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
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|---|---|---|---|---|---|
The graph of the function y = cos(x) undergoes the following transformations: a translation of π/3 units to the left, a horizontal compression by a factor of 1/2, and a vertical stretch by a factor of 3. Find the analytical expression of the resulting function. | y = 3cos(2x + π/3) | big_math | [
"Mathematics -> Precalculus -> Functions -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given three pairwise skew lines $a$, $b$, and $c$ in space, determine the number of lines that intersect all three lines $a$, $b$, and $c$. Provide your answer as a single numerical value. | 1 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the distance $d$ from the point $P(1,0)$ to the line $x-2y+1=0$ using the formula $d = \frac{|Ax_0+By_0+C|}{\sqrt{A^2+B^2}}$. Express your answer as a fraction with a radical, if necessary. | \frac{2\sqrt{5}}{5} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function y = x*sqrt(1-4*x^2) and the constraint 0 < x < 1/2, calculate the maximum value of y. Express your answer as a single number. | \frac{1}{4} | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
If 5 numbers are inserted between 8 and 5832 to form a geometric sequence, what is the value of the fifth term? Express your answer as a whole number. | 648 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
5 people are scheduled for a 5-day duty roster, with one person on duty each day. Each person can be scheduled for multiple days or not at all, but the same person cannot be scheduled on consecutive days. Calculate the total number of ways to arrange the duty roster. | 1280 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
For the fraction $\frac{1}{{x-5}}$ to be meaningful, what is the condition on the values of $x$? Express your answer as an inequality or a statement about the possible values of $x$. | x \neq 5 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Inequalities"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Given two vectors $\overrightarrow{a}=(2,m,3)$ and $\overrightarrow{b}=(4,-1,n)$, and $\overrightarrow{a} \parallel \overrightarrow{b}$, find the value of $mn$. Express your answer as a single integer. | -3 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
In an equilateral triangle ABC with a side length of 1, calculate the dot product of vectors AB and AC. Express your answer as a single numerical value. | \frac {1}{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $x > 0$, $y > 0$ and $x + y = 1$, find the minimum value of $\frac{4}{x} + \frac{9}{y}$. Express your answer as a single number. | 25 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Sixty apples and sixty pears are to be packed into boxes so that each box contains the same number of apples, and no two boxes contain the same number of pears. What is the largest possible number of boxes that can be packed in this way? Express your answer as a whole number. | 10 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Find the valid range of the independent variable x for the function y = 2 / sqrt(x-1). Express your answer as an inequality using mathematical notation. | x > 1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Among the six numbers $(-3)^2$, $-2^2$, $|-2|$, $(-1)^3$, $-|-2|$, $(-1)^{2n-1}$ (where $n$ is a positive integer), how many are negative? Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the arithmetic square root of 0.09. Express your answer as a decimal number. | 0.3 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given the hyperbola equation x^2/4 - y^2/12 = 1, calculate the distance from its foci to its asymptotes. Express your answer as a single numerical value. | 2\sqrt{3} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the real numbers \( x_i (i = 1, 2, \cdots, n) \) satisfy \( \sum_{i=1}^{n} x_i = n \), find the minimum value of \( \sum_{i=1}^{n} 3^{x_i} \). Express your answer in terms of \( n \). | 3n | big_math | [
"Mathematics -> Applied Mathematics -> Mathematics Word Problems"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Find the value of m for which there is no integer pair (x, y) satisfying the equation 3x^2 - 10xy - 8y^2 = m^19. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Find the slope angle of the line given by the equation $x - y + 1 = 0$. Express your answer in degrees, including the degree symbol (°). | 45^\circ | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Other"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
The line 3x + y + a = 0 passes through the center of the circle x^2 + y^2 + 2x - 4y = 0. Calculate the value of a using the given information. Express your answer as an integer. | 1 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
If the function f(x) is a power function and satisfies f(4)/f(2) = 3, find the value of f(1/2). Express your answer as a single value in the form \boxed{ }. | \frac{1}{3} | big_math | [
"Mathematics -> Algebra -> Functions -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Find the sum of all three-digit integers formed using only the digits 1 and/or 2. Express your answer as a single numerical value. | 1332 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Compute the result of $-1-2\times(-2)$. Express your answer as a single integer. | 3 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Given the set S={A_0,A_1,A_2,A_3} and the operation ⊕ defined as A_i⊕A_j=A_k, where k is the remainder of i+j divided by 4, find the number of ordered pairs (i,j) that make the relation ((A_i⊕A_i)⊕A_j)=A_0 true. | 4 | big_math | [
"Mathematics -> Discrete Mathematics -> Algebra -> Other"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $z(1+i) = i - 2$, find $\overline{z}$, the conjugate of $z$. Express your answer as a complex number in the form $a + bi$. | -\frac{1}{2} - \frac{3}{2}i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Point P(m-1, m+3) is on the y-axis of the Cartesian coordinate system. Find the coordinates of point P. Express your answer as a coordinate pair (x, y). | (0, 4) | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$, where $|\overrightarrow{a}|= \sqrt {2}$, $|\overrightarrow{b}|=2$, and $(\overrightarrow{a}-\overrightarrow{b})\perp \overrightarrow{a}$, calculate the angle between vector $\overrightarrow{a}$ and $\overrightarrow{b}$ in radians. Express your answer in the f... | \frac{\pi}{4} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Given the hyperbola x^2/a^2 - y^2/b^2 = 1 (a > 0, b > 0), its left and right vertices are A and B, respectively. The right focus is F, and the line l passing through point F and perpendicular to the x-axis intersects the hyperbola at points M and N. P is a point on line l. When angle APB is maximized, point P is exactl... | \sqrt{2} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
A certain quality indicator $Z$ of a particular product produced by a company follows a normal distribution $N(200,150)$. A user purchased $100$ of these products. Let $X$ denote the number of products among these $100$ with the quality indicator value within the interval $(187.8, 212.2)$. Calculate the expected value ... | 68.26 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the radius of circle O is 6cm, and the distance from the center O to line l is 5cm, determine the number of intersection points between line l and circle O. Express your answer as a whole number. | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Express 121 trillion yuan in scientific notation. Provide your answer in the form $a \times 10^{n}$ where $1 \leq a < 10$ and $n$ is an integer. | 1.21 \times 10^{14} | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
When asked whether they have visited cities \(A\), \(B\), and \(C\),
Student A said: I have visited more cities than Student B, but I have not visited city \(B\).
Student B said: I have not visited city \(C\).
Student C said: The three of us have visited the same city.
Based on this information, what is the name of... | A | big_math | [
"Mathematics -> Discrete Mathematics -> Logic"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Pablo buys popsicles for his friends. The store sells single popsicles for $1 each, 3-popsicle boxes for $2 each, and 5-popsicle boxes for $3 each. What is the greatest number of popsicles that Pablo can buy with $8? Express your answer as a whole number. | 13 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
What is the degree measure of the rotation of the hour hand on a clock from 3 o'clock to 6 o'clock? Express your answer in degrees (°) as a whole number. | 90 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x)= \begin{cases} \log _{2}x,x > 0\\ 2^{x},x\leqslant 0\\ \end{cases}$, find all possible values of $a$ such that $f(a)= \dfrac {1}{2}$. Express your answer as a list of values. | -1, \sqrt{2} | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Precalculus -> Functions"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = 2^x - a^2 - a, find the range of values for the positive real number a such that the function has a zero in the interval (-∞, 1]. Express your answer as an interval using standard interval notation. | (0, 1] | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Let $M=\{x\left|\right.x-1 \lt 0\}$, $N=\{y\left|\right.y=1-2x,x\in M\}$. Calculate the intersection of $M$ and $N$, expressing your answer in interval notation. | (-1, 1) | big_math | [
"Mathematics -> Algebra -> Set Theory -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
A is located 60 kilometers west of B. Individuals A and B depart from location A while individuals C and D depart from location B at the same time. A, B, and D all travel east, while C travels west. It is known that the speeds of A, B, C, and D form an arithmetic sequence, with A having the highest speed. After n hours... | 30 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Spot's doghouse has a regular hexagonal base that measures one yard on each side. He is tethered to a vertex with a two-yard rope. Calculate the area, in square yards, of the region outside of the doghouse that Spot can reach. Express your answer in terms of pi. | 3\pi | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\mathbf{a} = (2,1)$ and $\mathbf{b} = (-1,k)$, and the equation $\mathbf{a} \cdot (2\mathbf{a} - \mathbf{b}) = 0$, calculate the value of $k$. Express your answer as a single integer. | 12 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of integer solutions to the equation $(x^2+x-1)^{x+3} = 1$. | 4 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Given a positive increasing geometric sequence ${a_n}$, where the sum of the first $n$ terms is $S_n$, if $a_4 + a_7 = 20$ and $a_4 \cdot a_7 = 64$, calculate the value of $\frac{S_6}{S_9}$. Express your answer as a simplified fraction. | \frac{5}{21} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z=1+bi$ ($b\in\mathbb{R}$) and $z^{2}=-3+4i$, find the imaginary part of the complex conjugate of $z$ ($\overline{z}$). Express your answer as a real number. | -2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Find the minimum value of the function $f(x) = x^3 - 3x^2 + 1$. Express your answer as a single number. | -3 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = x^2 + 2x, where -2 ≤ x ≤ 1 and x ∈ ℤ, calculate the range of f(x). Express your answer as a set of values. | {-1, 0, 3} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Quadratic Functions"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
In a Cartesian coordinate system, find the maximum value of the distance d from the point P(cos θ, sin θ) to the line x - my - 2 = 0 as θ and m vary. Express your answer as a single numerical value. | 3 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Liam spent all his money buying 50 soda bottles at the corner shop for £1 each. He sold each bottle at the same higher price. After selling 40 bottles, he had £10 more than he started with. Calculate the total amount of money Liam had once all the bottles were sold. Express your answer in pounds (£). | £75 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given the definition of $a_n = \log_{n+1}(n+2)$ and the examples of products of terms $a_1a_2$ and $a_1a_2a_3a_4a_5a_6$, find the value of $m$ such that $a_1a_2 \dots a_m = 2016$. Express your answer as a single numerical value. | 2^{2016} - 2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
In space, if the edge length ratio of two regular tetrahedrons is 1:2, then what is the ratio of their volumes? Express your answer as a ratio, using a colon (:) to separate the values. | 1:8 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
In triangle \( \triangle ABC \), \( 3 \sin A + 4 \cos B = 6 \) and \( 4 \sin B + 3 \cos A = 1 \). Calculate the measure of angle \( \angle C \) in degrees. Express your answer in degrees, including the degree symbol. | 30^\circ | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Express the number 218,000,000 in scientific notation. Provide your answer in the format a × 10^b, where a is the coefficient and b is the exponent. | 2.18 × 10^8 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given $a > 0$ and $b > 0$, if $\sqrt{3}$ is the geometric mean of $3^a$ and $3^b$, find the maximum value of $ab$. Express your answer as a single number. | \frac{1}{4} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Given the quadratic equation $x^{2}-6x+k=0$ with roots $x_{1}$ and $x_{2}$, and the condition $\dfrac{1}{x_{1}}+\dfrac{1}{x_{2}}=3$, find the value of $k$. Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Niko counted a total of 60 birds perching in three trees. Five minutes later, 6 birds had flown away from the first tree, 8 birds had flown away from the second tree, and 4 birds had flown away from the third tree. He noticed that there was now the same number of birds in each tree. Calculate the original number of bir... | 22 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Let real numbers x and y satisfy the following equations:
x^3 = 3y^2x + 5 - sqrt(7)
y^3 = 3x^2y + 5 + sqrt(7)
Find the value of x^2 + y^2. Express your answer as a single number. | 4 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
The distance between the adjacent symmetry axes of the graph of the function f(x) = sqrt(3) * sin(ωx + φ) (ω > 0) is 2π. Determine the value of ω. Express your answer as a fraction. | ω = 1/2 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
A sample consists of 1, 3, 2, 2, a, b, c. It is known that the mode of this sample is 3, and the mean is 2. Calculate the variance of this sample. Express your answer as a fraction or decimal. | \frac{8}{7} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Find the area of the triangle formed by the tangent line to the curve $y=\frac{1}{3}x^3+x$ at the point $\left(1, \frac{4}{3}\right)$ and the coordinate axes. Express your answer as a simplified fraction. | \frac{1}{9} | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Given the line $3x-y+1=0$ has an inclination angle of $\alpha$, calculate the value of $\frac{1}{2}\sin 2\alpha+\cos^2\alpha$. Express your answer as a simplified fraction. | \frac{2}{5} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Let $\{a_n\}$ be an arithmetic sequence with the first term being 1 and the common difference being 5. If $a_n = 2016$, then find the value of $n$. Express your answer as a whole number. | 404 | big_math | [
"Mathematics -> Algebra -> Algebra -> Sequences and Series"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
The ratio of the number of oatmeal cookies to the number of chocolate chip cookies in a jar is 5:2. If there are 20 oatmeal cookies, find the number of chocolate chip cookies. Express your answer as a whole number. | 8 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence {a_n} that satisfies a_2 + a_8 = 12, find the value of a_5. Express your answer as a single integer. | 6 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
An export company plans to open four direct sales stores in five candidate cities, with no more than two stores in the same city. Calculate the total number of different plans the company can have for establishing its direct sales stores. Express your answer as a single integer. | 45 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given a cube, determine the number of pairs of skew lines that can be formed by one of its diagonals and the edges of the cube. Express your answer as a whole number. | 6 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x)=\frac{x^{3}+\sin x}{1+x^{2}}+3$, let $M$ and $n$ be the maximum and minimum values of $f(x)$ respectively. Find the value of $M+n$. Express your answer as a single integer. | 6 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $3(4x + 5\pi) = P$, simplify the expression $6(8x + 10\pi)$ in terms of $P$. Express your answer as a coefficient of $P$. | 4P | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
A shopping survey collected 500,000 valid questionnaires in total. The number of people who purchased the following four types of products are: 198,000 for clothing, shoes, and hats; 94,000 for household goods; 116,000 for cosmetics; and 92,000 for home appliances. It is known that 116 people were selected from the "co... | 94 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
A factory produces three different models of products: A, B, and C. The ratio of the quantities of these products is 3:4:7. Now, using stratified sampling, a sample is drawn, and it includes 15 units of model A. What is the total sample size n? Express your answer as a whole number. | 70 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Two squares have integer side lengths in centimeters. The difference between their areas is 19 cm\(^2\). Find the sum of their perimeters. Express your answer in centimeters as a whole number. | 76 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $z(\cos 60^\circ + i\sin 60^\circ) = -1 + \sqrt{3}i$, find the imaginary part of the conjugate of $z$. Express your answer as a single number. | -\sqrt{3} | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
On the number line, point A represents 3 and point B represents -7. Find the distance between A and B. Express your answer as a whole number. | 10 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Four identical trapezoids with bases 30 cm and 50 cm and non-parallel sides equal are combined to form a square with an area of 2500 cm^2, leaving a square "hole" in the middle. Find the area of one trapezoid. Express your answer in square centimeters (cm^2) as a whole number. | 400 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
In the polar coordinate system, find the distance from point $M(2, \frac{\pi}{3})$ to the line $l$: $\rho\sin(\theta + \frac{\pi}{4}) = \frac{\sqrt{2}}{2}$. Express your answer as a single value. | \frac{\sqrt{6}}{2} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the random variable X follows the normal distribution N(2, σ^2), and P(-1 < X ≤ 2) = 0.35, calculate the probability P(X ≥ 5). Express your answer as a decimal value between 0 and 1, rounded to 2 decimal places if necessary. | 0.15 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Given the inequality $x^2 - x \leq 0$, let $M$ be its solution set. Define set $N = \{x \mid -1 < x < 1\}$. Calculate the intersection $M \cap N$ and express your answer in interval notation. | [0, 1) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $a+bi = \frac{1}{i(1-i)}$ (where $a, b \in \mathbb{R}$, $i$ is the imaginary unit), calculate the value of $a+b$, the sum of the real parts of $a$ and $b$. Express your answer as a single number. | 0 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
From the four numbers -1, -2, 3, and 4, two numbers are randomly selected and multiplied together. Calculate the probability that the product is negative. Express your answer as a simplified fraction. | \dfrac{2}{3} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Find the arithmetic square root of 4. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $\cos 2a + 2\sin^2a$. Provide your answer as a simplified mathematical expression. | 1 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets $A=\{-2,-1,0,1,2\}$ and $B=\{x\mid 2^{x} > 1\}$, find the intersection $A∩B$. Express your answer as a set using standard set notation. | \{1,2\} | big_math | [
"Mathematics -> Set Theory -> Other"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Let f(x) be an odd function. When x > 0, f(x) = x^2 - 2. Calculate f(f(1)). Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a=4^{0.3}$, $b=( \frac {1}{2})^{-0.9}$, and $c=2\log_{6}2$, determine the ordering of $a$, $b$, and $c$ from smallest to largest. Express your answer using inequality symbols (e.g., a < b < c). | c < a < b | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions",
"Mathematics -> Algebra -> Intermediate Algebra -> Logarithmic Functions"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
The 'Penny's Puddings' company uses one tonne of rice to make twenty-five thousand cans of rice pudding. Each tonne of rice contains approximately fifty million grains of rice. Calculate the approximate number of grains of rice in a single can of Penny's rice pudding. Express your answer as a whole number. | 2000 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Given a > 0, b > 0, and a + 2b = 8, find the maximum value of ab. Express your answer as a single numerical value. | 8 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
The five symbols $@, *, \#, \&$, and $\wedge$ used in the equations below represent different digits.
\[ @ + @ + @ = * \]
\[ \# + \# + \# = \wedge \]
\[ * + \wedge = \& \]
Calculate the value of $\&$. Express your answer as a single digit. | 9 | big_math | [
"Mathematics -> Discrete Mathematics -> Logic"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Let $x_0 = [a], x_1 = [2a] - [a], x_2 = [3a] - [2a], x_3 = [3a] - [4a],x_4 = [5a] - [4a],x_5 = [6a] - [5a],..., $ where $a=\frac{\sqrt{2013}}{\sqrt{2014}}$. Calculate the value of $x_9$. Express your answer as an integer. | 1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
If 3 ones and 2 zeros are randomly arranged in a row, calculate the probability that the 2 zeros are not adjacent. Express your answer as a decimal value between 0 and 1, rounded to one decimal place if necessary. | 0.6 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Two lines $x+a^2y+1=0$ and $(a^2+1)x-by+3=0$ are perpendicular to each other, where $a, b \in \mathbb{R}$. Find the minimum value of $|ab|$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x)= \frac {2^{x}}{1+2^{x}}- \frac {1}{2}$, find the range of the function $y=[f(x)]+[f(-x)]$. Express your answer as a set of values. | \{-1,0\} | big_math | [
"Mathematics -> Precalculus -> Functions"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation (1+2i)z = 1-i, calculate the value of the complex number z. Express your answer in the form a + bi, where a and b are real numbers. | -\frac{1}{5} - \frac{3}{5}i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
The library's bookshelf has three layers. The first layer has 3 different math books, the second layer has 5 different Chinese books, and the third layer has 8 different English books. Now, if one book is randomly selected from the bookshelf, how many different ways are there to select it? Express your answer as a whol... | 16 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Let f_1(k) be the square of the sum of the digits of a natural number k, and let f_n(k) = f_1(f_{n-1}(k)). Compute the range of f_n(11) for n in N*. Express your answer as a set of values. | {4, 16, 49, 169, 256} | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation of line l: x-y+1=0, find the equation of the line with respect to the x-axis. Express your answer as a linear equation in the format Ax + By + C = 0. | x + y + 1 = 0 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the slope of line $l_1$ is $1$ and $l_1 \perp l_2$, find the angle of inclination of $l_2$. Express your answer in degrees as a whole number. | 135 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation x + y = 1, find the maximum value of xy. Express your answer as a decimal number. | 0.25 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given that 47^{100} is a 168-digit number, determine the number of digits in 47^{17}. Express your answer as a whole number. | 29 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
All vertices of a cube with an edge length of 2 are on the same sphere. Calculate the surface area of the sphere. Express your answer in terms of π. | 12\pi | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
In the tetrahedron \(ABCD\), \(AB = 2\), \(AC = 3\), \(AD = 4\), \(BC = \sqrt{13}\), \(BD = 2\sqrt{5}\), and \(CD = 5\). Calculate the volume of the tetrahedron \(ABCD\). Express your answer as a single number. | 4 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $\square \times 3a=-3a^{2}b$, solve for the algebraic expression that represents $\square$. Express your answer as a simplified algebraic expression. | -ab | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f\left(x\right)=\left\{\begin{array}{l}{\log_{2}(1-x), x<0}\\{2^{x}, x\geq0}\end{array}\right.$, calculate the value of $f\left(-3\right)+f(\log_{2}3)$. Express your answer as a single numerical value. | 5 | big_math | [
"Mathematics -> Algebra -> Functions"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Find the smallest positive integer $n$ that satisfies the inequality $\sqrt{n} - \sqrt{n-1} < 0.01$. Express your answer as a single integer. | 2501 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
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