instruction_seed stringlengths 40 1.39k | answer stringlengths 1 95 | source stringclasses 1
value | domain listlengths 1 3 | llama8b_solve_rate float64 0 0.98 | _source stringclasses 1
value |
|---|---|---|---|---|---|
Given a function y = f(x) whose graph passes through the point (0,1), find the point that the graph of the inverse function of f(4-x) passes through. Express your answer as an ordered pair (x,y). | (1,4) | big_math | [
"Mathematics -> Algebra -> Functions"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Khelen wants to color each quadrilateral in a tessellation of the plane such that no two quadrilaterals that meet (even at a point) have the same color. What is the smallest number of colors he needs to achieve this? Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Discrete Mathematics -> Graph Theory"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Given the inverse proportion function $y=\frac{k}{x}(k\neq 0)$, and the graph passes through the point $P(-3,2)$, find the value of $k$. Express your answer as a single number. | -6 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Find the vertex coordinates of the parabola given by the equation $y=2(x-3)^2+1$. Express your answer as an ordered pair (x, y). | (3,1) | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Quadratic Functions"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\alpha$ is an angle in the third quadrant and $\sin\alpha = -\frac{24}{25}$, calculate the value of $\tan\frac{\alpha}{2}$. Express your answer as a fraction, including a negative sign if applicable. | -\frac{4}{3} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the terminal side of angle α passes through the point (3,4), calculate the value of (sin α + cos α) / (sin α - cos α). Express your answer as a single number. | 7 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence {a_n} with a common difference d = 2 and a_4 = 3, find the value of a_2 + a_8. Express your answer as a single integer. | 10 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
In a 6x6 grid, where each row contains exactly one unshaded square, how many 1x1 squares are shaded? Express your answer as a whole number. | 30 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $f(x)$ is an odd function defined on $\mathbb{R}$, and when $x \in (0, +\infty)$, $f(x) = \log_2{x}$, find the value of $f(-8)$. Express your answer as a single number. | -3 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given two natural numbers p and q that do not end in zero, and their product p * q is a power of 10, if p > q, what is the last digit that cannot be the result of p - q? Express your answer as a single digit. | 5 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Roger uses his allowance each week to buy a movie ticket and a bottle of soda. The price of a movie ticket is 20% of the difference between his allowance and the price of the soda. The price of the soda is 5% of the difference between his allowance and the price of the movie ticket. What percentage of his allowance is ... | 23 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
The annual salaries of 25 employees in a certain company are as shown in the table below. Find the mode of the annual salaries of all employees in the company. Express your answer in dollars.
| Annual Salary (in $10,000) | $30$ | $14$ | $9$ | $6$ | $4$ | $3.5$ | $3$ |
|-----------------------------|------|------|-----... | 35,000 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $z$ is a complex number and $\overline{z}$ is the conjugate of $z$, where $\overline{z}=|z|-1+5i$, find the imaginary part of $z$. Express your answer as a single number. | -5 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the number of positive integers less than 1000 that cannot be divided evenly by 5 or 7. Express your answer as a single integer. | 686 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation (x-2)(x+1) = x^2 + nx - 2, find the value of n. Express your answer as a single integer. | -1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $x > 1$, find the minimum value of $y=x+ \frac {1}{x-1}$. Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Let ξ follow a binomial distribution B(n, P), with Eξ = 15 and Dξ = 11.25. Find the value of n. Express your answer as a whole number. | 60 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
If the complex number \( z \) satisfies \( |z|<1 \) and \( \left|\bar{z}+\frac{1}{z}\right|=\frac{5}{2} \), then find the magnitude \( |z| \). Express your answer as a fraction in the form \( \boxed{\frac{a}{b}} \), where \( a \) and \( b \) are integers. | \frac{1}{2} | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets $A = \{1, 2, 3\}$ and $B = \{1, 3, 4\}$, find the number of subsets of $A \cap B$. Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
In triangle $\triangle ABC$, given $a=1$, $b=2$, $C=\frac{{2π}}{3}$, calculate the length of side $c$ using the cosine rule. Express your answer as a numerical value, including the square root symbol if necessary. | \sqrt{7} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given a hyperbola $C:\\frac{x^2}{a^2}-\\frac{y^2}{b^2}=1$ with an asymptote equation $2x-y=0$, calculate the eccentricity of the hyperbola $C$. Express your answer as a square root if it is not a whole number. | \sqrt{5} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Alberto, Bernardo, and Carlos participated in a race, each running at a constant speed throughout the entire distance. When Alberto crossed the finish line, Bernardo and Carlos were 36 meters and 46 meters behind him, respectively. When Bernardo crossed the finish line, Carlos was 16 meters behind him. Calculate the le... | 96 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Let $M$ be a point inside $\triangle ABC$, and the area of $\triangle ABC$ is $2$. Define $f(M)=(m,n,p)$, where $m$, $n$, $p$ are the areas of $\triangle MBC$, $\triangle MCA$, $\triangle MAB$ respectively. If a moving point $P$ inside $\triangle ABC$ satisfies $f(P)=(1,x,y)$, find the minimum value of $\dfrac{1}{x}+ \... | 9 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Find the solution set for the inequality $\frac{x^{2}{+}2x{+}2}{x{+}2}{ > }1$. Express your answer as an interval or union of intervals. | (-2, -1) \cup (0, \infty) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
In an increasing geometric sequence $\{a_n\}$, the sum of the first 3 terms is 13 and the product of the first 3 terms is 27. Calculate the sum of the first 5 terms. Express your answer as a single numerical value. | 121 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Two coins of uniform quality are tossed simultaneously. Given that one coin is already known to be facing up, calculate the probability that both coins are facing up. Express your answer as a simplified fraction. | \frac{1}{3} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given a convex quadrilateral ABCD where the sides AD and BC are extended to meet at point E, and given the definitions of s and s' as in the original problem, what is the value of r = s/s'? Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
Find the line of symmetry for the function y = cos(x) - sin(x). Express your answer as an equation of a line in the form x = a, using radians for any angle measurements. | x = -\frac{\pi}{4} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
In the three-dimensional Cartesian coordinate system, find the coordinates of the point symmetric to P(3,-2,1) with respect to the coordinate plane xOz. Express your answer as a set of coordinates (x, y, z). | (3,2,1) | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number Z = (m+2i)/(1+i), where Z is an imaginary number, find the value of the real number m. Express your answer as a single integer. | -2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the sum of the natural numbers \( a \), \( b \), and \( c \) is \( S \), and the following conditions:
$$
a + b = 1014, \quad c - b = 497, \quad a > b,
$$
calculate the maximum value of \( S = a + b + c \). Express your answer as a single integer. | 2017 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation of the parabola y = 4x^2, find the equation of its directrix. Express your answer as an equation of a line in the form y =.... | y = -\dfrac{1}{16} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
A sequence of monomials is given by $x^{3}$, $-x^{5}$, $x^{7}$, $-x^{9}$, $x^{11}$, $\ldots$. Based on the pattern observed in the sequence, express the nth monomial as a function of n. | (-1)^{n-1}x^{2n+1} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $x=2$ is a root of the equation $x^{2}-5x-m=0$, find the other root of the equation. Express your answer as a single numerical value. | 3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = e^x, find its derivative f'(x). Express your answer as a function of x. | f'(x) = e^x | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Select 5 different letters from the word "equation" to arrange in a row, ensuring that "qu" are together and in the same order. Calculate the total number of different arrangements. Express your answer as a single integer. | 480 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given sets $M=\{x|0<x<3\}$ and $N=\{x|x>2\}$, calculate the intersection $M\cap (\complement_{\mathbb{R}}N)$. Express your answer in interval notation. | (0,2] | big_math | [
"Mathematics -> Set Theory -> Set Operations"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Find the constant term in the expansion of $(|x|+ \\frac {1}{|x|}-2)^{3}$ by first converting it into $\\frac {(|x|-1)^{6}}{|x|^{3}}$ and then identifying the term in the numerator $(|x|-1)^{6}$ that contains $|x|^{3}$ using the binomial expansion formula. Express your answer as a single integer. | -20 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
A sports stadium has four gates on the south side and three gates on the north side. If a person goes there for morning exercises, how many different combinations of entry and exit gates are possible? Provide your answer as a whole number. | 49 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Combinations"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the complex numbers $z_1$ and $z_2$ are symmetric about the imaginary axis in the complex plane, and $z_1 = 3 - i$, find the value of $\frac{z_1}{z_2}$. Express your answer as a complex number in the form $a + bi$. | -\frac{4}{5} + \frac{3}{5}i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of \((8 \times 4) + 3\). Express your answer as a single integer. | 35 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
What is the remainder when $29 \cdot 169^{1990}$ is divided by 11? Express your answer as a single integer. | 7 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
The real number sequence 1, a, 16 is a geometric sequence. Find the value of a. Express your answer as a mathematical expression, including all possible values. | \pm 4 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Find the sum of the squares of all real numbers that satisfy the equation \( x^{256} - 256^{32} = 0 \). Express your answer as a single integer. | 8 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Polynomial Operations"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Suppose the center of a moving circle lies on the parabola y = (1/12)x^2, and it is tangent to the line y + 3 = 0. Calculate the fixed point through which the circle always passes. Express your answer as a coordinate (x, y). | (0,3) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of x values that satisfy the equation (x^2-4)(x^2-1)=(x^2+3x+2)(x^2-8x+7). Express your answer as a whole number. | 3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Given two conditions p: $x>2$, $y>3$, and conditions q: $x+y>5$, $xy>6$, determine the nature of the relationship between condition p and condition q. Is condition p sufficient, necessary, both, or neither for condition q? Provide your answer in a clear statement regarding the sufficiency and necessity of condition p f... | Condition p is sufficient but not necessary for condition q. | big_math | [
"Mathematics -> Discrete Mathematics -> Logic"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
The sophomore class plans to select 3 students from 3 male students and 4 female students to attend a conference. Calculate the number of ways to choose 3 students such that both genders are represented. Express your answer as a single integer. | 30 | big_math | [
"Applied Mathematics -> Probability -> Combinations"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
A middle school radio station randomly selects two festivals from the five traditional Chinese festivals: Spring Festival, Lantern Festival, Qingming Festival, Dragon Boat Festival, and Mid-Autumn Festival. What is the probability that at least one of the Spring Festival and Dragon Boat Festival is selected? Express yo... | 0.7 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Combinations"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Two lines with slopes \(\frac{1}{4}\) and \(\frac{5}{4}\) intersect at \((1,1)\). What is the area of the triangle formed by these two lines and the vertical line \(x=5\)? Express your answer as a single number. | 8 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Evaluate the expression $\log_{2}3 \cdot \log_{3}4$. Provide your answer as a numerical value. | 2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Logarithmic Functions"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
In the algebraic expressions $-1$, $-\frac{2}{3}a^{2}$, $\frac{1}{6}x^{2}y$, $3a+b$, $0$, $\frac{{x-1}}{2}$, how many monomials are there? Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z= \frac {5-i}{1-i}$, find the imaginary part of $z$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Given the ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with left and right foci $F_1$ and $F_2$, respectively. The line $y = \frac{\sqrt{5}b}{3}$ intersects the ellipse $C$ at points $A$ and $B$. If the quadrilateral $ABF_2F_1$ is a rectangle, calculate the eccentricity of the ellipse $C$. Express y... | \frac{2}{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(n)=1+ \frac {1}{2}+ \frac {1}{3}+…+ \frac {1}{3n-1}$ for $n \in \mathbb{N}^{*}$, calculate the value of $f(n+1)-f(n)$. Express your answer as a mathematical expression in terms of n. | \frac {1}{3n}+ \frac {1}{3n+1}+ \frac {1}{3n+2} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Let $a > 0$. If the inequality $|x-a| + |1-x| \geq 1$ holds for any $x \in \mathbb{R}$, find the minimum value of $a$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Let \( n \) be the largest integer for which \( 14n \) has exactly 100 digits. Counting from right to left, what is the 68th digit of \( n \)? Express your answer as a single digit. | 1 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, \(ABCD\) is a square with a side length of 10. If \(AY = CX = 8\), calculate the area of the shaded region. Express your answer as a single numerical value. | 20 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Solve the equation (2^x - 4)^3 + (4^x - 2)^3 = (4^x + 2^x - 6)^3 for x, where x is a real number. Find the sum of all real solutions. Express your answer as a single number. | \frac{7}{2} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Five people participate in a relay game, each randomly choosing between two options: left path and right path. If more than 2 people choose the same path, each of them gets 1 point; if fewer than 3 people choose the same path, each of them gets 0 points. Let $\xi$ denote the score of a player named Xiaoqiang. Calculate... | \frac{11}{16} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given that vectors $\overrightarrow{a} = (x+1, 2)$ and $\overrightarrow{b} = (1, -1)$ are parallel, find the magnitude of their sum, $|\overrightarrow{a} + \overrightarrow{b}|$. Express your answer as a numerical value, including the square root symbol if necessary. | \sqrt{2} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Find the maximum value of the function y = x + sqrt(1-x) + 3, given that x <= 1 for the square root to be defined. Express your answer as a single number. | \frac{17}{4} | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Given a sequence $\{a_n\}$ that satisfies $2a_n + a_{n+1} = 0$ ($n \in \mathbb{N}^*$) and $a_3 = -2$, find the value of $a_8$. Express your answer as a single numerical value. | 64 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Consider a hyperbola given by the equation x²/a² - y²/b² = 1 (a > 0, b > 0). Let F be its right focus. Draw a perpendicular line from F to one of the asymptotes, intersecting it at point A and the other asymptote at point B. If FB = 2FA, find the eccentricity of this hyperbola. Express your answer as a single numerical... | 2 | big_math | [
"Mathematics -> Geometry -> Differential Geometry -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given two sets M and N, where M contains m elements and N contains n elements, determine the number of possible mappings from M to N. Express your answer in the form n^m. | n^m | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Let $x \in \mathbb{R}$. If the function $f(x)$ is a monotonically increasing function, and for any real number $x$, it holds that $f[f(x) - e^x] = e + 1$ (where $e$ is the base of the natural logarithm), then calculate the value of $f(\ln 2)$. Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
A computer executes the following program:
1. Initial values: x = 3, S = 0.
2. x = x + 2.
3. S = S + x.
4. If S >= 10000, go to step 5; otherwise, go back to step 2.
5. Print x.
6. Stop.
What is the value of x when the program stops? | 201 | big_math | [
"Mathematics -> Discrete Mathematics -> Algorithms"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Point M is located in the second quadrant, above the x-axis, with a distance of 2 to the x-axis and a distance of 1 to the y-axis. Find the coordinates (x, y) of point M. Express your answer as a coordinate pair (x, y), where x and y are integers. | (-1, 2) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of integer solutions x that satisfy the equation (x^2 - x - 1)^(x+2) = 1. | 4 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
In an acute triangle ABC, let D be a point on [AC] and E be a point on [AB] such that ∠ADB = ∠AEC = 90°. If the perimeter of triangle AED is 9, the circumradius of triangle AED is 9/5, and the perimeter of triangle ABC is 15, calculate the length of BC. Express your answer as a single value. | \frac{24}{5} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Let non-zero complex numbers x and y satisfy the equation x^2 + xy + y^2 = 0. Determine the value of the algebraic expression (x/(x+y))^1990 + (y/(x+y))^1990. Express your answer as a single numerical value. | -1 | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression: \( 2^{3} \times 2^{2} \times 3^{3} \times 3^{2} \). Express your answer in the simplest exponential form possible. | 6^5 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
How many integers greater than 10 and less than 100, written in base-10 notation, are increased by 9 when their digits are reversed? Express your answer as a single integer. | 8 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Arrange the five words "仁、义、礼、智、信" in a row such that "礼、义" are adjacent and "智、信" are adjacent. Calculate the total number of arrangements satisfying these conditions. Express your answer as a single integer. | 24 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Define the sets M and N as follows:
M = {u | u = 12m + 8n + 4l, m, n, l in Z}
N = {u | u = 20p + 16q + 12r, p, q, r in Z}
Describe the relationship between sets M and N using standard set notation and terminology. | M = N | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Six colored squares (R = red, B = blue, O = orange, Y = yellow, G = green, and W = white) are hinged together as shown, then folded to form a cube. What is the color of the face opposite the white face? Express your answer as a single letter representing the color. | B | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a \in \mathbb{R}$, the complex number $z = \frac{(a - i)(1 + i)}{i}$, if $\overline{z} = z$, calculate the value of $a$. Express your answer as an integer. | -1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Given the hyperbola $\frac{x^{2}}{9} - \frac{y^{2}}{m} = 1$ with an eccentricity of $\frac{\sqrt{14}}{3}$, determine the distance from the focus $F$ to the asymptote. Express your answer as a single value. | \sqrt{5} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function y=f(x) that satisfies both y=f(-x) and y=f(x+2), and f(1)=1, calculate the value of f(-1)+f(7). | 2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Let f(x) be an odd function and g(x) be an even function defined on the real numbers. Suppose that f'(x) * g(x) + f(x) * g'(x) > 0 for x < 0, and g(-3) = 0. Find the solution set for the inequality f(x) * g(x) < 0. Express your answer as an interval or union of intervals. | (-∞, -3) ∪ (0, 3) | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Calculus -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
In an isosceles triangle \( riangle ABC \), the length of the altitude to one of the equal sides is \( \sqrt{3} \) and the angle between this altitude and the base is \( 60^\circ \). Calculate the area of \( riangle ABC \). Express your answer as a numerical value. | \sqrt{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
A 0.1mm sheet of paper is folded in half four times. Calculate the thickness of the paper after these folds. Express your answer in millimeters. | 1.6 \text{ mm} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given the inequality -2x^2 + 5x - 2 > 0, evaluate the expression sqrt(4x^2 - 4x + 1) + 2|x - 2|. | 3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Chantelle had two candles, one of which was 32 cm longer than the other. She lit the longer one at 3 p.m. and lit the shorter one at 7 p.m. At 9 p.m., they were both the same length. The longer one was completely burned out at 10 p.m. and the shorter one was completely burned at midnight. The two candles burned at diff... | 52 \text{ cm} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Determine the range of values for $k$ that allow the equation $x^2 + y^2 + 4x + 2y + 4k + 1 = 0$ to represent a circle. Express your answer as an interval in standard interval notation. | (-\infty, 1) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x) = \begin{cases} f(x+2),x < 2 \\ (\frac{1}{3})^{x},x\geqslant 2 \end{cases}$, evaluate $f(-1+\log_{3}5)$. Express your answer as a simplified fraction. | \frac{1}{15} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given that \( a, c, d \) are integers, \( b \) is a positive integer, and the following conditions are satisfied:
\[ a + b = c \]
\[ b + c = d \]
\[ c + d = a \]
Calculate the maximum value of \( a + b + c + d \). Express your answer as a single integer. | -5 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given unit vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $\overrightarrow{a} \cdot \overrightarrow{b} = 0$. If the vector $\overrightarrow{c} = \overrightarrow{a} + \sqrt{3}\overrightarrow{b}$, calculate $\cos\langle \overrightarrow{b}, \overrightarrow{c} \rangle$. Express your answer as a fraction or d... | \frac{\sqrt{3}}{2} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Given complex numbers $z_{1}=1-2i$ and $z_{2}=1+bi$, if $\overline{{z_1}•{z_2}}=7-i$, calculate the real number $b$. Express your answer as a single real number. | 3 | big_math | [
"Mathematics -> Algebra -> Complex Numbers -> Other"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation 4^x = 8, find the value of x. Express your answer as a fraction or decimal. | \frac{3}{2} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given that \( Q(z) \) and \( R(z) \) are polynomials satisfying
\[ z^{2021} + 1 = \left( z^2 + z + 1 \right) Q(z) + R(z) \]
with the constraint \(\operatorname{deg} R(z) < 2\), determine the polynomial \( R(z) \). Express your answer as a polynomial in terms of \( z \). | -z | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
If the measure of an exterior angle of a regular polygon is 40°, then find the number of sides of the polygon. Express your answer as a whole number. | 9 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Polygons"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z = 1 - i$, calculate the value of the expression $\left| \frac {2}{z} + z^{2} \right|$. Express your answer as a numerical value. | \sqrt{2} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
A certain unit needs to participate in three meetings: A, B, and C. Meeting A requires 2 participants, while meetings B and C each require 1 participant. From a group of 10 people, 4 individuals are to be selected to attend these three meetings. Find the number of different arrangements possible. Express your answer as... | 2520 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Find the roots of the equation $(x^{2}-3x+2)(x)(x-4)=0$. Express your answer as a list of roots, separated by commas. | 0,1,2,4 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Polynomial Operations"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Given a polynomial equation \(z^{4} + a z^{3} + b z^{2} + c z + d = 0\) with real coefficients and all its roots lying on the unit circle in the complex plane centered at the origin, calculate the sum of the reciprocals of these roots. Express your answer as a single value in terms of the coefficients of the polynomial... | -a | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the result of the expression \( 2013 imes \frac{2015}{2014} + 2014 imes \frac{2016}{2015} + \frac{4029}{2014 imes 2015} \). Express your answer as a single number. | 4029 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
A company has 840 employees. Using systematic sampling, 42 people are selected for a questionnaire survey. The 840 people are randomly numbered from 1 to 840. Calculate the number of people among the 42 selected whose numbers fall within the interval [481, 720]. Express your answer as a whole number. | 12 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Find the range of the real number $\lambda$ such that the inequality $a^2+8b^2 \geq \lambda b(a+b)$ holds for any real numbers $a$ and $b$. Express your answer as a closed interval. | [-8,4] | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Find the sum of all prime numbers within 10. Express your answer as a single numerical value. | 17 | big_math | [
"Mathematics -> Number Theory -> Prime Numbers"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
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