instruction_seed stringlengths 40 1.39k | answer stringlengths 1 95 | source stringclasses 1
value | domain listlengths 1 3 | llama8b_solve_rate float64 0 0.98 | _source stringclasses 1
value |
|---|---|---|---|---|---|
In the convex quadrilateral ABCD, E and F are the midpoints of [AD] and [BC] respectively. Given |CD| = 12 and |AB| = 22, find the sum of all possible integer values of |EF| = x. | 121 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Consider a geometric sequence { a_n } where each term is positive, and it is given that a_3 * a_8 = 81. Find the value of log_3 a_1 + log_3 a_2 +... + log_3 a_10. Express your answer as a single integer. | 20 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = x^3 - x + 3, find the equation of the tangent line to the curve y = f(x) at the point (1, f(1)). Express your answer in standard form (e.g., Ax + By + C = 0). | 2x - y + 1 = 0 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Tian measured her steps and found that it took her 625 steps to walk 500 meters. If she walks 10,000 steps at this same rate, what distance will she walk in kilometers? Express your answer in kilometers, rounded to one decimal place if necessary. | 8 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Given the line $l_{1}$: $x+2ay-1=0$ and the line $l_{2}$: $\left(2a-1\right)x-ay-1=0$, find the value(s) of $a$ for which these lines are parallel. Express your answer as a numerical value(s). | 0 \text{ or } \frac{1}{4} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the derivative of the even function f(x) (x ≠ 0) is f'(x), and it satisfies f(1) = 0, when x > 0, xf'(x) < 2f(x). Find the range of x that makes f(x) > 0 true. Express your answer in interval notation. | (-1, 0) ∪ (0, 1) | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
A geometric sequence with 5 terms, where each term is a positive integer less than 100, has a sum of 211. Find the sum of the terms that are perfect squares. Express your answer as a single integer. | 133 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
In a certain exam, there are 50 questions. Correct answers earn 3 points each, incorrect answers deduct 1 point each, and unanswered questions score 0 points. Xiaolong scored 120 points. Calculate the maximum number of questions Xiaolong answered correctly. Express your answer as a whole number. | 42 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the number of ways to distribute 5 volunteers into 3 different Olympic venues for reception duties, given that each venue must have at least one volunteer. Express your answer as a single integer. | 150 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
From 2005 numbers, 20 numbers are to be sampled using the systematic sampling method. Calculate the sampling interval. | 100 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given a geometric sequence {a_n} where a_3 = 4 and a_6 = 1/2, find the common ratio q. Express your answer as a numerical value. | \frac{1}{2} | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
The sequence, formed from the sequence of primes by rounding each to the nearest ten, begins \(0,0,10,10,10,10,20,20,20,30, \ldots\). When continued, how many terms in this sequence are equal to 40? Express your answer as a whole number. | 3 | big_math | [
"Mathematics -> Number Theory -> Prime Numbers"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given the values of $a=\frac{1}{3}$, $b=\sin\frac{1}{3}$, and $c=\frac{1}{\pi}$, determine the correct order of these values using inequalities (e.g., 'a > b > c'). Express your answer as a string of inequalities, including all variables. | a > b > c | big_math | [
"Mathematics -> Precalculus -> Functions",
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Given the hyperbola equation $\frac{{x}^{2}}{4}-\frac{{y}^{2}}{12}=1$, if a point $P$ on the hyperbola is $8$ units away from its right focus, find the distance from point $P$ to its left focus. Express your answer in units, noting that there may be two possible values depending on the branch of the hyperbola. | 12 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function y = |x| and its domain A = {-1, 0, 1}, find the intersection of A and the range B of the function. Express your answer as a set. | {0, 1} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Determine the coefficient of $x^{5}$ in the expansion of $(x^{2}+x+1)^{5}$. Express your answer as a single integer. | 51 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
A random variable ξ follows a binomial distribution B(100,0.2). Calculate the variance of 4ξ + 3, denoted as D(4ξ+3). Provide your answer as a numerical value. | 256 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given a geometric sequence {a_n} with a_1 = a_8 = 3, find the sum of the first n terms, S_n. Express your answer as a formula in terms of n. | S_n = 3n | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Given $z \in \mathbb{C}$, and $|z + 2 - 2i| = 1$, calculate the minimum value of $|z - 2 - 2i|$. Express your answer as a single numerical value. | 3 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the domain of f(x) is (1, +∞), determine the domain of f(2x+1). Express your answer as an interval in standard notation. | (0, +∞) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $(1-i)z = 2i$, solve for $z$ and express your answer in the standard form $a + bi$. | -1 + i | big_math | [
"Mathematics -> Algebra -> Complex Numbers -> Other"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Find the distance from the point (1, 2) to the line x = -2. Express your answer as a whole number. | 3 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x)= \begin{cases} e^{x}+ax^{2}, & x > 0 \\ \frac {1}{e^{x}}+ax^{2}, & x < 0\end{cases}$, find the range of the real number a for which the function f(x) has four zeros. Express your answer as an interval. | (-\infty,- \frac {e^{2}}{4}) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Let \( a \) be a positive integer such that \( a < 100 \). Given that \( a^{3} + 23 \) is divisible by 24, determine how many such values of \( a \) exist. Express your answer as a whole number. | 5 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Find the inclination angle $\alpha$ of the line $x = 1$, where $\alpha$ is the angle between the line and the positive direction of the x-axis. Express your answer in degrees as a whole number. | 90^\circ | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z = \frac{1}{2} + i$, find the point corresponding to the conjugate of $z$ ($\bar{z}$) in the complex plane. Express your answer as an ordered pair (a, b), where a is the real part and b is the imaginary part of the conjugate. | (\frac{1}{2}, -1) | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Point P is on the ellipse with equation x^2/16 + y^2/9 = 1. F1 and F2 are the left and right foci of the ellipse, respectively. Given that |PF1| * |PF2| = 12, find the size of angle F1PF2 in degrees. | 60 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given the ellipse x^2/25 + y^2/16 = 1 and a point P on the ellipse at a distance of 4 from one focus, calculate the distance from P to the other focus of the ellipse. | 6 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Given the quadratic equation x^2 + (1 - 2i)x + 3m - i = 0, where m is a real number, and the equation has one real root, find the other root. Express your answer as a complex number in the form a + bi. | -\frac{1}{2} + 2i | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
A company had a total sales revenue of 10 million yuan last year, with annual costs consisting of 5 million yuan in production costs and 2 million yuan in advertising costs. If the annual profit is taxed at a rate of p% and any annual advertising expenses exceeding 2% of the annual sales revenue are also taxed at p%, w... | 25% | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
In the given convex quadrilateral ABCD, the diagonals intersect at point E, and point F is located on side BC. AB is parallel to CD, which is parallel to EF. The area of triangle BCE is denoted as S. How many triangles in the figure have an area equal to S? Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Given a cube with an edge length of $\sqrt{3}$, all of its vertices are on the surface of a sphere. Calculate the surface area of the sphere. Express your answer in terms of $\pi$. | 9\pi | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Surface Area"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
If the complex number $z={({\frac{{1+i}}{{1-i}}})^2}$, calculate the magnitude $|z|$. | 1 | big_math | [
"Mathematics -> Algebra -> Complex Numbers -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given \( a > 1 \) and \( b \) is a positive rational number, with the equation \( a^{b} + a^{-b} = 2 \sqrt{3} \). Calculate the value of \( a^{b} - a^{-b} \). Express your answer as a mathematical expression, including the square root symbol if necessary. | 2 \sqrt{2} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
In a four-level symbol pyramid, the symbols in the two bottom cells determine the symbol above them. If the symbols are the same, the cell above will be "+", otherwise, it will be "-". How many possible ways are there to fill the bottom row of four cells such that the top of the pyramid results in a "+"? Express your a... | 8 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
In the quadrilateral $ABCD$ in space, if $AC = AD$ and $BC = BD$, find the angle formed by $AB$ and $CD$. Express your answer in degrees. | 90 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the distribution of the random variable $\xi$ is $P(\xi=\frac{k}{5})=ak$ for $k=1,2,3,4,5$, calculate the probability $P(\frac{1}{10} < \xi < \frac{1}{2})$. Express your answer as a simplified fraction. | \frac{1}{5} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given that x and y are real numbers satisfying the following equations:
x + xy + y = 2 + 3√2 and x² + y² = 6,
find the value of |x + y + 1|. Express your answer as a boxed numerical value. | 3 + √2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
A cylinder with a base radius of $R$ is intersected by a plane that forms an angle of $\theta$ with its base. When $\theta$ is $30^\circ$, calculate the eccentricity of the resulting elliptical cross-section. Express your answer as a decimal or fraction. | \frac{1}{2} | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
Lil writes one of the letters P, Q, R, S in each cell of a 2x4 table. She does this in such a way that, in each row and in each 2x2 square, all four letters appear. How many distinct arrangements can she make? Express your answer as a single integer. | 24 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
A bag contains 6 white balls, 5 yellow balls, and 4 red balls. If a ball is randomly drawn from the bag, what is the probability of drawing a white ball? Express your answer as a simplified fraction. | \dfrac{2}{5} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Given a set of data $96$, $89$, $92$, $95$, $98$, what is the median of this set of data? Express your answer as a single number. | 95 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
In a plane, there is a segment AB with length 2 and a moving point P. If it satisfies $|PA| + |PB| = 8$, calculate the range of values for $|PA|$ and express your answer as an interval. | [3,5] | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given real numbers $a$, $b$, $c$, $d$, $e$ that satisfy the relations: $a+b+c+d+e=8$ and $a^2+b^2+c^2+d^2+e^2=16$, find the maximum value of the real number $e$. Express your answer as a single real number. | \frac{16}{5} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
The price of gasoline rose by 20% during January, fell by 20% during February, and rose by 25% during March. If the price at the end of April was the same as it had been at the beginning of January, what percentage decrease x occurred in April? Express your answer as a percentage rounded to the nearest integer. | 17 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
A woman skilled in weaving weaves faster day by day. Starting from the second day, she weaves an additional constant amount of fabric each day compared to the previous day. If she weaves 5 meters of fabric on the first day, and a total of 390 meters of fabric in a month (considering a month as 30 days), then what is th... | \dfrac{16}{29} | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
In isosceles right triangle $ABC$, with $\angle A = 90^{\circ}$ and $AB = AC = 2$, calculate the projection of the vector $\vec{AB}$ in the direction of $\vec{BC}$. Express your answer as a numerical value. | -\sqrt{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence {a_n}, it is known that a_1+a_2+a_3+a_4+a_5=20. Determine the value of a_3. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Given the universal set $U={x∈Z∣−5<x<5}$ and set $S={−1,1,3}$, if $∁_UP⊆S$, find the number of sets $P$. Express your answer as a whole number. | 8 | big_math | [
"Mathematics -> Discrete Mathematics -> Set Theory -> Other"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a$ is the geometric mean of $1+2b$ and $1-2b$, find the maximum value of the expression $$\frac {2ab}{|a|+2|b|}.$$ Express your answer as a fraction with a square root. | \frac{\sqrt{2}}{4} | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Applied Mathematics -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is $θ$, the "cross product" of $\overrightarrow{a}$ and $\overrightarrow{b}$, denoted as $\overrightarrow{a} × \overrightarrow{b}$, is defined as a vector with magnitude $|\overrightarrow{a} × \overrightarrow{b}| = |\overrightarrow{a}| \... | 2 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
If there is a point (-4,a) on the terminal side of an angle measuring 120 degrees, calculate the value of a. Express your answer as a single value, including the square root symbol if necessary. | 4\sqrt{3} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions -> Other"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence with a common difference of 2, and the sum of the first 100 terms (S100) is 10000, calculate the sum of the odd-numbered terms (a1 + a3 + a5 +... + a99). Express your answer as a single numerical value. | 4950 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given lines $l_1$: $x + y + 1 = 0$, $l_2$: $x + y - 1 = 0$, find the distance between $l_1$ and $l_2$. Express your answer as a numerical value. | \sqrt{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression \( \frac{\sqrt{6} + 4\sqrt{3} + 3\sqrt{2}}{(\sqrt{6} + \sqrt{3})(\sqrt{3} + \sqrt{2})} \). Express your answer as a simplified algebraic expression. | \sqrt{6}-\sqrt{2} | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Solve for $x$ in the linear equation $3x - 6 = 0$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Linear Equations"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
A rectangular field has an area of 864 square steps. The width is 12 steps less than the length. Denote the width as x steps. What is the equation that correctly represents the relationship between the width, length, and area of the rectangular field? Express your answer as an algebraic equation. | x(x + 12) = 864 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $f(x) = x^7 + ax^5 + bx - 5$, and $f(-3) = 5$, calculate the value of $f(3)$. Express your answer as a single integer in the form $\boxed{[answer]}$. | -15 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the intersection point of the parabola $y^{2}=2px$ is $(4,0)$, find the equation of its directrix. Express your answer in the form $x =...$. | x=-4 | big_math | [
"Mathematics -> Algebra -> Quadratic Functions"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given $\sin \left( \frac {\pi}{2}-\alpha \right) = \frac {3}{5}$, calculate $\cos (\pi-2\alpha)$. Express your answer as a fraction. | \frac{7}{25} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Given an isosceles triangle with one of its base angles being 80°, find the degree measure of the vertex angle of this isosceles triangle. Express your answer in degrees (°) as a whole number. | 20 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Express the mass of an oxygen atom, 0.000 000 000 000 000 000 000 000 02657kg, in scientific notation. Include the unit (kg) in your answer. | 2.657 \times 10^{-26} kg | big_math | [
"Mathematics -> Other -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the ellipse x^2/9 + y^2/4 = 1 and the point P(1,1) that bisects the chord AB, find the equation of the line containing AB. Express your answer in the form Ax + By + C = 0. | 4x + 9y - 13 = 0 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given the custom operation $\odot$ defined as $x \odot y = \frac{x}{2-y}$, find the range of real number $a$ such that the solution set of the inequality $(x-a) \odot (x+1-a) > 0$ is a subset of the set $\{x|-2 \leq x \leq 2, x \in \mathbb{R}\}$. Express your answer as an interval. | [-2, 1] | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
In the parallelogram ABCD, A(1, 2), B(-2, 0), $$\overrightarrow {AC}$$ = (2, -3). Calculate the coordinates of point D. Express your answer as a coordinate pair (x, y). | (6, 1) | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
In the Cartesian coordinate system, find the coordinates of the point A(-5,-2) after reflecting it across the x-axis. Express your answer as an ordered pair (x, y). | (-5,2) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given the following premises:
(1) The diagonals of a parallelogram are equal.
(2) A square is a parallelogram.
Using syllogistic reasoning, deduce a conclusion about the diagonals of a square. Express your answer as a direct conclusion based on the premises. | The diagonals of a square are equal. | big_math | [
"Mathematics -> Discrete Mathematics -> Logic"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Find the minimum value of the quadratic function $y=\left(x-1\right)^{2}-5$. Express your answer as a single integer. | -5 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Quadratic Functions"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of \( 8 + 2(3^2) \). Provide your answer as a single number. | 26 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Three identical squares form rectangle $W X Y Z$, as shown. The perimeter of $W X Y Z$ is $56 \mathrm{~m}$. Find the area of $W X Y Z$. Express your answer in square meters (m^2) as a whole number. | 147 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
A ball is floating on the surface of a frozen lake, and when it is removed, it leaves a hole with a diameter of 24 cm and a depth of 8 cm. Calculate the radius of the ball. Express your answer in centimeters (cm). | 13 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Jia, Yi, Bing, Ding, and Wu sit around a circular table to play cards. Jia has a fixed seat. If Yi and Ding cannot sit next to each other, how many different seating arrangements are possible? Express your answer as a whole number. | 12 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
The diameter of the Omicron variant virus is 120 nanometers, where 1 nanometer = 10^-9 meters. Calculate the radius of this coronavirus in scientific notation, including units of meters. | 6 \times 10^{-8} \text{ meters} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the function y=f(x) is increasing on (-∞, 2) and the axis of symmetry of y=f(x+2) is x=0, compare the values of f(-1) and f(3). Provide your answer as a direct comparison between these two function values. | f(-1) < f(3) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
The surface area of a cube is 96 cm². Find the volume of the cube in cubic centimeters (cm³). Express your answer as a whole number. | 64 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Volume"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
A sub-factory of an oil refinery refines crude oil into gasoline, which requires cooling and heating the crude oil. If in the x-th hour, the temperature of the crude oil T (in degrees Celsius) is given by the function f(x) = (1/3)x^3 - x^2 + 8 (0 ≤ x ≤ 5), then what is the minimum instantaneous rate of change of the cr... | -1 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the graph of the function $f(x) = 3 - a^{x+1}$ always passes through a fixed point P, find the coordinates of point P. Express your answer as an ordered pair (x, y). | (-1, 2) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Represent 110 billion in scientific notation. Express your answer in the format $a imes 10^{n}$, where $1 \leq |a| < 10$ and $n$ is an integer. | 1.1 \times 10^{10} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
From the five numbers $1$, $2$, $3$, $4$, $5$, if any two numbers are selected, calculate the probability that both numbers are even. Express your answer as a simplified fraction. | \frac{1}{10} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Find the coordinates of the vertex of the parabola given by the equation y = 2(x-5)^2 + 3. Express your answer as an ordered pair (x, y). | (5,3) | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Quadratic Functions"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Given that {a_n} is an arithmetic sequence, a_3 + a_8 = 22, and a_6 = 7, find the value of a_5. Express your answer as a single integer. | 15 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Let \( n \) be a natural number between 100 and 200. Determine how many values of \( n \) make \( 7n + 2 \) a multiple of 5. Express your answer as a single integer. | 20 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $y=16{{x}^{2}}$, find the coordinates of its focus. Express your answer as an ordered pair $(x, y)$, where $y$ is an exact fraction. | (0, \dfrac{1}{64}) | big_math | [
"Mathematics -> Algebra -> Quadratic Functions"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
In the right triangle ABC, side AC is divided into 8 equal parts. From each division point, a segment parallel to BC is drawn, resulting in a total of 7 segments. Given that BC = 10, find the sum of the lengths of these 7 segments. Express your answer as a whole number. | 35 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = x sin(x) + ax, and the derivative of f(x) at pi/2 equals 1, calculate the value of a. Express your answer as a single number. | 0 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given $x_1 + x_2 = a$ and $x_1 \cdot x_2 = b$, express $|x_1 - x_2|$ in terms of $a$ and $b$. Provide your answer as an algebraic expression, including any necessary absolute value and square root symbols. | \sqrt{a^2 - 4b} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence {a_n} where a_1 = 4 and a_2 = 6, calculate the sum of the first 4 terms, S_4. Express your answer as a single integer. | 28 | big_math | [
"Mathematics -> Algebra -> Algebra -> Sequences and Series"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Consider the line y = a and the function y = x^3 - 3x. Find the range of values for a such that there are three distinct intersection points between the graph of the line and the graph of the function. Express your answer as an interval (lower bound, upper bound). | (-2, 2) | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given a geometric sequence {a_n} that satisfies a_2 = 2 and a_5 = 1/4, find the common ratio q. Express your answer as a numerical value. | \frac{1}{2} | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets $M=\{x \mid -1 < x < 3\}$ and $N=\{x \mid x^2 - 6x + 8 < 0\}$, find the intersection $M \cap N$. Express your answer in interval notation, specifying the numerical bounds of the interval. | (2,3) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z= \frac{2-i}{2+i}$, where $i$ is the imaginary unit, calculate the value of $|z|$. Express your answer as a numerical value. | 1 | big_math | [
"Mathematics -> Algebra -> Complex Numbers -> Other"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Using a semicircular piece of paper with a diameter of 2 cm to make the lateral surface of a cone (not counting the overlapping part), find the radius of the base of the cone. Express your answer in centimeters (cm) as a decimal number. | 0.5 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Given a geometric sequence {a_n} that satisfies a_4a_5a_6=8 and a_2=1, find the value of a_2+a_5+a_8+a_{11}. Express your answer as a single numerical value. | 15 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Find the range of the function y=log_{1/2}(x^2-6x+17). Express your answer in interval notation if appropriate. | (-\infty, -3] | big_math | [
"Mathematics -> Algebra -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given that quadrilateral ABCD is a cyclic quadrilateral inscribed in circle O, with AC = sqrt(3)BD and angle ADC = 2*angle BAD. If AB*CD + BC*AD = 4*sqrt(3), calculate the radius of circle O. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given points M(x_1, f(x_1)) and N(x_2, g(x_2)) on the graphs of functions f(x) = sin(x) + (1/6)x^3 and g(x) = x - 1 respectively, where x_1 >= 0 and x_2 >= 0. If line MN is parallel to the x-axis, find the minimum distance between points M and N. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Calculus -> Applications of Derivatives"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
A certain middle school holds a math problem-solving competition. The scores of 5 participants are $70$, $85$, $90$, $75$, $95$. Find the upper quartile of these 5 scores. Express your answer as a single whole number. | 90 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
Given the solution set of the inequality (x+a)/(x^2+4x+3) > 0 is {x|-3 < x < -1 or x > 2}, calculate the value of the real number a. | -2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Two different numbers are randomly selected from the set {-3, -1, 0, 2, 4} and then multiplied together. What is the probability that the product of the two numbers chosen is 0? Express your answer as a simplified fraction. | \frac{2}{5} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Four points are on a line segment as shown. If \( AB: BC = 1:2 \) and \( BC: CD = 8:5 \), then find the ratio \( AB: BD \). Express your answer as a ratio in the simplest form. | 4:13 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
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