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 |
|---|---|---|---|---|---|
A sports palace has 5 seats in the first row, 7 seats in the second row, 9 seats in the third row, and so on. The number of seats in each row increases by 2 from the previous row. Find the number of seats in the 15th row. Express your answer as a whole number. | 33 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Xiao Pang starts practicing calculations on February 6, 2014 (Thursday), and continues until February 17, 2014. However, he does not practice on Saturdays and Sundays. He solves 1 problem on the first day, 3 problems on the second day, 5 problems on the third day, and so on, increasing the number of problems by 2 each ... | 64 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Given the ellipse C defined by the parametric equations x=2cosθ and y=√3sinθ, draw line l through the right focus F of C, intersecting C at points M and N. If |MF|=m and |NF|=n, calculate the value of 1/m + 1/n. Express your answer as a simplified fraction. | 1 | big_math | [
"Mathematics -> Geometry -> Analytic Geometry -> Conic Sections"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = sin(x) + cos(x), find the equation of the tangent line to the function at the point (0, f(0)). Express your answer in the standard form of a linear equation (Ax + By + C = 0). | x - y + 1 = 0 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Find the surface area of a tetrahedron with all edge lengths equal to 1. Express your answer as a numerical value with a square root if necessary. | \sqrt{3} | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the function y = f(x) is an even function, and the function y = f(x - 2) is monotonically decreasing on the interval [0,2], determine the ordering of the function values f(-1), f(0), and f(2). Express your answer using the less than (<) symbol to separate the values. | f(0) < f(-1) < f(2) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Let the sum of the first n terms of an arithmetic sequence {an} be Sn. If a5 = 2, then find the value of 2S6 + S12. | 48 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the complex number $z$ satisfies the equation $1 + (1 + 2z)i = 0$ ($i$ is the imaginary unit), find the imaginary part of the complex number $z$. Express your answer as a single number. | \frac{1}{2} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given two numbers whose sum is 6 and the absolute value of whose difference is 8, form the quadratic equation that has these numbers as its roots. Express your answer in the standard form ax^2 + bx + c = 0. | x^2 - 6x - 7 = 0 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Quadratic Functions"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
In an arithmetic sequence $\{a_n\}$, if $a_1 + a_4 + a_7 = 39$ and $a_3 + a_6 + a_9 = 27$, find the sum of the first $9$ terms, $S_9$. Express your answer as a single number. | 99 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
A has three different books, and B borrows at least one book. Find the total number of different borrowing methods. Express your answer as a single numerical value. | 7 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $x^{3}\cdot x^{3}$ by applying the rule of exponents for multiplication. Express your answer as a simplified algebraic expression. | $x^{6}$ | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.046875 | 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"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
The song "Five hundred and twenty-five thousand six hundred minutes" from the musical Rent refers to a length of time. Convert this time to years, rounding to the nearest whole number if necessary. | 1 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = 2sin(2x + φ) where |φ| < π/2, the graph of y = f(x) is shifted to the left by π/8 units, resulting in the function y = g(x). If the graph of g(x) is symmetric about the y-axis, find the minimum value of g(x) + g(x/2). Express your answer as a single numerical value. | -\frac{9}{4} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given $\sqrt{99225}=315$ and $\sqrt{x}=3.15$, calculate the value of $x$. Express your answer as a decimal value, rounded to 4 decimal places if necessary. | 9.9225 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Positive integers are classified according to the following pattern: (1) (2, 3) (4, 5, 6).... Find the third number in the 10th bracket. Express your answer as a single integer. | 48 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Find the coefficient of $x^2$ in the expansion of $(1+2x)^5$. Express your answer as a numerical value. | 40 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
In the xy-plane, find the number of lines passing through the point (4,3) with a prime number x-intercept and a positive integer y-intercept. Express your answer as a whole number. | 2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
A particle is placed on the parabola y = x^2 - x - 6 at a point P whose y-coordinate is 6. It is allowed to roll along the parabola until it reaches the nearest point Q whose y-coordinate is -6. Calculate the horizontal distance traveled by the particle (the numerical value of the difference in the x-coordinates of P a... | 4 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other",
"Mathematics -> Algebra -> Functions -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
In a class of 21 students, at least two of any three students are friends. What is the largest possible value of k such that there exists a student with at least k friends? Express your answer as a whole number. | 10 | big_math | [
"Mathematics -> Discrete Mathematics -> Graph Theory"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
In four independent repeated experiments, the probability of event A occurring in each experiment is the same. If the probability of event A occurring at least once is 65/81, calculate the expected value of the number of times event A occurs. Express your answer as a fraction. | \frac{4}{3} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Other"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence ${\{a_n\}}$ that satisfies $a_n = \log_{n+1}(n+2) \ (n \in \mathbb{N}^*)$, define $k \ (k \in \mathbb{N}^*)$ that makes the product ${a_1} \cdot {a_2} \cdot {a_3} \cdot \ldots \cdot {a_k}$ a positive integer as a "desired number." Calculate the sum of all "desired numbers" within the interval $[1, 20... | 2026 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
If the function f(x) = x^3 + 2x^2 - 1, find the value of the derivative f'(-1). Express your answer as a single number. | -1 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given the vector $\overrightarrow{a} = (2, -1, 2)$, find the vector $\overrightarrow{x}$ that is collinear with $\overrightarrow{a}$ and satisfies $\overrightarrow{a} \cdot \overrightarrow{x} = -18$. Express your answer as a vector in the form $\boxed{(x, y, z)}$. | (-4, 2, -4) | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
In the complex plane, four complex numbers are located at the vertices of a square, three of which are \(1+2i\), \(-2+i\), and \(-1-2i\). Find the fourth complex number that completes the square. Express your answer in the form \(a+bi\). | 2-i | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Let A, B, and C be three distinct points on line l, and let O be a point not on line l. If $p\vec{OA} + q\vec{OB} + r\vec{OC} = \vec{0}$ ($p, q, r \in \mathbb{R}$), then find the value of p+q+r. Express your answer as a single integer. | 0 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Other"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Given the parabola $y^{2}=8x$ with a focus at point $F$, a line $l$ is drawn through point $F$ intersecting the parabola at points $A$ and $B$. If the distance from the midpoint $E$ of segment $AB$ to the $y$-axis is $3$, find the length of chord $AB$. Express your answer as a single number. | 10 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given an ellipse $C$ with its center at the origin and its right focus at $F(1,0)$, and its eccentricity equals $\frac{1}{2}$, find the equation of $C$ in standard form. | \frac{x^2}{4}+ \frac{y^2}{3}=1 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
The arithmetic sequence 2, 5, 8,... has a term equal to 20. Calculate the position of this term in the sequence. Express your answer as a whole number. | 7 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Find the coefficient of $x^2$ in the expansion of $(x+2)^6$. Express your answer as a single integer. | 240 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Binomial Theorem"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = ax^3 + bx - 4, and f(2) = 6, calculate the value of f(-2). Express your answer as a single integer. | -14 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
A building has 4 units, and two people, A and B, live in this building. Find the probability that A and B live in the same unit. Express your answer as a simplified fraction. | \frac{1}{4} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Other"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
If the sum of 7 consecutive even numbers is 1988, then what is the largest of these numbers? Express your answer as a whole number. | 290 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Construct a complex number where the real part is the imaginary part of $-\sqrt{2}+7i$, and the imaginary part is the real part of $\sqrt{7}i+5i^{2}$. Express your answer in the form $a+bi$. | 7-5i | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
The product of N consecutive four-digit positive integers is divisible by 2010^2. What is the least possible value of N? Express your answer as a single integer. | 5 | big_math | [
"Mathematics -> Number Theory -> Divisibility"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
A tetrahedron with four equilateral triangular faces has a sphere inscribed within it and a sphere circumscribed about it. For each of the four faces, there is a sphere tangent externally to the face at its center and to the circumscribed sphere. A point P is selected at random inside the circumscribed sphere. Calculat... | 0.1 | big_math | [
"Mathematics -> Applied Mathematics -> Probability"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a} = (x-1, 2)$ and $\overrightarrow{b} = (4, y)$, if $\overrightarrow{a} \perp \overrightarrow{b}$, calculate the minimum value of $9^{x} + 3^{y}$. Provide your answer as a single numerical value. | 6 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of solutions to the equation $|x+1|+|x+9|+|x+2|=1992$. Express your answer as a whole number. | 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Students A and B each randomly choose to join one of the three clubs: "Chess", "Literature", or "Photography". What is the probability that both students join the same club? Express your answer as a fraction in its lowest terms. | \dfrac{1}{3} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given a point M(-3,3) and a circle C: (x-1)^2 + y^2 = 25, find the equation of the tangent line to the circle passing through point M. Express your answer in the standard form Ax + By + C = 0. | 4x - 3y + 21 = 0 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Given the definitions of a, b, and c as follows: a = 1/3, b = sin(1/3), and c = 1/π, determine the exact relationship between a, b, and c using standard comparison operators (>, <, =). Express your answer in the format "a [operator] b [operator] c", where [operator] is one of the comparison operators. | a > b > c | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
In a school journalism club, reporters want to take a photo of 5 volunteering students and 2 teachers from the Youth League committee. The 2 teachers must stand next to each other but not at the ends. Calculate the total number of different arrangements for the group. Express your answer as a single integer. | 960 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Permutations"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
When Mr. Gu and Mr. Jian play chess, the probability of Mr. Jian winning is 40%, and the probability of Mr. Gu not winning is 60%. Calculate the probability of the game ending in a draw. Express your answer as a percentage. | 20\% | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Solve for $n$ in the equation $\frac{1}{2n} + \frac{1}{4n} = \frac{3}{12}$. Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Let $a=\sin \frac{3\pi}{5}$, $b=\cos \frac{2\pi}{5}$, $c=\tan \frac{2\pi}{5}$. Express the ordering of a, b, and c from smallest to largest using less-than (<) and greater-than (>) symbols. | b < a < c | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given four non-collinear points O, A, B, and C on a plane, if $\overrightarrow{OA} - 4\overrightarrow{OB} + 3\overrightarrow{OC} = \overrightarrow{0}$, calculate the value of $\frac{|\overrightarrow{AB}|}{|\overrightarrow{BC}|}$. Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Given the parabola y = 4x^2, find the ordinate (y-coordinate) of a point M on the parabola that is at a distance of 1 from the focus. Express your answer as a fraction. | 15/16 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Find the distance from the point (1,1) to the line x+y-1=0. Express your answer as a fraction or decimal, including the square root symbol if necessary. | \frac{\sqrt{2}}{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given that {a<sub>n</sub>} is an arithmetic sequence, if a<sub>2</sub> = 2a<sub>3</sub> + 1 and a<sub>4</sub> = 2a<sub>3</sub> + 7, find the value of a<sub>5</sub>. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given a function \( f(x) \) that satisfies the condition \( f(x+2) = \frac{1}{f(x)} \) for any \( x \), and given that \( f(1) = -5 \), calculate the value of \( f[f(5)] \). Express your answer as a fraction, including a negative sign if applicable. | -\frac{1}{5} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the circle with center at $(2,-1)$ and radius $4$. Express your answer in the standard form of a circle's equation. | (x-2)^2 + (y+1)^2 = 16 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Given a > b and c ∈ ℝ, what is the resulting inequality after adding -c to both sides of the inequality a > b? Express your answer as a single inequality. | a - c > b - c | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Given \(\omega = \cos \frac{\pi}{5} + i \sin \frac{\pi}{5}\), find the polynomial equation with roots \(\omega, \omega^3, \omega^7, \omega^9\). Express your answer in the form of a polynomial equation, using the variable \(x\). | x^4 - x^3 + x^2 - x + 1 = 0 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, $\triangle PQR$ is right-angled at $Q$ and has $\angle QPR=54^{\circ}$. Also, point $S$ lies on $PQ$ such that $\angle PRS=\angle QRS$. Find the measure of $\angle RSQ$. Express your answer in degrees as a whole number. | 72 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(2x) = 2/(2+x), calculate 2f(x) in terms of x. Express your answer as a fraction or algebraic expression. | \frac{8}{4+x} | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Alexis took a total of 243,000 steps during the 30 days in the month of April. What was her mean (average) number of steps per day in April? Express your answer as a whole number. | 8100 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Laura earns $10 per hour and works 8 hours per day for 10 days. She first spends 25% of her pay on food and clothing, and then pays $350 in rent. Calculate the amount of her pay that she has left. Express your answer in dollars as a whole number. | 250 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
An extracurricular interest group has 5 members, including 3 female members and 2 male members. Now, 2 members are randomly selected for a study report. Let X denote the number of female members selected. Calculate the mean of X, expressing your answer as a fraction or decimal. Include all necessary calculations in you... | 6/5 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function y = a^(x - 1) + 3 with conditions a > 0 and a!= 1, find a point (x, y) that must lie on the graph of the function. Express your answer as an ordered pair. | (1, 4) | big_math | [
"Mathematics -> Algebra -> Functions"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
How many sets A satisfy A ∪ {-1, 1} = {-1, 0, 1}? Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
If $\left(5x-3\right)^{3}=\sqrt{64}$, then what is the value of $x$? Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the complex number $z$ satisfies the equation $(1+i) z=2$, find the imaginary part of the complex number $z$. Express your answer as a single number. | -1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, the side AB of triangle ABC is divided into n equal parts (n > 1990). Through the n-1 division points, lines parallel to BC are drawn intersecting AC at points Bi, Ci respectively for i=1, 2, 3,..., n-1. Calculate the ratio of the area of triangle AB1C1 to the area of the quadrilateral B1989B1990C1990C1... | 1:3979 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
A couple, both phenotypically normal, have a son who suffers from both hemophilia (an X-linked recessive condition) and phenylketonuria (an autosomal recessive condition). If they have another daughter, what is the probability that she will be phenotypically normal? Express your answer as a simplified fraction. | \frac{3}{4} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Determine the number of different pairs of digits (a, b) such that the product of 5a68 and 865b is divisible by 824. Provide your answer as a single integer. | 19 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
During a field practice of the "research and study" group of a certain school, it was found that x branches grew on one main stem of a plant, and each branch had x small branches. If the sum of the number of main stems, branches, and small branches on one main stem is 57, find the value of x. Express your answer as a w... | 7 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
A car travels 5 kilometers to the east, and it is recorded as 5 kilometers. How would traveling 5 kilometers to the west be recorded? Express your answer in kilometers. | -5 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
As shown in the figure, semicircles with diameters AB, AC, and CB are mutually tangent to each other. If CD is perpendicular to AB, calculate the ratio of the area of the shaded region to the area of the circle with radius CD. Express your answer as a simplified ratio in the form a:b. | 1:4 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
A residential community plans to convert a triangular piece of land into a green space. The two sides of the triangular land measure $32m$ and $68m$, and the angle between them is $30^{\circ}$. Given that the renovation cost is $50$ yuan/$m^{2}$, calculate the renovation cost for this triangular piece of land. Express ... | 27200 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
The side length of an equilateral triangle is \(a\). A new equilateral triangle is formed by connecting the midpoints of each side of the original triangle. The third equilateral triangle is similarly formed by connecting the midpoints of each side of the second triangle, and this process continues indefinitely. Calcul... | 6a | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Given that non-zero complex numbers \( x \) and \( y \) satisfy \( x^{2} + xy + y^{2} = 0 \), find the value of the algebraic expression \(\left(\frac{x}{x + y}\right)^{2005} + \left(\frac{y}{x + y}\right)^{2005}\). Express your answer as a single numerical value. | -1 | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets E = {x | x = cos(nπ/3), n ∈ Z} and F = {x | x = sin((2m-3)π/6), m ∈ Z}, determine the exact relationship between sets E and F. Express your answer as a set relationship (e.g., E = F, E ⊂ F, etc.). | E = F | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Given a geometric sequence {a_n} with real number terms, let S_n denote the sum of the first n terms. If S_{10} = 10 and S_{30} = 70, find S_{40}. Express your answer as a single numerical value. | 150 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given the set $M=\{x \in \mathbb{Z} | 1 \leq x \leq m\}$, if set $M$ has 4 subsets, then find the real number $m$. Express your answer as a real number. | 2 | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Given $2^{a}=5^{b}=m$, and $\frac{1}{a}+\frac{1}{b}=\frac{1}{2}$, calculate the value of $m$. Express your answer as a single value. | 100 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Let x, y, and z be three positive real numbers whose sum is 1. If none of the numbers is greater than twice the other two combined, find the minimum value of the product xyz. Express your answer as a single fraction. | \frac{1}{32} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression: the cube root of $\sqrt{9}$. Express your answer as a simplified radical expression. | $\sqrt[3]{3}$ | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Grandma Zhang received the newspapers she subscribed to from the postman. By chance, she pulled out a sheet and found that page 8 and page 21 were on the same sheet. Determine the total number of pages in this newspaper. Express your answer as a whole number. | 28 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given that tan(α) = 2, find the value of tan(α - 3π/4). Express your answer as a single number. | -3 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions -> Other"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Given a line passing through points A (-2, m) and B (m, 4) is parallel to the line $2x+y-1=0$, find the value of $m$. Express your answer as a single integer. | -8 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $\sqrt[3]{{{x^2}}}$ using exponent rules. Express your answer as a simplified expression in exponent notation. | x^{\frac{2}{3}} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
A bar graph shows the grades in a mathematics class for the last grading period. Grades A, B, C, and D are considered satisfactory. If the graph shows 5 students with grade A, 4 students with grade B, 3 students with grade C, 3 students with grade D, and 5 students with grade F, what fraction of the grades shown in the... | \frac{3}{4} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a > 0$, find the minimum value of $a$ such that the inequality $x + \frac{a}{x-1} \geqslant 5$ holds true for all $x \in (1, +\infty)$. Express your answer as a single number. | 4 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $y = \frac{kx^2}{k^2 + k - 2}$, if its graph is a hyperbola located in the second and fourth quadrants, find the value of $k$ that satisfies these conditions. Express your answer as a single numerical value. | -1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Find the range of values for $a$ such that the function $f(x) = (2-a^2)x + a$ is always positive on the interval $[0, 1]$. Express your answer as an inequality using standard mathematical notation. | 0 < a < 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
In a geometric sequence with positive terms {a_n}, a_1 and a_{99} are the two roots of the equation x^2-10x+16=0. Calculate the value of a_{40}a_{50}a_{60}. | 64 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
A special lecture on fire safety knowledge is to be held at Jixi Experimental Middle School and then at three other schools, with each school hosting the lecture once. Calculate the number of different possible orders in which the lecture can be given at the four schools, given that the lecture at Jixi Experimental Mid... | 6 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Permutations"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Count the number of integers from 1 to 100, inclusive, that have at least one digit equal to 6. Express your answer as a whole number. | 19 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
In triangle $\triangle ABC$, the sides opposite to the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $\cos A = \frac{\sqrt{6}}{3}$, $b = 2\sqrt{2}$, and $c = \sqrt{3}$, calculate the value of $a$ using the cosine rule. Express your answer as a single value, including the square root symbol if... | \sqrt{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow {a}=(1-t, 2t-1, 3)$ and $\overrightarrow {b}=(2, t, t)$, find the minimum value of $|\overrightarrow {a}- \overrightarrow {b}|$. Express your answer as a numerical value. | 2 \sqrt{2} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Solve the system of equations $$\begin{cases} \overset{x+y=1}{x-y=-1}\end{cases}$$ and express the solution set in set notation, in the format {(x, y)}. | \{(0, 1)\} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Let {a_n} be an arithmetic sequence with a common difference less than zero, and let S_n denote the sum of its first n terms. If S_8 = S_12, then for what value of n is S_n maximized? Express your answer as a single integer. | 10 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Let $\theta$ be an angle in the second quadrant, and let $\cos\left(\frac{\pi}{2} - \theta\right) = \frac{3}{5}$. Calculate the value of $\sin(2\theta)$. Express your answer as a fraction, including the negative sign if applicable. | -\frac{24}{25} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression \(2^{4}-2^{3}\). If the result is a power of 2, express your answer in that form; otherwise, provide the numerical value. | 2^3 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
In a positive geometric sequence {a_n}, if a_4a_7 = 4, then find the sum of the first 10 terms of the sequence {log_2 a_n}. Express your answer as a single numerical value. | 10 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
Given sets S={0, 1, 2} and T={0, 3}, find the number of proper subsets of P, where P=S∩T. Express your answer as a whole number. | 1 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics -> Other"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Given that the random variable ξ follows a normal distribution N(4, 6^2), and P(ξ ≤ 5) = 0.89, find the probability P(ξ ≤ 3). Express your answer as a decimal value between 0 and 1. | 0.11 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
In a survey of 198 people, 149 people supported the first plan, and 119 people supported the second plan. If exactly 29 people did not support either plan, calculate the number of people who supported both plans. | 99 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Other"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given the ellipse $\frac{x^2}{4} + \frac{y^2}{2} = 1$ with foci $F_1$ and $F_2$, and a point $M$ on the ellipse such that $MF_1 \perp x$-axis, calculate the distance from point $F_1$ to the line $F_2M$. Express your answer as a single numerical value. | \frac{2\sqrt{2}}{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
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