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 |
|---|---|---|---|---|---|
In the sequence $\sqrt{2}$, $\sqrt{5}$, $2\sqrt{2}$, $\sqrt{11}$, …, $2\sqrt{5}$, …, find the term position of $2\sqrt{5}$. Express your answer as a single integer. | 7 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation 2x + 6 = 16, find the value of x + 4. Express your answer as a single integer. | 9 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Given real numbers $m$ and $n$, if $m \geqslant 0, n \geqslant 0$, and $m + n = 1$, find the minimum value of $\frac{m^2}{m + 2} + \frac{n^2}{n + 1}$. Express your answer as a simplified fraction. | \frac{1}{4} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given the curve y = 3x - ln(x + a), the equation of the tangent at the point (0,0) is y = 2x. Find the value of a. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Two cones share a common base and the vertices of both cones and the circumference of the base are all on the same sphere. If the area of the base of the cone is $\frac{3}{16}$ of the area of the sphere, calculate the ratio of the volumes of the two cones. Express your answer as a simplified ratio (e.g., a:b). | 1:3 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Volume"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given that circle C is symmetrical to the circle with the equation (x - 1)^2 + y^2 = 1 with respect to the line y = -x, find the equation of circle C. Express your answer in the standard form (x - h)^2 + (y - k)^2 = r^2. | x^2 + (y + 1)^2 = 1 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
The arithmetic sequence is defined as -3, 1, 5,.... Find the 15th term of this sequence. | 53 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
A telephone exchange has phone numbers in the format of $139XXXXXXXX$, where the first six digits are fixed. The last five digits are composed of either $6$ or $8$. Determine the total number of such phone numbers. Express your answer as a single integer. | 32 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the angle between the plane vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is 120°, and $\overrightarrow{a} = \left(\frac{3}{5}, -\frac{4}{5}\right)$ with $|\overrightarrow{b}| = 2$, calculate the magnitude of $2\overrightarrow{a} - \overrightarrow{b}$. Express your answer as a numerical value, includ... | 2\sqrt{3} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given the cubic equation $x^{3}-ax^{2}+1=0$, where $a > 3$, determine the exact number of real roots in the interval $(0,2)$. | 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Point A(-3,-5) is moved 4 units up and then 3 units to the left to point B. Find the coordinates of point B. Express your answer as an ordered pair (x, y). | (-6, -1) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
In the sequence below, each △ represents a number, and starting from the third term, each number is the sum of the two preceding numbers. Calculate the sum of all 9 numbers listed: △, △, △, △, 7, △, △, △, 47. Express your answer as a single integer. | 122 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence \( \{a_n\} \) where the sum of the first \( n \) terms is \( S_n \), and \( a_3 + a_7 + a_{11} = 6 \), calculate the value of \( S_{13} \). Provide your answer as a single numerical value. | 26 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
A square floor is fully covered with square tiles of the same size. The two diagonals are covered with black tiles, and the rest are covered with white tiles. If there are 101 black tiles in total, what is the total number of tiles used to cover the floor? Express your answer as a whole number. | 2601 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
The Gauss family has three boys aged 7, a girl aged 14, and a boy aged 15. What is the mean (average) of the ages of the children? Express your answer as a whole number. | 10 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Roll a die twice. Let X denote the maximum of the two rolls. Calculate the expected value E(X). Express your answer as a numerical value. | 4.472 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Solve the inequality $x - 2 \geqslant 0$ to find the range of values of $x$ that make the square root $\sqrt{x-2}$ meaningful. Express your answer as an inequality in the form of $x \geqslant \text{value}$. | x \geqslant 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Inequalities"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence {a_n} with a common difference of 2, if a_1, a_2, a_5 form a geometric sequence, find the value of a_2. Express your answer as a single integer. | 3 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
In the binomial expansion of $(1+x)^n$ where $n \in \mathbb{N}^*$, the coefficient of $x^5$ is the largest. Find the value of $n$. Express your answer as a single integer. | 10 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
What is the maximum number of subsets that a set with 10 elements can have, such that none of these subsets is a subset of another? Express your answer as a whole number. | 252 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given that in triangle $\triangle ABC$, the sides opposite to the angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and it is given that $A+C=2B$ and $\sqrt{3}a^2+\sqrt{3}c^2-2ac\sin B=9\sqrt{3}$, calculate the value of $b$. Provide your answer as a numerical value. | 3 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
The system of equations $\left\{\begin{array}{l}x y + y z = 63 \\ x z + y z = 23\end{array}\right.$ has how many positive integer solutions? Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given $a > 1$, solve for $a$ in the equation $\int_{1}^{a} (2x - \frac{1}{x}) \, dx = 3 - \ln 2$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
A bank savings card has a 6-digit password. When someone tries to withdraw money from an ATM, they forget the last digit of the password but remember that the last digit is an even number. Calculate the probability that they will enter the correct password in no more than two attempts. Express your answer as a simplifi... | \frac{2}{5} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Evaluate the expression $2 \times 0 + 1 - 9$. Provide your answer as a single integer. | -8 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Suppose a complex number $z$ satisfies $i(\overline{z} + 3) = 3 - i^2$. Find the magnitude of $z$, denoted as $|z|$. Express your answer as a single number. | 5 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\vec{a}=\{\sin 2 \varphi, 1, \tan \varphi\}$, $\vec{b}=\{1, \cos \varphi, \cot \varphi\}$ with $\varphi \in[0,2 \pi)$, find the values of $\varphi$ that satisfy the equation $\vec{a} \cdot \vec{b}=1$. Express your answer in radians as a list of values within the interval [0, 2π). | \frac{\pi}{2}, \frac{3\pi}{2}, \frac{7\pi}{6}, \frac{11\pi}{6} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Ann made a 3-step staircase using 18 toothpicks. To complete a 5-step staircase, how many additional toothpicks does she need? Express your answer as a numerical value. | 22 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Find the solution set of the inequality $(x+1)(x-2) \leq 0$. Express your answer in interval notation, including any necessary brackets or parentheses to denote the type of interval. | [-1, 2] | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Given that sin(2α) > 0 and cos(α) < 0, determine the quadrant in which the angle α lies. Express your answer as the name of the quadrant. | Third Quadrant | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence {a_n}, the sum of five consecutive terms is given as a_3+a_4+a_5+a_6+a_7=450. Find the value of a_2+a_8. Express your answer as a single integer. | 180 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
The central angle of a sector is 2 radians, and the corresponding arc length is 4. Calculate the area of this sector. Express your answer as a single number. | 4 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, $\triangle PQR$ is isosceles with $PQ = PR$, and $\triangle PRS$ is isosceles with $PS = SR = x$. Also, the perimeter of $\triangle PQR$ is 22, the perimeter of $\triangle PRS$ is 22, and the perimeter of $PQRS$ is 24. Find the value of $x$. Express your answer as a whole number. | 6 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Given complex numbers $z_{1}=\cos x-i\sin x$ and $z_{2}=\sin x-i\cos x$, calculate the modulus of the product $z_{1}\cdot z_{2}$. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Given a sequence {a_n} with the sum of its first n terms denoted by S_n such that S_n = n^2 + 1, find the value of a_5. Express your answer as a single numerical value. | 9 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Determine the coefficient of the term containing $x^4$ in the expansion of the binomial $({x^2 - \frac{1}{x})^5}$. Express your answer as a single integer. | 10 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
A ream of paper containing 500 sheets is 5 cm thick. Approximately how many sheets of this type of paper would there be in a stack 7.5 cm high? Express your answer as a whole number. | 750 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Given the curves f(x) = cos x and g(x) = x^2 + bx + 1, find the value of coefficient b such that the curves have a common tangent line at the point of intersection (0, m). Express your answer as an exact integer. | 0 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\alpha$ is an obtuse angle, and the equation $3\sin 2\alpha = \cos\alpha$ holds, find the value of $\sin\alpha$. Express your answer as a fraction or decimal value. | \frac{1}{6} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number z = 1 + ai (a ∈ ℝ) that corresponds to a point in the first quadrant of the complex plane, and |z| = √5, find the imaginary part of z. Express your answer as a real number. | 2 | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
When 5 integers are arranged in ascending order, the median is 6. If the only mode of this data set is 8, then what is the possible maximum sum of these 5 integers? Express your answer as a single numerical value. | 31 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Evaluate the expression \((1)^{10} + (-1)^{8} + (-1)^{7} + (1)^{5}\) and provide your answer as a single integer. | 2 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given $x > 0$, $y > 0$, satisfying $\frac{4}{y} + \frac{1}{x} = 4$, find the minimum value of $x + y$. Express your answer as a single number. | \frac{9}{4} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given that \(a, b, c\) are distinct real roots of the polynomial equation \(x^3 - 3x + 1 = 0\), find the value of \(a^8 + b^8 + c^8\). Express your answer as a single integer. | 186 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the parametric equations of a line $$\begin{cases} x=1+2t \\ y=2-3t \end{cases}$$ (where $t$ is the parameter), calculate the slope of the line. Express your answer as a fraction. | -\frac{3}{2} | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Other"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
There are 2 doctors and 4 nurses assigned to perform physical examinations for students in 2 schools, with each school allocated 1 doctor and 2 nurses. Determine the total number of different assignment methods. Express your answer as a single numerical value. | 12 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
The sum of the first 4 terms of a geometric sequence is 5, and the sum of the first 12 terms is 35. What is the sum of the first 8 terms? Express your answer as a single integer. | 15 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence {a_n} with a non-zero common difference d, it is known that a_1 = 1, and a_2, a_4 - 2, a_6 form a geometric sequence. If positive integers m and n satisfy m - n = 10, then calculate the value of a_m - a_n. Express your answer as a single number. | 30 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = -x^3 + 3x + m, calculate the values of m that satisfy the condition of exactly two zeros. | -2, 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given set A = {-2, 1/2, 3, 4}, find the number of 2-element subsets it has. Express your answer as a whole number. | 6 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Combinations"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given that a complex number $z$ satisfies $z(1+i)=-2i$, find the value of $z$. Express your answer as a complex number in the form $a+bi$. | -1-i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation:
\[
(\log_{3} x)(\log_{x} 2x)(\log_{2x} y) = \log_{x} x^{2}
\]
Solve for y. | 9 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Logarithmic Functions"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given that distinct real numbers a, b, and c form an arithmetic sequence, and ca, ab, bc form a geometric sequence, and a+b+c=15, find the value of a. | 20 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the circle (x-2)^2 + y^2 = 4 and the line of symmetry y = sqrt(3)/3 * x, find the equation of the circle symmetric to the given circle with respect to the line. Express your answer as a circle equation in standard form. | (x-1)^2 + (y-sqrt(3))^2 = 4 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given that $|a+1|$ and $|b+2|$ are opposite in sign, find the value of $a+b$. Express your answer as a single integer. | -3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Given $\overrightarrow{a}=(1-t,2t-1,0)$ and $\overrightarrow{b}=(2,t,t)$, calculate the minimum value of $|\overrightarrow{b}-\overrightarrow{a}|$. Express your answer as a numerical value. | \sqrt{2} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $\sqrt{2-2\cos 4}$. Express your answer as a simplified trigonometric expression. | 2\sin 2 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Find the coefficient and degree of the monomial $-xy^2z^3$. Express your answer in the format "coefficient and degree," where both values are integers. | -1 \text{ and } 6 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of zeros of the function $f(x) = \ln x + 2x - 1$. Express your answer as a whole number. | 1 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Two chords of a circle intersect, one chord is divided into two segments of 12cm and 18cm, and the other chord is divided in the ratio 3:8. Find the length of the other chord. Express your answer in centimeters (cm) as a whole number. | 33 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Solve for $x$ in the equation $3^{x-1} = \\frac{1}{9}$. Express your answer as a single number. | -1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
In $\triangle ABC$, given that $b^2+c^2-a^2=\sqrt {3}bc$, use the cosine theorem to find the measure of angle A. Express your answer in degrees as a numerical value. | 30 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $f\left(x\right)$ is an odd function, and when $x \lt 0$, $f\left(x\right)=e^{x}$, calculate the value of $f\left(e\right)$. Express your answer as a single mathematical expression. | -e^{-e} | big_math | [
"Mathematics -> Algebra -> Functions -> Other"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Given the equations $a+b=3$ and $a-b=5$, find the value of $b^{2}-a^{2}$. Express your answer as an integer. | -15 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Given a sequence: $$\sqrt {2}, \sqrt {5}, 2\sqrt {2}, \sqrt {11}, \ldots$$, find the term number of $4\sqrt {2}$ in this sequence. Express your answer as a whole number. | 11 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Convert the binary number $11101_2$ to its decimal equivalent. Express your answer as a single integer. | 29 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
The hyperbola $\frac{x^2}{m} - \frac{y^2}{n} = 1 (mn
eq 0)$ has an eccentricity of $\sqrt{3}$. One of its foci coincides with the focus of the parabola $y^2 = 12x$. Calculate the value of $mn$. Express your answer as a single number. | 18 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Given an isosceles triangle $\triangle ABC$, the exterior angle adjacent to $\angle A$ is $130^{\circ}$. Find the possible measures of the vertex angle of this triangle. Express your answer(s) in degrees. | 50^{\circ} \text{ or } 80^{\circ} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
In the cube $ABCD-A_{1}B_{1}C_{1}D_{1}$, point $P$ is the midpoint of $C_{1}D_{1}$, and $\overrightarrow {AP}=\overrightarrow {AD}+x\overrightarrow {AB}+y\overrightarrow {AA_{1}}$. Find the value of the real number $x+y$. Express your answer as a single real number. | \dfrac{3}{2} | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
In a box containing 3 white balls and 2 black balls, two balls are drawn without replacement. Calculate the probability of drawing exactly one black ball. Express your answer as a simplified fraction. | \frac{3}{5} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Combinations"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $m+i=(1+2i)\cdot ni$, where $m,n\in \mathbb{R}$ and $i$ is the imaginary unit, find the value of $n-m$. Express your answer as a real number. | 3 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Find the value(s) of $a$ such that the constant term in the expansion of $(x+a)^{2}\left( \frac {1}{x}-1\right)^{5}$ is $-1$. Express your answer as a numerical value(s) of $a$, including all possible solutions. | 1, 9 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
A company sells a brand of cars in two locations, A and B, with profits (in ten thousand yuan) of $L_1 = -x^2 + 21x$ and $L_2 = 2x$, respectively, where $x$ is the sales volume (in units). If the company sells a total of 15 units in both locations, calculate the maximum profit it can achieve. Express your answer in ten... | 120 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
From a group of 4 boys and 3 girls, select 3 people to undertake three different tasks. Calculate the total number of possible arrangements if there is at least one girl among the selected individuals. | 186 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given sets $A=\{0,1\}$ and $B=\{-1,0,a+3\}$, and knowing that $A \subseteq B$, find the value of $a$ that satisfies this subset relationship. Express your answer as a single integer. | -2 | big_math | [
"Mathematics -> Set Theory -> Subset Relationships"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the increasing function $f(x) = ax + b$ intersects the x-axis at the point $(2, 0)$, solve the inequality $bx^2 - ax > 0$. Express your answer as an interval in interval notation. | (- \frac {1}{2}, 0) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
How many ways are there to place 3 indistinguishable red chips, 3 indistinguishable blue chips, and 3 indistinguishable green chips in the squares of a 3x3 grid so that no two chips of the same color are directly adjacent to each other, either vertically or horizontally? Express your answer as a single integer. | 36 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given tan(alpha) = 3, evaluate the following expression:
(sin(alpha - pi) + cos(pi - alpha)) / (sin(pi/2 - alpha) + cos(pi/2 + alpha))
Express your answer as a single numerical value. | 2 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
A box contains a total of 400 tickets that come in five colors: blue, green, red, yellow, and orange. The ratio of blue to green to red tickets is 1:2:4. The ratio of green to yellow to orange tickets is 1:3:6. What is the smallest number of tickets that must be drawn to ensure that at least 50 tickets of one color hav... | 196 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
The diagram shows a rhombus and two sizes of regular hexagon. The smaller hexagon and the rhombus share an edge of length x, and the larger hexagon has an edge length of 2x. Find the ratio of the area of the smaller hexagon to the area of the larger hexagon. Express your answer in the format 'a:b'. | 1:4 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
There are 10000 people taking a certain examination, and their scores X approximately follow a normal distribution N(100,13^2). Given that P(61 < X < 139)=0.997, calculate the approximate number of people scoring no less than 139 points in this exam. Express your answer as a whole number. | 15 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Find the domain of the function $f(x)=\frac{{\sqrt{9-{x^2}}}}{{x-1}}$. Express your answer in interval notation. | [-3,1) \cup (1,3] | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
What is the last digit when the decimal number 77 is converted to base-4? Express your answer as a single digit. | 1 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Given the ratios AB : BC = 1 : 2 and BC : CD = 8 : 5, calculate the ratio AB : BD. Express your answer as a ratio, including both parts of the ratio. | 4 : 13 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Delete all perfect squares from the sequence of positive integers 1, 2, 3,..., and obtain a new sequence. Find the value of the 2003rd term in this new sequence. | 2048 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
The Athenas are playing a 44-game season. Each game results in a win or a loss, and cannot end in a tie. So far, they have 20 wins and 15 losses. In order to make the playoffs, they must win at least 60% of all of their games. What is the smallest number of their remaining games that they must win to meet this requirem... | 7 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
If $2m-4$ and $3m-1$ are square roots of the same positive number, find the possible values of $m$. Express your answer as numerical values, listing all possible solutions. | -3, 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z= \frac {2+i}{-2i+1}$, find the real part of $z$. Express your answer as an integer. | 0 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Given $a=60.5$, $b=0.56$, $c=\log_{10}0.56$, order the magnitudes of $a$, $b$, and $c$ from smallest to largest. Express your answer as a sequence of the variables, separated by commas. | c,b,a | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Ben and David go to buy bagels. For every 1 yuan Ben spends, David spends 25 cents less. As a result, Ben spends 12.50 yuan more than David. Calculate the total amount they spend together. Express your answer in yuan, rounded to 2 decimal places. | 87.50 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the linear function that passes through the point $A(-2,-1)$ and is parallel to the line $y=2x-3$. Express your answer in slope-intercept form (y=mx+b), where m is the slope and b is the y-intercept. | y=2x+3 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of functions f defined on positive real numbers such that f(1) = 1 and f(x^2 y^2) = f(x^4 + y^4) for all x, y in R. Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Given that x and y are both real numbers randomly chosen from the interval [0, π/2], calculate the probability that y ≤ sin(x). Express your answer as a fraction or decimal in the form of \boxed{answer}. | \frac{4}{\pi^2} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets $M=\{-1,0,1\}$ and $N=\{-2,-1,0,2\}$, determine the intersection $M\cap N$. Express your answer as a set. | \{-1,0\} | big_math | [
"Mathematics -> Discrete Mathematics -> Set Theory -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
The total length of the Great Wall is approximately $6.7\\times 10^{6}$ meters in scientific notation. Convert this value into its original number, expressing your answer as a whole number in meters. | 6700000 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z=\\frac{2i}{1+i}$ (where $i$ is the imaginary unit), find the conjugate of $z$. Express your answer as a complex number in the form $a+bi$. | 1-i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow {a}=(1,0)$ and $\overrightarrow {b}=(-\dfrac{1}{2},\dfrac{\sqrt {3}}{2})$, find the angle between $\overrightarrow {a}$ and $\overrightarrow {b}$. Express your answer in degrees as a whole number. | 120^{\circ} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
If $a^{m+1}b^3$ and $(n-1)a^2b^3$ are like terms, and their sum is 0, find the values of m and n. Express your answer as a pair of values (m, n). | (1, 0) | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Given the series \( S = 2 + 4 + 6 + \cdots + 2N \), where \( N \) is the smallest positive integer such that \( S > 10^6 \), calculate the sum of the digits of \( N \). Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Given the arithmetic sequence {a_n}, where a_3 = 4 and a_5 = 8, find the value of a_11. Express your answer as a single integer. | 20 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
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