instruction_seed stringlengths 40 1.39k | answer stringlengths 1 95 | source stringclasses 1
value | domain listlengths 1 3 | llama8b_solve_rate float64 0 0.98 | _source stringclasses 1
value |
|---|---|---|---|---|---|
Calculate the value of $(-3)^{2}$. Express your answer as a single integer. | 9 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given the parabola $y^2=8x$, find the distance from the point on the parabola with an x-coordinate of 8 to the focus of the parabola. Express your answer as a whole number. | 10 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
For real numbers $a$ and $b$, the equation $3x - 5 + a = bx + 1$ has a unique solution $x$. Find the condition on $b$ that ensures a unique solution. Express your answer as a mathematical condition (e.g., $b \neq 3$). | b \neq 3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence {a_n}, the sum of the first n terms is S_n, a_4+a_7+a_{10}=9, S_{14}-S_{3}=77. Find the value of n that makes S_n minimum. Express your answer as a whole number. | 5 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
A rectangle with vertices (0,0), (100,0), (100,50), (0,50) has a circle with center (75,30) and radius 10 cut out of it. Find the slope of the line through the point (75,30) that divides the remaining area of the rectangle into two shapes of equal area. Express your answer as a simplified fraction. | \frac{1}{5} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $i$ is the imaginary unit, calculate the value of the complex number $i^{2015}$. Express your answer as a complex number, including the imaginary unit $i$ if necessary. | -i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $a\sqrt{1-b^2}+b\sqrt{1-a^2}=1$, calculate the value of $a^{2}+b^{2}$. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets $M=\{x|x^2=1\}$ and $N=\{x|ax=1\}$, if $N \subseteq M$, then find the set of values for the real number $a$. Express your answer as a set of values. | \{-1,0,1\} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given that the average ages of students in classes A, B, and C are 14, 13, and 12 years respectively, and the overall average age of the students in the three classes is 13 years, find the equation that represents the relationship between the number of students in classes A, B, and C, denoted as a, b, and c respectivel... | a = c | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
In the geometric sequence $\{a_n\}$, the sum of the first $n$ terms is given by $S_n=5^{n+1}+a$. Find the value of $a$. Express your answer as a single integer. | -5 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given $\cos \left(\alpha- \frac {2\pi}{9}\right)=- \frac { \sqrt {7}}{4}$, and $\alpha\in\left( \frac {\pi}{2},\pi\right)$, find the exact value of $\sin \left(\alpha+ \frac {7\pi}{9}\right)$. Express your answer as a single fraction or decimal value. | -\frac{3}{4} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Find the arithmetic square root of 16. Express your answer as a positive number. | 4 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the directrix of the parabola given by the equation $y^2 = 8x$. Express your answer as an equation of a line (e.g., x = a or y = b). | x = -2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Find the unit digit of \(33 \cdot 83^{1001} \cdot 7^{1002} \cdot 13^{1003}\). Express your answer as a single digit. | 9 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function y = (m-1)x + m^2 - 1, find the value of m that makes this function a direct proportion. A direct proportion function has a constant term of zero and a non-zero coefficient of x. Express your answer as a single value of m. | m = -1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Find the value of x that minimizes the function y = x + 1/(x-1) on the interval (1, +∞). Express your answer as a single value. | 2 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
A school has 600 students in the first grade, 500 in the second grade, and 400 in the third grade. If a stratified sampling method is used to extract n students from the school, and there are 30 students from the first grade in the sample, calculate the total number of students in the sample (n). Express your answer as... | n = 75 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
How many different ways are there to express 2016 as the sum of a natural number multiple of 2 and a natural number multiple of 3? Express your answer as a single integer. | 337 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Solve the system of equations $$\begin{cases} x+y=3 \\ x-y=1\end{cases}$$ and express the solution set as an ordered pair (x, y). | (2, 1) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Systems of Equations"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
A quality inspector sequentially selects products numbered 3, 13, 23,..., 93 from 100 products numbered 1 to 100 for inspection. What sampling method is used in this case? Provide your answer as the name of the specific sampling method. | Systematic sampling | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
The sides of the convex quadrilateral \(ABCD\) are \(AB = 3\), \(BC = 4\), \(CD = 12\), and \(DA = 13\). The angle \(\angle CBA\) is a right angle. Find the area of the quadrilateral. Express your answer as a whole number. | 36 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
The parameterized equations of a line are given by $\begin{cases} & x=t\sin {{50}^{0}}-1 \\ & y=-t\cos {{50}^{0}} \end{cases}$, where $t$ is a parameter. Determine the inclination angle of the line in degrees. | 140 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given the quadratic equation $\left(m-1\right)x^{2}+3x-1=0$, find the condition on $m$ such that the equation remains quadratic. Express your answer as an inequality or equality condition on $m$ using mathematical notation. | m \neq 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Find the minimum value of $a$ such that the inequality $x^{2}+ax+1\geqslant 0$ holds for all $x\in(0, \frac{1}{2}]$. Express your answer as a single number. | -\frac{5}{2} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
In triangle ABC, given that cosA = -$\frac{{\sqrt{2}}}{2}$ and tanB = $\frac{1}{3}$, calculate the value of tan(A-B). Express your answer as a single number. | -2 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Judy had a total of 35 hits during the softball season, consisting of 1 home run, 1 triple, and 5 doubles, with the rest being singles. Calculate the percentage of her hits that were singles. Express your answer as a percentage. | 80\% | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overset{⇀}{a}=(2,-1)$ and $\overset{⇀}{b}=(-4,2)$, calculate the result of the operation $\overset{⇀}{a}-2\overset{⇀}{b}$. Express your answer as an ordered pair (x, y). | (10,-5) | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
A flag is in the shape of a right-angled triangle, with the horizontal and vertical sides being of lengths 72 cm and 24 cm respectively. The flag is divided into 6 vertical stripes of equal width. Calculate the difference between the areas of any two adjacent stripes. Express your answer in square centimeters (cm²). | 48 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = (x + a)ln((2x - 1)/(2x + 1)), find the value of a such that f(x) is an even function. Express your answer as a single number. | 0 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
The line L crosses the x-axis at (-8,0). The area of the shaded region is 16. Find the slope of the line L. Express your answer as a fraction or decimal. | \frac{1}{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x+1)=2x-1$, derive the expression for $f(x)$. Express your answer as a function of $x$. | 2x-3 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
The terminal side of angle $\alpha$ passes through point P($b$, 4), and $\cos\alpha= -\frac{3}{5}$. Find the value of $b$. Express your answer as a single number. | -3 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
A sequence $\{a_n\}$ consists of 5 terms, where $a_1 = 0$, $a_5 = 2$, and $|a_{i+1} - a_i| = 1$ for $i=1, 2, 3, 4$. Calculate the number of different sequences that satisfy these conditions. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number z that satisfies the equation (1+2i)z = 1-i, calculate the value of | conjugate(z) |. Express your answer as a simplified fraction or radical. | \frac{\sqrt{10}}{5} | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $tan({θ+\frac{π}{4}})=2tanθ-7$, calculate the value of $\sin 2\theta$. Express your answer as a single value. | \frac{4}{5} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $\frac{(1-i)^2}{z}=1+i$, calculate the value of the complex number $z$. Express your answer in the form $a+bi$. | -1-i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
In a right triangle, the square of the hypotenuse is equal to twice the product of the legs. Find the measure of one of the acute angles of the triangle. Express your answer in degrees. | 45^\circ | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
In a community calligraphy and painting competition, the elderly participating account for 3/5, middle-aged people account for 1/5, and children account for 1/5. The probabilities of the elderly, middle-aged, and children winning prizes are 0.6, 0.2, and 0.1 respectively. Calculate the probability of selecting an award... | 0.42 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of \((\sqrt{1+\sqrt{1+\sqrt{1}}})^{4}\). Express your answer in simplest radical form if necessary. | 3+2\sqrt{2} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence {a_n} with the sum of the first n terms denoted as S_n, and S_7 = 14, calculate the value of a_3 + a_5. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a} = (2, 1)$ and $\overrightarrow{b} = (3, \lambda)$, if $\overrightarrow{a} \perp \overrightarrow{b}$, find the value of λ. Express your answer as a single number. | -6 | big_math | [
"Mathematics -> Algebra -> Vectors"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Find the remainder when $91^{91}$ is divided by $100$. Express your answer as a single integer. | 91 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
A pack of monosodium glutamate weighs 80g, a pack of salt weighs 500g, and a pack of laundry detergent weighs 420g. Calculate the total weight of these three items. Express your answer in grams (g) or kilograms (kg), ensuring to specify the unit clearly. | 1kg | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Given a geometric sequence {a_n} with positive terms, where S_n is the sum of the first n terms, and it is known that a_2a_4 = 1 and S_3 = 7, calculate the value of S_5. Express your answer as a simplified fraction. | \frac{31}{4} | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given $3^{m}=4$ and $3^{n}=6$, calculate the value of $3^{2m-n}$. Express your answer as a fraction. | \frac{8}{3} | big_math | [
"Mathematics -> Algebra -> Exponential Functions"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Mukesh scored 80% on a test that has a total of 50 marks. Calculate the number of marks he obtained. Express your answer as a whole number. | 40 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a} = (8, \frac{1}{2}x)$ and $\overrightarrow{b} = (x, 1)$, with $x > 0$, if $\overrightarrow{a} - 2\overrightarrow{b}$ is collinear with $2\overrightarrow{a} + \overrightarrow{b}$, find the value of $x$. Express your answer as a single numerical value. | 4 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Given $|\vec{a}|=2$ and $2\vec{a}-\vec{b} \perp \vec{a}$, calculate the projection of $\vec{b}$ in the direction of $\vec{a}$. Express your answer as a single number. | 4 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = x^2 + 4x - 5 with domain x ∈ [-3, 2], find the range of the function. Express your answer in standard interval notation (e.g., [a, b]). | [-9, 7] | big_math | [
"Mathematics -> Precalculus -> Functions"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Given y = 77, calculate the value of the expression (7y + 77)/77. Express your answer as a single number. | 8 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
A certain factory produces three types of products: Type A, Type B, and Type C. The ratio of quantity among the three types is 2:3:5. Now, by using stratified random sampling, a sample of size $n$ is drawn. If 24 units of Type B product are drawn, calculate the value of $n$. Express your answer as a whole number. | 80 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Given the circle equation $x^{2}+y^{2}-6x=0$, find the coordinates of the center and the radius of the circle. Express the center coordinates as an ordered pair (x, y) and the radius as a numerical value. | (3,0), 3 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression: $\dfrac {1}{x-2}+ \dfrac {2}{x+2}+ \dfrac {4}{4-x^{2}}$. Express your answer as a simplified algebraic expression in the form $\boxed{\text{expression}}$. | \dfrac {3}{x+2} | big_math | [
"Mathematics -> Algebra -> Rational Expressions -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given a triangle with sides in the ratio 3:4:5, calculate the ratio of the corresponding heights. Express your answer as a simplified ratio of three numbers. | 20 : 15 : 12 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Evaluate the expression (x)(x^2)(1/x) when x = -2. Provide your answer as a single number. | 4 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given the proposition $p$: there exists an $n \in \mathbb{N}$ such that $3^n > 2018$, what is the correct negation $\neg p$? Express your answer as a logical statement using the correct quantifier and mathematical notation. | \forall n \in \mathbb{N}, 3^n \leq 2018 | big_math | [
"Mathematics -> Discrete Mathematics -> Logic"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
From a set of 5 distinct balls, select 4 to place into 3 boxes. The first box should contain 1 ball, the second box should contain 2 balls, and the third box should contain 1 ball. Calculate the number of distinct arrangements. | 60 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of $\log_2 9 \cdot \log_3 8$. Express your answer as a whole number. | 6 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a + 2b + 3c = 12$ and $a^2 + b^2 + c^2 = ab + ac + bc$, calculate the value of $a + b^2 + c^3$. Express your answer as a single integer. | 14 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Solve the inequality |-a| > -a. Express your answer as an inequality or interval. | a > 0 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
The air quality records of the Anshan area in 2016 show that the probability of having good air quality for one day is 0.8, and the probability of having good air quality for two consecutive days is 0.6. If today's air quality is good, then what is the probability that tomorrow's air quality will also be good? Express ... | 0.75 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Kim's flight took off from Newark at 10:34 AM and landed in Miami at 1:18 PM. Both cities are in the same time zone. If her flight took $h$ hours and $m$ minutes, with $0 < m < 60$, what is the sum of $h$ and $m$? Express your answer as a whole number. | 46 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
In △ABC, there are 2008 arbitrary points that are not collinear, plus the three vertices A, B, and C, making a total of 2011 points. If these 2011 points are connected to form non-overlapping small triangles, calculate the total number of small triangles that can be formed. | 4017 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
In a wooden box, there are 3 red balls, 5 yellow balls, and 7 blue balls. If you blindly pick balls from the box, what is the minimum number of balls you need to pick to ensure that at least two of them are of the same color? Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence {a_n}, a_2 = 1 and a_4 = 5. Find the sum of the first 5 terms S_5. Express your answer as a single integer. | 15 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfying $|\overrightarrow{a}| = 1$, $|\overrightarrow{b}| = 2$, and $(\frac{5}{2}\overrightarrow{a} + \overrightarrow{b})$ is perpendicular to $(\overrightarrow{a} - \overrightarrow{b})$, find the angle $\theta$ between vectors $\overrightarrow{a}$ and $\ov... | \frac{2\pi}{3} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
The perimeter of a rectangle is 56. If the length of the rectangle is defined as \( l = x + 4 \) and the width is defined as \( w = x - 2 \), find the area of the rectangle. Express your answer as a whole number. | 187 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Determine the value of $y$ for which the sum of the coefficients of the odd power terms of $x$ in the expansion of $(x+y)(x+1)^4$ is 32. Express your answer as a single integer. | 3 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x) = \begin{cases} \log_{2}x, & x>1 \\ (\frac{1}{2})^{x}, & x\leq 1 \end{cases}$, calculate the value of $f(f(-\frac{1}{2}))$. Express your answer as a single number. | \frac{1}{2} | big_math | [
"Mathematics -> Precalculus -> Functions"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
There are 6 balls of each of the four colors: red, blue, yellow, and green. Each color of balls is numbered from 1 to 6. Calculate the number of ways to select 3 balls with distinct numbers, such that no two balls have the same color or consecutive numbers. Express your answer as a whole number. | 96 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given the equations \( x^{2} y z^{3} = 7^{3} \) and \( x y^{2} = 7^{9} \), calculate the product \( x y z \). Express your answer as a power of 7, including the base and exponent. | 7^4 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
A heptagon (seven-sided polygon) has all its interior angles less than 180 degrees. What is the minimum number of obtuse angles it must have? Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the angle between vectors $\overrightarrow {a}$ and $\overrightarrow {b}$ is 120°, and $|\overrightarrow {a}| = |\overrightarrow {b}| = 4$, find the magnitude of the sum of vectors $\overrightarrow {a}$ and $\overrightarrow {b}$. Express your answer as a single number. | 4 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Given that a = $$\frac{\ln 2}{2}$$, b = $$\frac{\ln 3}{3}$$, and c = $$\frac{\ln 5}{5}$$, determine the correct ordering of a, b, and c. Express your answer as a string representing the correct ordering, using less-than (<) symbols to denote ordering (e.g., 'c < a < b'). | c < a < b | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
In an opaque bag, there are 4 identical balls marked with the numbers 0, 1, 2, and 2. Player A draws a ball and then puts it back. Player B then draws a ball. The player who draws the ball with the higher number wins (if the numbers are the same, it's a draw). Under the condition that player A wins, calculate the proba... | \frac{2}{5} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x)= \begin{cases} 2^{x-1}, & x > 1 \\ \tan \left( \frac {\pi}{3}x \right), & x\leqslant 1 \end{cases}$, calculate the value of $f\left( \frac {1}{f(2)}\right)$. Express your answer as a single value, exact if possible. | \frac{\sqrt{3}}{3} | big_math | [
"Mathematics -> Precalculus -> Functions"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence {a_n} that satisfies a_2 + a_4 = 4 and a_3 + a_5 = 10, find the value of a_5 + a_7. Express your answer as a single integer. | 22 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
A giant panda is transporting bamboo from location A to location B. Each time, the panda can carry 50 bamboo sticks, but it eats 5 bamboo sticks both on the way from A to B and on the way back from B to A. There are currently 200 bamboo sticks at location A. What is the maximum number of bamboo sticks that the panda ca... | 165 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Solve the equation $x^{2}-4x+1=0$ using the completing the square method. Express your answer as a completed square equation. | (x-2)^{2}=3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Given the expressions $\frac{1}{x-2}$, $\frac{3xy}{π}$, $-\frac{2a{b}^{2}c}{3}$, $\frac{2x}{y}$, how many of these expressions qualify as fractions in the context of algebra, considering fractions to typically involve variables in the denominator? Provide your answer as a whole number. | 2 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Fractions"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Solve the equation (x+1)^2 = 0 for its roots. Express your answer in the format x1 = x2 = value. | x1 = x2 = -1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Let $C(A)$ denote the number of elements in a non-empty set $A$. Define
$$
A*B= \begin{cases}
C(A)-C(B), & \text{when } C(A) \geq C(B) \\
C(B)-C(A), & \text{when } C(A) < C(B)
\end{cases},
$$
If $A=\{x|x^2-ax-1=0, a\in \mathbb{R}\}$, $B=\{x||x^2+bx+1|=1, b\in \mathbb{R}\}$, let $S=\{b|A*B=1\}$. Find the number of elem... | 3 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $y=x^2-1840x+2009$ intersects the x-axis at points $(m,0)$ and $(n,0)$, calculate the value of $(m^2-1841m+2009)(n^2-1841n+2009)$. Express your answer as a single integer. | 2009 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
The complex number $z$ satisfies $(z-i)(2-i)=5$. Find the value of $z$ in the form $a+bi$. | 2+2i | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Given $a \in \mathbb{R}$, let set $A = \{x \mid |x-1| \leq 2a-a^2-2\}$. Determine the total number of subsets of set $A$. Express your answer as a single value. | 1 | big_math | [
"Mathematics -> Set Theory -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
From 2 male and 2 female students, two people are randomly chosen to participate in a charity event on Saturday and Sunday, with one person per day. Find the probability that a male student is arranged for Saturday and a female student for Sunday. Express your answer as a fraction or decimal. | \dfrac{1}{3} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
How many integers between 100 and 300 are multiples of both 5 and 7, but are not multiples of 10? Express your answer as a whole number. | 3 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Let the set $A=\{x | 1 < x < 4 \}$, and $B=\{x | x^{2}-2x-3\leqslant 0 \}$. Find the union of sets A and B, and express your answer in interval notation. | [-1,4) | big_math | [
"Mathematics -> Set Theory -> Set Operations"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the exact value of $(1-i)^{4}$. Express your answer as a single numerical value. | -4 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given a sector with a central angle of $\frac{4}{3}\pi$ and an area of $6\pi$, which represents the lateral surface of a cone, calculate the volume of the cone. Express your answer in terms of $\pi$ and include the appropriate units and precision. | \frac{4\sqrt{5}}{3}\pi | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(-1,2)$ and $\overrightarrow{b}=(x,3)$, where $\overrightarrow{a} \perp \overrightarrow{b}$, calculate the magnitude of vector $\overrightarrow{b}$, denoted as $|\overrightarrow{b}|$. Express your answer as a numerical value with a radical if necessary. | 3\sqrt{5} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
One morning, the temperature was $-3^{\circ}\mathrm{C}$. By noon, it rose by $6^{\circ}\mathrm{C}$, and then dropped by $7^{\circ}\mathrm{C}$ in the middle of the night. What was the temperature in the middle of the night? Express your answer in degrees Celsius (°C). | -4^{\circ}\mathrm{C} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Express the number 1,200,000,000 in scientific notation. Provide your answer in the format of coefficient times 10 to the power of exponent, like this: $\boxed{a \times 10^{b}}$. | 1.2 \times 10^{9} | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Find the exact value of $\tan {75}^{{o}}$. Express your answer as an exact algebraic expression, including any necessary radicals. | 2+\sqrt{3} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given \( x \in \mathbf{R} \) and \( a_n = \cos \left( x + \frac{2}{7} n \pi \right) \), calculate the value of \( a_0 + a_1 + a_2 + a_3 + a_4 + a_5 + a_6 \). Express your answer as a single number. | 0 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Given that the line $l$ passes through point P$(-2, 0)$ and intersects with circle C: $x^2 + y^2 - 4x - 2y + 1 = 0$ at two points A and B, and that $8\overrightarrow{PA} = 5\overrightarrow{AB}$, find the length of $|AB|$. Express your answer as a single value, including any necessary radicals. | \frac{8\sqrt{5}}{5} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of $\left(-3\right)^{3}+5^{2}-\left(-2\right)^{2}$. Express your answer as a single integer. | -6 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the complex number z satisfies the equation z(2-i) = 18+11i, find the real part of z. Express your answer as a single integer. | 5 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Determine how many of the integers \(19, 21, 23, 25, 27\) can be expressed as the sum of two prime numbers. Provide your answer as a single integer count. | 3 | big_math | [
"Mathematics -> Number Theory -> Prime Numbers"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
An isosceles triangle has two sides of lengths $3$ and $7$. Determine the perimeter of the triangle, ensuring that your solution satisfies the triangle inequality theorem. Express your answer as a whole number. | 17 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.28125 | 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.