectively. A circle with center $O$ and radius $OF\_1$ intersects the hyperbola at point $P$ in the first quadrant. If the area of $\triangle PF\_1F\_2$ is equal to $a^2$, fi... | \sqrt{2} | big_math | [
"Mathematics -> Geometry -> Analytic Geometry -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
If the coefficient of the term $x^3$ in the expansion of $(1+2x)^n$ is 8 times the coefficient of the term $x$, then find the value of $n$. Express your answer as a single integer. | 5 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Binomial Theorem"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Five students, A, B, C, etc., from a certain class participated in a speech competition, and the order of appearance was determined by drawing lots. Given that student A must appear before student B, calculate the probability that students A and B appear adjacent to each other. Express your answer as a simplified fract... | \frac{2}{5} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Permutations"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Given a point P(m, m+1) on the terminal side of angle α, and cosα = 3/5, calculate the value of m. Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $\frac{\log{a}}{p}=\frac{\log{b}}{q}=\frac{\log{c}}{r}=\log{x}$, all logarithms to the same base and $x \not= 1$, find the value of $y$ in the equation $\frac{b^2}{ac}=x^y$. Express your answer as a single mathematical expression. | 2q - p - r | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Logarithmic Functions"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Given the parabola y^2 = 8x, point M lies on the parabola such that the distance from M to the focus F of the parabola is 4 units. Find the distance from the origin O to point M. Express your answer as a numerical value, including any necessary square roots. | 2\sqrt{5} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Toss a fair die twice. Let event A be {the two outcomes are different}, and event B be {at least one outcome is a 3}. Calculate the conditional probability P(B|A). Express your answer as a simplified fraction. | \frac{1}{3} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a$ and $b$ are both positive real numbers, find the minimum value of $(a+ \frac {1}{b})(b+ \frac {4}{a})$. Express your answer as a single integer. | 9 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Find the opposite of -8. Express your answer as a single integer. | 8 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
A student rolls two dice simultaneously, with the resulting numbers being $a$ and $b$. Determine the probability that the eccentricity $e$ of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ satisfies $e > \frac{\sqrt{3}}{2}$. Express your answer as a simplified fraction. | \frac{1}{6} | big_math | [
"Mathematics -> Applied Mathematics -> Probability"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = x^3 + (a-1)x^2 + ax, where f(x) is an odd function, find the equation of the tangent line to the curve y = f(x) at the point (0,0). Express your answer in the form y = mx + b, where m is the slope and b is the y-intercept. | y = x | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Two interns each manufacture a part. The probability that a part is of first-class quality is $\frac{2}{3}$ for one intern and $\frac{3}{4}$ for the other. Assuming the quality outcomes of the two parts are independent, find the probability that exactly one of the two parts is of first-class quality. Express your answe... | \frac{5}{12} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.90625 | 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 -> Sequences and Series -> Other"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the vectors $\{\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}\}$ form a basis for a space, and the vectors $\{\overrightarrow{a}-\overrightarrow{b},\overrightarrow{a}+\overrightarrow{b},\overrightarrow{c}\}$ form another basis for the space. If the coordinates of vector $\overrightarrow{p}$ in the ... | (1,3,-1) | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Find the minimum positive period of the function f(x) = sin(x) * (4 * cos^2(x) - 1). Express your answer in radians as a fraction of pi. | \dfrac{2\pi}{3} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Three fair six-sided dice are rolled. What is the probability that the values shown on two of the dice sum to the value shown on the remaining die? Express your answer as a simplified fraction. | \frac{5}{24} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
A list of 5 integers repeats to form the pattern: $4,-3,2,-1,0,4,-3,2,-1,0,\ldots$. What is the sum of the first 23 integers? Express your answer as a single integer. | 11 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
A number line between 0 and 2 is divided into 8 equal parts. The numbers 1 and S are marked on the line, where S is the point immediately after 1.0. Calculate the value of S. Express your answer as a decimal value. | 1.25 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Decimals"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
A certain place has selected 4 outstanding teachers to conduct a one-year support teaching activity in three places, A, B, and C. Each person can only go to one place, and each place must have at least one person. Calculate the total number of different selection schemes for distributing the teachers among the three pl... | 36 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
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