instruction_seed stringlengths 40 1.39k | answer stringlengths 1 95 | source stringclasses 1
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In the Cartesian coordinate system xOy, the vertices of angles α and β coincide with the origin O, and their initial sides coincide with the non-negative half-axis of the x-axis. Their terminal sides are symmetric about the x-axis. Given that cos α = 2/3, find the value of cos β. Express your answer as a ratio or fract... | 2/3 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Given the numbers 3, 5, 7, 4, 6, calculate the standard deviation of this sample. Express your answer as a mathematical expression. | \sqrt{2} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given the universal set $U=\{x|x\text{ is a positive integer less than 9}\}$, set $M=\{1,2,3\}$, and set $N=\{3,4,5,6\}$, find the set resulting from $(\complement_UM) \cap N$. Express your answer as a set, with elements separated by commas and enclosed in curly brackets {}. | \{4,5,6\} | big_math | [
"Mathematics -> Discrete Mathematics -> Logic"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Find the smallest positive period of the function y = tan(2x - π/3). Express your answer as a fraction of π, if applicable. | \frac{\pi}{2} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions -> Other"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Determine the range of values for the real number a such that the function f(x) = (a-1)^x is a decreasing function on (-∞, +∞). Express your answer as an inequality in terms of a. | 1 < a < 2 | big_math | [
"Mathematics -> Algebra -> Exponential Functions"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
A truncated cone with a height of 8 contains a sphere O1 with a radius of 2. The center of sphere O1 is on the axis of the truncated cone, and sphere O1 is tangent to the upper base and the lateral surface of the truncated cone. Another sphere O2 with a radius of 3 can be placed inside the truncated cone such that sphe... | 2 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the roots of the quadratic equation \( x^2 + p x + q = 0 \) are the cubes of the roots of the quadratic equation \( x^2 + m x + n = 0 \), derive an expression for p in terms of m and n. | p = m^3 - 3mn | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Select two numbers $a$, $b$ sequentially (without replacement) from the four numbers $1$, $2$, $3$, $4$. Calculate the probability that $a^{2} \geqslant 4b$. Express your answer as a simplified fraction. | \frac{1}{2} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Using simple random sampling to draw a sample of size 3 from a population of 6 individuals, what is the probability of each individual being selected? Express your answer as a simplified fraction. | \frac{1}{2} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Combinations"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
A hospital has 120 male doctors and 180 female doctors. The hospital needs to select a team of 15 doctors to support rural healthcare services. Using stratified sampling, how many male doctors should be selected for the team? Express your answer as a whole number. | 6 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the terminal side of angle $\theta$ passes through the point $\left(-\frac{3}{5}, \frac{4}{5}\right)$, calculate the value of $\sin\left(\frac{\pi}{2}+\theta\right)+\cos(\pi-\theta)+\tan(2\pi-\theta)$. Express your answer as a single fraction in the form $\boxed{\frac{a}{b}}$. | \frac{4}{3} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Postman Pete has a pedometer to count his steps. The pedometer records up to 99999 steps, then flips over to 00000 on the next step. Pete plans to determine his mileage for a year. On January 1 Pete sets the pedometer to 00000. During the year, the pedometer flips from 99999 to 00000 forty-four times. On December 31 th... | 2472 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
If $(1-mi)(m+i) < 0$, where $i$ is the imaginary unit, then find the value of $m$ as a real number. | -1 | big_math | [
"Mathematics -> Algebra -> Complex Numbers -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, each of the two circles has center O. Also, OP: PQ = 1:2. If the radius of the larger circle is 9, calculate the area of the shaded region. Express your answer in terms of pi (π). | 72\pi | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Given $ab < 0$, simplify the expression $a^2|b| - b^2|a| + ab(|a| - |b|)$. Express your answer as a single numerical value or expression. | 0 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
In the expansion of $(x^2 + 3x + 2)^5$, find the coefficient of $x$. Express your answer as a single integer. | 240 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the set $A=\{x|2x+a>0\}$ ($a\in \mathbb{R}$), and $1 \notin A$, $2 \in A$, find the range of values for $a$. Express your answer as an inequality. | -4 < a \leq -2 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Inequalities"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given the geometric sequence {a<sub>n</sub>}, where a<sub>2</sub>a<sub>3</sub>a<sub>4</sub> = 1 and a<sub>6</sub>a<sub>7</sub>a<sub>8</sub> = 64, find the value of a<sub>5</sub>. Express your answer as a single numerical value. | 2 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
The relationship between the amount of pollutants P (mg/L) in the exhaust gas generated by a factory after filtration and the time t (h) during the filtration process is given by P = P0e^(-kt). If 20% of the pollutants were eliminated in the first 2 hours, calculate the time required to reduce the pollutants by 50%. Ex... | 6 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence $a_n = \frac{1}{n} + \frac{1}{n+1} + \frac{1}{n+2} + \ldots + \frac{1}{n^2}$, find the value of $a_2$. Express your answer as a sum of fractions. | \frac{1}{2} + \frac{1}{3} + \frac{1}{4} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Given a sequence of 99 terms with a Cesàro sum of 1000, calculate the Cesàro sum of the new 100-term sequence formed by adding 1 to the beginning of the original sequence. Express your answer as a single integer. | 991 | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Class 1 and Class 2 of the senior year at a certain school are holding New Year's Eve parties in their respective classes on December 30th. All 6 teachers are assigned to these two classes to join the celebration with the students, with each class having no more than 4 teachers. Find the number of different arrangement... | 50 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
A rectangular building block has a square base $ABCD$. Its height is 8 units. If the block has a volume of 288 cubic units, what is the side length of the base? Express your answer in units as a whole number. | 6 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Volume"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the sum of the first n terms of an arithmetic sequence is S_n, and the sum of the first n terms of a geometric sequence is T_n, and a_3 = b_3, a_4 = b_4, and (S_5 - S_3) / (T_4 - T_2) = 5, calculate the ratio (a_5 + a_3) / (b_5 + b_3). Express your answer as a simplified fraction. | -\frac{3}{5} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Let complex numbers $z_1$ and $z_2$ correspond to points in the complex plane that are symmetric about the origin, and $z_1 = 2-i$. Calculate the product $z_1 \cdot z_2$. Express your answer as a complex number in the form $a+bi$. | -3+4i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Person A and person B independently decode a password. The probability of person A decoding the password successfully is $\frac{1}{3}$, and the probability of person B decoding the password successfully is $\frac{1}{4}$. Calculate the probability that both person A and person B successfully decode the password. Express... | \frac{1}{12} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
The radius of the Earth's orbit around the sun is approximately 149000000 kilometers. Express this number in scientific notation in the form $a \times 10^{b}$, where $a$ is the coefficient and $b$ is the exponent. | 1.49 \times 10^{8} | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given $3\sin 2\theta = 4\tan \theta$, and $\theta \neq k\pi$ ($k \in \mathbb{Z}$), calculate $\cos 2\theta$. Express your answer as a fraction or decimal. | \frac{1}{3} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Trigonometric Functions"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Arrange 2016 coins into a triangle. The 1st row has 1 coin, the 2nd row has 2 coins, and the Nth row has N coins. Find the sum of the digits of N, where N is the number of rows in the triangle. Express your answer as a single integer. | 9 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of solution-pairs in the positive integers for the equation $3x + 5y = 501$. Express your answer as a single integer. | 33 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given a differentiable function $f(x)$ defined on the interval $(0,+\infty)$, with its derivative denoted by $f'(x)$, and satisfying $xf'(x) + 2f(x) > 0$, find the solution set of the inequality $\frac{(x+2016)f(x+2016)}{5} < \frac{5f(5)}{x+2016}$. Express your answer in interval notation. | (-2016, -2011) | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the equations 2^a = 3 and 8^b = 1/6, calculate the value of (a+3b+1)^3. Express your answer as a single integer. | 0 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Given a hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ with an eccentricity of $2$, where $a > 0$ and $b > 0$, and a parabola $y^2 = 4x$, if the right focus of the hyperbola coincides with the focus of the parabola, calculate the value of $\frac{a}{b}$. Express your answer as a simplified fraction. | \frac{\sqrt{3}}{3} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
In a certain month, three of the Sundays have dates that are even numbers. Determine the day of the week for the tenth day of this month. Express your answer as a specific day of the week (e.g. Monday, Tuesday, etc.). | Monday | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a$ and $b$ are two positive real numbers, and $\frac{1}{2^a} \cdot \frac{1}{2^b} = \left( \frac{1}{2^a} \right)^b$, find the minimum value of $ab$. Express your answer as a single numerical value. | 4 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Lorri took a 240 km trip to Waterloo. On her way there, her average speed was 120 km/h. She was stopped for speeding, so on her way home her average speed was 80 km/h. What was her average speed, in km/h, for the entire round-trip? Express your answer as a whole number. | 96 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given the universal set $U=\{1,2,3\}$ and $\complement_{U}A=\{2\}$, find the number of proper subsets of set $A$. Express your answer as a whole number. | 3 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
The current population of a certain township is 50,000. With the long-standing implementation of the national family planning policy, the annual number of births and deaths is 4.3% and 1.3% of the population at the beginning of the year, respectively. Calculate the approximate population of the town after 2 years. Expr... | 53045 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Ali is arranging the books on his bookshelves. He puts half his books on the bottom shelf and two-thirds of what remains on the second shelf. Finally, he splits the rest of his books over the other two shelves so that the third shelf contains four more books than the top shelf. There are three books on the top shelf. D... | 30 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
If the length of a rectangle is increased by $p\%$, in order to keep the area of the rectangle unchanged, by what percentage should the width of the rectangle decrease? Express your answer as a formula in terms of $p$. | \frac{100p}{100+p} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
A person walks a total distance of 378 li over 6 days. The distance walked each day is halved from the previous day due to foot pain. Calculate the distance walked on the last day. | 6 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation of the circle $(x+2)^2 + (y-1)^2 = 1$, find the coordinates of its center. Express your answer as an ordered pair $(h,k)$. | (-2,1) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Let set P = {1,2,3,4}, set Q = {3,4,5}, and let the universal set U = R (real numbers). Find the intersection of set P and the complement of set Q with respect to U, denoted as P ∩ C_U Q. Express your answer as a set of numbers using standard set notation. | P ∩ C_U Q = {1, 2} | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
For the geometric sequence \\(\\{a_n\\}\\) where all terms are positive, and \\(a_5a_6 + a_4a_7 = 18\\), find the value of \\(\\log_3a_1 + \\log_3a_2 + \\cdots + \\log_3a_{10}\\). Express your answer as a single numerical value. | 10 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function \(f(n) = i^n - i^{-n}\) where \(n\) is a positive integer (\(n \in \mathbb{N}^*\)), find the number of elements in the set \(\{f(n)\}\). Express your answer as a single integer. | 3 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Given \( m > 2 \), the lines \( l_1: y = \frac{m-2}{m} x + 2 \) and \( l_2: y = -x + 2m \) form a triangle with the y-axis, and the area of this triangle is 30. Find the value of \( m \). Express your answer as a single number. | 6 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
The sequence $\{a_n\}$ satisfies $a_{n+1}= \frac {1}{1-a_{n}}$, and $a_{1}=2$. Calculate the value of $a_{2016}$. Express your answer as a single value. | \frac{1}{2} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Given the expressions:
a = 2^(-1/2)
b = (1/2)^(-2)
c = log10(1/2)
Determine the correct relationship between a, b, and c, from greatest to least. Express your answer as the relationship between the variables (e.g., a > b > c). | b > a > c | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Given that 148 students have scores ranging from 100 to 120 (inclusive), with scores being positive integers, and students with the same score are given the same rank, what is the smallest possible number x such that each rank can have at most x students? Express your answer as a single integer. | 8 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
On a dark and stormy night Snoopy suddenly saw a flash of lightning. Ten seconds later he heard the sound of thunder. The speed of sound is 1088 feet per second and one mile is 5280 feet. Calculate the distance Snoopy was from the flash of lightning. Express your answer in miles, rounded to the nearest half-mile. | 2 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Given the parabola equation y^2 = 2px (p > 0) and the circle equation x^2 + y^2 + 2x - 3 = 0, find the value of p such that the length of the line segment cut by the circle from the directrix of the parabola is 4. Express your answer as a single numerical value. | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the complex number expression $\frac{{{{(1-i)}^2}}}{{1+i}}$ and express your answer as a simplified complex number in the form $a + bi$, where $a$ and $b$ are real numbers. | -1 - i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Three non-zero numbers \(a, b, c\) form an arithmetic sequence. When \(a\) is increased by 1 or \(c\) is increased by 2, the three numbers form a geometric sequence. Find the value of \(b\). Express your answer as a single integer. | 12 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given two non-zero vectors $\overrightarrow{a}$ and $\overrightarrow{b}$, if $\overrightarrow{a}+ \overrightarrow{b}=(-3,6)$ and $\overrightarrow{a}- \overrightarrow{b}=(-3,2)$, calculate the value of $\overrightarrow{a}^{2}- \overrightarrow{b}^{2}$. Express your answer as a single integer. | 21 | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Linear Algebra -> Vectors"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Let the random variable X follow the normal distribution N(3, 4). If P(X < 2a+3) = P(X > a-2), then find the value of a. Express your answer as a numerical value. | \frac {5}{3} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
The six edges of a tetrahedron $ABCD$ measure $7, 13, 18, 27, 36$ and $41$ units. If the length of edge $AB$ is $41$, then find the length of edge $CD$. Express your answer as a whole number. | 13 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given sets $A=\{x|\log _{2}x \lt 1\}$ and $B=\{x|x \gt -1, x \lt 1\}$, find the intersection $A\cap B$. Express your answer in interval notation. | (0,1) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Given the set $M=\{x \in \mathbb{Z} | 1 \leq x \leq m\}$, if set $M$ has 4 subsets, then find the value of the real number $m$. Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Mr. Li lives at point A in a town and is planning to drive to his workplace at point B, passing through a total of 6 intersections along the way without detours. Assume that the probability of a traffic jam occurring at each intersection is 1/6. What is the expected value E(ξ) of the number of times Mr. Li encounters a... | 1 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Use Horner's rule to compute the value of the function f(x) = 2x^4 + 3x^3 + 5x - 4 when x = 2. Count the number of times the multiplication operation is performed. Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Point M on the number line first moves 7 units to the left and then moves 4 units to the right, ending exactly at the origin. What is the initial position of point M? Express your answer as a single integer. | 3 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets A = {-1, 2} and B = {x | mx + 1 = 0}, find the set of values of the real number m that satisfy the condition A ∪ B = A. Express your answer as a set of values. | \left\{- \frac{1}{2},0,1\right\} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
In the Maths Premier League, teams get 3 points for a win, 1 point for a draw, and 0 points for a loss. Last year, my team played 38 games and got 80 points. We won more than twice the number of games we drew and more than five times the number of games we lost. Calculate the number of games my team drew. Express your ... | 11 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
There are 2 employees in Department A, 4 employees in Department B, and 3 employees in Department C. Calculate the total number of different ways to select one person from the employees of these three departments to participate in a customer reception activity. Express your answer as a single integer. | 9 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
The library of a school bought 140 new books, ending up with $\frac{27}{25}$ times the number of books it had before the purchase. Find the original number of books in the library. Express your answer as a whole number. | 1750 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Solve for $\square$ in the equation: $\square \times 2ab = 4a^{2}b$. Express your answer as a simplified algebraic expression. | 2a | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given two vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ in a plane with an angle of $30^{\circ}$ between them, $\overrightarrow{a}=(1,0)$, and $|\overrightarrow{b}|= \sqrt {3}$, calculate the magnitude of the vector $|\overrightarrow{a}- \overrightarrow{b}|$. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of positive divisors of the integer 23232. Express your answer as a single integer. | 42 | big_math | [
"Mathematics -> Number Theory -> Factorization"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $i$ is the imaginary unit, if the complex number $(a+i)^{2}i$ is a positive real number, find the value of the real number $a$. Express your answer as a single real number. | -1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Find the last four digits of $5^{2017}$. Express your answer as a 4-digit number. | 3125 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Given the expression \((x^{2}-3x+1)^{5}=a_{0}+a_{1}x+a_{2}x^{2}+\ldots+a_{10}x^{10}\), calculate the sum of the coefficients \(a_{1}+a_{2}+a_{3}+\ldots+a_{10}\). Express your answer as a single integer. | -2 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given $sin\alpha = -\frac{5}{13}$ and $\alpha$ is an angle in the fourth quadrant, calculate the value of $tan(\alpha + \frac{\pi}{4})$. Express your answer as a simplified fraction. | \frac{7}{17} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Costa is building a new fence in his garden. He uses 25 planks of wood, each of which is \(30 \mathrm{~cm}\) long. He arranges these planks so that there is the same slight overlap between any two adjacent planks. The total length of Costa's new fence is \(6.9\) metres, or \(690 \mathrm{~cm}\). Find the length in centi... | 2.5 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
A regular hexagon $ABCDEF$ has its side midpoints connected to form a smaller regular hexagon. Find the ratio of the area of the smaller hexagon to the area of the larger hexagon. Express your answer as a simplified fraction. | \frac{3}{4} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Polygons"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Given two vectors in the plane, $\overrightarrow{a} = (1, -2)$ and $\overrightarrow{b} = (-2, x)$, find the value of $x$ such that $\overrightarrow{a}$ and $\overrightarrow{b}$ are collinear. Express your answer as a single number. | 4 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Given the system of linear equations in two variables x and y:
\[
\left\{
\begin{array}{l}
3x+2y=k+1 \\
2x+3y=k
\end{array}
\right.
\]
If the solution satisfies x + y = 3, calculate the value of k. | 7 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Given a sequence \(\{a_n\}\) where \(a_1 = -1\) and \(a_{n+1} = S_nS_{n+1}\), with \(S_n\) denoting the sum of the first \(n\) terms, find a general formula for \(S_n\) in terms of \(n\). Express your answer using appropriate mathematical notation. | -\dfrac{1}{n} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the geometric sequence {a_n}, where a_4 = 2 and a_7 = 16, find the sum of the first 10 terms of the sequence { log_2(a_n) }. Express your answer as a single integer. | 25 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Sequences and Series"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
The digits 2, 4, 6, and 8 are each used once to create two 2-digit numbers. What is the largest possible difference between the two 2-digit numbers? Express your answer as a whole number. | 62 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Find the area of the triangle formed by the lines $y=5$, $y=1+x$, and $y=1-x$. Express your answer as a whole number. | 16 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Find the term number that contains the constant term in the expansion of $\left(3x+\dfrac{1}{x}\right)^{8}$. Express your answer as a single integer. | 5 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
The second class starts at 9:25, and each class lasts for 35 minutes. Find the time when the second class ends. Express your answer in 12-hour clock format (HH:MM). | 10:00 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Eleven points are marked from left to right on a straight line. The sum of all the distances between the first point and the other points is 2018 cm. The sum of all the distances between the second point and the other points, including the first one, is 2000 cm. Find the distance between the first and second points. Ex... | 2 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Find the sufficient and necessary condition for the proposition "$\forall x\in \left[1,2\right]$, $x^{2}-a\leqslant 0$" to be true. Express your answer as an inequality. | a \geqslant 4 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
The ages of Grannie's seven grandchildren are consecutive positive integers. The youngest three grandchildren have a mean age of 6. Calculate the mean age of the oldest three grandchildren. Express your answer as a whole number. | 10 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Given that α and β belong to the interval [-π/2, π/2] and αsin(α) - βsin(β) > 0, describe the relationship between α^2 and β^2 that satisfies this inequality. Express your answer using mathematical notation. | α^2 > β^2 | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given line $l_1: y-2=(k-1)x$, which always passes through the fixed point $(0,2)$, find the fixed point $(x,y)$ that line $l_2$ passes through, where line $l_2$ is symmetric to line $l_1$ about the line $y=x+1$. Express your answer as an ordered pair $(x,y)$. | (1,1) | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given a parabola ${x}^{2}=2py(p>0)$ and a hyperbola $\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1(a>0,b>0)$, the directrix of the parabola and the asymptotes of the hyperbola form an isosceles right triangle with an area of $1$. Find the value of $p$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
For the final round of a math competition, 500 students are numbered from 001 to 500. A systematic sampling method is used to select a sample of 50 students, and the randomly selected number is 003. The students are divided among three examination centers: from 001 to 200 at the first center, from 201 to 355 at the sec... | 16 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. Given that $A=45^{\circ}$, $a=6$, and $b=3 \sqrt {2}$, find the magnitude of angle $B$ in degrees. | 30 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Given the coordinates of three vertices of a rectangle as (1,1), (5,1), and (1,7), find the coordinates of the fourth vertex. Express your answer as a coordinate pair (x, y). | (5,7) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Find the smallest positive integer n that satisfies the inequality \\( \\sqrt{n} - \\sqrt{n-1} < 0.01 \\). Express your answer as a single integer. | 2501 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $P(AB)=\frac{3}{10}$ and $P(B|A)=\frac{1}{2}$, calculate the value of $P(A)$. Express your answer as a fraction. | \frac{3}{5} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
What is the name of an equation that contains two variables and whose terms with variables are of degree 1? Provide your answer as the specific type of equation (e.g., linear equation in two variables). | linear equation in two variables | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Linear Equations"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
What is the time 2017 minutes after 20:17? Express your answer in 24-hour clock format (HH:MM). | 05:54 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Claire measures the interior angles of a convex polygon and calculates their sum to be 2017 degrees. However, she realizes that she forgot to include one of the angles in her calculation. Find the measure of the angle she forgot. Express your answer in degrees as a whole number. | 143 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Randomly select a number x in the interval [0,2π]. Calculate the probability of the event "cos(x + π/3) + √3sin(x + π/3) ≥ 1" occurring. Express your answer as a simplified fraction. | \frac{1}{3} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
The sum of three numbers is 98. The ratio of the first to the second is 2/3, and the ratio of the second to the third is 5/8. Calculate the value of the second number. Express your answer as a whole number. | 30 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given that in the plane region {(x, y) | |x| ≤ 1, |y| ≤ 1} the inequality ax - 2by ≤ 2 always holds, what is the area of the region formed by the moving point P(a, b)? Express your answer as a single integer. | 8 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Find the maximum value of the real number \( k \) such that the inequality \( \sqrt{x-3} + \sqrt{6-x} \geq k \) has a solution for \( x \). Express your answer as a single mathematical expression. | \sqrt{6} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
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