instruction_seed stringlengths 40 1.39k | answer stringlengths 1 95 | source stringclasses 1
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Three machines P, Q, and R working together can complete a task in x hours. If machine P works alone, it requires x + 6 hours to complete the task. Machine Q needs x + 1 hours to complete the task alone, and machine R requires 2x hours to complete the task alone. Find the value of x. Express your answer as a fraction o... | \frac{2}{3} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
On a grid paper, there are two points $A$ and $B$. If a Cartesian coordinate system is established with point $B$ as the origin, the coordinates of point $A$ are $\left(-3,4\right)$. Find the coordinates of point $B$ if a Cartesian coordinate system is established with point $A$ as the origin. Express your answer as an... | \left(3,-4\right) | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
In triangle △ABC, a=2, ∠A=π/6, and b=2√3. Using the Law of Sines and the properties of triangles, calculate the exact value of angle C in radians. | \frac{\pi}{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function y = a^(x-1) + 3, where a > 0 and a!= 1, find the coordinates of the fixed point P that all such graphs pass through. Express your answer as an ordered pair (x, y). | (1,4) | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = sin(x) + 2x, find the minimum value of the real number t such that for any x1, x2 in the interval [-π, π], the condition |f(x1) - f(x2)| ≤ t is satisfied. Express your answer as a single mathematical expression. | 4π | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x) = \\begin{cases} \\log_{9}x, x > 0 \\\\ 4^{-x} + \\frac{3}{2}, x \\leqslant 0 \\end{cases}$, calculate the value of $f(27) + f(-\\log_{4}3)$. Express your answer as a single number. | 6 | big_math | [
"Mathematics -> Precalculus -> Functions"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
There is a 5-cent coin, four 2-cent coins, and eight 1-cent coins. Calculate the number of ways to make 9 cents using these coins. Express your answer as a whole number. | 7 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Let $a$ be a real number. If $\frac{1+2i}{a+i}$ is a real number, find the value of $a$. Express your answer as a single value. | \frac{1}{2} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Stephen had a 10:00 a.m. appointment 60 km from his home. He averaged 80 km/h for the trip and arrived 20 minutes late for the appointment. At what time did he leave his home? Express your answer in 12-hour clock format (e.g., 9:00 a.m.). | 9:35 a.m. | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
A class consists of 26 boys and 24 girls. If one student is to be chosen as the mathematics representative, how many different ways are there to select this student? Express your answer as a whole number. | 50 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given a rhombus with diagonals that are the real roots of the quadratic equation $x^{2}-10x+m=0$, and an area of 11, calculate the length of the side of the rhombus. Express your answer in simplified radical form if necessary. | \sqrt{14} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Solve the system of linear equations $\left\{{\begin{array}{l}{x+y=6}\\{x=2y}\end{array}}\right.$ and express your answer as an ordered pair (x, y). | (4,2) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Systems of Equations"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
For the inequality system about $x$: $\left\{{\begin{array}{l}{-\frac{1}{3}x>\frac{2}{3}-x,}\\{\frac{1}{2}x-1<\frac{1}{2}(a-2)}\end{array}}\right.$, find the maximum value of $a$ such that there are exactly three integer solutions for $x$. Express your answer as a single integer. | 5 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, point \( P \) is on the number line at 3 and \( V \) is at 33. The number line between 3 and 33 is divided into six equal parts by the points \( Q, R, S, T, U \). Find the sum of the lengths of \( P S \) and \( T V \). Express your answer as a whole number. | 25 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Given $$\overrightarrow {a} = (\cos \frac {2}{3}\pi, \sin \frac {2}{3}\pi)$$, $$\overrightarrow {OA} = \overrightarrow {a} - \overrightarrow {b}$$, $$\overrightarrow {OB} = \overrightarrow {a} + \overrightarrow {b}$$, and that $\triangle OAB$ is an isosceles right triangle with $O$ as the right angle vertex, calculate ... | 1 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Let set $A=\{1,2,3,5,7\}$, $B=\{x\in\mathbb{Z}|1<x\leq6\}$, and the universal set $U=A\cup B$. Find the intersection of $A$ and the complement of $B$ in $U$, denoted as $A\cap C_{U}B$. Express your answer as a set of integers using standard set notation. | \{1,7\} | big_math | [
"Mathematics -> Discrete Mathematics -> Logic"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Given that \( x = 4 \) and \( y = 3x \), calculate the value of \( y \). Express your answer as a single integer. | 12 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Simple Equations"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
In the sequence {a_n}, a_1 = 1 and a_n - a_{n-1} = n, where n is a positive integer. Find the value of a_8. | 36 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Given \( x, y, z \in \mathbf{Z}_{+} \) and \([x, y] = 72\), \([x, z] = 600\), \([y, z] = 900\), where \([m, n]\) represents the least common multiple of integers \( m \) and \( n \), find the number of ordered triples \((x, y, z)\) satisfying these conditions. Express your answer as a single integer. | 15 | big_math | [
"Mathematics -> Number Theory -> Least Common Multiples (LCM)"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the solution set of the quadratic inequality $x^{2}+bx-a < 0$ is ${x|-2 < x < 3}$, find the value of $a+b$. Express your answer as a single integer. | 5 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Quadratic Functions"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Let $a \in \mathbb{Z}$, and $0 \leq a \leq 13$. If $51^{2012} + a$ is divisible by 13, calculate the exact value of $a$. Express your answer as a single integer. | 12 | big_math | [
"Mathematics -> Number Theory -> Congruences"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Find the smallest positive period of the function \( y = \frac{7 \sin \tau \tan x}{\sec x \cos 2 x \left(1 - \tan^2 x\right)} \). Express your answer in terms of pi. | \pi | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
If the arc length corresponding to a central angle of 2 radians is 4 cm, then calculate the area of the sector formed by this central angle. Express your answer in square centimeters (cm^2). | 4 \text{ cm}^2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
How many subsets consisting of 5 elements can be selected from the set $\{1,2,3,4,\ldots,10\}$ such that the sum of any two numbers within these 5 elements is not equal to 11? Express your answer as a whole number. | 32 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
An arithmetic sequence has 2n+1 terms. Find the ratio of the sum of the odd terms to the sum of the even terms. Express your answer as a ratio of two algebraic terms (e.g., n, n+1). | \frac{n+1}{n} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given the quadratic function y = 2x^2, if its graph is shifted up by 1 unit, what is the expression of the new function? Express your answer as a function of x. | y = 2x^2 + 1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Quadratic Functions"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(1,-2)$ and $\overrightarrow{b}=(m,m-2)$, find the value of $m$ such that $\overrightarrow{a}\bot \overrightarrow{b}$. Use the condition that the dot product of perpendicular vectors is $0$ to solve the equation and find $m$. Express your answer as a single numerical value. | 4 | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Linear Algebra -> Vectors"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
The non-zero digits \( p, q \) and \( r \) are used to make up the three-digit number \( pqr \), the two-digit number \( qr \), and the one-digit number \( r \). The sum of these numbers is 912. Find the value of \( q \). Express your answer as a single digit. | 5 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Find the greatest common divisor (GCD) of the two integers 1908 and 4187. Express your answer as a single integer. | 53 | big_math | [
"Mathematics -> Number Theory -> Greatest Common Divisors (GCD)"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Two circles are given: the first has center A(5,3) and radius 12, and the second has center B(2,-1) and radius 6. Find the shortest distance between the two circles. Express your answer as a whole number. | 1 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the geometric sequence $\{a_n\}$ satisfies $a_n a_{n+1} = 16^n$, find the common ratio. Express your answer as a single number. | 4 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of real solutions to the equation $x^{\ln3} + x^{\ln4} = x^{\ln5}$. Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Exponential Functions"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
A school arranges four classes in the morning, each lasting 40 minutes. The first class starts from 8:00 to 8:40, with a 10-minute break between classes. If a student arrives late, and he arrives at the classroom between 9:10 and 10:00, what is the probability that he listens to the second class for no less than 10 min... | \dfrac{1}{5} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given the polynomials \(x^{2} - 3x + 2\) and \(x^{2} - 5x + k\), find the sum of all possible real values of \(k\) such that these polynomials have a common root. Express your answer as a single integer. | 10 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
A person has a probability of 0.6 to hit the target with one shot. Calculate the probability that this person hits the target exactly twice in three shots. Express your answer as a simplified fraction. | \frac{54}{125} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Combinations"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the result of $1^{2009} + (-1)^{2009}$. Express your answer as a single integer. | 0 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of $\cos 105^{\circ}\cos 45^{\circ}+\sin 45^{\circ}\sin 105^{\circ}$. Express your answer as a fraction. | \dfrac {1}{2} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
The sequence $\{a_n\}$ satisfies the following conditions:
$$
a_{n+1} = \begin{cases}
2a_n, & 0 \leq a_n < \frac{1}{2} \\
2a_n - 1, & \frac{1}{2} \leq a_n < 1
\end{cases}
$$
If $a_1 = \frac{3}{5}$, then calculate $a_{2014}$. Express your answer as a fraction in the form $\boxed{\frac{a}{b}}$. | \frac{1}{5} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors a = (1, sqrt(2)) and b = (cos(theta), sin(theta)), where theta is in the interval (0, 2*pi), and the dot product of a and b equals the magnitude of a, find the value of tan(theta). Express your answer as a numerical value. | \sqrt{2} | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Geometry -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given $2^{a}=3$ and $8^b=\frac{1}{6}$, calculate the value of $\left(a+3b+1\right)^{3}$. Express your answer as a single integer. | 0 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Given that sin(a) + cos(a) = sqrt(2), and a in (-pi/2, pi/2), find the value of tan(a). Express your answer as a single numerical value. | 1 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
A school schedules four classes in the morning, each lasting 40 minutes. The first class starts at 8:00 AM and ends at 8:40 AM, with a 10-minute break between classes. A student arrives late for class, and if they arrive during the period from 9:10 AM to 10:00 AM, what is the probability that they will attend at least ... | \frac{1}{5} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence {a_n}, it is known that a_5 + a_6 = 20. Find the value of (a_4 + a_7)/2. | 10 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Find the square root of 4. If the number has both a positive and a negative square root, include both in your answer. | \pm 2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
A chord of a circle has a length equal to the radius of the circle. Find the measure of the central angle corresponding to this chord in radians. | \frac{\pi}{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Given x > 0 and y > 0, and lg 2^x + lg 8^y = lg 2, find the minimum value of 1/x + 1/(3y). Express your answer as a single numerical value. | 4 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Given the curve y = e^x, find the slope of the tangent line at the point A(0,1). Express your answer as a single numerical value. | 1 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
The probability of good air quality in a day is 0.8, and the probability of good air quality for two consecutive days is 0.68. Given that the air quality is good on a certain day, find the probability that the air quality will be good on the following day. Express your answer as a decimal value, rounded to 3 decimal pl... | 0.85 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given that f(x) is an even function defined on the real numbers R that is not identically zero, and for any real number x, xf(x+1) = (x+1)f(x), find the value of f(9/2). Express your answer as a single number. | 0 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
A mixture is prepared by adding 50.0 mL of 0.200 M NaOH to 75.0 mL of 0.100 M NaOH. Calculate the [OH^-] in the mixture. Express your answer in M (molarity) with appropriate precision. | 0.140 M | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the longest chord in circle $\odot O$ is $8$ cm, what is the radius of circle $\odot O$? Express your answer in centimeters (cm) as a whole number. | 4 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Paula the painter had just enough paint for 30 identically sized rooms. Unfortunately, on the way to work, three cans of paint fell off her truck, so she had only enough paint for 25 rooms. How many cans of paint did she use for the 25 rooms? Express your answer as a whole number. | 15 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation of a circle: (x+5)^2 + (y-12)^2 = 14^2, find the minimum value of x^2 + y^2. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given the operation defined by $p \times s - q \times r$ on a matrix
| p | q |
| :--- | :--- |
| r | s |
calculate the result of this operation on the matrix
| 4 | 5 |
| :--- | :--- |
| 2 | 3 |
. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Matrices"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
A small block is placed along a 10 cm ruler. The left edge of the block is at the 3 cm mark, and the right edge is between the 5 cm and 6 cm marks. Estimate the length of the block. Express your answer in centimeters (cm), rounded to one decimal place. | 2.5 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Five circles are each tangent to lines L1 and L2 and are sequentially tangent to each other. If the radius of the largest circle is 18 and the radius of the smallest circle is 8, what is the radius of the middle circle? Express your answer as a single numerical value. | 12 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given the hyperbola $C$: $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ with $a > 0$ and $b > 0$, one of its asymptotes has a slope of $2$ and the focal length is $2\sqrt{5}$. Find the value of $a$. | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets $M=\{x|\ln x > 0\}$ and $N=\{x|x^{2}\leqslant 4\}$, find the intersection $M\cap N$. Express your answer in interval notation. | (1,2] | big_math | [
"Mathematics -> Algebra -> Inequalities",
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $f(x)$ is a function with a derivative $f^{\prime}(x)$, and it satisfies $f(x) = 2xf^{\prime}(e) - \ln x$, find the value of $f^{\prime}(e)$. Express your answer as a fraction or decimal value. | \frac{1}{e} | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
The graphs of the functions y=a^(x+1) and y=log_a(x+1) (where a>0 and a!=1) are symmetric with respect to a line. Find the equation of this line of symmetry. | y=x+1 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the following functions: $y=x^{3}$; $y=4x^{2}$; $y=x^{5}+1$; $y=\left(x-1\right)^{2}$; $y=x$, how many of these functions are power functions according to the definition $y = x^k$, where $k$ is a constant? Provide your answer as a whole number. | 2 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Evaluate the expression $12-7\times(-32)+16\div(-4)$ and provide your answer as a single number. | 232 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
The sequence ${a_n}$ satisfies ${a_{n+1}}={a_n}+\frac{n}{2}$, where $n \in \mathbb{N}^*$, and $a_1=2$. Determine the value of $a_{20}$. Express your answer as a single integer. | 97 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Given the point $P(-3,2)$ in the second quadrant, find the distance from point $P$ to the $y$-axis. Express your answer as a whole number. | 3 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given $a=\sqrt{3}$, $b=3$, and $B=2A$, calculate the length of side $c$. Express your answer as a numerical value, including the square root symbol if necessary. | 2\sqrt{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
The parabola $y^{2}=2px$ passes through the point $(1,\sqrt{3})$. Find the focus of the parabola. Express your answer as an ordered pair (x, y). | (\frac{3}{4}, 0) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Given the hyperbola equation mx^2 - y^2 = 1, where the asymptotes are y = ±3x, find the value of m. Express your answer as a single number. | 9 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $a$, $b$, $c$, where any two of them are not collinear, but $a+b$ is collinear with $c$, and $b+c$ is collinear with $a$, find the vector $a+b+c$. Express your answer as a vector, and if the answer is the zero vector, specify it as 0 or \boxed{0}. | 0 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
If the octal representation of a perfect square is $ab3c$ (i.e., $n^{2}=$ $(ab3c)_{8}=8^{3}a+8^{2}b+8 \cdot 3+c$), where $a \\neq 0$, then what is the value of $c$? Express your answer as a single digit. | 1 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(x,-1)$ and $\overrightarrow{b}=(1, \sqrt {3})$, if $\overrightarrow{a}\perp \overrightarrow{b}$, calculate the magnitude of $\overrightarrow{a}$. Express your answer as a single numerical value. | 2 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
In the quadratic equation $x^{2}-5x-2=0$, what is the coefficient of the linear term? Provide your answer as a single number. | -5 | big_math | [
"Mathematics -> Algebra -> Algebra -> Equations and Inequalities"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Given non-zero vectors $m$ and $n$ satisfying $4|m|=3|n|$ and $\cos \langle m,n \rangle =\frac{1}{3}$, calculate the value of the real number $t$ such that $n \perp (tm+n)$. Express your answer as a single real number. | -4 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
A school conducted a survey on the weight of eighth-grade students by randomly sampling some students' weights and grouping them appropriately before drawing a frequency distribution histogram. Given that the frequency of one group of data is 50 and the frequency rate is 0.5, calculate the sample size of the surveyed s... | 100 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given points A(-3, 0), B(0, 4), and M is a moving point on the circle C: $x^2+y^2-4x=0$, calculate the minimum area of triangle $\triangle MAB$. Express your answer as a single whole number. | 5 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
In the Cartesian coordinate system, find the distance from point P(4,-3) to the x-axis. Express your answer as a single numerical value. | 3 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given circle O: x^{2}+y^{2}=4 (where O is the origin) passes through the endpoints of the minor axis and the two foci of the ellipse C: \dfrac {x^{2}}{a^{2}}+ \dfrac {y^{2}}{b^{2}}=1 (a > b > 0), find the standard equation of ellipse C. Express your answer in the standard form of an ellipse equation, including the valu... | \dfrac {x^{2}}{8}+ \dfrac {y^{2}}{4}=1 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
Find the monotonic increasing interval of the function y = log2(4 + 3x - x^2). Express your answer as an interval, using interval notation. | (-1, 3/2) | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence $\{a_n\}$ satisfies $a_1=1, a_2=1, a_{n+1}=|a_n-a_{n-1}|(n\geqslant 2)$, calculate the sum of the first $2017$ terms of this sequence. | 1345 | big_math | [
"Mathematics -> Discrete Mathematics -> Sequences and Series"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Given \( a > b > c > d \) and the inequality \( \\frac{1}{a-b} + \\frac{1}{b-c} + \\frac{1}{c-d} \\geqslant \\frac{n}{a-d} \), derive the maximum possible value of \( n \). Provide your answer as a single number. | 9 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
In an exam, candidates are required to choose 6 out of 9 questions to answer, with the condition that at least 3 of the first 5 questions must be included. Find the number of different ways the candidates can choose their questions. Express your answer as a numerical value. | 74 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Combinations"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the complex number $z$ satisfies the equation $i \cdot \left(z+i\right)=1$, find the imaginary part of $z$. Express your answer as a single number. | -2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given the circle x^2 + y^2 = 10, find the equation of the line containing the chord with midpoint P(1, 1). Express your answer as a linear equation in the form Ax + By + C = 0. | x + y - 2 = 0 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Represent the number 15300000000 in scientific notation. Express your answer in the form a × 10^b, where a is the coefficient and b is the exponent. | 1.53 × 10^10 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given positive real numbers x, y satisfying 4x + y + 12 = xy, find the minimum value of xy. Express your answer as a single numerical value. | 36 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities",
"Mathematics -> Calculus -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Xiao Ming bought a football at a 20% discount and saved 25 yuan. Calculate the original price of the football in yuan. | 125 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Given $tan\alpha=\frac{2}{3}$, calculate the value of $\sin 2\alpha -\cos \left(\pi -2\alpha \right)$. Express your answer as a single fraction. | \frac{17}{13} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Determine how many prime divisors the number \(17! - 15!\) has, where \(n! = 1 \cdot 2 \cdots n\) is the factorial of \(n\), i.e., the product of all natural numbers from 1 to \(n\) inclusive. Express your answer as a whole number. | 7 | big_math | [
"Mathematics -> Number Theory -> Prime Numbers"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
If there are a total of 81 different mappings from set P to set Q={a, b, c}, then find the number of different mappings that can be made from set Q to set P. Express your answer as a whole number. | 64 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
From 4 type A and 5 type B television sets, randomly select 3, with at least one of each type required. Calculate the number of different ways to do this. Express your answer as a numerical value. | 70 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Combinations"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
The line $l$ is the tangent to the circle $x^2+y^2=4$ at the point $(-1, \sqrt {3})$, and the point $P$ is a moving point on the circle $x^2-4x+y^2+3=0$. Calculate the minimum distance from the point $P$ to the line $l$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ that satisfy $|\overrightarrow{a}| = 2$, $|\overrightarrow{b}| = 3$, and $(\overrightarrow{a} - \overrightarrow{b}) \cdot \overrightarrow{a} = 7$, find the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$, denoted as $\theta$, in radians. Express yo... | \frac{2\pi}{3} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $a$ and $b$ that satisfy $|a|=2$, $|b|=1$, and $a \cdot b = 1$, find the angle between vector $a$ and vector $(a-b)$. Express your answer in radians. | \dfrac{\pi}{6} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Compute the result of $\frac{1+i}{i}$, where $i$ is the imaginary unit. Express your answer as a complex number in the form $a+bi$. | 1-i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of $\frac{(3!)!}{3!}$. Express your answer as a single integer. | 120 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the circle with center at (0, 4) and passing through the point (3, 0). Express your answer in standard form. | x^2 + (y - 4)^2 = 25 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Chris and Pat go on a 30 km run. They both usually run at 10 km/h. If Chris runs at half his usual running speed, and Pat runs at 1.5 times her usual speed, calculate the difference in time, in hours, it takes Chris to complete the run compared to Pat. | 4 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
A function $f(x)$ satisfies $f(m+n) = f(m) \cdot f(n)$ for any positive integers $m, n$, and $f(1) = 2$. Calculate the following expression:
$$
\frac{f(2)}{f(1)} + \frac{f(4)}{f(3)} + \frac{f(6)}{f(5)} + \dots + \frac{f(2016)}{f(2015)}
$$
Express your answer as a single integer. | 2016 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Given the probability distribution rule $p(x=n)= \frac{a}{(n+1)(n+2)}$ $(n=1,2,3,4)$, where $a$ is a constant, calculate the value of $P( \frac{1}{2} < X < \frac{5}{2})$. Express your answer as a fraction in lowest terms. | \frac{3}{4} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given a sequence {a<sub>n</sub>} with a<sub>1</sub>=1 and 3S<sub>n</sub>=a<sub>n+1</sub>-1 for n∈N<sup>*</sup>, where S<sub>n</sub> is the sum of the first n terms of the sequence. If a<sub>k</sub>=1024, find the value of k. Express your answer as a single integer. | 6 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Ringo has a certain number of marbles that leave a remainder of 4 when divided by 6. Paul has a certain number of marbles that leave a remainder of 3 when divided by 6. When they pool their marbles and divide them into bags of 6, what is the remainder? Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Number Theory -> Congruences"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
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