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
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|---|---|---|---|---|---|
Given the piecewise function $f(x)=\begin{cases} x^{2} & (-1 \leqslant x \leqslant 0) \\ 1 & (0 < x \leqslant 1) \end{cases}$, calculate the value of $\int_{-1}^{1}f(x)dx$. Express your answer as a single fraction or decimal. | \dfrac{4}{3} | big_math | [
"Mathematics -> Calculus -> Integral Calculus -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Eight classes in the first year of high school negotiate to form a grade basketball team, which needs a total of 10 players. Each class must contribute at least 1 player. Calculate the total number of different composition methods to form the basketball team. | 36 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Combinations"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Distribute 8 different playing cards into two stacks alternately: one to the left, one to the right, one to the left, one to the right... Then place the left stack on top of the right stack and repeat this process (do not flip the cards over). What is the minimum number of times this process must be repeated to restore... | 3 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given sets A = {0, 1, 2} and B = {2, 3}, if x is not in A and x is in B, then what is the value of x? Express your answer as a single integer. | 3 | big_math | [
"Mathematics -> Discrete Mathematics -> Logic"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $(x-2y+3=8)$, find the value of the polynomial $(x-2y)$. Express your answer as a single integer. | 5 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Simple Equations"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the complex number z satisfies z(3i-4) = 25i, where i is the imaginary unit, calculate the conjugate of z. Express your answer as a complex number in the form a + bi. | -4 + 3i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence \(\{a_n\}\) with the sum of the first \(n\) terms \(S_n=2n^2+3n-1\), calculate the value of \(a_5\). Express your answer as a single number. | 21 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Among the given numbers $+4$, $\pi$, $-\frac{1}{3}$, $0$, $-3.142$, $-0.5$, $0.4$, $-3.2626626662\ldots$, count how many are rational numbers. A rational number is any number that can be expressed as the quotient or fraction $\frac{p}{q}$ of two integers, with the denominator $q$ not equal to zero. Express your answer ... | 6 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $\frac{10i}{1-2i}$, find its equivalent form in the format $a+bi$. Express your answer with the values of $a$ and $b$. | -4+2i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Given a complex number $z$ satisfies $z(1-2i)=|3+4i|$ (where $i$ is the imaginary unit), calculate the imaginary part of the complex number $z$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
A five-digit number is divisible by 37 and 173, and its thousands digit is 3. Find the hundreds digit of this number. Express your answer as a single digit. | 2 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Two classmates went together to apply for a job at an organization. Before the interview, the person in charge of the organization told them, "We will hire 3 people from those interviewed, and the probability of both of you being hired together is 1/70." Calculate the total number of people interviewed. Express your an... | 21 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Combinations"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
In the geometric sequence {a_n}, if a_3 = -1, find the value of a_1a_2a_3a_4a_5. Express your answer as a single numerical value. | -1 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Find the minimum value of the function $y=\frac{6}{2^{x}+3^{x}}$ within the domain $-1 \leqslant x \leqslant 1$. Express your answer as a single value, using the boxed notation. | \frac{6}{5} | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the real axis of a hyperbola is twice the length of its imaginary axis, calculate the eccentricity of the hyperbola. Express your answer as a fraction with a square root. | \frac{\sqrt{5}}{2} | big_math | [
"Mathematics -> Geometry -> Non-Euclidean Geometry -> Other"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Let $i$ be the imaginary unit. Calculate the imaginary part of the complex number $\frac {5i}{2-i}$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Let $f(x) = x^3$. Find the derivative of $f(a-bx)$. Express your answer as a mathematical expression. | -3b(a-bx)^2 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Find the minimum value of the quadratic expression $x^2 - 4x - 2019$. Express your answer as an integer. | -2023 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Quadratic Functions"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Let \( a, b \in \mathbf{R}_{+} \) and satisfy the equation:
\[
\sqrt{\lg a} + \sqrt{\lg b} + \lg \sqrt{a} + \lg \sqrt{b} = 100,
\]
where \( \sqrt{\lg a}, \sqrt{\lg b}, \lg \sqrt{a}, \lg \sqrt{b} \in \mathbf{Z}_{+} \). Calculate the value of \( ab \). Express your answer as an exponential of 10. | 10^{164} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the line $4x-3y-2=0$ and the circle $x^2+y^2-2ax+4y+a^2-12=0$, find the range of values for $a$ such that the line intersects the circle at two distinct points. Express your answer as an interval. | (-6, 4) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given point A(2,5), find the coordinates of point B, which is symmetric to point A about the y-axis. Express your answer as an ordered pair (x, y). | (-2,5) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Catherine's car gas tank is initially 1/8 full. When 30 litres of gas are added, the tank becomes 3/4 full. If the gas costs $1.38 per litre, how much will it cost her to fill the remaining quarter of the tank? Express your answer in dollars, rounded to two decimal places. | 16.56 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
A carpet is on sale with its price reduced by 10% every 15 minutes. Initially, at 9:00 a.m., the carpet costs $10.00. At what time will the price of the carpet fall below $8.00? Express your answer in the format HH:MM a.m./p.m. | 9:45 a.m. | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
The 'Belt and Road Initiative' involves 65 countries along the route, with a total population of approximately 4,500,000,000. Express 4,500,000,000 in scientific notation in the form $a \times 10^{b}$. | 4.5 \times 10^{9} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z= \frac{1-i}{1+3i}$, calculate the imaginary part of $z$. Express your answer as a fraction. | -\frac{2}{5} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
In a certain game, two "electronic dogs," one black and one yellow, start from vertex A of a cube with edge length 1, labeled ABCD-A<sub>1</sub>B<sub>1</sub>C<sub>1</sub>D<sub>1</sub>, and crawl along the edges. Completing one edge is referred to as "completing one segment." The black "electronic dog" follows the path ... | 1 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given that line l passes through the point (0,3) and is perpendicular to the line x+y+1=0, find the equation of line l. Express your answer in the standard form of a line equation (e.g., Ax + By + C = 0). | x - y + 3 = 0 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Linear Equations"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
Given a geometric sequence $\{a_n\}$ with positive terms, where $a_1a_{100} + a_3a_{98} = 8$, find the sum of the logarithms of all terms, $\log_2 a_1 + \log_2 a_2 + \ldots + \log_2 a_{100}$. Express your answer as a single integer. | 100 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the slope of the line passing through points A(0,√3) and B(3,0) using the formula k_AB = (y2 - y1) / (x2 - x1). Express your answer as a numerical value. | -\frac{\sqrt{3}}{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of integer solutions to the equation \((x+1)^{2}+(y-2)^{2}=1\). Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Given the expansion $(1+x)^{10} = a + a_1(1-x) + a_2(1-x)^2 + \ldots + a_{10}(1-x)^{10}$, find the value of the coefficient $a_8$. Express your answer as an integer. | 180 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given sets $A=\{-2,-1,1,2,4\}$ and $B=\{x|(x+2)(x-3) < 0\}$, find the intersection $A \cap B$. Express your answer as a set of numbers, including all elements of the intersection. | \{-1, 1, 2\} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Given the universal set $U=\{2,4,6,8,10\}$, set $A=\{2\}$, and set $B=\{8,10\}$, find the complement of the union of sets A and B, denoted as $\complement_U(A\cup B)$. Express your answer as a set of elements using standard set notation. | \{4,6\} | big_math | [
"Mathematics -> Discrete Mathematics -> Set Theory -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given the universal set U = {x | x is a positive integer less than 9}, set M = {1, 2, 3}, and set N = {3, 4, 5, 6}, find the intersection of the complement of M in U and N. Express your answer as a set of integers. | {4, 5, 6} | big_math | [
"Mathematics -> Discrete Mathematics -> Logic -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence $\left\{a_{n}\right\}$ that satisfies $3 a_{n+1}+a_{n}=4(n \geqslant 1)$ and $a_{1}=9$, the sum of the first $n$ terms is $S_{n}$. Find the smallest integer $n$ that satisfies the inequality $\left|S_{n}-n-6\right|<\frac{1}{125}$. | 7 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Convert the binary number $10011_{(2)}$ to a decimal number. Express your answer as a single integer. | 19 | big_math | [
"Mathematics -> Other -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Determine the number of distinct real roots of the equation 3^x + 5^x + 7^x = 11^x. | 1 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Given sets A = {a, b, c} and B = {-1, 0, 1}, and a mapping f from A to B that satisfies f(a) + f(b) + f(c) = 0, how many such mappings f exist? Provide your answer as a single integer. | 7 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence $\{a_n\}$ defined by: $$a_n=\begin{cases} (4-a)n-10,(n\leqslant 7) \\ a^{n-6},(n > 7) \end{cases}$$ and the condition that $\{a_n\}$ is an increasing sequence, determine the range of the real number $a$ that satisfies these conditions. Express your answer as an interval or range of values for $a$. | (2, 4) | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the operation defined by $A \\oplus B = \\left\\{ x \\,|\\, x = \\frac{m}{n}, m \\in A, n \\in B \\right\\}$ and the sets $A = \\{1,2,4\\}$ and $B = \\{2,4,8\\}$, calculate the number of unique elements in $A\\oplus B$. Express your answer as a whole number. | 5 | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given a sector with a radius of $3$ and a central angle of $\dfrac{2\pi}{3}$, calculate the arc length of the sector. Express your answer as a numerical value, including $\pi$ if necessary. | 2\pi | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Given that AD, BE, and CF are the three altitudes of an acute triangle ∆ABC, with H being the orthocenter, count the total number of right-angled triangles in the figure. Express your answer as a whole number. | 12 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.015625 | 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+b)^2 - c^2 = 4 and angle C = 60 degrees, find the value of ab. Express your answer as a single numerical value in the form \boxed{answer}. | \dfrac{4}{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Five people are lined up in a row. How many different arrangements are possible where person A is not next to person B, and person A is also not next to person C? Express your answer as a single integer. | 48 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
A town's vehicle license plate numbers consist of four characters. The first character must be a vowel (A, E, I, O, U), the second and third characters must be different non-vowel letters, and the fourth character must be a digit from 0 to 9. Calculate the probability that a randomly chosen license plate meeting these ... | \frac{1}{21000} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Given the general term formula of the sequence $\{a_n\}$ as $a_n = \frac{1}{\sqrt{n+1}+\sqrt{n}}$ ($n=1,2,\ldots,$), and $S_n$ is the sum of the first $n$ terms of the sequence $\{a_n\}$, derive the expression for $S_n$. Express your answer as a mathematical expression, including all necessary mathematical notation and... | \sqrt{n+1}-1 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Solve the inequality $2|x-10|+3|x-20| \leq 35$ and express the solution set as an interval. | [9, 23] | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the measurement result ξ follows a normal distribution N(1, σ^2) (σ > 0) and the probability of ξ falling within the interval (0, 1) is 0.4, calculate the probability of ξ taking a value within the interval (0, 2). Express your answer as a decimal value between 0 and 1. | 0.8 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given the hyperbola $\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{3}=1$ with $a＞0$, and it passes through the point $\left(2,3
ight)$, find the equation of its asymptotes in the form $y=±kx$. | y=±√3x | big_math | [
"Mathematics -> Geometry -> Differential Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
The ratio of the number of students in the first, second, and third grades of a high school is 2:3:5. If a stratified sampling method is used to draw a sample of size 200, find the number of students to be drawn from the third grade. Express your answer as a whole number. | 100 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Given set A defined by the inequality x^2 + x - 6 > 0 and set B defined by the interval -2 < x < 4, calculate the intersection of sets A and B. Express your answer as an interval in standard notation. | (2,4) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Evaluate the expression $(x^x)^{(x^x)}$ when $x = 2$. Provide your answer as a single integer. | 256 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Exponential Functions"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation sin(x) = 3*sin(x - pi/2), calculate the value of cos(x)*cos(x + pi/2). Express your answer as a fraction. | \frac{3}{10} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $\left(-ab\right)^{3}(-3b)^{2}$. Express your answer as a simplified algebraic expression, including all necessary variables and coefficients. | -9a^{3}b^{5} | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
In triangle ABC, it is known that a = 1, b = 2, and angle C = π/3. Using the cosine theorem, calculate the length of side c. Express your answer as a numerical value, potentially including a square root. | \sqrt{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
If National Day in 2019 was on Tuesday, what day of the week will National Day be in 2020? Express your answer as the full day of the week. | Thursday | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the line that passes through the center of the circle $x^2 + 2x + y^2 = 0$ and is perpendicular to the line $x + y = 0$. Express your answer in the standard form of a line equation (Ax + By + C = 0). | x - y + 1 = 0 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the solutions of the equation $x-2y+b=0$ are points $(x,y)$ lying on the line $y=\frac{1}{2}x+b-1$, find the value of the constant $b$. Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\sqrt{96n}$ is an integer, find the smallest value of the positive integer $n$. Express your answer as a single integer. | 6 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence {a_n} with common difference d > 0, a_{2009} and a_{2010} are the roots of the equation x^2 - 3x - 5 = 0. S_n is the sum of the first n terms of the sequence {a_n}. Determine the smallest natural number n such that S_n > 0. | 4018 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a$ and $b$ belong to the set of real numbers ($\mathbb{R}$) and $a + b = 3$, find the minimum value of $2^a + 2^b$. Express your answer as a mathematical expression in the form $\boxed{[answer]}$. | 4\sqrt{2} | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Given the hyperbola equation x^2/16 - y^2/9 = 1 and point P(4, 4), count the number of lines that pass through P and intersect the hyperbola at only one point. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Geometry -> Non-Euclidean Geometry -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ that satisfy: $|\overrightarrow{a}| = |\overrightarrow{b}| = 1$, and $|k\overrightarrow{a} + \overrightarrow{b}| = \sqrt{3}|\overrightarrow{a} - k\overrightarrow{b}| (k > 0)$, find the maximum value of the angle between vectors $\overrightarrow{a}$ and $\overr... | \frac{\pi}{3} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Three dice, each showing numbers 1 to 6, are colored red, blue, and yellow, respectively. Each of the dice is rolled once. Calculate the total number of different ways the numbers rolled can sum up to 10. | 27 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the line passing through points $A(-2,m)$ and $B(m,4)$ is parallel to the line passing through points $C(m+1,1)$ and $D(m,3)$, find the value of $m$. Express your answer as a single integer. | -8 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Given the universal set U = {1, 3, 5, 7} and the complement of M in U is {5, 7}, find the set M. Express your answer as a set of elements enclosed in curly brackets, separating elements with commas if necessary. | {1, 3} | big_math | [
"Mathematics -> Discrete Mathematics -> Logic -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
There are 3 white balls, 4 black balls, and 5 red balls in a bag. Three people, A, B, and C, each draw a ball from the bag one by one without replacement. If each ball has an equal chance of being drawn, what is the probability that at least two of the balls drawn have the same color? Express your answer as a simplifie... | \dfrac{8}{11} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Suppose $S = \{1, 2, 3, x\}$ is a set with four distinct real numbers for which the difference between the largest and smallest values of $S$ is equal to the sum of elements of $S.$ Calculate the value of $x.$ Express your answer as a fraction or decimal, including a negative sign if applicable. | -\frac{3}{2} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number z = (a-1) + (a+1)i, where a is a real number and i is the imaginary unit, find the condition on a such that the point corresponding to z is located in the third quadrant. Express your answer as an inequality involving a. | a<-1 | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x) = \begin{cases} 1+\log_{2}(2-x), & x < 1 \\ 2^{x}, & x\geqslant 1 \end{cases}$, calculate the value of $f(-2)+f(2)$. Express your answer as a single number. | 7 | big_math | [
"Mathematics -> Algebra -> Functions"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Given $|\vec{a}|=5, |\vec{b}|=4$, and the angle $\theta=120^{\circ}$ between $\vec{a}$ and $\vec{b}$, find the projection of vector $\vec{b}$ onto vector $\vec{a}$. Express your answer as a single number. | -2 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\cos x = \frac{12}{13}$ and $x$ is an angle in the fourth quadrant, calculate the value of $\tan x$. Express your answer as a fraction, including a negative sign if applicable. | -\frac{5}{12} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(1,2)$ and $\overrightarrow{b}=(1,-1)$, calculate the dot product $\overrightarrow{a} \cdot \overrightarrow{b}$. Express your answer as a single integer. | -1 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Let function f(x) defined on R satisfy the following three conditions: (1) f(x+3) = -1/f(x), (2) for any 3 <= x_1 < x_2 <= 6, it holds that f(x_1) < f(x_2); (3) the graph of y=f(x+3) is symmetric about the y-axis. Compare the values of f(3), f(4.5), and f(7) using inequalities. Express your answer in the format f(a) < ... | f(3) < f(4.5) < f(7) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
In the Cartesian coordinate system, the origin $O$ is the midpoint of the line segment $AB$. If the coordinates of point $A$ are $(-1,2)$, calculate the coordinates of point $B$. Express your answer as an ordered pair $(x, y)$. | (1, -2) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Given that the derivative of a function f(x) is f'(x), if for any x in R, f'(x) - f(x) < 1, and f(0) = 2022, find the solution set of the inequality f(x) + 1 > 2023e^x. | (-\infty, 0) | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence ${a_n}$ where $a_1=2$, $a_2=5$, and $a_{n+1} = a_{n+2} + a_{n}$, calculate the value of $a_6$. Provide your answer as a single integer. | -3 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence with a first term of -1 and a common difference of -3, calculate the value of the 8th term (a_8). Express your answer as a single integer. | -22 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given the inequality $\frac{x}{3}+2m \lt -3$ and that the real number $3$ is a solution to this inequality, find the largest integer value that $m$ can take. Express your answer as an integer. | -3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
A factory has several workshops. Using stratified sampling, a sample of 128 items is drawn from the 2048 products produced that day to check their quality. If one workshop produced 256 items on that day, calculate the number of items drawn from this workshop. Express your answer as a whole number. | 16 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Using the digits 0, 1, 2, 3 without repeating any digit, how many three-digit numbers can be formed, given that the highest digit cannot be 0? Express your answer as a single integer. | 18 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Permutations"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence {a_n} with the sum of the first n terms denoted as S_n, and given that a_5 = 8 and S_3 = 6, find the value of the common difference d. Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
5 people stand in a row. Calculate the total number of ways to arrange them so that person A and person B are not adjacent to each other. Express your answer as a whole number. | 72 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given the universal set $U=\{-3,-2,-1,0,1,2,3\}$, set $A=\{-3,-2,2,3\}$, and $B=\{-3,0,1,2\}$, find the intersection of the complement of $A$ in $U$ and $B$, denoted as $(\complement _{U}A)\cap B$. Express your answer as a set of elements using standard set notation. | \{0,1\} | big_math | [
"Mathematics -> Discrete Mathematics -> Set Theory -> Other"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the complex number Z satisfies Z(2+i)=-3+i, find the conjugate of Z. Express your answer in the form a+bi, where a and b are real numbers. | -1 - i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Find the domain of the function $f(x)=\frac{1}{{\sqrt{1-2x}}}$ in interval notation. | (-\infty, \frac{1}{2}) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
What is the maximum number of points at which a circle can intersect a triangle? Express your answer as a whole number. | 6 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Find the solution set of the inequality $|x| \cdot (1 - 2x) > 0$. Express your answer in interval notation. | \left(- \infty, 0 \right) \cup \left(0, \frac{1}{2} \right) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given that \( x \) and \( y \) are both negative integers satisfying the equation \( y = \frac{10x}{10 - x} \), find the maximum value of \( y \). Express your answer as a single integer, including the negative sign if applicable. | -5 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $\left(\cos \theta +i\sin \theta \right)^{n}=\cos n\theta +i\sin n\theta$, apply it to the specific case where $\theta = \frac{\pi}{9}$ and $n=6$. Then, given that $z\cdot\left(\cos\frac{\pi}{9}+i\sin\frac{\pi}{9}\right)^{6}=2$, calculate the magnitude of z, |z|. Express your answer as a single numbe... | 2 | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Find the value of the real number $a$ given that the coefficient of $x^{7}$ in the expansion of $(x-a)^{10}$ is $15$. Express your answer as a single real number. | -\frac{1}{2} | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Niko counted a total of 60 birds perching in three trees. Five minutes later, 6 birds had flown away from the first tree, 8 birds had flown away from the second tree, and 4 birds had flown away from the third tree. He noticed that there was now the same number of birds in each tree. Calculate the number of birds origin... | 22 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets $A=\{1,3,5,7,9\}$ and $B=\{0,3,6,9,12\}$, find the intersection of sets A and B. Express your answer as a set of elements, separated by commas and enclosed in curly brackets. | \{3,9\} | big_math | [
"Mathematics -> Discrete Mathematics -> Set Theory -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence {a_n}, the sum of the first n terms is S_n. If a_2 + a_8 = 10, find the value of S_9. Express your answer as a single numerical value. | 45 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
An ellipse passes through the origin and has foci at ${F_1}(1,0)$ and ${F_2}(3,0)$. Determine the length of its semi-minor axis. Express your answer as a numerical value, including the square root symbol if necessary. | \sqrt{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Given that the derivative of the even function f(x) (x ≠ 0) is f'(x), and it satisfies f(1) = 0, when x > 0, xf'(x) < 2f(x). Find the range of x that makes f(x) > 0 true. Express your answer as an interval or union of intervals. | (-1, 0) ∪ (0, 1) | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\overrightarrow{e_{1}}$ and $\overrightarrow{e_{2}}$ are two unit vectors with an angle of $60^{\circ}$ between them, and $\overrightarrow{a} = \overrightarrow{e_{1}} + \overrightarrow{e_{2}}$, $\overrightarrow{b} = -4\overrightarrow{e_{1}} + 2\overrightarrow{e_{2}}$, find the angle between $\overrightarrow... | 120 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the real and imaginary parts of the complex number $\frac{1+ai}{2+i}$ are equal, find the value of the real number $a$. Express your answer as a single real number. | 3 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Convert the binary number $110011_{(2)}$ to a decimal number. Express your answer as a single decimal number. | 51 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
A paper strip has 9 consecutive sections that can be colored either red or white. If no two consecutive sections can be white, how many valid colorings are possible? Express your answer as a single integer. | 89 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics -> Other"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
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