instruction_seed stringlengths 40 1.39k | answer stringlengths 1 95 | source stringclasses 1
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|---|---|---|---|---|---|
Han Xin counted the number of soldiers in a troop. When grouped by three, there were two left over; when grouped by five, there were three left over; and when grouped by seven, there were four left over. What is the least number of soldiers in the troop? Express your answer as a whole number. | 53 | big_math | [
"Mathematics -> Number Theory -> Congruences"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Convert the binary number $1\,010\,011_{(2)}$ to its equivalent octal number. Express your answer in octal format with a subscript (e.g., $123_{(8)}$). | $123_{(8)}$ | big_math | [
"Mathematics -> Other -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $f(x)=\sin (2x+\varphi)+ \sqrt {3}\cos (2x+\varphi)$ is an even function on $\mathbb{R}$ and $0 < \varphi < \pi$, find the value of $\varphi$. Express your answer in radians. | \dfrac {\pi}{6} | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Evaluate the expression x^{\ln \ln x} - (\ln x)^{\ln x} given that x > 1. Express your answer as a single value. | 0 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Given $\overrightarrow{a}=(2,-3,1)$ and $\overrightarrow{b}=(4,-6,x)$, if $\overrightarrow{a} \perp \overrightarrow{b}$, find the value of $x$. Express your answer as a single integer. | -26 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Other"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Given the hyperbola $\frac{x^2}{4} - \frac{y^2}{m^2} = 1 (m > 0)$, the slope of its asymptote is $\sqrt{2}$. Find the eccentricity of this hyperbola. | \sqrt{3} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the polynomial $x^2 - kx - 6$ factors into $(x - 2)(x + 3)$, find the value of $k$. Express your answer as a single integer. | -1 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Polynomial Operations"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
What is the area of the part of the xy-plane within which x^3 y^2 - x^2 y^2 - x y^4 + x y^3 >= 0 and 0 <= x <= y? Express your answer as a single number. | \frac{1}{4} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $(2x+1)^3 = a + a_1x + a_2x^2 + a_3x^3$, find the value of $-a + a_1 - a_2 + a_3$. Provide your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
In $\triangle ABC$, given $a=5$, $b=3$, and $C=120^\circ$, find the length of side $c$. Express your answer as a whole number. | 7 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given a mapping f: {1, 2, 3} → {1, 2, 3} that satisfies f(1) < f(2) < f(3) and f(f(x)) = 3x, find the value of f(1). Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence ${{a_n}}$ that satisfies ${{a_n}=26-2n}$, find the value of $n$ that makes the sum of the first $n$ terms ${{S_n}}$ maximum. If multiple values of $n$ maximize the sum, specify both. | 12 \text{ or } 13 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given a right-angled triangle with one leg measuring eight paces and the other measuring fifteen paces, find the probability that beans thrown into this triangle will land inside the inscribed circle. Express your answer as a simplified fraction involving pi. | \dfrac{3\pi}{20} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
The frequencies of the digits after the decimal point of pi up to 100 places are given in the following table:
| Digit | $0$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ |
|-------|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|
| Frequency | $8$ | $8$ | $12$ | $11$ | $10$ | $8$ | $9$ | $8$ | $12... | 9 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = ax^5 + b*sin(x) + c, find the value of c such that f(-1) + f(1) = 2. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the function f(x) has two zero points (x1, x2) where x1 < 0 < x2 < 1, and the function g(x) = x - ln(x^2), find the maximum number of real roots of the equation f[g(x)] = 0. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Applied Mathematics -> Other",
"Mathematics -> Algebra -> Other",
"Mathematics -> Calculus -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
When a fair die is thrown twice, calculate the probability that the sum of the numbers facing up is 6. Express your answer as a simplified fraction in the form $\boxed{\dfrac{a}{b}}$. | \dfrac{5}{36} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
In a regular tetrahedron ABCD with edge length 1, G is the centroid of triangle BCD, and M is the midpoint of line segment AG. Calculate the surface area of the circumscribed sphere of the pyramid M-BCD. Express your answer in terms of pi. | \frac{3}{2}\pi | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the formula $N=8 \cdot 10^{8} \cdot x^{-\frac{3}{2}}$, find the minimum income x for the top 800 earners. Express your answer as a power of 10. | 10^4 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Using Horner's method, calculate the value of $v_1$ in the polynomial $f(x)=4x^{5}+2x^{4}+3.5x^{3}-2.6x^{2}+1.7x-0.8$ when $x=5$. Express your answer as a single numerical value. | 22 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the internal angles of a regular triangle and a dodecagon are 60° and 150° respectively, according to the conditions for plane tiling, the sum of the internal angles at a vertex must be 360°. Determine the number of possible ways to tile a plane using regular triangles and dodecagons. Express your answer as ... | 1 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Polygons"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
There are three squares: large, medium, and small, all with side lengths in integer centimeters. The perimeter of the small square is less than the side length of the medium square. When these two squares are placed on the large square, the exposed area of the large square is 10 square centimeters. Find the area of the... | 36 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
In order to understand the learning situation of 800 students, a systematic sampling method is used to select a sample of size 40. Calculate the interval between segments using the formula: $k = \frac{\text{Population Size}}{\text{Sample Size}}$. Express your answer as a whole number. | 20 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
If $n$ is any integer, the value of $\left(n+11\right)^{2}-n^{2}$ can always be divided by a specific value of $k$. What is the value of $k$ that makes this true for any integer $n$? Express your answer as a single numerical value. | 11 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
The diameter of a spherical cell is $5\times 10^{2}$ nanometers. Calculate the length of a cell chain formed by $2\times 10^{3}$ such cells. Express your answer in nanometers using scientific notation. | 1\times 10^{6} | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given non-zero vectors $\overrightarrow{m}$ and $\overrightarrow{n}$ satisfying $|\overrightarrow{m}| = 2|\overrightarrow{n}|$ and $\cos\langle\overrightarrow{m}, \overrightarrow{n}
angle = \frac{1}{3}$, find the value of the real number $t$ such that $\overrightarrow{m}$ is perpendicular to $t\overrightarrow{n}+\overr... | -6 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Convert the decimal number 0.000048 to scientific notation. Express your answer in the form $a \times 10^b$, where $a$ is the coefficient and $b$ is the exponent. | 4.8 \times 10^{-5} | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Given that \( E \) is the midpoint of side \( BC \) of the square \( ABCD \), and perpendicular lines are drawn from points \( B \) and \( D \) to \( AE \) with feet of the perpendiculars at points \( F \) and \( G \) respectively, calculate the measure of \( \angle FBG \) in degrees. | 45^\circ | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $x, y \in (0, +\infty)$ and they satisfy $\frac{1}{x} + \frac{1}{2y} = 1$, find the minimum value of $x + 4y$. Express your answer as a single mathematical expression. | 3 + 2\sqrt{2} | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of real solutions to the equation $\left|x^{2}-3 x+2\right|+\left|x^{2}+2 x-3\right|=11$. Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a} = (x-1, 2)$ and $\overrightarrow{b} = (2, 1)$, find the value of x such that $\overrightarrow{a} \perp \overrightarrow{b}$. Express your answer as a single numerical value. | 0 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence {a_n} where a_1 = 5 and S_{n-1} = a_n for n >= 2, derive the general formula for S_n. Express your answer as a formula in terms of n. | 5 \cdot 2^{n-1} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
How many triples of positive integers (a, b, c) are there such that a! + b^3 = 18 + c^3? Express your answer as a single integer. | 0 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Two triangles are congruent, and the perimeter of one triangle is 5. Find the perimeter of the other triangle. Express your answer as a whole number. | 5 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Find the angle of inclination (in radians) corresponding to the slope of the line given by the equation $x + y + 1 = 0$. Express your answer as an exact value in radians. | \frac{3\pi}{4} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
A tangent is drawn from a point on the line $y=x+1$ to the circle $(x-3)^{2}+y^{2}=1$. Calculate the minimum distance from that point to the point of tangency. | \sqrt{7} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the curve y = x^3 - 2x + 4, find the angle of inclination (in degrees) of the tangent line at the point (1, 3). Express your answer in degrees, including the degree symbol (°). | 45° | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets $M=\{1,2,3\}$ and $N=\{2,3,4\}$, find the intersection of sets M and N, denoted as $M\cap N$. Express your answer as a set using standard set notation. | \{2,3\} | big_math | [
"Mathematics -> Discrete Mathematics -> Logic"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence {a_n} with a common difference d!= 0, and a_5^2 + a_7^2 + 16d = a_9^2 + a_11^2, calculate the sum of the first 15 terms S_15 of the sequence. | 15 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors a = (1, 2) and b = (x, -2), and the scalar product a · b = -3, find the magnitude of a + b. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
For the quadratic equation $x^{2}+mx-2=0$, given that one of its solutions is $x=1$, find the value of the other solution. Express your answer as a single number. | -2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Quadratic Functions"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
In the fraction (x+y)/(x^2+y^2), if the values of x and y are both enlarged by a factor of 3, describe the resulting change in the fraction's value. Specify if the value is enlarged or reduced and by what factor. Express your answer in the format "enlarged/reduced by a factor of [factor]". | reduced by a factor of 3 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given the system of equations:
\[
x + 1 = y - 8
\]
\[
x = 2y
\]
Find the value of \( x + y \). Express your answer as a single integer. | -27 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of integer solutions x that satisfy the equation (x^2 - x - 1)^(x+2) = 1. | 4 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
If $$\int_{1}^{a} (2x + \frac{1}{x}) \, dx = 3 + \ln 2$$ and $a > 1$, then find the value of the real number $a$. Express your answer as a single real number. | 2 | big_math | [
"Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $f(x)$ is an even function and $\int^{6} f(x) \, dx = 8$, calculate the value of $\int_{-6}^{6} f(x) \, dx$. Express your answer as a single number. | 16 | big_math | [
"Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z=1+i$ (where $i$ is the imaginary unit), calculate the conjugate of $\frac{2}{z}-z^{2}$. Express your answer as a complex number in the form $a+bi$. | 1+3i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
The mean (average) of the four integers $78, 83, 82$, and $x$ is 80. Calculate the value of $x$. Express your answer as a single integer. | 77 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Find the minimum value of the function f(x) = 2cos^2 x + sin x. Express your answer as a single number. | -1 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = ax^2 - (2a+1)x + a + 1, find the range of real number x such that the inequality f(x) < 0 holds true for all a in [-1,1]. Express your answer as an interval or a range of values. | (1,2) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Write the number "seven thousand twenty-two" in standard numerical form. | 7022 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $| \overrightarrow{a}|=3$, $| \overrightarrow{b}|=2$, and $| \overrightarrow{a}- \overrightarrow{b}|= \sqrt {19}$, calculate the projection of $\overrightarrow{a}$ on $\overrightarrow{b}$. Express your answer as a fraction. | - \frac {3}{2} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the random variable $\\xi$ satisfies $D\\xi=2$, calculate the variance of $2\\xi+3$ ($D(2\\xi+3)$). Express your answer as a single number. | 8 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
If the ratio of the radii of three spheres is 1:2:3, then how many times is the volume of the sphere with the largest radius compared to the sum of the volumes of the other two spheres? Express your answer as a single numerical value. | 3 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Volume"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Let $U=R$, $A=\{x|x \gt 0\}$, $B=\{x|x\leqslant 1\}$. Find the intersection of $A$ and the complement of $B$ in $U$, denoted as $A$⋂$(\complement _{U}B)$. Express your answer as a set in the format {x|condition}. | \{x|x \gt 1\} | big_math | [
"Mathematics -> Discrete Mathematics -> Logic"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Find the value of k for which the equation (x-1)/(x-2) = (x-k)/(x-6) has no solution. Express your answer as a single integer. | 5 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Let the function $f(x) = \begin{cases} 2^{x}, & x<0 \\ g(x), & x>0 \end{cases}$. Given that $f(x)$ is an odd function, calculate the value of $g(3)$. Express your answer as a single value, including the negative sign if applicable. | -\frac{1}{8} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Given a sequence {a_n} where a_n^2 + a_{n+1}^2 = 1 for any positive integer n, and the first term a_1 = 1, calculate the sum of the maximum and minimum values of the sum of the first 2014 terms, S_{2014}. Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
In a geometric sequence, the first term $a_{1}=\dfrac{1}{2}$, the common ratio $q=\dfrac{1}{2}$, and the nth term $a_{n}=\dfrac{1}{32}$. Find the value of $n$. Express your answer as a whole number. | 5 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Points \(M\) and \(N\) are the midpoints of two sides of the rectangle. What fraction of the rectangle's area is shaded? Express your answer as a simplified fraction. | \frac{1}{4} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
What is the remainder when $29 \cdot 169^{1990}$ is divided by 11? Express your answer as a single integer. | 7 | big_math | [
"Mathematics -> Number Theory -> Congruences"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Let the function $f(x)=2a^{x}-b^{x}$, where $b\geqslant 2a > 0$. Find the interval where the root of $f(x)$ is located. Express your answer using standard interval notation. | (0,1] | big_math | [
"Mathematics -> Calculus -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given that 8 kilometers is approximately 5 miles, convert 1.2 kilometers to miles. Express your answer in miles, rounded to two decimal places if necessary. | 0.75 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given the real numbers $x$ and $y$ that satisfy the equation $2^x + 2^y = 1$, find the maximum value of $x + y$. Express your answer as a single numerical value. | -2 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
In $\triangle ABC$, it is known that $A=30^{\circ}$, $B=45^{\circ}$, and $a=1$. Use the Law of Sines to calculate the length of side $b$. Express your answer as a simple radical if necessary. | \sqrt{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\{a_{n}\}$ is an arithmetic sequence, if $a_{30}=100$ and $a_{100}=30$, calculate the common difference of $\{a_{n}\}$. Express your answer as a single numerical value. | -1 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the sum of the first n terms of an arithmetic sequence {a_n} is S_n, and a_3 + a_7 = 10, find the value of S_9. Express your answer as a single numerical value. | 45 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Sequences and Series"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Given that on the interval (0, +∞), the function f(x) is a decreasing function, and when x > 0, f(x) > 0. If 0 < a < b, analyze the relationship between bf(a) and af(b). Express your answer as an inequality. | bf(a) > af(b) | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
There are seven seats in a row, with persons A and B taking a seat. The condition is that there must be at least one empty seat between A and B. Find the number of different seating arrangements that satisfy this condition. Express your answer as a whole number. | 30 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Solve for the complex number z that satisfies the equation (1+2i)z = 4+3i. Express your answer in the form a+bi, where a and b are real numbers. | 2-i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
During the Reading Month activity, each person needs to choose several books to read from 5 social science books and 4 natural science books, with the requirement that the number of social science books chosen must be one more than the number of natural science books. Calculate the total number of different book select... | 121 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
When 0 < x < 1, find the maximum value of the function y = x(1-x). Express your answer as a single number. | \frac{1}{4} | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
A disease control station is conducting a health survey among students, using stratified sampling to draw a sample. A high school has a total of 2000 students, and a sample with a size of 200 is drawn. If there are 103 males in the sample, calculate the total number of female students in this high school. Express your ... | 970 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
A fair six-sided die is rolled twice, and the resulting numbers are recorded as $a$ and $b$. Calculate the probability that the quadratic equation $ax^2 + bx + 1 = 0$ has real solutions. Express your answer as a simplified fraction. | \frac{19}{36} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
For the function $f(x)$ which satisfies $f(2+x) = f(2-x)$ for all real numbers $x$, and $f(x)$ has exactly four distinct zeros, calculate the sum of these zeros. Express your answer as a single integer. | 8 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
In an arithmetic sequence {a_n}, if a_2 + a_8 = 10, find the value of a_5. Express your answer as a single numerical value. | 5 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Find the coefficient of $x^2$ in the expansion of $(x+1)^5(x-2)$. Express your answer as a single integer. | -15 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
How many prime factors does the number 2008006 have? Express your answer as a whole number. | 6 | big_math | [
"Mathematics -> Number Theory -> Factorization"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the tangent line to the curve y = xln(x) at x = e. Express your answer in the form y = mx + b, where m is the slope and b is the y-intercept. | y = 2x - e | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
If $x^{2}+(x+1)^{7}=a_{0}+a_{1}(x+2)+a_{2}(x+2)^{2}+\ldots+a_{7}(x+2)^{7}$, calculate the value of $a_{2}$. Express your answer as an integer. | -20 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $(x-1)^5+5(x-1)^4+10(x-1)^3+10(x-1)^2+5(x-1)$. Express your answer as a simplified algebraic expression. | x^5-1 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.03125 | 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 -> Surface Area"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the tangent line to the curve y = x^3 - 2x at the point (1, -1). Express your answer in standard form (Ax + By + C = 0). | x - y - 2 = 0 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Farmer Fatima rears chickens and goats. Today she returned from market and said, "I sold 80 animals, and now there are 200 fewer legs on my farm than before!" Calculate the number of goats she sold. | 20 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\frac{b}{a}=\frac{5}{13}$, find the value of $\frac{a-b}{a+b}$. Express your answer as a simplified fraction. | \frac{4}{9} | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence {a_n} with the sum of the first n terms as S_n, a_5 and a_7 are the two roots of the equation x^2 - 4x + k = 0 with respect to x. Calculate the sum of the first 11 terms, S_11. | 22 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Given the circumference of the circle $C$: $(x-1)^{2}+(y-2)^{2}=4$, find the maximum distance between point $P(3,3)$ and the moving point $M$ on the circle $C$. Express your answer as a numerical value, including the square root symbol if necessary. | \sqrt{5}+2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the result of $2^{0}-1$. Express your answer as a single integer. | 0 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given a geometric sequence with a common ratio of 2, and the sum of the first 4 terms is 1, find the sum of the first 8 terms of this geometric sequence. Express your answer as a single numerical value. | 17 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Find the domain of the function $f(x)=\log_{2}(1-2x) + \frac{1}{x+1}$. Express your answer in interval notation. | (-\infty,-1)\cup(-1, \frac{1}{2}) | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
The function f(x) = 5^{|x|} is translated 1 unit to the right, resulting in the graph of y = g(x). Find the line of symmetry for g(x). Express your answer as an equation of a line (e.g., x = a). | x = 1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given a = 5 - 2√6, calculate the value of a^2 - 10a + 1. Express your answer as a single number in the form \boxed{[answer]}. | 0 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the result of $\left(-3\right)\times 2$. Express your answer as an integer. | -6 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation of the parabola y = -2x^2, calculate the exact coordinates of the focus. Express your answer as an ordered pair (x, y). | (0, -\frac{1}{8}) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Given that the geometric sequence $(b_{n})$ with common ratio $2$ satisfies $\log_{2}b_{n} \cdot \log_{2}b_{n+1} - 2 = n^2 + 3n$, find the first term of the sequence $(b_{n})$. Express your answer as a single numerical value. | 4 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
In an acute triangle ∆ABC, the lengths of the sides opposite angles A and B are a and b respectively. Given that 2a sin(B) = √3b, calculate the measure of angle A in degrees. | 60 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Solve the equation |x-1| + |x-5| = 4 and express the solution set for x in interval notation or as an inequality. | [1, 5] | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Given the point G(x, x-5) is in the third quadrant of the rectangular coordinate system, determine the range of x that satisfies this condition. Express your answer as an inequality. | x < 0 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Find the minimum value of the function $y=\frac{6}{2^{x}+3^{x}}$ within the domain $-1 \leqslant x \leqslant 1$. Express your answer as a single value. | \frac{6}{5} | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Express the number 1.5 million in scientific notation. Provide your answer in the format a × 10^b. | 1.5 × 10^6 | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
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