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 |
|---|---|---|---|---|---|
There are 4 students to be divided into class A and class B. Each class can accept no more than 3 students. Given that student A will not go to class A, determine the number of different distribution methods. Express your answer as a whole number. | 7 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
The time difference between Paris and Beijing is -7 hours. If the time in Beijing is 5:00 on October 26, calculate the time in Paris. Express your answer in the format HH:MM and specify the date (October 25 or 26). | 22:00 on October 25 | big_math | [
"Applied Mathematics -> Math Word Problems"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Five numbers form a geometric sequence with a common ratio of $-2$. Calculate the ratio of the sum of the odd terms to the sum of the even terms. Express your answer as a simplified fraction. | -\dfrac{21}{10} | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Sequences and Series"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Solve the inequality |x-1|-|x-5| < 2 and express the solution set in interval notation. | (-\infty,4) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
There are 5 different textbooks, including 2 Chinese books, 2 math books, and 1 physics book. If these books are placed on the same shelf in a row such that books of the same subject are not adjacent, how many arrangements are possible? Express your answer as a whole number. | 48 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
In a square ABCD with side length 8, a circle passes through points A and D and is tangent to BC. Find the radius of the circle. Express your answer as a numerical value with a possible square root. | 4\sqrt{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Among the real numbers $\frac{{22}}{7}$, $\sqrt{5}$, $0$, $3.1415$, $\sqrt{16}$, $3\pi$, $3.1010010001\ldots$ (where the number of zeros between adjacent ones increases by $1$ each time), how many are irrational numbers? Express your answer as a whole number. | 3 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
Point P(2,-5) is symmetric with respect to the x-axis. Find the coordinates of this symmetric point. Express your answer as an ordered pair (x, y). | (2,5) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
A circle with radius $r$ is tangent to sides $AB, AD$, and $CD$ of rectangle $ABCD$ and passes through the midpoint of diagonal $AC$. Calculate the area of the rectangle in terms of $r$. Express your answer in the form of $Ar^2$, where $A$ is a numerical coefficient. | 8r^2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = -x^3 + 2x, calculate its derivative. Express your answer as a polynomial function, including all terms and coefficients. | f'(x) = -3x^2 + 2 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression:
$$
\left(1+\frac{1}{2}\right)\left(1+\frac{1}{3}\right)\left(1+\frac{1}{4}\right)\left(1+\frac{1}{5}\right)\left(1+\frac{1}{6}\right)\left(1+\frac{1}{7}\right)\left(1+\frac{1}{8}\right)\left(1+\frac{1}{9}\right)
$$
Express your answer as a single number. | 5 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
For the parabola $y^2=2px$ ($p>0$), describe the positional relationship between the circle with the focal radius of the parabola as its diameter and the y-axis. Express your answer as a geometric term. | Tangent | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function y = 1 / (x + 3), what is the domain of the independent variable x? Express your answer as an inequality or equality statement about x, using mathematical notation (e.g., ≠, >, <) as necessary. | x ≠ -3 | big_math | [
"Mathematics -> Precalculus -> Functions"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given the ellipse $\frac {x^{2}}{16}+ \frac {y^{2}}{9}=1$, find the equation of the line containing the chord with midpoint $A(2, \frac {3}{2})$. Express your answer as a linear equation in the form $Ax + By + C = 0$. | 3x+4y-12=0 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the line $2ax-by+2=0$ ($a > 0, b > 0$) passes through the center of the circle $x^2+y^2+2x-4y+1=0$, find the minimum value of $\frac{1}{a}+\frac{1}{b}$. Express your answer as a single number. | 4 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $f'(x_0) = 2$, calculate the limit $$\lim_{k \to 0} \frac {f(x_{0}-k)-f(x_{0})}{2k}.$$ Express your answer as a single number. | -1 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a > 0$ and $b > 0$, with $m=\frac{a}{\sqrt{b}}+\frac{b}{\sqrt{a}}$, $n=\sqrt{a}+\sqrt{b}$, and $p=\sqrt{a+b}$, determine the order of magnitude of $m$, $n$, and $p$. Express your answer as an inequality using \geqslant or > notation. | m \geqslant n > p | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
If \( y = 3 \), calculate the value of \( \frac{y^{3} + y}{y^{2} - y} \). Express your answer as a single number. | 5 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $\sin 50^{\circ}\sin 70^{\circ}-\cos 50^{\circ}\sin 20^{\circ}$ using trigonometric identities. Express your answer as a simplified fraction, if possible. | \frac{1}{2} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Mary sells machines, selling 1 machine on the first day, 3 machines on the second day, 5 machines on the third day, and so on, with each day selling 2 more machines than the previous day. Calculate the total number of machines she has sold after 20 days. Express your answer as a whole number. | 400 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
In acute triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $a = 2b$ and $\sin B = \frac{\sqrt{3}}{4}$, calculate the measure of angle $A$ in radians. | \frac{\pi}{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Given the arithmetic sequence $\{2-3n\}$, find the common difference $d$. Express your answer as a single integer. | -3 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
What is the degree measure of the smaller angle formed by the hands of a clock at 10 o'clock? Express your answer in degrees, including the degree symbol (°). | 60° | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
A line l passes through the point M(1,2) and intersects the circle C: (x-2)^2 + y^2 = 9 at points A and B. C is the center of the circle. Find the equation of line l when the distance from point C to line l is maximized. Express your answer as a line equation in standard form (e.g., Ax + By + C = 0). | x - 2y + 3 = 0 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a > 0$ and $b > 0$, find the minimum value of $6 \sqrt {ab}+ \frac {3}{a}+ \frac {3}{b}$. Express your answer as a single number. | 12 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given the expression $x^{2}-8x+a$, find the value of $a$ that allows it to be written as a perfect square trinomial. Express your answer as a single integer. | 16 | big_math | [
"Mathematics -> Algebra -> Algebra -> Quadratic Functions"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of the expression \( 43 - 41 + 39 - 37 + 35 - 33 + 31 - 29 \). Express your answer as a single integer. | 8 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Find the real part of the complex number $$\frac {i+5}{1+i}$$, where $i$ is the imaginary unit. Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Given a sample with a capacity of 32, and knowing that the frequency rate of a certain group of samples is 0.25, calculate the frequency of that group. Express your answer as a whole number. | 8 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Find the numbers that satisfy the condition y^2 = 4. Express your answer as \\pm value. | \\pm 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
The constant term of the product of the binomial $(x-m)(x+7)$ is 14. Find the value of $m$. Express your answer as a single integer. | -2 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
What is the largest prime p that makes \sqrt{17p + 625} an integer? Express your answer as a single integer. | 67 | big_math | [
"Mathematics -> Number Theory -> Prime Numbers"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Two poles of heights 20 and 80 units are 100 units apart. Lines are drawn from the top of each pole to the base of the other. Find the height of the intersection point of these lines. Express your answer in units as a whole number. | 16 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression: $\frac{bx(a^2x^2 + 2a^2y^2 + b^2y^2) + ay(a^2x^2 + 2b^2x^2 + b^2y^2)}{bx + ay}$. Express your answer as a simplified algebraic expression. | (ax + by)^2 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Solve for the number that should be placed in the box to make the following statement true: \(8+\frac{7}{\square}+\frac{3}{1000}=8.073\). Express your answer as a whole number. | 100 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Fractions"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
There are $n$ identical electronic components connected in parallel in a circuit. The probability that each electronic component works normally is $0.5$. Calculate the minimum value of $n$ such that the probability of the entire circuit working normally is not less than $0.95$. Express your answer as a single integer. | 5 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Given that point A is symmetric to point B (1, 2) about the line x + y + 3 = 0, find the coordinates of point A. Express your answer as an ordered pair (x, y). | (-5, -4) | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given that x≥0, y≥0, and x+3y=9, find the maximum value of x^2y. Express your answer as a single number. | 36 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the random variable ξ ~ N(5, σ^2), and P(3 ≤ ξ ≤ 7) = 0.4, calculate the probability P(ξ > 7). Express your answer as a decimal value between 0 and 1, rounded to 1 decimal place. | 0.3 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given an ellipse $\frac{x^{2}}{6} + \frac{y^{2}}{5} = 1$ and a point $P(2, -1)$ inside it, if a chord passing through point $P$ is bisected by $P$, find the equation of the line containing this chord. Express your answer as a linear equation in the form $Ax + By + C = 0$. | 5x - 3y - 13 = 0 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = sin(x) + sqrt(3)*cos(x), the graph of the function y = f(x + phi) is symmetric about the line x = 0. Find the exact value of phi in radians. | \dfrac{\pi}{6} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
How many three-digit numbers are perfect squares? Express your answer as a whole number. | 22 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
If four people, A, B, C, and D, line up in a row, calculate the total number of arrangements in which B and C are on the same side of A. Provide your answer as a single integer. | 16 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Arrange the integers from 1 to 2019 that can be divided by 5 with a remainder of 2 and by 7 with a remainder of 2 in ascending order to form a sequence. Calculate the number of terms in this sequence. | 58 | big_math | [
"Mathematics -> Number Theory -> Congruences"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Given a complex number $z$ satisfies $\left(1+i\right)z=3+2i$, find the imaginary part of $\overline{z}$. Express your answer as a single number. | \frac{1}{2} | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
A student has a probability of $\dfrac {1}{3}$ to pass a computer test. If the student takes the test $3$ consecutive times, what is the probability that he passes exactly $1$ time? Express your answer as a simplified fraction. | \dfrac{4}{9} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z=\frac{{1+{i^4}}}{{1-i}}$, calculate the conjugate of $z$, denoted as $\overline{z}$. Express your answer as a complex number in the form $a + bi$. | 1-i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
In an algebra class, five students scored 100 points each on a test, and every student scored at least 60 points. If the mean score of the class was 76 points, what is the smallest possible number of students in the class? Express your answer as a whole number. | 13 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
If x and log10(x) are real numbers and log10(x) < 0, determine the range of x. Express your answer as an inequality. | 0 < x < 1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\overrightarrow{a}$ and $\overrightarrow{b}$ are unit vectors, and the angle between them is 60°, calculate the magnitude of the vector $3\overrightarrow{a} + \overrightarrow{b}$. Express your answer as a numerical value, including the square root symbol if necessary. | \sqrt{13} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
In the complex plane, find the coordinates of the point corresponding to the complex number $\frac{1+i}{i}$ ($i$ is the imaginary unit). Express your answer in the format (x, y). | (1,-1) | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z$ such that $zi=-1+i$, calculate the value of $z$. Express your answer as a complex number in the form $a+bi$, and include the boxed notation around your answer. | 1+i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the line passing through the point (1, 3) with a slope of 1. Express your answer in the standard form of a linear equation (Ax + By + C = 0). | x-y+2=0 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function y = |log2(x)| with a domain of [1/m, n] (m, n are positive integers) and a range of [0, 2], calculate the number of integer pairs (m, n) that satisfy these conditions. | 7 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Using the digits 4, 7, and 9, how many different 3-digit whole numbers can be formed, assuming that no digit can be repeated in a number? Express your answer as a whole number. | 6 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Permutations"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Among the four real numbers $-\sqrt{3}$, $0$, $2$, $-3$, what is the smallest number? Express your answer as a single number. | -3 | big_math | [
"Mathematics -> Other -> Other"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Given \(f(x) = x^{2} + 2xf''(1) - 6\), calculate \(f''(1)\). Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Let $f_{0}(x) = \cos x$, $f_{1}(x) = f'_{0}(x)$, $f_{2}(x) = f'_{1}(x)$, $f_{n+1}(x) = f'_{n}(x)$ $(n \in \mathbb{N})$. Find the function $f_{2012}(x)$. Express your answer as a trigonometric function of x. | \cos x | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Given a geometric sequence {a_n} where all terms are positive, vectors $$\\overrightarrow {a}=(a_{4}，a_{3})$$ and $$\\overrightarrow {b}=(a_{7}，a_{8})$$ are known, and $$\\overrightarrow {a}\\cdot \\overrightarrow {b}=4$$. Calculate the value of log_2 a_1 + log_2 a_2 + … + log_2 a_10. | 5 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
A parabola is given by the equation $y^{2}=2px (p > 0)$, with a focus at point $F$. A line with a slope angle of $\frac{\pi}{4}$ passes through the focus $F$, intersecting the parabola at points $A$ and $B$. If the perpendicular bisector of chord $AB$ passes through the point $(0, 2)$, calculate the value of $p$. Expre... | \frac{4}{5} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the number of trailing zeros in the factorial of 2006. Provide your answer as a single integer. | 500 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
In a pocket, there are 3 white balls and 2 black balls of the same size. If one ball is randomly drawn without replacement for two consecutive times, and the first ball drawn is white, what is the probability of drawing a white ball the second time? Express your answer as a simplified fraction. | \frac{1}{2} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the probability of a shuttlecock's weight being less than 4.8g is 0.3 and the probability of its weight being less than 4.85g is 0.32, calculate the probability that the weight is within the range [4.8, 4.85] (g). Express your answer as a decimal value. | 0.02 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given the set M={y|y=1-sinx, x∈R} and N={x|y=ln(2-x)}, find the union M∪N. Express your answer in interval notation. | (-∞, 2] | big_math | [
"Mathematics -> Precalculus -> Functions"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Find the interval in which the function $y=\log_{\frac{1}{2}}(x^{2}-5x+6)$ is strictly increasing. Express your answer as an interval. | (-\infty,2) | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Find the imaginary part of the complex number $i(1+i)$. Express your answer as a numerical value. | 1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Given $a+b+c=3$, and $a^2+b^2+c^2=3$, find the value of $a^{2008}+b^{2008}+c^{2008}$. Express your answer as a single integer. | 3 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the inverse proportion function y = 4/x and three points on its graph (x1, y1), (x2, y2), (x3, y3), where x1 < x2 < 0 and x3 > 0, determine the relationship between y1, y2, and y3. Express your answer using inequality symbols (e.g., y1 < y2 < y3). | y2 < y1 < y3 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
The function $f(x)$ satisfies $f(x) = f(2-x)$, where $x \in \mathbb{R}$, and when $x \leq 1$, $f(x) = x^3 - x^2 - 4x + 4$. Calculate the sum of all real roots of the equation $f(x) = 0$. Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Polynomial Equations"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
An equilateral triangle and a regular hexagon have perimeters of the same length. Calculate the ratio of the area of the equilateral triangle to the area of the regular hexagon. Express your answer as a simplified ratio. | \frac{2}{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Polygons"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given that {a_n} is an arithmetic sequence and a_2 + a_8 = 12, find a_5. Express your answer as a single number. | 6 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
If the axial section of a cone is an equilateral triangle with an area of $\sqrt{3}$, calculate the total surface area of the cone. Express your answer in terms of $\pi$. | 3\pi | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Given that x > 0, y > 0, and x + y = 2, find the minimum value of the expression 1/x + 4/y. Express your answer as a single number. | \frac{9}{2} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
The equation of hyperbola $C$ is: $x^{2}-y^{2}=a^{2}(a > 0)$. The distance from one of the foci of hyperbola $C$ to an asymptote is $2$. Find the value of the real number $a$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Given a complex number $z$ that satisfies the equation $\frac{z+i}{-2{{i}^{3}}-z}=i$, determine the value of $|\overline{z}+1|$. Express your answer as a fraction with a square root. | \frac{\sqrt{2}}{2} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression \( \left( 7 + 4 \sqrt{3}
ight)^{\frac{1}{2}} - \left( 7 - 4 \sqrt{3}
ight)^{\frac{1}{2}} \). Express your answer as a simplified algebraic expression. | 2 \sqrt{3} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Find the ratio of the radius of the circumscribed circle to the radius of the inscribed circle of a regular hexagon. Express your answer as a simplified ratio, using a colon (:) to separate the two parts of the ratio. | 2: \sqrt{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Polygons"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Find the coefficient of the $x$ term in the expansion of $(1-x)(1+x)^{5}$. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Given that the product of two numbers x and y is equal to their sum, express x in terms of y. Provide your answer in the form of an algebraic expression. | \frac{y}{y-1} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
If \( x \) is a complex number satisfying \( x^{2} + x + 1 = 0 \), what is the value of \( x^{49} + x^{50} + x^{51} + x^{52} + x^{53} \)? Express your answer as a single integer. | -1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number w satisfies w-1=(1+w)i (where i is the imaginary unit), find the value of w. Express your answer as a complex number in standard notation (e.g., a+bi). | i | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Given the quadratic function $y=2(x-1)^2-3$, find the coordinates of its vertex. Express your answer as an ordered pair $(h, k)$. | (1, -3) | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Quadratic Functions"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function \( f(x) = a \sin x - b \sqrt[3]{x} + c \ln \left(x + \sqrt{x^2 + 1}\right) + 4 \), where \( a, b, c \) are real numbers, and \( f(\lg \log_{3} 10) = 5 \), find the value of \( f(\lg \lg 3) \) using the properties of odd functions. Express your answer as a single integer. | 3 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
The imaginary unit is $i$, and the complex number $\frac {2i}{1-i}$ corresponds to a point in the complex plane. Find the distance from this point to the origin. Express your answer as a numerical value. | \sqrt{2} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
In the regular triangular prism ABC-A1B1C1, there are six diagonals on the sides: AB1, A1B, BC1, B1C, C1A, CA1. Given that AB1 ⊥ BC1, how many other pairs of these diagonals are also perpendicular to each other? Express your answer as a whole number. | 5 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $\left(\cos \theta +i\sin \theta \right)^{n}=\cos n\theta +i\sin n\theta$, if a complex number $z$ satisfies $z\cdot\left(\cos\frac{\pi}{9}+i\sin\frac{\pi}{9}\right)^{6}=2$, calculate the magnitude of $z$, $|z|$. Express your answer as a numerical value. | 2 | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Lighthouses A and B are both a km away from the ocean observation station C. Lighthouse A is located 20° north of east from station C, and lighthouse B is located 40° south of east from station C. Calculate the distance between lighthouse A and lighthouse B. Express your answer in kilometers (km). | \sqrt{3}a \text{ km} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x) = \log_2 (x + 1)$, if $f(\alpha) = 1$, then find the value of $\alpha$. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Logarithmic Functions"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given the parametric equations of a line, \( \begin{cases} x=2-3t \\ y=1+2t \end{cases} \), where \(t\) is a parameter, derive the general form of the line's equation. Express your answer in the form \(Ax + By + C = 0\), where \(A\), \(B\), and \(C\) are integers. | 2x + 3y - 7 = 0 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Evaluate the expression \((1)^{10} + (-1)^{8} + (-1)^{7} + (1)^{5}\). Provide your answer as a single integer. | 2 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Determine the number of common tangent lines for the following two circles:
$$x^{2}+y^{2}=9$$
$$x^{2}+y^{2}-8x+6y+9=0$$
Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Find the coefficient of the term containing x in the expansion of (1-2x)^4. Express your answer as a single integer. | -8 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
An even function f(x) is monotonic on the interval [0, a] (a > 0) and satisfies f(0) * f(a) < 0. Calculate the total number of zeros of the function f(x) in the interval [-a, a]. | 2 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
A body's motion equation is given by $s = t^2 - t + 2$ (where $s$ is in meters, $t$ is in seconds). Find the instantaneous velocity of the object at $t = 4$ seconds. Express your answer in meters/second as a whole number. | 7 \text{ meters/second} | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given the equations of two lines, $4x + y = 6$ and $3x + 2y = 7$, find the coordinates of their intersection point. Express your answer as an ordered pair (x, y). | (1, 2) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence {a_n}, the sequence {b_n} is defined by: b_1 = a_1, b_2 = a_2 + a_3, b_3 = a_4 + a_5 + a_6, etc. If the limit of b_n/n^3 as n approaches infinity is 2, find the common difference d of the sequence {a_n}. | 4 | big_math | [
"Mathematics -> Sequences and Series -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
In the chart, each number below the top row is the positive difference of the two numbers to the right and left in the row immediately above it. Calculate the value of x. Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Discrete Mathematics -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
How many integer values of x satisfy the compound inequality:
\[
\frac{x-1}{3} < \frac{5}{7} < \frac{x+4}{5}
\]
Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
From 4 boys and 3 girls, 4 people are to be selected to participate in a seminar. If the selection must include both boys and girls, calculate the total number of different selections. Express your answer as a whole number. | 34 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Combinations"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Find the minimum value of $n+d$ for an arithmetic sequence $\{a_n\}$ with a common difference $d$ and positive integral terms, given that $a_1 = 1$ and $a_n = 51$. Express your answer as a single integer. | 16 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
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