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 |
|---|---|---|---|---|---|
A set of numbers is given as {-3, -1, 0, 2, 4}. Two different numbers are randomly selected from this set and then multiplied together. What is the probability that the product of the two numbers chosen is 0? Express your answer as a fraction or decimal. | \frac{2}{5} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
A 10 by 10 grid is created using 100 points. Point \( P \) is given. One of the other 99 points is randomly chosen to be \( Q \). Find the probability that the line segment \( PQ \) is vertical or horizontal. Express your answer as a simplified fraction. | \frac{2}{11} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given $a=-\dfrac{1}{4}$, $b=-2$, and $c=-2\dfrac{3}{4}$, calculate the value of $|a|+|b|-|c|$. Express your answer as a simplified fraction. | -\dfrac{1}{2} | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
As shown in the diagram, the houses of Xiaohong, Xiaolan, and Xiaolu are located along a circular road where one can only travel in a counterclockwise direction. It takes Xiaohong 30 meters to walk to Xiaolan's house, Xiaolu 26 meters to walk to Xiaohong's house, and Xiaolan 28 meters to walk to Xiaolu's house. Calcula... | 54 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
The table below shows the per capita Gross Domestic Product (GDP) of 10 provinces in China in the year 2022.
| Province Number | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|-----------------|------|------|------|------|------|------|------|------|------|------|
| Per Capita GDP (in ten thous... | 0.4 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Find the coefficient of the x^3 term in the expansion of the algebraic expression (1-x^3)(1+x)^10. Express your answer as a single integer. | 119 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence {a_n}, a_2 = 3 and a_5 + a_7 = 10. Find the value of a_1 + a_10. Express your answer as a numerical value. | 9.5 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Let $m$ and $n$ be any two odd numbers, with $n$ less than $m$. Find the largest integer that divides all possible numbers of the form $m^2-n^2$. Express your answer as a single integer. | 8 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
When testing 5 high-quality items and 3 defective items, which are identical in shape and size but have different labels, and drawing 2 items sequentially without replacement, calculate the probability of drawing a high-quality item on the second draw given that the first draw was a high-quality item. Express your answ... | \dfrac{4}{7} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
A piece of string fits exactly once around the perimeter of a square whose area is 144. Calculate the area of the largest circle that can be formed from the piece of string, rounded to the nearest whole number. | 183 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given the system of inequalities $\begin{cases} & x\geqslant 1, \\\\ & y\geqslant 2, \\\\ & x+y\leqslant 4, \\end{cases}$ and the point $P(2a+b,3a-b)$ within the region represented by this system, find the range of values for the expression $\frac{b+2}{a-1}$. Express your answer as an interval, including the endpoints. | [-12,-\frac{9}{2}] | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
4 students and 1 teacher stand in a row for a photo. The students ask the teacher to stand in the middle. How many different ways can they stand? Express your answer as a whole number. | 24 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Permutations"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the domain of the function f(x) is R, and f(2x+1) is an even function and f(x+2) is an odd function, derive the value of f(2022) using these properties. | 0 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the constant term in the expansion of ${(ax-\frac{1}{\sqrt{x}})}^{9}$ is $672$, find the value of $a$. Express your answer as a single value. | 2 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Given an infinite geometric progression with a sum to infinity of 6 and the sum of the first two terms being 4.5, find the exact value(s) of the first term of the progression. Express your answer as exact numerical values, separating multiple values with 'or' if necessary. | 3 or 9 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given the planar vectors $\overset{→}{a}$ and $\overset{→}{b}$, with $|\overset{→}{a}| = \sqrt{3}$, $|\overset{→}{b}| = 2$, and an angle $θ = \frac{π}{6}$ between $\overset{→}{a}$ and $\overset{→}{b}$. If $(\overset{→}{a} - m\overset{→}{b}) ⊥ \overset{→}{a}$, find the value of $m$. Express your answer as a single numbe... | 1 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Given the planar vectors $\overrightarrow {a}$ and $\overrightarrow {b}$, where $\overrightarrow {a}$ = (2cosα, 2sinα) and $\overrightarrow {b}$ = (cosβ, sinβ), if the minimum value of $|\overrightarrow {a} - λ\overrightarrow {b}|$ for any positive real number λ is $\sqrt{3}$, then calculate the value of $|\overrightar... | \sqrt{3} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given that a light ray is emitted from point M(2,1), reflects off the x-axis, and then reaches point N(4,5), calculate the total distance the light ray travels. Express your answer in simplest radical form. | 2\sqrt{10} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the year 500 BC is represented as -500 years, how would you represent the year 2023 AD using the same system? Express your answer as a signed integer, including the sign (+ or -). | +2023 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence {a_n}, the difference between the sum of the first 7 terms and the sum of the first 2 terms is 450, i.e., S_7 - S_2 = 450. Find the sum of the 2nd term and the 8th term, a_2 + a_8. | 180 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given that points \( P \) and \( Q \) are inside triangle \( \triangle ABC \), and
\[
\begin{array}{l}
\overrightarrow{PA} + 2\overrightarrow{PB} + 3\overrightarrow{PC} \\
= 2\overrightarrow{QA} + 3\overrightarrow{QB} + 5\overrightarrow{QC} = \mathbf{0}.
\end{array}
\]
Calculate the ratio \( \frac{|\overrightarrow{PQ}|... | \frac{1}{30} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Select 4 students from 7 students (including 4 boys and 3 girls) to participate in an environmental knowledge contest. If among these 4 people there must be both boys and girls, calculate the number of different selection methods. Express your answer as a whole number. | 34 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Combinations"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Laura and Dina have a running race. Laura runs at a constant speed and Dina runs n times as fast where n > 1. Laura starts s meters in front of Dina. Find the distance, in meters, that Dina runs before she overtakes Laura. Express your answer as a formula in terms of n and s. | \frac{ns}{n-1} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
Given the constraints x >= 0, y >= 0, and x + 2y = 1/2, determine the maximum value of the function u = log_{1/2}(8xy + 4y^2 + 1). Express your answer as a numerical value. | 0 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given the vectors $\overset{→}{a}=(k,3)$, $\overset{→}{b}=(1,4)$, and $\overset{→}{c}=(2,1)$, and the condition that $\left(2\overset{→}{a}-3\overset{→}{b}
ight)\perp\overset{→}{c}$, determine the value of the real number $k$. Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Estimate the value of $\sqrt{30}$ by finding two consecutive integers $a$ and $b$ such that $a < \sqrt{30} < b$. Express your answer in the format $a < \sqrt{30} < b$. | 5 < \sqrt{30} < 6 | big_math | [
"Mathematics -> Precalculus -> Functions"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Given a point (3, M) and that its distance to the line x + sqrt(3)y - 4 = 0 is 1, find the value(s) of m. Express your answer using any necessary mathematical notation (e.g., sqrt(3)). | \sqrt{3} \text{ or } -\frac{\sqrt{3}}{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
In a certain city, the monthly gas fee is charged according to the following regulations: The monthly gas usage is calculated in whole cubic meters; if the monthly gas usage does not exceed 60 cubic meters, the charge is 0.8 yuan per cubic meter; if it exceeds 60 cubic meters, the excess part is charged at 1.2 yuan per... | 66 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
A copper pentagonal prism with a base area of $16\\text{cm}^2$ and a height of $4\\text{cm}$ is melted and cast into a cubic copper block (without considering any loss). Calculate the edge length of the resulting cubic copper block. Express your answer in centimeters (cm). | 4\\text{cm} | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Volume"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(t) = (log_{2} 3)^t - (log_{5} 3)^t, and knowing that log_{2} 3 > 1 and 0 < log_{5} 3 < 1, determine the correct inequality relationship between x and y such that f(x) >= f(-y). Express your answer in the form of the correct inequality, such as "x + y >= 0". | x + y >= 0 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the linear equation $mx+2x-12=0$, where the solutions are positive integers, find the number of possible values of the positive integer $m$. Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Let point O be located inside a regular tetrahedron ABCD, and let G be the centroid of the tetrahedron. The line OG intersects each face of the tetrahedron at points A1, B1, C1, and D1. Find the value of the sum of the ratios of A1O to A1G, B1O to B1G, C1O to C1G, and D1O to D1G. | 4 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given sets A = {0, 1} and B = {-1, 0, 1}, and a mapping f from A to B, count the number of mappings that satisfy the condition f(0) > f(1). | 3 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence ${a_n}$, the sum of three consecutive terms is given as $a_{1007} + a_{1008} + a_{1009} = 18$. Find the sum of the first 2015 terms of the sequence. Express your answer as a whole number. | 12090 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
A piece of clothing is marked at 132 yuan. If it is sold at a 10% discount, it still yields a 10% profit. What is the cost price of this piece of clothing? Express your answer in yuan as a whole number. | 108 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Find the area enclosed by the curve $y=\cos x$ $(0 \leqslant x \leqslant \frac{3\pi}{2})$ and the coordinate axes. Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
If $n$ is an odd number, calculate the remainder of $7^n+C_n^17^{n-1}+C_n^27^{n-2}+\ldots+C_n^{n-1}7$ when divided by 9. Express your answer as a single integer. | 7 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(0,-2 \sqrt {3})$ and $\overrightarrow{b}=(1, \sqrt {3})$, calculate the projection of vector $\overrightarrow{a}$ in the direction of vector $\overrightarrow{b}$. Express your answer as a single number. | -3 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Find the coefficient of the $x^{3}$ term in the expansion of $(1+x)(1- \sqrt {x})^{6}$. Express your answer as a single integer. | 16 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $$f(x)= \begin{cases} \log_{2}x &,x>0 \\ -3^{x}+1 &,x\leq 0 \end{cases}$$, find the value of $f(f(1))$. Express your answer as a single number. | 0 | big_math | [
"Mathematics -> Precalculus -> Functions"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Let the set $M=\{x|-x^2-5x+6>0\}$, and $N=\{x||x+1|<1\}$. Calculate the intersection of sets $M$ and $N$, expressing your answer in standard interval notation. | (-2,0) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
After 4 o'clock, from the first time the hour hand and the minute hand form a 90° angle to the second time they form a 90° angle, how many minutes does it take? Express your answer in minutes, rounded to the nearest whole number. | 33 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
There are six people, and their zodiac signs are as follows: A, B, C are dragons, D, E are tigers, and F is a monkey. What is the probability that if three people are randomly selected from these six, their zodiac signs will all be different? Express your answer as a simplified fraction. | \frac{3}{10} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Combinations"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
One corner of a square is folded to its center to form an irregular pentagon. The area of the square is 1 unit greater than the area of the pentagon. Find the area of the square. Express your answer as a whole number. | 8 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
A function $f(x)$ defined on $\mathbb{R}$ satisfies $f(x+1)=2f(x)$, and when $x \in (0,1)$, $f(x)=4^{x}$. Calculate the value of $f(5.5)$. Express your answer as a single number. | 64 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Three letters are distributed into two mailboxes. What is the probability that each mailbox contains at least one letter? Express your answer as a simplified fraction. | \dfrac{3}{4} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Given two non-zero vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ that satisfy $|\overrightarrow{a} - \overrightarrow{b}| = |\overrightarrow{a} + 2\overrightarrow{b}|$, and the cosine of the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $-\frac{1}{4}$, calculate the ratio $\frac{|\overrightarrow... | 2 | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Geometry -> Other",
"Mathematics -> Linear Algebra -> Vectors"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Expand $\left(x^{2}-2xy+y^{2}\right)^{7}$ and find the sum of its coefficients. Express your answer as a single integer. | 0 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the function f''(x) is the derivative of an odd function f(x) (where x ∈ ℝ), and when x > 0, ln(x) * f'(x) < -1/x * f(x), find the range of values of x that satisfies the inequality (x^2 - 1) * f(x) > 0. Express your answer as an interval or union of intervals using interval notation. | (-∞, -1) ∪ (0, 1) | big_math | [
"Mathematics -> Calculus -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given x > 0 and y > 0, and the equation 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"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
The Fibonacci sequence is defined by the recursion formula $F_{n}= $$\begin{cases} \left.\begin{matrix}1，_{ }n=1，2 \\ F_{n-1}+F_{n-2}，_{ }n≥3.\end{matrix}\right.\end{cases}$. Calculate $F_{7}$ using this formula. Express your answer as a single integer. | 13 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
The number of different integer solutions (x,y) for |x|+|y|=1 is 4, and it increases by 4 as the right side of the equation increases by 1. Find the number of different integer solutions (x,y) for |x|+|y|=20. Express your answer as a single integer. | 80 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a$ is a real number, and $\frac {a+i}{1-i}$ is a pure imaginary number, find the value of $a$. Express your answer as a single real number. | 1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Let $a=\sqrt{2}$, $b=\sqrt[3]{3}$, $c=\log_{3}2$. Determine the order of a, b, and c, expressing your answer as an inequality (e.g., a < b < c). | c < a < b | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Let the complex numbers $z_1 = 2 - i$ and $z_2 = a + 2i$ (where $i$ is the imaginary unit and $a \in \mathbb{R}$). If $z_1z_2 \in \mathbb{R}$, find the value of $a$. Express your answer as a single real number. | 4 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
At points \(A\), \(B\), and \(C\) on a certain road, there are traffic lights. The green light durations at these three points within a minute are \(25\) seconds, \(35\) seconds, and \(45\) seconds, respectively. Calculate the probability that a car driving on this road does not stop at all three points. Express your a... | \dfrac{35}{192} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
To obtain the graph of the function $y=\sin 3x+\cos 3x$, the graph of the function $y=\sqrt{2}\cos 3x$ should be shifted by how many radians and in which direction? Express your answer in radians and specify the direction of the shift (left or right). | \frac{\pi}{4}\text{ units to the left} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Let $\overline{z}$ be the conjugate of the complex number $z$. Given that $z + \overline{z} = 3$ and $(z - \overline{z}) = 3i$ (where $i$ is the imaginary unit), find the sum of the real part and imaginary part of $z$. Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Let \(a_{1}, a_{2}, \cdots, a_{n}\) be all the irreducible proper fractions with a denominator of 60. Calculate the value of \(\sum_{i=1}^{n}\left(\cos \frac{a_{i} \pi}{2}\right)^{2}\). Express your answer as a single integer. | 8 | big_math | [
"Mathematics -> Number Theory -> Other",
"Mathematics -> Trigonometry -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
In traditional Chinese music, instruments are generally classified into "eight sounds." This is the earliest method in China to classify instruments based on their manufacturing materials, first seen in the book "Zhou Li $\cdot$ Chun Gong $\cdot$ Da Shi." The classification includes "metal, stone, earth, leather, silk,... | \frac{1}{5} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Combinations"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Three balls labeled 1, 2, and 3 are placed in a jar. A ball is drawn from the jar, its number is recorded, and then the ball is replaced back into the jar. This process is repeated three times. Each ball has an equal probability of being drawn in each trial. If the sum of the recorded numbers is 6, what is the probabil... | \frac{1}{7} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
In a data set of 80 samples, the maximum value of the samples is 145, the minimum value is 50, and the interval for grouping is 10. Calculate the number of groups the data can be divided into. | 10 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Toothpicks are used to form squares in a pattern: four toothpicks are used to form one square, seven to form two squares, and so on. If this pattern continues, how many toothpicks will be used to form 10 squares in a row? Express your answer as a whole number. | 31 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
The sum of nine consecutive positive integers is 99. Find the largest of these integers. Express your answer as a positive integer. | 15 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the graph of the function y=f(x) is symmetric about the line x=3, and f(-1)=320 and cos(x) - sin(x) = 3*sqrt(2)/5, calculate the value of f(15*sin(2x)/cos(x + pi/4)). | 320 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $i$ is the imaginary unit, $\overline{z}$ represents the complex conjugate of the complex number $z$. If $z=1+i$, then evaluate $\frac{z}{i}+i\cdot \overline{z}$. Express your answer as a simplified complex number in the form $\boxed{a+bi}$, where $a$ and $b$ are real numbers. | 2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $(10a + b)(10a + c) = 100a(a + 1) + bc$, where $a, b, c$ are positive integers less than $10$, find the condition for this equation to hold. Express your answer as an equation or condition using mathematical notation. | b+c=10 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.3125 | 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"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
In triangle \( \triangle ABC \), \( AB=AC \) and \( \angle A=80^{\circ} \). Points \( D, E \), and \( F \) are on sides \( BC, AC \), and \( AB \) respectively, and \( CE=CD \) and \( BF=BD \). Find the measure of \( \angle EDF \) in degrees. | 50^\circ | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of the expression $(\lg 2)^{2}+0.064\;^{- \frac {1}{3}}+\lg 5\lg 20$. Express your answer as a single number. | 3.5 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Logarithmic Functions"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Two identical equilateral triangles, A and B, both have a line segment drawn connecting the midpoints of two sides. A and B are positioned symmetrically left and right, but the internal line segments of A and B are asymmetrical. Starting from the position shown in the diagram, triangle A moves horizontally to the right... | 11 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Polygons"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
A rectangular pyramid has a base length and width of 7 feet and 5 feet, respectively, and a height of 8 feet. One of its edges is perpendicular to the base. What is the surface area of the circumscribed sphere of this pyramid? Express your answer in square feet, including π. | 138\pi | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Surface Area"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Select 5 distinct letters from the word "equation" and arrange them in a row such that the sequence "qu" remains together and in the same order. Find the total number of different permutations containing "qu". Express your answer as a numerical value. | 480 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Permutations"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
In the number "20220420", calculate the frequency of the digit "2" as a simplified fraction. | \frac{1}{2} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given that a, b, and c are positive real numbers, and the following equations hold:
\[ a(b + c) = 152 \]
\[ b(c + a) = 162 \]
\[ c(a + b) = 170 \]
Calculate the product abc. Express your answer as a single integer. | 720 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given a circle $x^2+y^2-2x+6y+1=0$ has two points symmetric about the line $2x+y+c=0$, calculate the value of $c$ that satisfies this condition. Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence $\sqrt{3}, 3, \sqrt{15}, \sqrt{21}, 3\sqrt{3}, \ldots$, find the position of the term $\sqrt{75}$ in the sequence. Express your answer as a whole number. | 13 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
A sequence t_{1}, t_{2},..., t_{n},... is defined as follows:
t_{1} = 14
t_{k} = 24 - 5t_{k-1} for each k >= 2
For every positive integer n, t_{n} can be expressed as t_{n} = p * q^{n} + r, where p, q, and r are constants. Calculate the sum of the constants p, q, and r. | -3 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
In a sequence, each term starting from the third is the sum of the two immediately preceding terms. If the second term is 1 and the fifth term is 2005, what is the sixth term? Provide your answer as a single integer. | 3008 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
A contest began at noon one day and lasted for 1000 minutes. At what time did the contest end? Express your answer in 12-hour clock format (e.g., 4:40 a.m.). | 4:40 \text{ a.m.} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $3C$ $(_{n-1}^{n-5})$ $=5A_{n-2}^{2}$, where $n \in \mathbb{N}^{*}$, calculate the value of n. Express your answer as a single positive integer. | 9 | big_math | [
"Mathematics -> Combinatorics -> Permutations",
"Mathematics -> Combinatorics -> Combinations"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
In triangle ABC, angle CBA = 72 degrees, E is the midpoint of side AC, D is on side BC such that 2BD = DC, and AD intersects BE at F. Find the ratio of the areas of triangle BDF to quadrilateral FDCE. Express your answer as a ratio of areas. | \frac{1}{5} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given 4 points in a plane, what is the maximum number of triangles that can be formed by connecting these points with line segments? Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics -> Other"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence \( 10^{\frac{1}{11}}, 10^{\frac{2}{11}}, 10^{\frac{3}{11}}, \ldots, 10^{11} \), find the smallest positive integer \( n \) such that the product of the first \( n \) terms of this sequence exceeds 100,000. | 11 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Let $a$ and $b$ be two non-perpendicular skew lines. Planes $\alpha$ and $\beta$ are constructed passing through lines $a$ and $b$ respectively. Consider the following four situations: $b \parallel \alpha$, $b \perpendicular \alpha$, $\alpha \parallel \beta$, $\alpha \perpendicular \beta$. How many of these situations ... | 3 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Drawing randomly from a bag containing several red, white, and yellow balls of the same size, the probabilities of drawing a red, white, or yellow ball are respectively $\dfrac{1}{2}$, $\dfrac{1}{3}$, and $\dfrac{1}{6}$. If a ball is drawn randomly from the bag, its color is noted and then it is put back, and this proc... | \dfrac{5}{12} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
In the complex plane, $O$ is the origin, and the complex number corresponding to the vector $\overrightarrow{OA}$ is $2+i$. The symmetric point of point $A$ about the imaginary axis is point $B$. What is the complex number corresponding to the vector $\overrightarrow{OB}$? Express your answer as a complex number in the... | -2+i | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=\left(x,2,-2\right)$ and $\overrightarrow{b}=\left(3,-4,2\right)$, if $\overrightarrow{a}\bot \overrightarrow{b}$, calculate the value of $x$. Express your answer as a single number. | 4 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number z = sqrt(3)/2 + 1/2 i, calculate the value of z^2016. Express your answer as a single complex number or a real number if the imaginary part is zero. | 1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Each face of a cube must be painted with exactly one colour, such that no two faces that share an edge are painted with the same colour. What is the smallest number of colours needed to paint the cube under these constraints? Express your answer as a single integer. | 3 | big_math | [
"Mathematics -> Discrete Mathematics -> Graph Theory -> Coloring"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
What is the radian measure of 15°? Express your answer as a fraction of π. | \frac{\pi}{12} | big_math | [
"Mathematics -> Precalculus -> Functions -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
The fixed telephone local call charge rule is: the first three minutes cost 0.22 yuan (if less than three minutes, it is charged as three minutes), and thereafter, each minute costs 0.11 yuan (if less than one minute, it is charged as one minute). Calculate the charge for a local call that lasts 550 seconds. Express yo... | 0.99 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the line passing through the point (-1,3) and parallel to the line y = (1/2)x + 1. Express your answer in standard form (Ax + By + C = 0). | x - 2y + 7 = 0 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
A positive number $x$ has the property that $x\%$ of $x$ is $4$. What is the value of $x$? Express your answer as a whole number. | 20 | big_math | [
"Mathematics -> Algebra -> Simple Equations"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, $\triangle P Q R$ is equilateral and is made up of four smaller equilateral triangles. If each of the smaller triangles has a perimeter of $9 \mathrm{~cm}$, what is the perimeter of $\triangle P Q R$? Express your answer in centimeters as a whole number. | 18 \text{ cm} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $\frac{2-bi}{1+2i}$, where $i$ is the imaginary unit and $b$ is a real number, and knowing that the real and imaginary parts of this complex number are opposites of each other, calculate the value of $b$. Express your answer as a real number. | -\frac{2}{3} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given that point M is the intersection point of line l: 2x - y - 4 = 0 with the x-axis, find the equation of a perpendicular line to l passing through point M. Express your answer in the standard form of a line equation (Ax + By + C = 0). | x + 2y - 2 = 0 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
Given $a, b \in \mathbb{R}$, and $a^2 + 2b^2 = 6$, find the minimum value of $a + b$. Express your answer as a single real number. | -3 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Brenda received 36 votes in a recent election, which represents 3/10 of the total votes cast. Find the total number of votes cast in the election. Express your answer as a whole number. | 120 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Find the area enclosed by the line $y=2x+3$ and the parabola $y=x^{2}$. Express your answer as a fraction or decimal value. | \frac{32}{3} | big_math | [
"Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
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