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 |
|---|---|---|---|---|---|
The mean score of the students who took a mathematics test was 6. Exactly 60% of the students passed the test with a mean score of 8. Calculate the mean score of the students who failed the test. | 3 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Given sets $M=\{x|(x-1)(x-4)=0\}$ and $N=\{x|(x+1)(x-3)<0\}$, calculate the intersection $M \cap N$. Express your answer using proper set notation. | \{1\} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
If 7 numbers form a geometric sequence in order, and the common ratio is -2, then find the ratio of the sum of the even terms to the sum of the odd terms. Express your answer as a simplified fraction. | -\dfrac{42}{85} | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Jill starts a 5-hour trip with an odometer reading of 13831 km. If she never drives faster than 80 km/h and her final odometer reading is a palindrome, calculate her greatest possible average speed. Express your answer in km/h, rounded to the nearest whole number. | 62 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
What is the maximum number of rectangular wooden blocks with dimensions 20 cm x 30 cm x 40 cm that could fit into a rectangular box with inner dimensions 40 cm x 60 cm x 80 cm? Express your answer as a whole number. | 8 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Let \( f(x)=\log _{2}\left(2^{x}-1\right) \). Solve the equation \( f(2 x)=f^{-1}(x) \) for \( x \). Express your answer as a single numerical value. | 1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Logarithmic Functions"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Use Horner's method to find the value of V_2 for the polynomial function f(x) = 2x^5 - 3x^3 + 2x^2 - x + 5 when x = 2. Express your answer as a single integer. | 5 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given a geometric sequence $\{a_n\}$ with $a_n > 0$ and $q > 1$, where $a_3 + a_5 = 20$ and $a_2 \cdot a_6 = 64$, find the sum of the first six terms of the sequence, $S_6$. Express your answer as a single integer. | 63 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors AB = (3, -1) and n = (2, 1), and n dot AC = 7, calculate the value of n dot BC. Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Find the area of a triangle with integer side lengths and a perimeter of 8. Express your answer in simplest radical form if necessary. | 2\sqrt{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Consider a positive arithmetic sequence {a_n} whose sum of the first n terms is denoted as S_n. If S_{2017} = 4034, then find the minimum value of 1/a_9 + 9/a_{2009}. Express your answer as a single number. | 4 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
A rectangular solid has side lengths of 1 meter, 20 centimeters, and 50 centimeters. Calculate the volume of the solid in cubic centimeters. Express your answer as a whole number. | 100000 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Volume"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. If $a$, $b$, and $c$ form a geometric sequence, and $c = 2a$, then find the value of $\cos B$. Express your answer as a fraction or decimal. | \dfrac{3}{4} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the function f(x) is monotonic on R and satisfies f[f(x) - 3^x] = 4 for any x in R, find the value of f(2). Express your answer as a single number. | 10 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets $M = \{0, 2, 4, 8\}$ and $N = \{x | x = 2a, a \in M\}$, find the intersection $M \cap N$. Express your answer as a set, using curly brackets {} and separating elements with commas. | \{0, 4, 8\} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
For \( x \in \mathbb{R} \), the function satisfies \( f(x+2) + f(x-2) = f(x) \). It is a periodic function. Find one period of the function. Express your answer as a single integer. | 12 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $f′(x)=a(x-1)(x-a)$ is the derivative of the function $f(x)$, if $f(x)$ attains a maximum value at $x=a$, find the range of the real number $a$. Express your answer as an interval or range using proper mathematical notation. | (0,1) | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Given the parabola $y^2 = 4x$ with focus F at (1,0), and points A, B, and C on the parabola such that $\vec{FA} + \vec{FB} + \vec{FC} = \vec{0}$, calculate the sum of the distances $|\vec{FA}| + |\vec{FB}| + |\vec{FC}|$. Express your answer as a single numerical value. | 6 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of solutions (x, y) to the equation 3x + y = 100, where x and y are positive integers. Express your answer as a whole number. | 33 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $F_1$ and $F_2$ are the left and right foci of the hyperbola $C: x^2 - y^2 = 1$, and point P lies on the hyperbola with $\angle F_1PF_2 = 90°$, find the product $|PF_1| \cdot |PF_2|$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Geometry -> Non-Euclidean Geometry -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
A sample of size 20 has data grouped into intervals with the following frequencies: $(10,20]$, 2; $(20,30]$, 3; $(30,40]$, 4; $(40,50]$, 5; $(50,60]$, 4; $(60,70]$, 2. Calculate the frequency ratio of the sample in the interval $(10,50]$. Express your answer as a decimal value rounded to one decimal place. | 0.7 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Frequency Distribution"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Suppose hops, skips and jumps are specific units of length. If $b$ hops equals $c$ skips, $d$ jumps equals $e$ hops, and $f$ jumps equals $g$ meters, then express the number of skips equivalent to one meter in terms of $b$, $c$, $d$, $e$, $f$, and $g$. Provide your answer as a fraction or ratio. | \frac{cef}{bdg} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x)=\frac{2{(x+2)}^{2}+\ln (\sqrt{1+9{{x}^{2}}}-3x)\cos x}{{{x}^{2}}+4}$, and $f(2017)=2016$, find the exact value of $f(-2017)$. Express your answer as a single integer. | -2012 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
A train is moving on a straight track and encounters an emergency situation. The train then emergency brakes and comes to a stop. The velocity of the train during this braking process is given by $v(t)=5-t+\frac{55}{1+t}$ (where $t$ is in seconds and $v$ is in meters per second). Determine the distance the train contin... | 55\ln 11 | big_math | [
"Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Let the random variable ξ follow a normal distribution N(2,σ^2). Given that P(ξ > 4) = 0.1, calculate the probability P(ξ < 0). Express your answer as a decimal value between 0 and 1, rounded to 1 decimal place if necessary. | 0.1 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\sin\left(\theta - \frac{\pi}{4}\right) = \frac{1}{5}$, find the value of $\cos\left(\theta + \frac{\pi}{4}\right)$. Express your answer as a single value, possibly a fraction or decimal. | -\frac{1}{5} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given the polynomial x^4 - bx - 3 = 0 with roots a, b, c, and d, find the polynomial whose roots are -(1/a), -(1/b), -(1/c), and -(1/d). Express your answer as a polynomial equation in the form Ax^4 + Bx^3 + C = 0. | 3x^4 - bx^3 - 1 = 0 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given the hyperbola $C$: $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ $(a>0, b>0)$ with a focal length of $2\sqrt{5}$, and the point $P(2,1)$ lies on the asymptote of $C$, find the equation of $C$. Express your answer in the standard form of a hyperbola equation. | \frac{x^2}{4}-y^2=1 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Find the value of m that makes the fractional equation $\dfrac{3x}{x+1}=\dfrac{m}{x+1}+2$ have no solution. Express your answer as a single integer. | -3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Three equally spaced parallel lines intersect with a circle, resulting in three chords of lengths 38, 38, 34 units. Calculate the distance between two adjacent parallel chords. Express your answer in units. | 6 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
The function f(x) defined on ℝ is symmetric about the line x = 2, and f(x) satisfies: for any x1, x2 ∈ (-∞, 2] (x1 ≠ x2), it holds that (f(x1) - f(x2)) / (x1 - x2) < 0, and f(4) = 0. Find the solution set of the inequality f(x) / x < 0 with respect to x. Express your answer as an interval or a union of intervals in int... | (-∞, 0) ∪ (0, 4) | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Precalculus -> Functions"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given that the arithmetic mean of the real numbers $m$ and $2n$ is $4$, and the arithmetic mean of the real numbers $2m$ and $n$ is $5$, find the arithmetic mean of $m$ and $n$. Express your answer as a single numerical value. | 3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Let $U=\{1,2,3,4,5\}$, $A=\{1,2,3\}$, $B=\{2,3,4\}$. Evaluate the intersection of $A$ and the complement of $B$ with respect to $U$, denoted as $A \cap \complement_U B$. Express your answer as a set. | \{1\} | big_math | [
"Mathematics -> Discrete Mathematics -> Set Theory -> Other"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
In a geometric sequence with positive terms, let S_n be the sum of its first n terms. Given that S_3 = 3 and S_9 = 39, find the value of S_6. Express your answer as a single numerical value. | 12 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Find the increasing interval of the function y = x - e^x. Express your answer as an interval, using standard interval notation (e.g., (-∞, 0)). | (-∞, 0) | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given that \( \alpha \in \left(0, \frac{\pi}{2}\right) \) and \( \sin^2 \alpha + \cos \left( \frac{\pi}{2} + 2\alpha \right) = \frac{3}{10} \), find the exact value of \( \tan \alpha \). | 3 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
In square ABCD, M and N are the midpoints of BC and CD respectively. Find sin(theta), where theta is the angle between AM and AN. Express your answer as a numerical value. | \frac{3}{5} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Let the function $y=f(x)$ be defined on $\mathbb{R}$ and be an even function, and let $f(x)$ be increasing on $(-\infty, 0)$. If $x_1 < 0 < x_2$, and $x_1 + x_2 > 0$, compare the values of $f(-x_1)$ and $f(x_2)$. Provide your answer as a direct comparison of the function values, using the format $\boxed{f(-x_1) > f(x_2... | f(-x_1) > f(x_2) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
The function f(x) is an odd function defined on the domain of real numbers. When x is greater than or equal to 0, f(x) = x^2 + 2x. Calculate the value of f(-2) using the definition of the function and the odd function property. Express your answer as a single integer. | -8 | big_math | [
"Mathematics -> Algebra -> Functions"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given that \(S_{n}\) is the sum of the first \(n\) terms of an arithmetic sequence \(\{a_{n}\}\), if \(a_{9}=4\) and \(S_{15}=30\), calculate the value of \(a_{15}\). Express your answer as a single numerical value. | 16 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Convert the binary number 111.11 to decimal. Express your answer as a decimal number. | 7.75 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the random variable ζ follows a normal distribution N(4, 6^2), and P(ζ ≤ 5) = 0.89, determine P(ζ ≤ 3). Express your answer as a decimal value between 0 and 1, rounded to 2 decimal places if necessary. | 0.11 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Given a sequence $\{a_n\}$, satisfying $a_{n+1}=\frac{1}{1-a_n}$, if $a_1=\frac{1}{2}$, calculate the value of $a_{2009}$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Giselle had run 50 metres, which is 25% of the total race distance. Find the total distance of the race in metres. Express your answer as a whole number. | 200 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function y = 3^|x| - 1 with a domain of [-1, 2], find the range of the function. Express your answer as an interval, including the endpoints. | [0, 8] | big_math | [
"Mathematics -> Precalculus -> Functions"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Given ${(2x+\frac{a}{x})}^{5}$, the coefficient of $x$ in its expansion is $80$. Find the value of $a$. Express your answer as a numerical value, including both positive and negative possibilities. | \pm 1 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
The product of three consecutive odd numbers is 9177. Find the sum of these numbers. Express your answer as a whole number. | 63 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Ari and Bri are riding bicycles on a 30 km trip. Ari rides at an average speed of 20 km/h, and Bri rides at an average speed of 15 km/h. If they begin at the same time, how many minutes after Ari finishes the trip will Bri finish? Express your answer in minutes as a whole number. | 30 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Xiao Zhang practices calligraphy. The number of characters he writes each day is twice the total number of characters he has written in all previous days. If by the end of the 5th day, Xiao Zhang has completed one-third of the total task, then on which day should he finish the predetermined task? Provide your answer as... | 6 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $i$ is the imaginary unit, calculate the modulus of the complex number $z= \frac {1-i}{1+i}$. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
In a geometric sequence $\{a_n\}$ where all terms are positive, $a_1=3$, and $a_1+a_2+a_3=21$, find the value of $a_4+a_5+a_6$. Express your answer as a single integer. | 168 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
The integers from 1 to \( n \), inclusive, are equally spaced in order around a circle. The diameter through the position of the integer 7 also goes through the position of 23. Find the value of \( n \). Express your answer as a whole number. | 32 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the sum of the first $n$ terms of a geometric sequence $\{a_n\}$ is $S_n$, and $q=3$, calculate the value of $$\frac {S_{4}}{a_{4}}.$$ Express your answer as a simplified fraction. | \frac{40}{27} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
The line $l$ is the tangent to the circle $x^2+y^2=4$ at the point $(-1, \sqrt {3})$, and the point $P$ is a moving point on the circle $x^2-4x+y^2+3=0$. Calculate the minimum distance from the point $P$ to the line $l$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given two vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ with an angle of $120^{\circ}$ between them, and $\overrightarrow{a}=(1,- \sqrt{3})$, $| \overrightarrow{b}|=1$, find the magnitude of the vector $\overrightarrow{a}+\overrightarrow{b}$. Express your answer as a single value, including the square root symb... | \sqrt{3} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Given unit vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|3\overrightarrow{a}-2\overrightarrow{b}|=\sqrt{7}$, calculate the magnitude of $3\overrightarrow{a}+\overrightarrow{b}$. Express your answer as a numerical value, including the square root symbol if necessary. | \sqrt{13} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
If $-x^2y^n$ and $3yx^2$ are like terms, then what is the value of $n$? Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $i$ is the imaginary unit, the complex number $(1 + 3i)(a - i)$ corresponds to a point in the fourth quadrant on the complex plane. Find the range of values for $a$ that satisfies this condition. Express your answer as an interval, including the endpoints if they are included in the solution. | \left(-3, \frac{1}{3}\right) | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
At a road intersection with a traffic light, the red light lasts for 30 seconds, the yellow light lasts for 5 seconds, and the green light lasts for 40 seconds. Calculate the probability of not seeing the red light when arriving at the intersection. Express your answer as a simplified fraction. | \frac{3}{5} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Given a random variable X that follows a binomial distribution B(4, 1/3), calculate the probability P(X=1). Express your answer as a fraction. | \frac{32}{81} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Find the domain of the function $f(x)=\ln (x-1)+ \frac{1}{ \sqrt {2-x}}$. Express your answer in interval notation. | (1, 2) | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
In a high school senior grade with 900 students numbered 1, 2, 3, …, 900, a sample of 45 students is selected using systematic sampling. Calculate the number of students in the selected sample whose numbers fall within the interval [481, 720]. | 12 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the even function f(x) is monotonically increasing in the interval [0,+\u221e), determine the range of x that satisfies the inequality f(2x-1) < f(1/3). Express your answer as an interval. | \left( \frac{1}{3}, \frac{2}{3}\right) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $||\overrightarrow{OA}||=||\overrightarrow{OB}||=2$, point $C$ is on line segment $AB$, and the minimum value of $||\overrightarrow{OC}||$ is $1$, find the minimum value of $||\overrightarrow{OA}-t\overrightarrow{OB}||$ for all real numbers $t$. Express your answer as a single number. | \sqrt{3} | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Geometry -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
A square \(ABCD\) is bisected by its diagonals. The vertices of the square are colored either red or yellow, and any triangle formed by vertices of the same color is called a homogeneous triangle. Determine the number of distinct ways to color the vertices such that at least one homogeneous triangle is formed. | 10 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence ${a_n}$, where $a_1=1$ and $a_n=\frac{a_{n-1}}{2a_{n-1}+1}$, find the value of $a_{12}$. Express your answer as a fraction in simplest form. | \frac{1}{23} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
There are 81 cars in the CMC parking lot, which are all Acuras, Beetles, or Camrys. There are half as many Acuras as Beetles. The number of Camrys is 80% of the number of Acuras and Beetles together. Calculate the number of Beetles in the parking lot. Express your answer as a whole number. | 30 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of intersection points between the curve \( z^4 + z = 1 \) and the circle \( |z| = 1 \) in the complex plane. Express your answer as a single integer. | 0 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence ${a_n}$ with the sum of its first $n$ terms denoted as $S_n$, where $S_{m-1}=16$, $S_{m}=25$, and $S_{m+2}=49$ ($m\geq 2$ and $m\in\mathbb{N}$), determine the value of $m$. Express your answer as a whole number. | 5 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
There are 20 students participating in an after-school program offering classes in yoga, bridge, and painting. Each student must take at least one of these three classes, but may take two or all three. Given that 10 students take yoga, 13 take bridge, 9 take painting, and 9 take at least two classes, calculate the numb... | 3 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Let $f(x) = a\sin(\pi x + \alpha) + b\cos(\pi x + \beta) + 4$, where $a$, $b$, $\alpha$, and $\beta$ are non-zero real numbers. If $f(1988) = 3$, then find the value of $f(2013)$. Express your answer as a single number. | 5 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Let the sets $A=\{ x|x^{2}-4x+3 < 0\}$ and $B=\{ x|2x-3 > 0\}$. Calculate the intersection $A \cap B$ and express your answer in interval notation, using parentheses to denote open intervals. | \left( \dfrac{3}{2},3 \right) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Given $\triangle ABC \cong \triangle A'B'C'$, and the perimeter of $\triangle ABC$ is 20, with $AB=8$, $BC=5$, calculate the length of $A'C'$. Express your answer as a whole number. | 7 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Xiaoming looks at the clock and sees the time is 3:30. The hour hand points midway between the numbers 3 and 4, and the minute hand points at 6. Calculate the angle formed by the hour and minute hands. Express your answer in degrees as a whole number. | 75 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Five children (Tian Tian, Shi Tou, Kimi, Cindy, Angela) need to switch fathers, with each child able to choose any one of the other four fathers except their own. Find the total number of different combinations of choices for the five children. Express your answer as a single integer. | 44 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
A cone has a base radius that is half the length of its slant height. The value of its lateral surface area is $\frac{1}{2}$ of its volume. Determine the base radius of the cone. Express your answer as a numerical value, including the square root symbol if necessary. | 4\sqrt{3} | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Between the years 1000 and 2000 (inclusive), how many years satisfy the following two properties:
(a) The year is a palindrome number, meaning it reads the same backward as forward.
(b) The year can be factored into the product of a two-digit palindrome prime and a three-digit palindrome prime.
Express your answer as a... | 4 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the polynomial p = x^4 + 6x^3 + 11x^2 + 3x + 31, find the number of integer values of x for which p is a perfect square. Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
If $z=\frac{2+i}{i}$, find the coordinates of the point corresponding to $z$ in the complex plane. Express your answer as an ordered pair (x, y). | (1, -2) | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence {a_n}, the sum of the first n terms is S_n, and it is known that 2S_3 - 3S_2 = 15. Find the common difference of the sequence. Express your answer as a single numerical value. | 5 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $(ax^2+bx-5)(ax^2+bx+25)+c=(ax^2+bx+10)^2$, find the value of $c$ that makes the equation true. Express your answer as a single integer. | 225 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Given that set A consists of three elements: 0, m, and m^2-3m+2, and 2 is an element of set A, find the value of the real number m that satisfies the uniqueness condition of elements in set A. | 3 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.78125 | 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 -> Statistics -> Probability"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the sum of 0.8 and 0.02. Express your answer as a decimal value. | 0.82 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Decimals"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
A sample data of capacity 20 is grouped, and the group intervals and frequencies are as follows: [10,20), 2; [20,30), 3; [30,40), 4; [40,50), 5; [50,60), 4; [60,70], 2. Calculate the relative frequency of the data in the interval [10,50). Express your answer as a decimal value. | 0.7 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex equation $$\frac {1+i}{3i+z} = i$$ (where $i$ is the imaginary unit), calculate the imaginary part of the complex number $z$. Express your answer as a single numerical value. | -4 | big_math | [
"Mathematics -> Algebra -> Complex Numbers -> Other"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
A school track team has 28 male athletes and 21 female athletes. Using stratified sampling, 7 athletes are selected from the entire team to form a training squad. After a period of training, 3 athletes from the training squad are chosen to represent the school in a competition. Find the number of ways to select these 3... | 30 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Combinations"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
At a certain primary school in my city, there are two 3rd-grade classes, Class A and Class B. Class A has 30 boys and 20 girls, while Class B has 25 boys and 25 girls. Now, each class needs to randomly select 20% of the students from both boys and girls for a survey. Calculate the total number of boys selected from bot... | 11 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
In a high school senior class, there are 4 boys and 3 girls in the extracurricular speech group. They need to select 3 boys and 2 girls, who will then give speeches one by one in the class. Calculate the number of ways for the 2 girls not to give speeches consecutively. Express your answer as a whole number. | 864 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence {a_n}, the sum of the first n terms is denoted as S_n. If a_2 and a_4 are the two real roots of the equation x^2-x-1=0, then find the value of S_5. Express your answer as a numerical value in the form \boxed{ }. | \frac{5}{2} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
A stationery store sold two scientific calculators for 120 yuan each. One was sold at a 20% profit, and the other at a 20% loss. Calculate the overall profit or loss incurred by the stationery store. Express your answer in yuan, clearly indicating whether it is a profit or loss. | -10 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
A summer camp organizes 5 high school students to visit five universities including Peking University and Tsinghua University, requiring each student to choose one university to visit. Find the total number of different ways in which exactly 2 students choose Peking University. | 640 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Combinations"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Find the value of $\lim_{\Delta x\to 0} \frac{f(x_0) - f(x_0 - 2\Delta x)}{\Delta x}$, given that the equation of the tangent line to the function $y=f(x)$ at the point $(x_0, y_0)$ is $y=2x+1$. Express your answer as a single number. | 4 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
In a box containing 3 white balls and 2 black balls (they are identical except for color), 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"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Given the expressions $a=0.5^{2}$, $b=2^{0.5}$, and $c=\log _{0.5}2$, determine the correct order of the values of $a$, $b$, and $c$ from smallest to largest. Express your answer using less than (<) symbols to denote the order (e.g., a < b < c). | c < a < b | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation \( \sin^{x} \alpha + \cos^{x} \alpha = 1 \) where \( 0 < \alpha < \frac{\pi}{2} \), determine the exact number of solutions for \( x \). Provide your answer as a whole number. | 1 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
A five-legged Martian has a drawer full of socks, each of which is red, white or blue, and there are at least five socks of each color. The Martian pulls out one sock at a time without looking. What is the minimum number of socks the Martian must remove from the drawer to be certain there will be 5 socks of the same co... | 13 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $x > 0$, $y > 0$, and $2 \lg (x - 2y) = \lg x + \lg y$, find the ratio $x:y$. Express your answer as a numerical ratio (x:y). | 4:1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Given \(\cos \alpha = -\frac{3}{5}\) and \(\alpha \in \left(\frac{\pi}{2}, \pi\right)\), \(\sin \beta = -\frac{12}{13}\) and \(\beta\) is an angle in the third quadrant, find \(\cos (\beta - \alpha)\). Express your answer as a fraction. | -\frac{33}{65} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence {a_n} with the sum of the first n terms denoted as S_n, and it is known that S_{2n} - S_{2n-1} + a_2 = 424, where n is a positive integer, calculate the value of a_{n+1}. | 212 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.03125 | 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.