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 |
|---|---|---|---|---|---|
Calculate the value of $123\,456 \times 9 + 7$. Express your answer as a single integer. | 1\,111\,111 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence {a_n}, a_5 + a_7 = 16 and a_3 = 4. Find the value of a_9. Express your answer as a single numerical value. | 12 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.75 | 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"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence {an} that satisfies a1 = 1 and an+2 - an = 6, what is the value of a11? Express your answer as a single integer. | 61 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Given an even function f(x) defined on R satisfying f(x) = -1/f(x+3) and f(4) = -2018, find the value of f(2018). Express your answer as an integer. | -2018 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the value of the expression $x-2y$ is $3$, find the value of the expression $1-2x+4y$. Express your answer as a single integer. | -5 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
If $x$ and $y$ are each chosen from the integers -10, -9,..., -1, 0, 1,..., 10, then how many points $(x, y)$ are in the second quadrant? Express your answer as a single integer. | 100 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Person A first drives 50 miles on the highway and then 10 miles on the coastal road. The speed at which person A drives on the highway is 3 times the speed on the coastal road. If person A spent 30 minutes driving on the coastal road, how much time did the entire trip take in minutes? Express your answer as a whole num... | 80 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given that {a_{n}} is an increasing geometric sequence, and satisfies a_{3} = 4, 1/a_{1} + 1/a_{5} = 5/8, find the value of a_{7}. Express your answer as a single numerical value. | 16 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Find the fixed point that the graph of the function $y=a^{x+2}+1$ (where $a>0$ and $a\neq 1$) always passes through. Express your answer as an ordered pair $(x, y)$. | (-2, 2) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the result of $x^{3}\cdot x^{2}$. Express your answer as a single variable term with an exponent. | $x^{5}$ | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
The total length of the Great Wall, a World Heritage site, is approximately 6,700,000 meters. Express this number in scientific notation, including units (m). | 6.7 \times 10^6 \text{ m} | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the tangent line to the function y = xe^x at the point (1, e). Express your answer in slope-intercept form (y = mx + b). | y = 2ex - e | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the result of $818^{0}$. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(-1,2)$ and $\overrightarrow{b}=(\lambda,1)$, find the value of $\lambda$ such that $\overrightarrow{a}+ \overrightarrow{b}$ is parallel to $\overrightarrow{a}$. Express your answer as a single value. | -\frac{1}{2} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
In the expansion of \( (x^{2} - \\frac{1}{x})^{n} \), the constant term is 15. Determine the value of \( n \). Express your answer as a single integer. | 6 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the fraction $\frac{{-x+y}}{{-x-y}}$. Express your answer as a simplified fraction. | \frac{x-y}{x+y} | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, each of the two circles has center O. Also, OP: PQ = 1:2. If the radius of the larger circle is 9, calculate the area of the shaded region. Express your answer in terms of pi (π). | 72\pi | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
A savings card has a 6-digit password, with each digit being able to be selected from 0 to 9. A person forgot the last digit of the password when withdrawing money from an ATM. Calculate the probability of pressing the correct last digit within 2 attempts. Express your answer as a fraction. | \dfrac{1}{5} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given a point P on the hyperbola x^2/16 - y^2/20 = 1, let F_1 and F_2 be the left and right foci of the hyperbola, respectively. If |PF_1| = 9, calculate the value of |PF_2|. | 17 | big_math | [
"Mathematics -> Geometry -> Non-Euclidean Geometry -> Hyperbolic Geometry"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the arithmetic sequence {a_n} where the sum of the first 11 terms is 22, find the sum of a_3, a_5, and a_{10}. Express your answer as a single integer. | 6 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Given a point P(2,1) inside the circle (x-1)^{2}+y^{2}=4, find the equation of the line containing the diameter that passes through point P. Express your answer as an equation of a line in standard form (e.g., Ax + By + C = 0). | x - y - 1 = 0 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the line with a slope of 3 and a y-intercept of 4. Express your answer in the standard form of a line equation (Ax + By + C = 0). | 3x - y + 4 = 0 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Linear Equations"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Given the ellipse $\frac{x^2}{2} + y^2 = 1$, a line $l$ with an inclination angle of 45° passes through one of the foci of the ellipse and intersects the ellipse at points A and B. Let O be the origin of the coordinates. Calculate the dot product $\overrightarrow {OA} \cdot \overrightarrow {OB}$. | -\frac{1}{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $(5 - i) - (3 - i) - 5i$. Express your answer in the form a + bi, where a and b are real numbers. | 2 - 5i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
A 3x3 grid is to be filled with the digits 1, 2, and 3, such that each number appears once in each row and each column. How many different ways are there to fill the grid? Express your answer as a whole number. | 12 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the function f(x) satisfies f(x+2)f(x) = 1 for any real number x, and f(2) = 2, calculate the value of f(2016). | \frac{1}{2} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Given the set $A=\{x|x=a_0+a_1\times3+a_2\times3^2+a_3\times3^3\}$, where $a_k\in\{0,1,2\}$ ($k=0,1,2,3$), and $a_3\neq0$, calculate the sum of all elements in set $A$. Express your answer as a single integer. | 2889 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
There are 6 people standing in two rows, with 3 people in each row. Among them, person A does not stand in the front row, and person B does not stand in the back row. Calculate the total number of ways they can stand. Express your answer as a single integer. | 216 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given that positive numbers x and y satisfy the equation 4x^2 + 9y^2 + 3xy = 30, find the maximum value of xy. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
An archery target has 3 regions, each worth a different value if it is hit. Three archers shoot two arrows each and record scores as follows:
First archer: 1 arrow in C and 1 arrow in A for a score of 15 points
Second archer: 1 arrow in C and 1 arrow in B for a score of 18 points
Third archer: 1 arrow in B and 1 a... | 16 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the sum of two interior angles of a parallelogram is 70 degrees, find the largest interior angle of this parallelogram. Express your answer in degrees as a whole number. | 145 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Find the distance from the point on the parabola $y^2 = -4x$ with an x-coordinate of -6 to its focus. Express your answer as a whole number. | 7 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the function f(x) is an even function defined on R and is increasing on [0,+∞), find the range of the real number a such that f(a) ≤ f(2). Express your answer as an interval. | [-2,2] | big_math | [
"Mathematics -> Precalculus -> Functions"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Given that $2x^{2a-b}-y^{a+b-1}=3$ is a linear equation in $x$ and $y$, find the value of $\left(a-2b\right)^{2023}$. Express your answer as a single integer value. | -1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Given $\alpha \in \left(0, \frac{\pi}{2}\right)$, and $\sin^2\alpha + \cos\left(\frac{\pi}{2} + 2\alpha\right) = \frac{3}{10}$, find the value of $\tan\alpha$. Express your answer as a single numerical value. | 3 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given a line passing through the point (1, 0) and perpendicular to the line x - 2y - 2 = 0, find the equation of this line in the standard form Ax + By + C = 0, where A, B, and C are integers. | 2x + y - 2 = 0 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Gail is a server at a restaurant. On Saturday, Gail gets up at 6:30 a.m., starts work at \( x \) a.m. and finishes at \( x \) p.m. Calculate the total number of hours Gail works on Saturday. Express your answer as a simple algebraic expression or numerical value in hours. | 12 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets $$M=\{x\in\mathbb{N}|\,-2\leq x<4\},\;N=\{x|\,\frac{x+1}{3-x}\geq0\}$$, calculate the number of elements in the set $M\cap N$. Express your answer as a single integer. | 3 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Place five balls numbered 1 to 5 into five boxes also numbered 1 to 5, with each box holding one ball. If exactly three of the balls match the numbers of their boxes, calculate the number of different ways to place the balls. Express your answer as a single integer. | 10 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
There are 5 people queuing, and we need to find the number of ways for A, B, and C to queue in the order A, B, C. Calculate the exact number of ways for this arrangement to occur. Express your answer as a single integer. | 20 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Permutations"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of distinct sequences of prime numbers \( p_{1}, p_{2}, p_{3} \) that form an arithmetic sequence with a common difference of 80. Express your answer as a whole number. | 1 | big_math | [
"Mathematics -> Number Theory -> Prime Numbers"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Find the sum of the minimum and maximum values of the function f(x) = x^2 / (2x + 2) on the interval [-1/3, 1]. Express your answer as a single fraction. | \frac{1}{3} | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
The measures of the six interior angles of a convex hexagon form an arithmetic sequence of natural numbers. Let \( m^{\circ} \) be the degree measure of the largest interior angle of this hexagon. Calculate the maximum possible value of \( m^{\circ} \). Express your answer in degrees, including the degree symbol. | 175^\circ | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Select 4 representatives from 5 male students and 4 female students to participate in an activity, where there must be at least two males and at least one female. Calculate the total number of ways to make this selection. | 100 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Combinations"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Solve for x in the equation: $\frac{{x-1}}{{x+2}} = 0$. Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Find the cube root of -8. Express your answer as a single integer. | -2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
The price of 2 cherries is the same as the price of 3 apples, and an apple is 12 times the size of a cherry. If a mother buys a box of apples using the money for 1 box of cherries, and the box of apples has the same size as the box of cherries, how many boxes of apples can she buy? Express your answer as a whole number... | 18 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x) = \begin{cases} \log_{2}x + 2017, &x > 0 \\ -f(x+2), &x \leqslant 0 \end{cases}$, find the value of $f(-2016)$. Express your answer as an integer in the form $\boxed{[answer]}$. | -2018 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
A circle has the equation x^2+y^2+Dx+Ey+F=0 and is symmetric with respect to the lines x-y+4=0 and x+3y=0. Find the value of D+E. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Geometry -> Other",
"Mathematics -> Algebra -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Given the parametric equations of a line $\begin{cases} x=1+3t \\ y=2-\sqrt{3}t \end{cases}$ (where $t$ is the parameter), calculate the angle of inclination of the line in degrees. Ensure the angle is within the range [0°, 180°) and express your answer in degrees (°). | 150° | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Kieran the Kangaroo takes 6 seconds to make 4 jumps. How long does it take him to make 30 jumps? Express your answer in seconds. | 45 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
A man walks x miles due west, turns 150° to his left and walks 3 miles in the new direction. If he finishes at a point √3 miles from his starting point, calculate the value of x. Specify if the value of x is uniquely determined or not. | The value of x is not uniquely determined. | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the directrix of the parabola y = -1/4x^2. Express your answer as an equation of a line (e.g. y =...). | y = 1 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
A store makes a profit of $300, which is recorded as $+300. How would a loss of $300 be recorded in this system? Express your answer as a dollar amount. | -300 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Let $i$ be the imaginary unit, and $a\in\mathbb{R}$. If $\frac{1-ai}{1+i}$ is a pure imaginary number, find the value of $a$. Express your answer as a single real number. | 1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
When expanding \((a-b)^n \) using the binomial theorem \(( n \geqslant 2, a b \neq 0 )\) and if \( a = k b \) (where \( k \) is a positive integer), the sum of the second and third terms is 0. Find the value of \( n \) that satisfies this condition. Express your answer as a single integer or expression for n. | 2k + 1 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Let $a$ be a positive integer less than 100, and $a^3 + 23$ is divisible by 24. Find the number of possible values of $a$. Express your answer as a single integer. | 5 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
In triangle ABC, angle A = 100°, angle B = 50°, and angle C = 30°. Altitude AH and median BM are drawn. Find the measure of angle MHC. Express your answer in degrees as a whole number. | 30 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $2 \cdot (3^8)^2 \cdot (2^3)^2 \cdot 3 = 2^x \cdot 3^y$, simplify and find the values of x and y. Express your answer as an ordered pair (x, y). | (7, 17) | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Exponential Functions"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation 2^a * 3^b = 8 * (6^10), where a and b are integers, find the value of b - a. Express your answer as an integer. | -3 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
A school's wind band consists of 100 boys and 80 girls, and its string band consists of 80 boys and 100 girls. In total, 230 students participated in the two bands, with 60 boys participating in both. Calculate the number of girls who participated in the wind band but not in the string band. Express your answer as a si... | 10 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
In tetrahedron P-ABC, lines are formed by connecting the midpoints of each edge. Calculate the probability that a randomly selected line from these lines intersects plane ABC. Express your answer as a simplified fraction. | \frac{3}{5} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function y = xln(x), find the equation of the tangent line to the function at the point x = 1. Express your answer in the form y = mx + b, where m is the slope and b is the y-intercept. | y = x - 1 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
In the complex plane, the point represented by the complex number $z= \frac {2}{1-i}-2i^{3}$ (where $i$ is the imaginary unit) has coordinates $(x, y)$. Calculate the values of $x$ and $y$. Express your answer as an ordered pair $(x, y)$. | (1,3) | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Given $a^{m}=9$ and $a^{n}=3$, calculate the value of $a^{m-n}$. Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Given the product sequence $\dfrac{3}{2}\cdot \dfrac{4}{3}\cdot \dfrac{5}{4}\cdot \dfrac{6}{5}\cdot \ldots\cdot \dfrac{a}{b} = 9$, calculate the sum of $a$ and $b$. Express your answer as a single integer. | 35 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Solve for the complex number z in the equation (z - i)(2 - i) = 5. Express your answer in the form a + bi, where a and b are real numbers. | 2 + 2i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number \(z = a + b \mathrm{i}\) where \(a, b \in \mathbb{R}\), solve the inequality \(|z - \mathrm{i}| > |z + \mathrm{i}|\) for \(b\). Express your answer as an inequality involving \(b\). | b < 0 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Given that one base angle of an isosceles triangle is 30 degrees greater than the vertex angle, find the degree of the vertex angle. Express your answer in degrees as a whole number. | 40 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $e_1$ and $e_2$ are the eccentricities of a standard ellipse and a standard hyperbola with foci on the x-axis and common foci $F_1$ and $F_2$. $O$ is the coordinate origin, and $P$ is a common point on both curves satisfying $2|\overrightarrow {OP}|=|\overrightarrow {F_{1}F_{2}}|$. Calculate the value of $\f... | \frac{\sqrt{2}}{2} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence {a_n}, if a_3 = 6 and a_6 = 3, find the value of a_9. Express your answer as a single integer. | 0 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Find the coordinates of the focus of the parabola given by the equation $y^2 = 4x$. Express your answer as an ordered pair $(x, y)$. | (1, 0) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
The temperature at noon one day was -2℃, and the temperature at night dropped by 4℃ compared to noon. What was the temperature at night? Express your answer in degrees Celsius (℃) as an integer, including a negative sign if applicable. | -6 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Arrange the numbers 10000, 1, 10, 100, and 1000 from largest to smallest. What is the middle number in this ordered sequence? | 100 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given that cosα=2(1+sinα), and α≠2k$π- \frac {π}{2}$, k∈Z, calculate the value of tanα. Express your answer as a simplified fraction. | \frac {3}{4} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the tangent line to the curve y = sin(x) + e^x at the point (0,1). Express your answer in standard form (Ax + By + C = 0). | 2x - y + 1 = 0 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Given the line $\begin{cases} x=1-2t, \\ y=2+3t \end{cases}$ (where $t$ is a parameter) and the line $4x+ky=1$, find the constant $k$ that makes the two lines perpendicular. Express your answer as a single integer or fraction. | -6 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
A solid is composed of 7 identical cubes, where one central cube has each of its faces covered by one face of an identical cube. Given that the volume of the solid is 875 cm³, calculate the surface area of the solid in square centimeters (cm²). | 750 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Surface Area"
] | 0.09375 | 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 -> Discrete Mathematics -> Combinatorics -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Compute the real part of the complex number $z_{1}= \frac {1-2i}{2+i^{5}}$. Express your answer as a single number. | 0 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
A granary is open for collecting grain, and someone brings in 1536 dan of rice. Upon inspection, it is found that there is grain mixed in with the rice. A sample of the rice is taken, and in 256 grains, there are 18 grains of mixed grain. Calculate the amount of mixed grain in this batch of rice. Express your answer in... | 108 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Convert 6 minutes to hours. Express your answer as a decimal. | 0.1 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the edge length of a cube is $\frac {2 \sqrt {3}}{3}$, find the surface area of its circumscribed sphere. Express your answer in terms of $\pi$. | 4\pi | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence ${a_n}$, the middle term between $a_3$ and $a_8$ in their arithmetic difference is 10. Find the sum of the first 10 terms of the sequence, $S_{10}$. Express your answer as a numerical value. | 100 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression: $\frac{\cos 85^{\circ} + \sin 25^{\circ} \cos 30^{\circ}}{\cos 25^{\circ}}$. Express your answer as a simplified fraction. | \frac{1}{2} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Find the coefficient of $x^3$ in the expansion of $\left(x^3 + \frac{1}{x} - 1\right)^5$. Express your answer as a single integer. | 15 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Suppose that \( k>0 \) and that the line with equation \( y=3kx+4k^2 \) intersects the parabola with equation \( y=x^2 \) at points \( P \) and \( Q \). If \( O \) is the origin and the area of \( \triangle OPQ \) is 80, calculate the slope of the line. Express your answer as a single number. | 6 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Given a curve with the parametric equations
\[
\begin{cases}
x=3\cos \theta, \\
y=4\sin \theta
\end{cases}
\]
where $\theta$ is the parameter, $(0\leqslant \theta\leqslant \pi)$, and a point $P$ on the curve such that the line $OP$ (from $P$ to the origin $O$) has an inclination angle of $\dfrac{\pi}{4}$, calculate t... | \left( \dfrac {12}{5}, \dfrac {12}{5}\right) | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = 3sin(ωx + π/6) - 2 (ω > 0), the graph of the function is shifted to the right by 2π/3 units and overlaps with the original graph. Find the minimum value of ω. | 3 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given the piecewise function f(x) = { x + 1/2, x ∈ [0, 1/2) ; 2(1-x), x ∈ [1/2, 1] }, define f_n(x) = f(f_{n-1}(x)), where f_1(x) = f(x). Find f_{27}(1/5). Express your answer as a fraction. | \frac{4}{5} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Derive the relationship between the height of a regular tetrahedron and the radius of its inscribed sphere, analogous to the relationship in an equilateral triangle. Express the height of the tetrahedron as a multiple of the radius of its inscribed sphere and provide the numerical value of this multiplier. | 4 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a} = (1, -3)$ and $\overrightarrow{b} = (-2, 4)$, if the directed line segments representing vectors $4\overrightarrow{a}$, $3\overrightarrow{b} - 2\overrightarrow{a}$, and $\overrightarrow{c}$ can be connected end to end to form a triangle, calculate the vector $\overrightarrow{c}$. Expr... | (4, -6) | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $m - n = 1$, evaluate the expression $\left(m - n\right)^{2} - 2m + 2n$. Provide your answer as a single integer. | -1 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the arithmetic mean of $$\lg(\sqrt{3} - \sqrt{2})$$ and $$\lg(\sqrt{3} + \sqrt{2})$$. Express your answer as a single number. | 0 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given the vector $\overrightarrow{a} = (\sqrt{3}\sin\theta + \cos\theta + 1, 1)$, $\overrightarrow{b} = (1, 1)$, and $\theta \in [\frac{\pi}{3}, \frac{2\pi}{3}]$, calculate the maximum value of the projection $m$ of vector $\overrightarrow{a}$ onto vector $\overrightarrow{b}$. Express your answer as a single numerical ... | 2\sqrt{2} | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Geometry -> Other",
"Mathematics -> Calculus -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
In a soccer league with 6 teams (P, Q, R, S, T, W), each team must eventually play each other team exactly once. Given that P has played 1 match, Q has played 2 matches, R has played 3 matches, S has played 4 matches, and T has played 5 matches, determine the number of matches that W has played so far. | 3 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Find the derivative of the function $y=x+\frac{1}{x}$. Express your answer as a simplified mathematical expression. | 1-\frac{1}{x^2} | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the line $y=kx$ intersects with the curve $y=\ln x$, determine the maximum value of $k$. Express your answer as a single value in the boxed format. | \frac{1}{e} | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
The displacement S (in meters) of an object varies with time t (in seconds) according to the equation S(t) = 3t - t^2. Find the instantaneous velocity of the object at t = 2 seconds. Express your answer in meters per second (m/s). | -1 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
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