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 $\{a_n\}$ is an arithmetic sequence, and $a_3 + a_8 = 22$, calculate the sum of the first 10 terms of the sequence. | 110 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
If $(1+x)^n = 1+6x+15x^2+20x^3+15x^4+6x^5+x^6$, then find the value of $n$. Express your answer as a single integer. | 6 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Binomial Theorem"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x) = \begin{cases} -x^{2} & ;x < 0 \\ g(x) & ;x > 0 \end{cases}$, where $f(x)$ is an odd function, calculate the value of $g(2)$. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Algebra -> Functions"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
We write one of the numbers 0 and 1 into each unit square of a chessboard with 40 rows and 7 columns. If any two rows have different sequences, what is the maximum number of 1's that can be written into the unit squares? Express your answer as a single integer. | 198 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Solve the quadratic equation $x^{2}-4=0$. Express your answer as a solution set (e.g., $x_{1}=a$, $x_{2}=b$), including all solutions. | x_{1}=2, x_{2}=-2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Quadratic Functions"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x)=\begin{cases} 2^{x}+1, & x < 1 \\ x^{2}+ax, & x\geqslant 1 \end{cases}$, find the value of $a$ such that $f(f(0))=4a$. Express your answer as a single real number. | 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
To analyze the math scores of 400 students from 8 classes in their first mock college entrance examination in the senior year, it was decided to randomly select 12 test papers from each class for analysis. What is the sample size in this problem? Express your answer as a numerical value. | 96 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
A closed rectangular prism with height \(8 \text{ cm}\) is standing on a face with dimensions \(2 \text{ cm}\) by \(5 \text{ cm}\). The prism contains water with a depth of \(6 \text{ cm}\). When the prism is tipped so that it stands on a face with the greatest area, calculate the new depth of the water. Express your a... | 1.5 \text{ cm} | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Volume"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $(a+i)(1+i) = 2i$, where $a \in \mathbb{R}$ and $i$ is the imaginary unit, find the exact value of $a$. Express your answer as a single real number. | 1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Given the expression $\frac{(3+i)(a+2i)}{1+i}$, find the value of the real number $a$ such that the expression is a real number. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
For each natural number \( n \), the parabola \( y = (n^2 + n) x^2 - (2n + 1) x + 1 \) intersects the x-axis at points \( A_n \) and \( B_n \). Let \( |A_n B_n| \) represent the distance between these two points. Calculate the value of \( \sum_{i=1}^{1992} |A_i B_i| \). Express your answer as a single fraction. | \frac{1992}{1993} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Quadratic Functions"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Given that f(x) is an odd function with a domain of R, and it satisfies the property f(x+4) = f(x) for all x in R, find the value of f(43) if f(1) = 3. Express your answer as a single number. | -3 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Given a cube $ABCD\_1\_1\_1\_1$, a point is randomly selected within it. Calculate the probability that the point lies within the pyramid $O\_ABCD$ (where $O$ is the intersection point of the diagonals of the cube). Express your answer as a simplified fraction. | \frac{1}{6} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Find the equivalent radian measure for 60 degrees, given that pi radians equal 180 degrees. Express your answer in terms of pi. | \frac{\pi}{3} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $C_{20}^{2x-1} = C_{20}^{x+3}$, find the value(s) of x. Express your answer as a list of values separated by 'or'. | 4 or 6 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given two numbers whose sum is 10 and product is 20, find the sum of their reciprocals. Express your answer as a simplified fraction. | \frac{1}{2} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Given that a line with slope equal to 1 and a line with slope equal to 2 intersect at the point P(1,6), calculate the area of triangle PQR formed by the intersection of these lines with the x-axis. Express your answer as a whole number. | 9 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Find the absolute value of -9. Express your answer as a non-negative integer. | 9 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Find an angle that has the same terminal side as -80°. Express your answer in degrees (°) as a positive value less than 360°. | 280 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions -> Angle Measures"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Given \(a < 0\), the terminal side of angle \(\alpha\) passes through point \(P(3a,-4a)\). Find the value of \(\sin \alpha + 2\cos \alpha\). Express your answer as a simplified fraction. | -\dfrac{2}{5} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the cosine value of the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\frac{3}{5}$, and $| \overrightarrow{a}|=| \overrightarrow{b}|=1$, find the magnitude of the sum of vectors $\overrightarrow{a}$ and $\overrightarrow{b}$. Express your answer as a decimal value, including the squa... | \frac{4 \sqrt{5}}{5} | big_math | [
"Mathematics -> Linear Algebra -> Vectors"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the tangent line to the curve y = 1/x that passes through the point (2, 0). Express your answer in the format of a linear equation (e.g., x + y - 2 = 0). | x + y - 2 = 0 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
A palindrome is a positive integer whose digits are the same when read forwards or backwards. For example, 2002 is a palindrome. What is the smallest positive integer that can be added to 2002 to produce a larger palindrome? Express your answer as a positive integer. | 110 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence {a_n}, a_4 = 4 and a_3 + a_8 = 5. Find the value of a_7. Express your answer as a single numerical value. | 1 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Alicia had two containers. The first was $\tfrac{5}{6}$ full of water and the second was empty. She poured all the water from the first container into the second container, at which point the second container was $\tfrac{3}{4}$ full of water. What is the ratio of the volume of the first container to the volume of the s... | \frac{9}{10} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Two congruent pentagons are each formed by removing a right-angled isosceles triangle from a square of side-length 1. The two pentagons are then fitted together as shown. Calculate the length of the perimeter of the octagon formed. | 6 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Let the sum of the first n terms of a geometric sequence {a_n} be S_n, satisfying a_n > 0, q > 1, and a_3 + a_5 = 20, a_2 * a_6 = 64. Find S_6, the sum of the first 6 terms of the sequence. Express your answer as a single numerical value. | 63 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $\lg 8 + 3\lg 5$. Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Logarithmic Functions"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the triangle sides \(a, b, c\) are all integers and \(a \leqslant b \leqslant c\), with \(b = 10\), calculate the number of such triangles that exist. Express your answer as a single integer. | 55 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
The slant height of a cone is 2, and its lateral surface unfolds into a semicircle. Find the surface area of this cone. Express your answer in terms of pi. | 3\pi | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given two vectors $\overrightarrow {a}$ and $\overrightarrow {b}$ in a plane with an angle of 120° between them, $\overrightarrow {a}=(1,0)$, and $|\overrightarrow {b}|=1$, find the value of $|\overrightarrow {a}+2\overrightarrow {b}|$. Express your answer as a single value, using radical notation if necessary. | \sqrt{3} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = ln(x) + x^2, find the minimum value of the slope of the tangent line to the curve y = f(x) at any point P. Express your answer as a single numerical value. | 2\sqrt{2} | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = ax^7 - bx^5 + cx^3 + 2, and f(-5) = m, calculate the value of f(5) + f(-5). Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overset{→}{a}=(1,-2)$ and $\overset{→}{b}=(3,0)$, if $(2\overset{→}{a}+ \overset{→}{b}) \parallel (m\overset{→}{a}- \overset{→}{b})$, find the value of $m$. Express your answer as a single numerical value. | -2 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Find the smallest positive period of the function $f(x) = \sin x + \cos x$. Express your answer in terms of $\pi$. | 2\pi | big_math | [
"Mathematics -> Precalculus -> Functions -> Other"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
Represent the number 3.5 million in scientific notation, following the format $a\times10^{n}$, where $1\leq|a|<10$, and $n$ is an integer. | 3.5\times10^{6} | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Let the function f(x), defined on R, satisfy f(x) + f(y) = f(x+y) for any real numbers x and y, and f(3) = 4. Calculate the value of f(0) + f(-3). Express your answer as a single integer. | -4 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
A certain unit has 120 business staff, 24 management staff, and 16 logistics staff. Using the stratified sampling method, a sample of size $n$ is drawn from the staff of the unit. It is known that 3 management staff members are chosen. Calculate the value of $n$. Express your answer as a whole number. | 20 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Given two vectors in the plane, $\overrightarrow{a} = (3, 1)$ and $\overrightarrow{b} = (t, -3)$, and $\overrightarrow{a} \perp \overrightarrow{b}$, find the value of t. | 1 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence {a_n}, a_5 = 1, and a_8 + a_10 = 16. Find the value of a_13. Express your answer as a single integer. | 15 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Find the cube root of 64. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
From a group of 4 boys and 3 girls, select 3 students to attend a student forum. The selection must include both boys and girls. Calculate the total number of different selections that meet this criterion. | 30 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Combinations"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given that α is an acute angle, and sin(α-10°) = √3/2, find the value of α in degrees. | 70^\circ | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence \\(\\{a_n\\}\\) satisfies \\(a_{n+1}+a_n=n\\) and \\(a_1=2\\), calculate the value of \\(a_4-a_2\\). Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Count the number of positive integers between 10 and 1000 whose digits sum to 3. Express your answer as a whole number. | 9 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
A fair and uniform coin is tossed consecutively n times. Determine the minimum value of n such that the probability of the event "at least one head appears" is greater than or equal to 15/16. Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Point A has coordinates (1,-1). After moving point A up by 2 units and then left by 3 units, we get point B. What are the coordinates of point B? Express your answer as an ordered pair (x, y). | (-2, 1) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Zara has a collection of 4 marbles: an Aggie, a Bumblebee, a Steelie, and a Tiger. She wants to display them in a row on a shelf, but does not want to put the Steelie and the Tiger next to one another. Calculate the number of ways she can do this. | 12 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
If $f(x)= \begin{cases} x^{3}+\sin x, & -1\leqslant x\leqslant 1 \\ 2, & 1 < x\leqslant 2\end{cases}$, then find the value of $\int_{-1}^{2}f(x)dx$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
What is the result of adding 2 to -1? Express your answer as an integer. | 1 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given that a function f(x) satisfies f(m+n) = f(m) * f(n) for any positive integers m and n, and f(1) = 2, calculate the sum of f(2)/f(1) + f(4)/f(3) + f(6)/f(5) +... + f(2016)/f(2015). Express your answer as a single integer. | 2016 | big_math | [
"Mathematics -> Algebra -> Abstract Algebra -> Other"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Given the quadratic function y = 3x^2 - 6x + 5, if its vertex remains unchanged, the direction of its opening is reversed, and then it is translated along the axis of symmetry, a new parabola is obtained. This new parabola intersects the line y = -x - 2 at the point (a, -4). Find the equation of the new parabola in the... | y = -3x^2 + 6x - 4 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Quadratic Functions"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given a random variable X that follows a binomial distribution B(n, 0.8), and the variance D(X) is 1.6, calculate the value of n using the formula for variance of a binomial distribution. Express your answer as a whole number. | n = 10 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
A rectangular solid has dimensions length = 2, width = 4, and height = 8. A cube has a volume equal to that of the rectangular solid. Calculate the length of each edge of the cube. Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Volume"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence {a_n}, it is known that a_2 + a_{10} = 16. Find the value of a_4 + a_8. Express your answer as a single integer. | 16 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, \( P Q = 19 \), \( Q R = 18 \), and \( P R = 17 \). Point \( S \) is on \( P Q \), point \( T \) is on \( P R \), and point \( U \) is on \( S T \) such that \( Q S = S U \) and \( U T = T R \). Find the perimeter of \(\triangle P S T\). Express your answer as a whole number. | 36 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, \( P Q = 10 \) and \( Q R = x \). Given that \( \angle QSR = \angle QRS \) and \( \angle SPQ = 90^\circ \) and \( \angle PQS = 60^\circ \), find the value of \( x \). Express your answer as a numerical value. | 20 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $x[x\\{x(2-x)-4\\}+10]+1$. Express your answer as a simplified polynomial in standard polynomial order. | -x^4+2x^3-4x^2+10x+1 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Xiaoming has 2 one-yuan meal tickets, 2 two-yuan meal tickets, and 1 five-yuan meal ticket in his pocket. If he randomly draws 2 tickets from his pocket, what is the probability that the sum of their values is not less than four yuan? Express your answer as a simplified fraction. | \dfrac{1}{2} | big_math | [
"Mathematics -> Applied Mathematics -> Probability"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $\sin 2t = - \int_{0}^{t} \cos x \,dx$ where $t \in (0,\pi)$, calculate the exact value of $t$ in radians. | t = \frac{2\pi}{3} | big_math | [
"Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
The mean (average) of the three integers \( p, q \) and \( r \) is 9. The mean of the two integers \( s \) and \( t \) is 14. Calculate the mean of the five integers \( p, q, r, s \), and \( t \). Express your answer as a single number. | 11 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
The sequence $\{a_n\}$ satisfies $a_{n+1} = \frac{1}{2-a_n}$, with $a_1 = 0$. Derive a general formula for the $n$-th term of the sequence $\{a_n\}$ using inductive reasoning. Express your answer as a clear and concise formula for $a_n$. | \frac{n-1}{n} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Nasrin paddled to her camp, which was 4.5 km away, and then returned. The trip to the camp took 2 hours and 30 minutes, while the return trip took 1/3 of that time. Calculate Nasrin's mean (average) speed for the round trip. Express your answer in km/h. | 2.7 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the exact value of the product \(\left(1-\frac{1}{3}\right)\left(1-\frac{1}{4}\right)\left(1-\frac{1}{5}\right)\). Express your answer as a simplified fraction. | \frac{2}{5} | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
In the complex plane, the complex numbers $\frac {1}{1+i}$ and $\frac {1}{1-i}$ (where $i$ is the imaginary unit) correspond to points A and B, respectively. Calculate the complex number corresponding to point C, which is the midpoint of line segment AB. Express your answer as a complex number in the form $a + bi$. | \frac {1}{2} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets $M=\{x \mid -1 < x < 3\}$ and $N=\{x \mid x^2 - 6x + 8 < 0\}$, find the intersection $M \cap N$ in interval notation. | (2,3) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Given that the complex number $z$ satisfies the equation $1 + (1 + 2z)i = 0$ ($i$ is the imaginary unit), find the imaginary part of the complex number $z$. Express your answer as a single number. | \frac{1}{2} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = cos(x) - sin(x), find the value of f'(π/2). Express your answer as a single number. | -1 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Given the hyperbola C: $$\frac{x^2}{m} - y^2 = 1$$ with one of its foci at (2, 0), determine the eccentricity of C. Express your answer as a fraction or decimal in simplest form. | \frac{2\sqrt{3}}{3} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the exact value of $\sin 105^{\circ} \sin 15^{\circ}$. Express your answer as a fraction. | \dfrac{1}{4} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\overrightarrow {a}$ and $\overrightarrow {b}$ are two mutually perpendicular unit vectors, and $\overrightarrow {c}\cdot \overrightarrow {a}= \sqrt {3}$, $\overrightarrow {c}\cdot \overrightarrow {b}=1$, calculate the value of $| \overrightarrow {b}+ \overrightarrow {c}|$. Express your answer as a single n... | \sqrt{7} | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Geometry -> Other",
"Mathematics -> Linear Algebra -> Vectors"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Point C is the golden section point of segment AB, where AB = 8cm and AC > BC. Calculate the length of segment AC. Express your answer in centimeters (cm) and simplify your answer to include the square root of 5. | 4(\sqrt{5}-1)cm | big_math | [
"Mathematics -> Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Mikhail has $10,000 in $50 bills. Calculate the number of $50 bills he has. Express your answer as a whole number. | 200 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
In the three-dimensional Cartesian coordinate system, the point $A(1,2,3)$ is symmetric with respect to the $xOy$ plane at point $B$. Find the coordinates of point $B$. Express your answer as a coordinate $(x, y, z)$. | (1,2,-3) | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
The physical education teacher has 9 identical footballs to be placed into three boxes labeled 1, 2, and 3, with the requirement that the number of footballs in each box is not less than its own label number. Determine the total number of different ways to distribute the footballs into the boxes under these constraints... | 10 | big_math | [
"Mathematics -> Applied Mathematics -> Combinatorics"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow {a}=(-1,4)$ and $\overrightarrow {b}=(x,2)$, if $\overrightarrow {a}$ is parallel to $\overrightarrow {b}$, find the value of $x$. Express your answer as a single number. | -\frac{1}{2} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Find the minimum positive period of the function f(x) = (sin x + sin 3x) / (cos x + cos 3x). Express your answer in terms of pi. | \pi | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
There are 18 teams participating in the opening ceremony of a competition. When entering, the 1st team has 27 members, the 2nd team has 26 members, and the 18th team has 10 members. If they enter in a single file, and all 18 teams' members are assigned numbers from 1 to 333 in the order they enter, then how many teams ... | 9 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Find the minimum value of the function f(x) = -x^2 + 4x + 5 within the closed interval [1, 4]. Express your answer as a single number. | 5 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
You have 5 gold medals, 3 silver medals, and 2 bronze medals. You randomly select 2 medals, and one of the selected medals is a gold medal. What is the probability that the other medal is also a gold medal? Express your answer as a simplified fraction. | \frac{2}{7} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Combinations"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given the definitions of \( r! = r \cdot (r-1) \cdots \cdot 1 \) and \( C_{j}^{k} = \binom{j}{k} = \frac{j!}{k!(j-k)!} \), where \( r, j, k \) are positive integers and \( k < j \), find the value of \( n > 3 \) such that \( C_{n}^{1}, C_{n}^{2}, C_{n}^{3} \) form an arithmetic sequence. Express your answer as a single... | 7 | big_math | [
"Mathematics -> Applied Mathematics -> Combinatorics"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Three rugs have a combined area of 200 m^2. By overlapping the rugs to cover a floor area of 140 m^2, the area which is covered by exactly two layers of rug is 24 m^2. Calculate the area of the floor that is covered by exactly three layers of rug. Express your answer in square meters (m^2). | 18 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given the hyperbola x^2/a^2 - y^2/b^2 = 1 has an asymptote that intersects the parabola y = x^2 + 1 at only one point, calculate the eccentricity of the hyperbola. Express your answer as a single value. | \sqrt{5} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Given the hyperbola $\dfrac {x^{2}}{a^{2}}- \dfrac {y^{2}}{b^{2}}=1$ ($a > 0$, $b > 0$) has an asymptote parallel to the line $l$: $y=2x+10$, and one of the foci of the hyperbola lies on the line $l$, find the equation of the hyperbola. Express your answer in the standard form of a hyperbola equation. | \dfrac {x^{2}}{5}- \dfrac {y^{2}}{20}=1 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given points $S$, $A$, $B$, $C$ on the surface of a sphere $O$, where $SA$ is perpendicular to plane $ABC$, $AB$ is perpendicular to $BC$, $SA = AB = 1$, and $BC = \sqrt{2}$, calculate the surface area of sphere $O$. Express your answer as a multiple of $\pi$. | 4\pi | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Surface Area"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given that point P (3, -1) and Q (-1, 2) are on the same side of the line $ax+2y=1$, find the range of the real number $a$. Express your answer as an interval. | (1, 3) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Find the value of α such that the domain of the function y = x^α is all real numbers (ℝ) and it is an odd function, satisfying f(-x) = -f(x). Express your answer as a single value of α. | α = 3 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence {a_n}, where a_1 = 2 and a_n - a_n-1 = 2n (n ≥ 2), derive the explicit formula for a_n in terms of n. Express your answer using standard mathematical notation. | a_n = n^2 + n | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Find the coordinates of the symmetric point of (-1, 2) with respect to the line x + y + 4 = 0. Express your answer as an ordered pair (x, y). | (-6, -3) | big_math | [
"Mathematics -> Geometry -> Coordinate Geometry -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Let the complex number $z= \frac {1+i}{1-i}$ (where $i$ is the imaginary unit). Calculate the magnitude of the conjugate of $z$, denoted as $|\overline{z}|$. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Let $P=\{x|\ln x > 0\}$ and $Q=\{x|-1 < x < 2\}$. Find the intersection $P \cap Q$ and express your answer as an interval. | (1,2) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the vectors $\overset{→}{BA}=(3,-2)$ and $\overset{→}{AC}=(0,6)$, calculate the magnitude of vector $\overset{→}{BC}$. Express your answer as a single number. | 5 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
The ages of Jo, her daughter, and her grandson are all even numbers. The product of their three ages is 2024. Find Jo's age. Express your answer as a whole number. | 46 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
There are 3 units hiring employees from 4 university graduates. Each unit must hire at least 1 person (it's not guaranteed that all 4 graduates will be hired). Calculate the total number of different hiring methods. Express your answer as a single integer. | 60 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Combinations"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
When the polynomial $x^9 - x$ is factored as completely as possible into polynomials and monomials with integral coefficients, how many factors are there? Express your answer as a single integer. | 5 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations -> Other"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
In an arithmetic sequence $\{a_n\}$, given that $2a_7=a_8+5$, calculate the sum of the first 11 terms, $S_{11}$. Express your answer as a single integer. | 55 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence $\{a_n\}$ with $a_2=2$, and $a_{n+2}+(-1)^{n+1}a_n = 1 + (-1)^n$ for $n\in\mathbb{N}^*$, find the sum of the first 100 terms of the sequence, $S_{100}$. Express your answer as a single number. | 2550 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given that the graph of y = f(x+1) passes through the point (3, 1), find the point on the graph of y = f^(-1)(x+1). Express your answer as an ordered pair (x, y). | (0, 4) | big_math | [
"Mathematics -> Algebra -> Functions"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given that f(x) is an even function defined on \mathbb{R}, and it is monotonically increasing in the interval (-\infty,0), find the range of values for the real number a that satisfies f(2^{|a-1|}) > f(-\sqrt{2}). Express your answer as an interval. | (\frac{1}{2}, \frac{3}{2}) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the absolute value of -5. Express your answer as a single number. | 5 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
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