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 the system of equations:
\[
x + 1 = y - 8
\]
\[
x = 2y
\]
Find the value of \( x + y \). Express your answer as a single integer. | -27 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ that satisfy $|\overrightarrow{a}| = 1$ and $|\overrightarrow{b}| = 2$, and the projection of $\overrightarrow{a}$ onto $\overrightarrow{b}$ is equal to the projection of $\overrightarrow{b}$ onto $\overrightarrow{a}$. Calculate the magnitude of the difference... | \sqrt{5} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation y = x + 1/x, express the polynomial x^4 + x^3 - 4x^2 + x + 1 = 0 in terms of y. Use the format x^2(y^2 + y + n) = 0 and find the exact value of n. | -6 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
In the geometric sequence {a_n}, the sum of the first two terms is 40 (a_1 + a_2 = 40), and the sum of the third and fourth terms is 60 (a_3 + a_4 = 60). Find the sum of the seventh and eighth terms (a_7 + a_8). Express your answer as a single integer. | 135 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
There are three people: Xiao Huang, Xiao Hao, and Xiao Xuan. They need to be hidden in 6 different cabinets, with each cabinet able to hide at most 2 people. Calculate the total number of different ways to hide them. Provide your answer as a whole number. | 210 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
There is a 3-hour time difference between Vancouver and Toronto, with Vancouver being behind Toronto. For example, when it is 1:00 p.m. in Vancouver, it is 4:00 p.m. in Toronto. What time is it in Vancouver when it is 6:30 p.m. in Toronto? Express your answer in 12-hour clock format with a.m./p.m. designation. | 3:30 p.m. | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Two people independently attempt to decrypt a password. Their respective probabilities of success are $\frac{1}{5}$ and $\frac{1}{4}$. Find the probability that the password is decrypted. Express your answer as a decimal value. | 0.4 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Given that point $(m, n)$ lies on the curve $\begin{cases} x= \sqrt {6}\cos\alpha \\ y= \sqrt {6}\sin\alpha \end{cases}$ (where $\alpha$ is a parameter), and point $(x, y)$ lies on the curve $\begin{cases} x= \sqrt {24}\cos\beta \\ y= \sqrt {24}\sin\beta \end{cases}$ (where $\beta$ is a parameter), find the maximum val... | 12 | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Given five points, no three of which are collinear, calculate the minimum number of convex polygons that can be formed using these points. Express your answer as a single integer. | 16 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Polygons"
] | 0.296875 | 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 |
Given the sequence $S_n = \sin \frac{π}{7} + \sin \frac{2π}{7} +... + \sin \frac{nπ}{7} \quad (n \in \mathbb{N}^+)$, find the number of zero values among $S_1$, $S_2$,..., $S_{2017}$. Express your answer as a single integer. | 288 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Evaluate the expression $\frac{2\sin 80{}^\circ -\sin 20{}^\circ }{\cos 20{}^\circ }$. Simplify your answer, which may involve square roots. | \sqrt{3} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
If one side of a square is increased by 3.5 times and the other side is decreased by 30 cm, resulting in a rectangle that has twice the area of the square, calculate the area of the original square in square centimeters. | 4900 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Given the equations $P^2 - PQ = 1$ and $4PQ - 3Q^2 = 2$, calculate the value of $P^2 + 3PQ - 3Q^2$. Express your answer as a single integer. | 3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Let the increasing sequence of positive terms \( a_{1}, a_{2}, \cdots, a_{n} \) be the simplest proper fractions with the denominator 60. Calculate the sum \( \sum_{i=1}^{n} \cos(a_{i} \pi) \). Express your answer as a single number. | 0 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
A bank teller has some stacks of bills. The total value of the bills in each stack is $1000. Every stack contains at least one $20 bill, at least one $50 bill, and no other types of bills. If no two stacks have the same number of $20 bills, what is the maximum possible number of stacks that the teller could have? Provi... | 9 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
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. Express your answer as a single integer. | 110 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of the expression $(\log_{3}2+\log_{3}5) \cdot \lg 9$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Logarithmic Functions"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
The variables x and y have a linear correlation. When x takes the values of 16, 14, 12, and 8, the observed values of y are 11, 9, 8, and 5 respectively. Using the reference formulas for linear regression, predict the approximate value of x when y = 10. Express your answer as a single numerical value. | 15 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Find the minimum value of the expression \( x^2 - 6x + 13 \). Provide your answer as a single number. | 4 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation x^2 + ax + 1 = 0, and knowing that x = 1 is a solution, find the value of a. Express your answer as a single integer. | -2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Quadratic Functions"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Solve the inequality $-x^{2}+3x-2 \geqslant 0$ and express the solution set in interval notation. | [1,2] | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Given a hyperbola C: $$\\frac{x^{2}}{a^{2}}$$ - $$\\frac{y^{2}}{b^{2}}$$ = 1 (a > 0, b > 0), one of its asymptotic line equations is y = - $$\\frac{\\sqrt{5}}{2}$$x. Calculate the eccentricity of the hyperbola C. Express your answer as a fraction or decimal value. | \\frac{3}{2} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
A sample of size 1000 is divided into 10 groups for a frequency distribution table. The frequencies of the first 3 groups form an arithmetic sequence with a common difference of 0.05. Given that the sum of the frequencies of the last 7 groups is 0.79, calculate the frequency number of the group with the smallest freque... | 20 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given sin(alpha + pi/3) + cos(alpha - pi/2) = -4*sqrt(3)/5, where -pi/2 < alpha < 0, calculate the value of cos(alpha + 2*pi/3). Express your answer as a single fraction or decimal value. | \frac{4}{5} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Jia has three different books, and Yi borrows at least one book. Find the total number of different borrowing methods. Express your answer as a single integer. | 7 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given that tetrahedron ABCD is inscribed in sphere O, and AD is the diameter of sphere O. If triangles ABC and BCD are equilateral triangles with side length 1, what is the volume of tetrahedron ABCD? Express your answer as a fraction with a square root. | \frac{\sqrt{3}}{12} | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z = \frac{10i}{3+i}$, calculate the modulus $|z|$. Express your answer as a numerical value. | \sqrt{10} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the terminal side of angle a passes through point P(1,-2), calculate the value of 2*sin(a)/cos(a). Express your answer as a single number. | 4 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=|\overrightarrow{b}|=2$ and the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $60^{\circ}$, calculate the projection of $\overrightarrow{a}$ on $\overrightarrow{a}+ \overrightarrow{b}$. Express your answer as a si... | \sqrt{3} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\overline{z}$ is the complex conjugate of complex number $z$, if $z = 1 + i$, calculate the complex number $\frac{z^2}{\overline{z}}$. Express your answer as a complex number in the form $a + bi$, where $a$ and $b$ are the real and imaginary parts of the complex number, respectively. | -1 + i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $i$ is the imaginary unit, find the conjugate of $\dfrac{1+i}{i^{3}}$. Express your answer as a complex number in the form $a+bi$, where $a$ and $b$ are real numbers. | -1-i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Two men and two women are lined up in a row. Find the total number of ways they can be arranged alternately by gender. Express your answer as a whole number. | 8 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Given the inequality sequence \( 1-x > (1-x)^{2} > (1-x)^{3} > (1-x)^{4} > \cdots \), determine the range of values for \( x \). Express your answer as an interval or range of values. | (0, 1) | big_math | [
"Mathematics -> Algebra -> Inequalities"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
The intersection point P of two lines $y=x+2a$ and $y=2x+a$ is inside the circle $(x-1)^2+(y-1)^2=4$. Calculate the range of the real number $a$. Express your answer in interval notation. | (-\frac{1}{5}, 1) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities",
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\sin2α + \sinα = 0, α ∈ (\frac{π}{2}, π)$, calculate the value of $\tan(α + \frac{π}{4})$. Express your answer as a single expression involving radicals. | -2 + \sqrt{3} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given two sequences {a_n} and {b_n} satisfying a_1=1, a_2=2, b_1=2, and for any positive integers i, j, k, l, when i+j=k+l, it holds that a_i+b_j=a_k+b_l. Calculate the value of 1/2013 * sum(a_i+b_i) from i=1 to 2013. Express your answer as a single integer. | 2015 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Using 9 matchsticks to form triangles by connecting them end to end, how many different triangles can be formed? Express your answer as a whole number. | 3 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics -> Counting Methods"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Convert the decimal number 0.09 to scientific notation. Express your answer in the form $a \times 10^b$, where $a$ is the coefficient and $b$ is the exponent. | 9 \times 10^{-2} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Given a square ABCD folded along its diagonal AC to form a triangle ADC, let β be the angle between AD and the plane ABC. When β reaches its maximum value, find the size of the dihedral angle B-AC-D. Express your answer in degrees (°). | 90° | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
In an arithmetic sequence, given that $a_2 + 4a_7 + a_{12} = 96$, calculate the value of $2a_3 + a_{15}$. Express your answer as a single integer. | 48 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation (1-i)^2z = 4-2i, calculate the conjugate of z, denoted as \\overline{z}, in the form a+bi. | \\overline{z} = 1 - 2i | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of integer solutions to the equation \(x^{1988} + y^{1988} + z^{1988} = 7^{1990}\). Express your answer as a single integer. | 0 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.25 | 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 simplified fraction. | \dfrac{1}{3} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation x^2 + y^2 = 14x + 6y + 6, find the maximum value of 3x + 4y. Express your answer as a single integer. | 73 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Given the operation $1 \\& 1 = 2$, $m \\& n = k$, $m \\& (n + 1) = k + 3$ ($m, n, k \\in \\mathbb{N}^*$), calculate the output of $1 \\& 2004$. Express your answer as a single integer. | 6011 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
One lap of a standard running track is 400 meters. Calculate the number of laps each athlete runs in a 5000 meter race. Express your answer as a mixed number or decimal, including units if necessary. | 12.5 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
In a decreasing arithmetic sequence {a_n}, if a_1 + a_100 = 0, then find the value of n when the sum of the first n terms, S_n, is maximized. Express your answer as a whole number. | 50 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the modulus of the complex number (a + 2i)(1 + i) is 4, calculate the value of the real number a. Express your answer as a real number, including the ± symbol if applicable. | \pm 2 | big_math | [
"Mathematics -> Algebra -> Complex Numbers -> Other"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Beatrix places dominoes on a 5x5 board, either horizontally or vertically, so that each domino covers two small squares. She stops when she cannot place another domino. What is the largest possible number of squares that may still be uncovered? Express your answer as a single integer. | 7 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Three dice are rolled once each. Let event A be "the three numbers are all different", and event B be "at most one odd number appears". Calculate the probability P(A∩B). Express your answer as a simplified fraction. | \frac{5}{18} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of the expression \(2^{3}+2^{2}+2^{1}\). Provide your answer as a single integer. | 14 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Liam spent all his money buying 50 soda bottles at the corner shop for £1 each. He sold each bottle at the same higher price. After selling 40 bottles, he had £10 more than he started with. He then sold all the remaining bottles. Calculate the total amount of money Liam had once all the bottles were sold. Express your ... | £75 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
The y-intercepts of three parallel lines are 2, 3, and 4. The sum of the x-intercepts of the three lines is 36. What is the slope of these parallel lines? Express your answer as a numerical value. | -\frac{1}{4} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation
$$
\frac{1}{x^{2}-x}+\frac{k-5}{x^{2}+x}=\frac{k-1}{x^{2}-1}
$$
with an extraneous root x = 1, calculate the value of k. Express your answer as a single integer. | 3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a \in \mathbb{R}$, if the complex number $z = \frac{a-2i}{1+i}$ is a pure imaginary number, then find the magnitude of $1+ai$. Express your answer as a numerical value. | \sqrt{5} | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Algebra -> Complex Numbers"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a} = (3, 1)$ and $\overrightarrow{b} = (-2, 5)$, calculate the result of $2\overrightarrow{a} + \overrightarrow{b}$. Express your answer as an ordered pair (x, y). | (4, 7) | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Given the three vertices of a triangle A(2, -1, 4), B(3, 2, -6), and C(5, 0, 2), find the length of the median through point A. Express your answer as a simplified radical expression. | 2 \sqrt{11} | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Find the greatest common divisor (GCD) of 1037 and 425. Express your answer as a single numerical value. | 17 | big_math | [
"Mathematics -> Number Theory -> Greatest Common Divisors (GCD)"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
In the square shown, find the value of x. Express your answer in degrees as a whole number. | 45 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the random variable ξ follows a normal distribution N(1, σ^2), and P(ξ < 2) = 0.6, determine P(0 < ξ < 1). Express your answer as a decimal value. | 0.1 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Arrange three identical black balls and three identical white balls from left to right into a row. A "valid arrangement" is defined as an arrangement where, starting from any position (including this position) to the right, up to the last ball, the number of black balls is greater than or equal to the number of white b... | \frac{1}{4} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
If \( x \) and \( y \) are positive integers such that \( xyz = 240 \), \( xy + z = 46 \), and \( x + yz = 64 \), calculate the value of \( x + y + z \). Express your answer as a single integer. | 20 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Find the coefficient of the $x^5$ term in the expansion of the binomial $(x+2)^7$. Express your answer as a single integer. | 84 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of solutions for the system of equations:
\[
\left\{
\begin{array}{l}
(x+y-1) \sqrt{x-1} = 0 \\
x^{2}+y^{2}+2 x-4=0
\end{array}
\right.
\]
Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given a rectangular parallelepiped of volume 12 with all its vertices on the surface of a sphere centered at O, and the height of the parallelepiped is 4, calculate the minimum surface area of sphere O. Express your answer in terms of π. | 22\pi | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Find the limit as x approaches 0 from the right of the function [x]/tan(x), where [x] is the greatest integer function. Express your answer as a single number. | 0 | big_math | [
"Mathematics -> Calculus -> Limits -> Other"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
The figure shown can be folded along the lines to form a rectangular prism with dimensions length = 10 cm, width = 6 cm, and height = 5 cm. Calculate the surface area of the rectangular prism. Express your answer in square centimeters (cm²) as a whole number. | 280 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Surface Area"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
The complex number \( z \) satisfies \( |z + 5 - 12i| = 3 \). Find the maximum value of \( |z| \). Express your answer as a single integer. | 16 | big_math | [
"Mathematics -> Geometry -> Other",
"Mathematics -> Algebra -> Other",
"Mathematics -> Other"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
The graph of the function f(x) = 2sin(3x + φ) is shifted to the right by π/12 units and the resulting graph is symmetric about the y-axis. Calculate the smallest possible value of |φ|. Express your answer in radians. | \frac{\pi}{4} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given the exponential equation $b^3=a$, where $b>0$ and $b\neq 1$, find its corresponding logarithmic form. Express your answer as a logarithmic equation, specifying the base and the result. | \log_ba=3 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions"
] | 0.171875 | 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"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given that tan(α) = 2, find the value of tan(α - 3π/4). Express your answer as a single numerical value. | -3 | big_math | [
"Mathematics -> Trigonometry -> Trigonometric Functions"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
If $f(x)$ is an odd function with a period of 5 on $\mathbb{R}$, and satisfies $f(1) = 1$ and $f(2) = 2$, calculate the value of $f(3) - f(4)$ using the properties of odd and periodic functions. | -1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z= \frac {a}{2-i}+ \frac {3-4i}{5}$, the sum of its real part and imaginary part is $1$. Find the value of the real number $a$. Express your answer as a single real number. | 2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = -2x^2 + 1, find the value of f(-1). Express your answer as a single integer. | -1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Quadratic Functions"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given the quadratic inequality \\(a{x}^{2}+2ax+1 > 0\\), let proposition A be that its solution set is the real number set \\(R\\), and let proposition B be that \\(0 < a < 1\\). Describe the logical relationship between propositions A and B, specifying whether A is necessary, sufficient, both, or neither for B to hold... | \text{Necessary but not sufficient} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Find the smallest positive integer $n$ such that $\sqrt{18n}$ is an integer. Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Number Theory -> Factorization"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Find the absolute value of -2022. Express your answer as a positive integer. | 2022 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
The digits 2, 4, 6, and 8 are each used once to create two 2-digit numbers. What is the largest possible difference between the two 2-digit numbers? Express your answer as a whole number. | 62 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
The equation of an ellipse is given by $\frac {x^{2}}{5}$+ $\frac {y^{2}}{4}$$=1$. Find the eccentricity of this ellipse. Express your answer as a fraction or decimal. | \frac{\sqrt{5}}{5} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
In the Cartesian coordinate system, point P(3,2) is translated 2 units to the right, and then symmetrically reflected about the origin. Calculate the coordinates of the point after these transformations. Express your answer as an ordered pair (x, y), including negative signs if applicable. | (-5, -2) | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\tan\alpha=3$, $\alpha\in\left(0,\, \frac {\pi}{2}\right)$, calculate the value of $\sin2\alpha+\cos(\pi-\alpha)$. Express your answer as a fraction or decimal, and include the boxed notation around the final answer. | \frac{6-\sqrt{10}}{10} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Determine the number of proper subsets of the set \\(\\{x \mid 0 < |x-1| < 3, x \in \mathbb{Z}\\}\\). Express your answer as a whole number. | 15 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
In a certain school, 20 students were late on Monday, 13 students were late on Tuesday, and 7 students were late on Wednesday. If there are 30 students who were late at least once during these three days, what is the maximum possible number of students who were late all three days? Express your answer as a whole number... | 5 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the distance from the point (2, 1) to the line y = (1/2)x + 1. Express your answer as a simplified fraction with a square root if necessary. | \frac{2 \sqrt{5}}{5} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Toss a fair die twice. Let event A be {the two outcomes are different}, and event B be {at least one outcome is a 3}. Calculate the conditional probability P(B|A). Express your answer as a simplified fraction. | \frac{1}{3} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
In rectangle \( ABCD \), point \( F \) is the midpoint of side \( CD \), and the length of side \( BC \) is three times the length of \( BE \). Calculate the ratio of the area of the rectangle to the area of the shaded portion. Express your answer as a single numerical value. | 3 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Let p be a prime number. The sum of all positive divisors of p^4 is a perfect square. Find the number of prime numbers p that satisfy this condition. Express your answer as a whole number. | 1 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given a hyperbola x^2/4 - y^2/m = 1 with an eccentricity equal to the radius of a circle that is centered at the right focus and is tangent to the asymptote of the hyperbola, determine the value of m. Express your answer as a fraction. | \frac{4}{3} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, \(PQRS\) is a trapezoid with an area of 12. \(RS\) is twice the length of \(PQ\). Find the area of \(\triangle PQS\). Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Given the inequality series \( 1-x > (1-x)^{2} > (1-x)^{3} > (1-x)^{4} > \cdots \), determine the range of values for \( x \). Express your answer as an interval. | (0, 1) | big_math | [
"Mathematics -> Algebra -> Inequalities"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
The area of a rectangle is 12 square metres. The lengths of the sides, in metres, are whole numbers. Calculate the greatest possible perimeter of the rectangle. Express your answer as a whole number in metres. | 26 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
If person A and person B independently solve a problem, and their respective probabilities of not being able to solve it are $a$ and $b$, then what is the probability that both A and B solve the problem? Express your answer as an equation using the variables $a$ and $b$. | (1-a)(1-b) | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Given the real number x satisfies x^3 + 1/x^3 = 18, calculate the sum of the fractional parts of x and 1/x, denoted as {x} + {1/x}. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
A sequence of positive integers starts with one 1, followed by two 2's, three 3's, and so on. (Each positive integer n occurs n times.) How many of the first 105 numbers in this sequence are divisible by 3? Express your answer as a single integer. | 30 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given $0 < x < 1$, find the value of $x$ that maximizes the expression $x(3-3x)$. Express your answer as a fraction. | \dfrac{1}{2} | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Solve for n in the equation 3n = 9 + 9 + 9. Express your answer as a single integer. | 9 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Find the derivative of the function $f(x)=\frac{1}{x}$. Express your answer as a mathematical expression in the form $\boxed{f'(x)}$. | -\frac{1}{x^2} | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
In a group stage of a football match, each pair of teams must play one home game and one away game. A total of 6 stadiums are used by the organizers, with 4 games scheduled at each stadium per day. If it takes 10 consecutive days to ensure that all group stage matches are completed, calculate the number of participatin... | 16 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.