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 |
|---|---|---|---|---|---|
Determine the value of $f(-3)$ for the function
$$
f(x) =
\begin{cases}
f(x+2), & \text{if } x < 2 \\
2^{-x}, & \text{if } x \geq 2
\end{cases}
$$
Express your answer as a fraction or decimal value. | \frac{1}{8} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Find the square root of $\sqrt{81}$. Express your answer with both positive and negative values. | \pm 3 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
In a certain information transmission process, each piece of information can only be a permutation of a 6-digit number composed of 0 or 1 (digits can be repeated and the first digit can be 0). Different permutations represent different information. Calculate the probability that a randomly selected piece of information... | \frac{15}{64} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation z(1+i) = 2-i, calculate the value of \\overline{z} \\cdot z. Express your answer as a simplified fraction. | \\frac{5}{2} | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence \(\left\{a_{n}\right\}\) defined by \( a_{1} = 0 \) and \( a_{n+1} = a_{n} + 1 + 2 \sqrt{1 + a_{n}} \) for \( n = 1, 2, \ldots \), calculate the value of \( a_{2009} \). Express your answer as a single integer. | 4036080 | big_math | [
"Mathematics -> Sequences and Series -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of elements in the set of points $\left\{(x, y) \left\lvert\, \lg \left(x^{3}+\frac{1}{3} y^{3}+\frac{1}{9}\right)=\lg x+\lg y\right.\right\}$. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
In the Cartesian coordinate system $xOy$, it is known that quadrilateral $ABCD$ is a parallelogram, $\overrightarrow{AB}=(1,-2)$, $\overrightarrow{AD}=(2,1)$. Calculate the value of $\overrightarrow{AD} \cdot \overrightarrow{AC}$. Express your answer as a single integer. | 5 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets A and B as follows:
\[
A = \left\{x^2 + x + 1, -x, -x-1\right\}, \quad B = \left\{-y, -\frac{y}{2}, y+1\right\}
\]
and knowing that A = B, find the value of x^2 + y^2. | 5 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $\frac{3-bi}{2+i}$ $(b \in \mathbb{R})$, find the value of $b$ such that its real part is equal to its imaginary part. Express your answer as a single integer. | -9 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x)=\begin{cases}e^{x}-1, & x\leqslant 1 \\ \ln x, & x > 1\end{cases}$, calculate the value of $f(\ln 2)$. Express your answer as a numerical value. | 1 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
A house worth $9000 is sold by Mr. A to Mr. B at a 10% loss. Mr. B then sells the house back to Mr. A at a 10% gain. Calculate Mr. A's net loss or gain after both transactions. Express your answer as a dollar value. | 810 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given that translating the graph of the function y = f(x) 2 units to the right along the x-axis results in y = |x - 2|, find the expression for y = f(x) before the translation. Express your answer as a function of x. | y = |x| | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function y = 2x^2 - 2x + 1, find its derivative y'. Express your answer as a function of x. | y' = 4x - 2 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = \begin{cases} 4\log_{2}(-x), & \text{if } x < 0 \\ |x^2 + ax|, & \text{if } x \geq 0 \end{cases}, and f[f(-\sqrt{2})]=4, find the value of f(a) given that a!= 0. Express your answer as a single number. | 8 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Given $x > -2$, find the minimum value of the expression $x + \frac{1}{x+2}$. Express your answer as a single number. | 0 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(1,-2)$ and $\overrightarrow{b}=(2,m)$, where $\overrightarrow{a} \perp \overrightarrow{b}$, calculate the magnitude of $\overrightarrow{b}$. Express your answer as a numerical value. | \sqrt{5} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence, the sum of the first m terms is 30, and the sum of the first 2m terms is 100. Find the sum of the first 3m terms. | 210 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z= \\frac {1+ai}{2-i}$, where $i$ is the imaginary unit, find the value of the real number $a$ such that $z$ is a pure imaginary number. Express your answer as a single real number. | 2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Let S_n be the sum of the first n terms of the arithmetic sequence {a_n}, with a_4 = 4 and S_5 = 15. Find the value of m such that the sum of the first m terms of the sequence {1/(a_n*a_(n+1))} is 10/11. Express your answer as a whole number. | 10 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions",
"Mathematics -> Algebra -> Sequences and Series"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, calculate the value of angle x using the given angle measurements and geometric properties. Express your answer as a single integer in degrees. | 60 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
A box contains chips, each of which is red, white, or blue. The number of blue chips is at least half the number of white chips, and at most one third the number of red chips. The number which are white or blue is at least 55. Find the minimum number of red chips that satisfies all these conditions. Express your answer... | 57 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $2^m = 3$ and $2^n = 4$, calculate the value of $2^{m-n}$. Express your answer as a simplified fraction. | \frac{3}{4} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Given the functions y = 2x - 1 and y = kx + k, where k is an integer, find the number of possible values for k such that the graphs of the functions intersect at a lattice point. | 4 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
In Jen's baseball league, each team plays exactly 6 games against each of the other teams in the league. If a total of 396 games are played, how many teams are in the league? Express your answer as a whole number. | 12 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Given the algebraic expression $1-k$, find the range of $k$ that satisfies the conditions $1-k>-1$ and $1-k\leq3$. Express your answer as an interval. | [-2, 2) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
In a family with two children, one of the children is a girl. What is the probability that the other child is also a girl? Express your answer as a fraction. | \dfrac{1}{3} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Given $a-b+c=0$, find one root of the quadratic equation $ax^2+bx+c=0$ ($a \neq 0$). Express your answer as a single numerical value or expression. | -1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Quadratic Functions"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $x=1$ is a solution to the equation $2-3(a+x)=2x$, find the value of $a$. Express your answer as a single integer or fraction. | -1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Simple Equations"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
The parabola C:y^2 = mx passes through the point (-2, sqrt(3)). Find the equation of the directrix of the parabola C. Express your answer as an equation of a line (e.g., x = a). | x = 3/8 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $\left(m+2\right)x^{|m|}=0$ is a quadratic equation in $x$, find the value of $m$ that satisfies the conditions $|m|=2$ and $m+2\neq 0$. Express your answer as a single value of $m$. | 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the line passing through the point P(0, 2) with a slope of 2. Express your answer in the standard form Ax + By + C = 0. | 2x - y + 2 = 0 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
In the geometric sequence {a_n}, the first term a_1 is 8 and the fourth term a_4 is 64. Find the common ratio q. Express your answer as a single numerical value. | 2 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
When conducting market research, what statistical measure of shoe models should a shoe store focus on in order to increase sales? Provide your answer as a single word or brief phrase. | Mode | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Four families, A, B, C, and D, each have two children, making a total of 8 individuals. They plan to use the Didi taxi service for an outing, with two cars, labeled as Alpha and Beta, available to them. Each car can seat 4 individuals, and the arrangement of the 4 children in the same car does not consider the seating ... | 24 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
The taxi fare standard in Wenzhou is as follows: the starting price is 10 yuan (i.e., a fare of 10 yuan is charged for a distance of up to 4 kilometers), and for every additional kilometer beyond 4 kilometers, an extra 1.5 yuan is charged (distances less than 1 kilometer are rounded up to 1 kilometer). If someone trave... | 10 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{m}=( \sqrt{3}\sin A,\sin B )$ and $\overrightarrow{n}=( \cos B,\sqrt{3}\cos A )$, and the condition $\overrightarrow{m}\cdot \overrightarrow{n}=1+\cos ( A+B )$, calculate the value of angle $C$ in degrees. Express your answer in the format $\boxed{[answer]°}$. | 120° | big_math | [
"Mathematics -> Algebra -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Factor the polynomial $a^2 + 2b - b^2 - 1$. Express your answer as a factored polynomial. | (a+b-1)(a-b+1) | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the function f(x) is an even function with a domain of R, and when x > 0, f'(x) < 0, find the solution set of the inequality f(x^2-x) - f(x) > 0. Express your answer as an interval. | (0, 2) | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Four students participate in three different extracurricular activities. If each student can freely choose to participate in one of them, find the probability that each activity has at least one student participating. Express your answer as a simplified fraction. | \dfrac {4}{9} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given two sets \( M = \left\{ z \mid z = \frac{t}{1+t} + i \frac{1+t}{t}, t \in \mathbf{R}, t \neq -1, t \neq 0 \right\} \) and \( N = \{ z \mid z = \sqrt{2}[\cos (\arcsin t) + i \cos (\arccos t)], t \in \mathbf{R}, |t| \leq 1 \} \), determine the number of elements in the intersection \( M \cap N \). Express your answ... | 0 | big_math | [
"Mathematics -> Complex Analysis -> Other"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
If the simplest quadratic radical \\sqrt{a+2} is of the same type as the simplest quadratic radical \\sqrt{3a}, find the value of a that satisfies this condition. Express your answer as a single integer value. | 1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\overrightarrow{a}$ and $\overrightarrow{b}$ are both unit vectors, and the dot product $(2\overrightarrow{a} + \overrightarrow{b}) \cdot (\overrightarrow{a} - 2\overrightarrow{b}) = -\frac{3\sqrt{3}}{2}$, calculate the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ in degrees. | 30 | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Geometry -> Other",
"Mathematics -> Discrete Mathematics -> Other"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets $P=\{2,3,4,5,6\}$ and $Q=\{3,5,7\}$, find the number of subsets of $M=P\cap Q$. Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
In a 400-meter race entering the final sprint stage, contestant A is 30 meters ahead of contestant B, contestant C is 60 meters behind contestant D, and contestant B is 20 meters ahead of contestant C. Calculate the distance between the two contestants at the front. Express your answer in meters. | 10 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given a complex number $z=\\frac{{5-i}}{{1+i}}$ (where $i$ is the imaginary unit), calculate the conjugate of $z$, denoted as $\\overline{z}$. Express your answer in the form $\\boxed{a+bi}$. | 2+3i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
A triangle has a base of 15 inches. Two lines are drawn parallel to the base, terminating in the other two sides, and dividing the triangle into three equal areas. Find the length of the parallel closer to the base. Express your answer in inches, including the square root symbol if necessary. | 5\sqrt{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence, the sum of the first four terms is 26, the sum of the last four terms is 110, and the sum of all terms in the sequence is 187. Determine the total number of terms in the sequence. Express your answer as a whole number. | 11 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given a sequence of "integer pairs" arranged according to the following pattern: \\((1,1)(1,2)(2,1)(1,3)(2,2)(3,1)(1,3)(2,2)(3,1)(1,4)(2,3)(3,2)(4,1)\\), \\(\ldots\\), find the 2017th integer pair. Express your answer as an ordered pair (a, b) in the form \\(\boxed{(a, b)}\\). | (1, 64) | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given the vectors $\overrightarrow{a} = (1,3)$ and $\overrightarrow{b} = (3,x)$, if $\overrightarrow{a}$ is perpendicular to $\overrightarrow{b}$, calculate the value of the real number $x$. Express your answer as a single real number. | -1 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
In the polar coordinate system, find the distance from the point $(2, \frac{\pi}{3})$ to the line $\rho(\cos \theta + \sqrt{3}\sin \theta) = 6$. Express your answer as a single whole number. | 1 | big_math | [
"Mathematics -> Geometry -> Differential Geometry -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
The variables S and i are initialized with the values S=0 and i=1. Then, the following two statements are executed consecutively three times: S=S+i, i=i+2. What is the value of S after these executions? Provide your answer as a single integer. | 9 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Let unit vectors $\overrightarrow{e_1}$ and $\overrightarrow{e_2}$ have an angle of 120° between them. If $\overrightarrow{a} = 2\overrightarrow{e_1} - \overrightarrow{e_2}$, then find the magnitude of $\overrightarrow{a}$, denoted as $|\overrightarrow{a}|$. Express your answer as a numerical value, including the squar... | \sqrt{7} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
In a school, 30 boys and 20 girls entered the Fermat competition. Certificates were awarded to 10% of the boys and 20% of the girls. Calculate the percentage of students who received certificates out of the total number of students who participated. Express your answer as a percentage. | 14\% | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets $M=\{-1,0,1,2\}$ and $N=\{x | x < 0 \text{ or } x > 1\}$, find the number of elements in $M \cap N$. Express your answer as a whole number. | 2 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
On a $6 \times 6$ grid, place Go pieces in all squares, with one piece in each square. The number of white pieces in each row must be different from one another, and the number of white pieces in each column must be the same. Calculate the total number of black Go pieces on the grid. | 18 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given $z= \frac{1}{i-1}$, calculate the magnitude of the complex conjugate of $z$, denoted $| \overline{z} |$. Express your answer as a numerical value in simplest radical form. | \frac{\sqrt{2}}{2} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = sqrt(e^x + (e-1)x - a), where e is the base of the natural logarithm and a is a real number, and given that there exists b in [0,1] such that f(b) = b, find the range of a. Express your answer as an interval. | [1, 2e-2] | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
When 100 is divided by a positive integer x, the remainder is 10. Find the remainder when 1000 is divided by x. Express your answer as a single integer. | 10 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Find the arithmetic square root of the real number 9. Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Given x > 0, y > 0, and 4x + 9y = 6, find the maximum value of xy. Express your answer as a single fraction. | \frac{1}{4} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence {a_n}, it is known that a_1 = 2, and a_2 + a_3 = 13. Find the value of a_5. Express your answer as a single numerical value. | 14 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x)=x^{5}-3x^{3}-6x^{2}+x-1$, calculate the value of $f(-2)$ using the Horner's method. Express your answer as a single integer. | -35 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
In the interval [0,4], two real numbers x and y are randomly selected such that x + 2y ≤ 8. Calculate the probability that x and y satisfy this inequality. Express your answer as a simplified fraction. | \frac{3}{4} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Geometric Probability"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
In triangle \( \triangle ABC \), \( AM \) and \( CN \) are the medians to sides \( BC \) and \( AB \), respectively, intersecting at point \( O \). \( P \) is the midpoint of \( AC \), and \( MP \) intersects \( CV \) at \( Q \). If the area of \( \triangle OMQ \) is \( n \), calculate the area of \( \triangle ABC \) i... | 24n | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
If set $A = \{x | (2x + 1)(x - 3) < 0\}$, and $B = \{x | x \in \mathbb{N}^*, x \leq 5\}$, then find the intersection $A \cap B$. Express your answer as a set of numbers. | \{1, 2\} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Grid lines are drawn on three faces of a rectangular prism as shown. A squirrel walks from point P to point Q along the edges and grid lines in such a way that she is always getting closer to Q and farther away from P. Count the different paths from P to Q that the squirrel can take. Express your answer as a single int... | 14 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\vec{a} = (x, 1)$ and $\vec{b} = (4, 2)$, if $\vec{a}$ is parallel to $\vec{b}$, calculate the value of $\vec{a} \cdot (\vec{b} - \vec{a})$. Express your answer as a single number. | 5 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of integer solutions to the inequality $\log_{3}|x-2| < 2$. | 17 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Inequalities"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(1,2)$ and $\overrightarrow{b}=(x,1)$, if $(2\overrightarrow{a}-\overrightarrow{b})\perp\overrightarrow{b}$, find all possible values of $x$. Express your answer as a numerical value or values. | -1, 3 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
A certain type of cell divides from one into two every 30 minutes. After 3 hours, how many cells will there be in total? Express your answer as a numerical value. | 64 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
The ratio of the areas of two concentric circles is 1:3. If the radius of the smaller circle is r, find the difference between the radii of the two circles. Express your answer as a multiple of r, rounded to two decimal places. | 0.73r | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Given the line y = 3x - 5 and the square PQRS with R at (2009, 2009), find the number of lattice points on the line that are inside the square. A lattice point is a point with integer coordinates. Express your answer as a single integer. | 670 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the remainder when \( 7^{2008} + 9^{2008} \) is divided by 64. Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Find the largest possible difference between two different prime numbers that sum to 126. | 100 | big_math | [
"Mathematics -> Number Theory -> Prime Numbers"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $i$ is the imaginary unit, and $z(1-i) = 1+i$, find the conjugate of the complex number $z$. Express your answer as a complex number. | -i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Find the conjugate of the complex number $\frac{2+i}{1-2i}$. Express your answer as a complex number in its simplest form. | -i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the domain of the function $f(x)$ is $(1,3)$, determine the domain of the function $f(2x-1)$. Express your answer in interval notation. | (1, 2) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Suppose f is a polynomial function such that for all real values of x, f(x^2 + 1) = x^4 + 5x^2 + 3. Find f(x^2 - 1). Express your answer as a polynomial. | x^4 + x^2 - 3 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
In the cube $ABCD - A_1B_1C_1D_1$, find the angle between line $BC_1$ and the cross-section $BB_1D_1D$. Express your answer in radians. | \frac{\pi}{6} | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Find the remainder when $2^{33}$ is divided by 9. Express your answer as a single integer. | 8 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Given $a=\pi^{-2}$, $b=a^{a}$, $c=a^{a^{a}}$, determine the order of magnitude for $a$, $b$, and $c$ from greatest to least. Express your answer as a sequence of variables using greater-than symbols (>) to indicate ordering. | b > c > a | big_math | [
"Mathematics -> Algebra -> Exponential Functions"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given the direction vector of line l is (-1, sqrt(3)), calculate the inclination angle of line l. Express your answer in radians as a fraction of pi. | \frac{2\pi}{3} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $m$ and $n$ are two roots of the equation $x^{2}+3x-1=0$, find the value of $m^{2}+4m+n$. Express your answer as a single integer. | -2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Assign 2 teachers and 4 students into 2 groups, to be sent to locations A and B for social practice activities. Each group consists of 1 teacher and 2 students. Calculate the total number of different arrangements. Express your answer as a whole number. | 12 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $3x + y = 4$, express $y$ in terms of $x$. Provide your answer as an algebraic expression in the form $\boxed{y = \text{expression}}$. | y = 4 - 3x | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence ${a_n}$, where $a_1=2008$, $a_2=2017$, and $a_{n+2}=a_{n+1}-a_{n} (n∈N_{+})$, calculate the value of $a_{2018}$. Express your answer as a single integer. | 2017 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Given $\cos \left( \theta- \frac{\pi}{6} \right)+\sin \theta= \frac{4 \sqrt{3}}{5}$, find the value of $\sin \left( \theta+ \frac{7\pi}{6} \right)$. Express your answer as a single value. | -\frac{4}{5} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence {a_n} defined as: a_1 = 5, a_2 = 9, a_3 = 14, a_4 = 20,..., where the differences between consecutive terms increase by 1 each step, derive a general formula for the sequence and calculate the value of a_{624}. Express your answer as a single integer. | 196250 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the hyperbola $\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1$ ($a > 0$, $b > 0$) with an eccentricity of $2$, and one of its foci coincides with the focus of the parabola ${y}^{2}=4x$. Calculate the value of $ab$. Express your answer as a single value in the form $\boxed{[answer]}$. | \frac{\sqrt{3}}{4} | big_math | [
"Mathematics -> Geometry -> Non-Euclidean Geometry -> Other"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Given the polynomial g(x) = x^5 + x^4 + x^3 + x^2 + x + 1, find the remainder when the polynomial g(x^{12}) is divided by g(x). Express your answer as a single number. | 6 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Three people, A, B, and C, are applying to three universities, A, B, and C, respectively. Each person can only apply to one university. Let event A be "the three people apply to different universities," and event B be "A applies to a university different from the other two." Calculate the conditional probability P(A|B)... | \frac{1}{2} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Evaluate the integral $\int_{2}^{3}{\frac{x-2}{\left( x-1 \right)\left( x-4 \right)}dx}$. Express your answer as a single expression, including any necessary logarithmic or algebraic terms. | -\frac{1}{3}\ln 2 | big_math | [
"Mathematics -> Calculus -> Integral Calculus -> Techniques of Integration"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
In the Cartesian coordinate plane, a line $l: ax+y-1=0$ passing through a fixed point $P$ intersects with a line $m: x-ay+3=0$ passing through a fixed point $Q$ at point $M$. Given that $P(0,1)$ and $Q(-3,0)$, calculate the value of $|MP|^{2}+|MQ|^{2}$. Express your answer as a single numerical value. | 10 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Find the coordinates of the fixed point that the line kx - y + 3k - 2 = 0 always passes through. Express your answer as an ordered pair (x, y). | (-3, -2) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(\cos x,\sin x)$ and $\overrightarrow{b}=(\sqrt{3}\sin x,\sin x)$, and that $\overrightarrow{a} \perp \overrightarrow{b}$, calculate the possible values of $\cos 2x$ for $x\in \left[0,\pi \right]$. Express your answer as a numerical value or a list of numerical values. | 1, -\frac{1}{2} | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Geometry -> Other",
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
The line y = -x + 1 is moved down 3 units. Find the analytical expression of the new line in slope-intercept form (y = mx + b). | y = -x - 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Find the degree of the polynomial $x^{2}y+\pi x^{2}y^{2}-xy^{3}$. Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations -> Other"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
The side lengths of a cyclic quadrilateral are 25, 39, 52, and 60. Find the diameter of the circle. Express your answer as a whole number. | 65 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
The integer \( n \) is the mean of the three numbers 17, 23, and \( 2n \). Find the sum of the digits of \( n \). Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Find the difference between the reciprocal of -4 and its opposite. Express your answer as a fraction. | -\frac{17}{4} | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Fractions"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
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