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 |
|---|---|---|---|---|---|
Simplify the complex fraction: $\dfrac{1+3i}{1-i}$. Express your answer in the form a+bi, including both real and imaginary parts. | -1+2i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
In the complex plane, determine the quadrant in which the point corresponding to the complex number $i(2-i)$ is located. Express your answer as the name of the quadrant (e.g. first, second, third, fourth). | first | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Let the quadratic function f(x) = ax^2 - 2ax + c be monotonically decreasing in the interval [0,1], and f(m) ≤ f(0). Find the range of the real number m. Express your answer using standard interval notation. | [0,2] | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Quadratic Functions"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given a sequence {a_n} where a_1 = 4/5, and a_{n+1} is defined by
$$
a_{n+1}=
\begin{cases}
2a_n, & \text{if } 0 \leq a_n \leq \frac{1}{2} \\
2a_n - 1, & \text{if } \frac{1}{2} < a_n \leq 1
\end{cases}
$$
find the value of a_{2015}. Express your answer as a fraction in simplest form. | \frac{1}{5} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
In a school of 100 students, 39 participated in a Chinese competition, 49 participated in a Math competition, and 41 participated in an English competition. Fourteen students participated in both the Chinese and Math competitions, thirteen participated in both the Math and English competitions, nine participated in bot... | 6 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
A certain school, in order to understand the intentions of students to participate in social practice activities, decided to use stratified sampling to select 200 students from the first, second, and third year students. It is known that there are 1300 first-year students, 1200 second-year students, and 1500 third-year... | 75 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Given two vectors in the plane $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfying $\overrightarrow{a} \cdot (\overrightarrow{a} + \overrightarrow{b}) = 5$ and $|\overrightarrow{a}| = 2$, $|\overrightarrow{b}| = 1$, calculate the angle between vector $\overrightarrow{a}$ and $\overrightarrow{b}$. Express your answ... | \dfrac{\pi}{3} | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Geometry -> Other",
"Mathematics -> Linear Algebra -> Vectors"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Select 4 people from 6 to visit Paris, London, Sydney, and Moscow, respectively, with the restriction that person A and person B cannot visit Paris. Each city is visited by one person, and each person visits only one city. Calculate the total number of different selection schemes. | 240 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $D$ is a point on the side $BC$ of $\triangle ABC$, and $\overrightarrow{BC}=3\overrightarrow{BD}$, express the vector $\overrightarrow{AD}$ in terms of vectors $\overrightarrow{AB}$ and $\overrightarrow{AC}$. Provide your answer as a vector equation. | \frac{2}{3}\overrightarrow{AB} + \frac{1}{3}\overrightarrow{AC} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, \( PR \) and \( QS \) meet at \( X \). Also, \(\triangle PQX\) is right-angled at \(Q\) with \(\angle QPX = 62^\circ\) and \(\triangle RXS\) is isosceles with \( RX = SX \) and \(\angle XSR = y^\circ\). Find the value of \( y \) in degrees. | 76 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Integers x and y with x > y > 0 satisfy the equation x + y + xy = 80. What is the value of x? Express your answer as a single integer. | 26 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
If $p(m, n)$ is a point on the terminal side of a $600^\circ$ angle, calculate the value of $\frac{n}{m}$. Express your answer as a single value, possibly with a square root. | \sqrt{3} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
Given that point $M$ is a point on the number line, and the distance from point $M$ to the origin is $1$. If point $M$ moves $3$ units to the right along the number line to point $N$, calculate the number or numbers represented by point $N$. Express your answer as a numerical value or values. | 4 \text{ or } 2 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\angle 1$ and $\angle 2$ are complementary, $\angle 2$ and $\angle 3$ are complementary, and $\angle 1 = 40^{\circ}$, find the measure of $\angle 3$. Express your answer in degrees as a whole number. | 40 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Solve for x in the equation x + 2 = 7. Express your answer as a single integer. | x = 5 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
In triangle $\triangle ABC$, given $BC=6$, $AC=4$, $\sin A=\frac{3}{4}$, calculate the measure of angle $B$ in radians. Express your answer using $\pi$ and simplify the fraction if possible. | \frac{\pi}{6} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given $(1-2x)^{2016} = a_0 + a_1(x-2) + a_2(x-2)^2 + \ldots + a_{2015}(x-2)^{2015} + a_{2016}(x-2)^{2016}$ ($x \in \mathbb{R}$), find the value of $a_1 - 2a_2 + 3a_3 - 4a_4 + \ldots + 2015a_{2015} - 2016a_{2016}$ after differentiating both sides with respect to $x$ and setting $x=1$. Express your answer as a single num... | 4032 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Assume the random variable $X\sim N(1,\delta^2)$. Given that $P(X > 2) = 0.3$, calculate the probability $P(X > 0)$. Express your answer as a decimal value between 0 and 1. | 0.7 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Find the domain of the function $f(x)= \dfrac {1}{ \sqrt {1-x}}$ and express your answer in interval notation. | (-\infty,1) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
For any positive integers n and k where k ≤ n, let f(n, k) represent the number of positive integers less than or equal to ⌊n/k⌋ that are coprime with n. Calculate f(100, 3). Express your answer as a single integer. | 14 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Given two circles C1: x^2 + y^2 = 1 and C2: x^2 + y^2 - 6x - 8y + m = 0 that are externally tangent, find the value of m. Express your answer as a single integer. | 9 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
In a high school, there are 700 students in the freshman year, 500 students in the sophomore year, and 300 students in the senior year. A sample of size n is taken from the high school students by stratified sampling, and it is known that 14 freshmen are sampled. Calculate the total sample size n. Express your answer a... | 30 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z=3-4i$, calculate the expression $\frac{{\overline{z}}}{{|z|}}$. Express your answer as a complex number in the form $a + bi$, where $a$ and $b$ are the real and imaginary parts, respectively. | $\frac{3}{5} + \frac{4}{5}i$ | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of terms in the arithmetic sequence -1, 1, …, 89. Express your answer as a whole number. | 46 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of $\left(\frac{1}{4}\right)^{-\frac{1}{4}}$. Express your answer as a single value, possibly with a radical. | \sqrt{2} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Given two positive integers where one is twice the other, and their sum is less than 100, calculate the maximum possible value of their sum. Express your answer as a single integer. | 99 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
There is a set of points M on a plane and seven different circles C1, C2,..., C7. Circle C7 passes through exactly 7 points in M, circle C6 passes through exactly 6 points in M, and so on, with circle C1 passing through exactly 1 point in M. Find the minimum number of points in M. Express your answer as a whole number. | 12 | big_math | [
"Mathematics -> Geometry -> Other",
"Mathematics -> Combinatorics -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given $\log_{7}[\log_{3}(\log_{2}x)]=0$, calculate the value of $x^{-\frac{1}{2}}$. Express your answer as a fraction, if possible. | \frac{\sqrt{2}}{4} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Logarithmic Functions"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
In a positive geometric sequence $\{a_n\}$, $a_1=1$, and the sum of the first $n$ terms is $S_n$. If $-a_3$, $a_2$, and $a_4$ form an arithmetic sequence, calculate the value of $S_7$. Express your answer as a single integer. | 127 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given sets $A=\{x|x^{2}-2x-3 \lt 0\}$ and $B=\{x|2^{x-2} \gt 1\}$, find the intersection of $A$ and the complement of $B$ in the real numbers, denoted as $A\cap (\complement _{R}B)$. Express your answer as an interval or set notation. | (-1,2] | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Inequalities"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Find the coefficient of $x^2$ in the expansion of $(x+2+\frac{1}{x})^5$. Express your answer as a single integer. | 120 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
If the complex number $\left(a+i\right)\left(1-ai\right)=2$, where $a\in R$, then find the value of $a$. Express your answer as a single real number. | 1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Given $z=4 + 3i$, find the value of $\frac{\overline{z}}{|z|}$. Express your answer as a complex number in the form $a + bi$. | \frac{4}{5} - \frac{3}{5}i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, the two circles are centered at \( O \). Point \( S \) is on the larger circle. Point \( Q \) is the point of intersection of \( OS \) and the smaller circle. Line segment \( PR \) is a chord of the larger circle and touches (that is, is tangent to) the smaller circle at \( Q \). Note that \( OS \) is t... | 6.5 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Circles"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
In the geometric sequence $\{a_n\}$, $a_3a_4a_5 = 3$ and $a_6a_7a_8 = 24$. Calculate the product of $a_9a_{10}a_{11}$. Express your answer as a single integer. | 192 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
A quadrilateral is inscribed in a circle. If an angle is inscribed into each of the four segments outside the quadrilateral, find the sum of these four angles, expressed in degrees. | 540 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given $a, b \in \mathbb{R}$, and $a^2 + b^2 = 4$, find the range of values for $3a + 2b$. Express your answer as a closed interval, including the endpoints. | [-2 \sqrt{13}, 2 \sqrt{13}] | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the sequence {a<sub>n</sub>} is a geometric sequence, a<sub>7</sub> = 4, a<sub>5</sub> + a<sub>9</sub> = 10, find the value of a<sub>3</sub> + a<sub>11</sub>. Express your answer as a single integer. | 17 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Let \(P\) be a point on the curve \(C\): \(y=x^{2}-2x+3\), and the range of the slope angle of the tangent line at point \(P\) on curve \(C\) is \(\left[0, \frac{\pi}{4}\right]\). Calculate the range of the x-coordinate of point \(P\). Express your answer in interval notation. | \left[1, \frac{3}{2}\right] | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ with an angle of $120^{\circ}$ between them, and $|\overrightarrow{a}|=1$, $|\overrightarrow{b}|=2$, calculate the projection of the vector $\overrightarrow{a}+ \overrightarrow{b}$ in the direction of vector $\overrightarrow{a}$. Express your answer as a numer... | 0 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Given a square ABCD with side length 2, and E is the midpoint of CD, calculate the dot product of vectors AC and BE. | 2 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Given the values of $a$, $b$, and $c$ as follows:
$a = (99)^0$,
$b = (-0.1)^{-1}$,
$c = (-\frac{5}{3})^{-2}$,
determine the order of the three numbers $a$, $b$, and $c$. Express your answer in the format 'a < b < c' or similar. | b < c < a | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a$ is the largest negative integer, $b$ is the integer with the smallest absolute value, and $c$ is the smallest positive integer, find the value of $|a+b-c|$. Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
The function $f(x)$ is an odd function defined on $\mathbb{R}$. When $x > 0$, $f(x)=-x+1$. Determine the expression of $f(x)$ when $x < 0$. Express your answer as a function of $x$, including all necessary mathematical operations and constants. | -x-1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence $\{\frac{2^n+1}{2^n}\}$, let $T_n$ be the sum of the first n terms. Find the minimum integer value of n such that $n > T_{10} + 1013$. | 1024 | big_math | [
"Mathematics -> Sequences and Series -> Other"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function y = (m^2 + 2m)x + m^2 + m - 1 + (2m - 3) is a linear function of x, find the value of the constant m. Express your answer as a single numerical value. | 1 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given the line $y=x+b$, where $b\in[-2,3]$, find the probability that the line's y-intercept is greater than $1$. Express your answer as a simplified fraction. | \dfrac {2}{5} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given that sin(74°) = m, use the cofunction and half-angle identities to express cos(8°) in terms of m. | \sqrt{\frac{1+m}{2}} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions -> Other"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
A 2 x 2 x 2 cube has three faces partially shaded, with each shaded face being half-shaded. What fraction of the total surface area of the cube is shaded? Express your answer as a simplified fraction. | \frac{1}{4} | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Surface Area"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the result of $(-2) + 1$. Express your answer as a single integer. | -1 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Solve for x in the equation $10^{x}\\cdot 100^{2x}=1000^{5}$. Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
A certain unit has 35 young workers, 25 middle-aged workers, and 15 older workers. To understand the health conditions of the workers in the unit, a stratified sampling method is used to draw a sample. If the sample contains 7 young workers, find the total sample size. Express your answer as a whole number. | 15 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\vec{a}=(x,y), \vec{b}=(-1,2)$, and $\vec{a}+\vec{b}=(1,3)$, calculate the value of $|\vec{a}-2\vec{b}|$. Express your answer as a single whole number. | 5 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Find the minimum positive period of the function y = sin(x) * |cos(x)|. Express your answer in terms of π. | 2\pi | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
If $x^2 + 14x + c$ is a perfect square, find the value of the constant $c$. Express your answer as a single integer. | 49 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given sin(π/2 + 2α) = -4/5, where α is in the interval (π/2, π), find the value of tan(α + π/4). Express your answer as a single fraction or decimal, including the negative sign if applicable. | -\frac{1}{2} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the radius of circle $\odot O$ is $3cm$, calculate the length of the longest chord in the circle, which is its diameter. Express your answer in centimeters. | 6cm | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equations x^2 y z^3 = 7^3 and x y^2 = 7^9, calculate the product x y z. Express your answer as a numerical value with base and exponent. | 7^4 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given a geometric sequence {a_n}, where a_3 and a_{15} are the two roots of the equation x^2 - 6x + 1 = 0, find the product a_7a_8a_9a_{10}a_{11}. Express your answer as a single numerical value. | 1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Let the constant term in the expansion of $(\\frac{1}{x}+x^{2})^{3}$ be $a$. Find the area of the figure enclosed by the line $y=ax$ and the curve $y=x^{2}$. Express your answer as a single number. | \\frac{9}{2} | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Given a function y = f(x), the graph of the function y = f(x-19) and the graph of the function y = f(99-x) are symmetric with respect to a certain line. Find the equation of this symmetry line. Express your answer in the form x = a. | x = 59 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Suppose $a$ is the number obtained by throwing a die. Calculate the probability that the equation $x^2 + ax + 2 = 0$ has two distinct real roots. Express your answer as a simplified fraction. | \frac{2}{3} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
In the figure, \(AB \parallel EF \parallel CD\), given \(AB = 20\), \(CD = 80\), and \(BC = 100\). Find the value of \(EF\). Express your answer as a whole number. | 16 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Let the sum of the first n terms of an arithmetic sequence {a_n} be S_n. Given that a_2 + a_4 = 6, calculate the value of S_5. Express your answer as a single numerical value. | 15 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Given that sin(2α) = 2/3, calculate the value of cos²(α + π/4). Express your answer as a fraction. | \frac{1}{6} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x)= \begin{cases} x^{2}+2x, x < 0 \\ \frac {x+1}{e}, x\geqslant 0 \end{cases}$, calculate the number of "sister point pairs" that $f(x)$ has, where a "sister point pair" is defined as two distinct points $A$ and $B$ that satisfy:
1) Points $A$ and $B$ both lie on the graph of the function $f(x)$;
... | 2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the random variable ξ follows a normal distribution N(1,1), and P(ξ < 3) = 0.977, find the probability P(-1 < ξ < 3). Express your answer as a decimal value between 0 and 1, rounded to 3 decimal places. | 0.954 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
A cylindrical glass with a base radius of 2 contains water with a volume of V. If a glass ball (with a radius equal to the base radius of the cylindrical glass) is placed in it and the water just submerges the glass ball, calculate the original volume of water, V. Express your answer as a fraction with pi (π) if necess... | \frac{16\pi}{3} | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Volume"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation of the hyperbola $\dfrac{x^2}{9} - y^2 = 1$ and the equation of the line $y = \dfrac{1}{3}(x+1)$, calculate the number of intersection points between the line and the hyperbola. Express your answer as a whole number. | 1 | big_math | [
"Mathematics -> Geometry -> Analytic Geometry -> Other"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Given $9^{m}=10$, $a=10^{m}-11$, and $b=8^{m}-9$, determine the relationship between $a$ and $b$ in terms of inequalities, including $0$ if applicable. Express your answer as a clear inequality relationship. | a > 0 > b | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
A baseball league consists of two four-team divisions. Each team plays every other team in its division N games. Each team plays every team in the other division M games with N > 2M and M > 4. Each team plays a 76 game schedule. Calculate the number of games a team plays within its own division. | 48 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given a sequence $\{a_n\}$ ($n\geq1$) satisfying $a_{n+2}=a_{n+1}-a_n$, and $a_2=1$. If the sum of the first 2011 terms of the sequence is 2012, then calculate the sum of the first 2012 terms. Express your answer as a single integer. | 2013 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
A magazine printed photos of three celebrities along with three photos of the celebrities as babies. The baby pictures did not identify the celebrities. Readers were asked to match each celebrity with the correct baby pictures. What is the probability that a reader guessing at random will match all three correctly? Exp... | \frac{1}{6} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Other"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Given $x + x^{-1} = 3$, calculate the value of $x^2 + x^{-2}$. Express your answer as a single numerical value. | 7 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Consider an ellipse x^2/m + y^2 = 1 (m > 1) and a hyperbola x^2/n - y^2 = 1 (n > 0) that share the same foci F1 and F2. Let P be a point of intersection between the two curves. Calculate the area of the triangle F1PF2. | 1 | big_math | [
"Mathematics -> Geometry -> Non-Euclidean Geometry -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Amoebas reproduce by splitting and their numbers double every three minutes. If one amoeba is placed in a bottle, then after 1 hour, the bottle is filled with amoebas. If initially, two amoebas are placed in the bottle, calculate the time it takes for the bottle to be filled with amoebas. Express your answer in minutes... | 57 | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
The height h (in meters) of a certain kind of fireworks above the ground is given by the formula h = -3.6t^2 + 28.8t, where t is the time in seconds. Find the moment when the fireworks burst after shooting up. Express your answer in seconds, rounded to the nearest tenth if necessary. | 4 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
In a sequence, each term after the first two terms is the mean of all the terms which come before that term. Given that the first term is 8 and the tenth term is 26, find the value of the second term. | 44 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Given the plane vectors $a=(1,m)$, $b=(2,5)$, $c=(m,3)$, and the condition that $(a+c) \parallel (a-b)$, calculate the value(s) of $m$. Express your answer using any necessary mathematical notation (e.g., ±, √). | \dfrac{3\pm \sqrt{17}}{2} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Find the distance between the parallel lines $x-2y+1=0$ and $2x-4y-3=0$. Express your answer as a fraction with a square root. | \frac{\sqrt{5}}{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Let $S_n$ be the sum of the first $n$ terms of a geometric sequence $\{a_n\}$. Given that $a_2 - 8a_5 = 0$, find the value of $\frac{S_8}{S_4}$. Express your answer as a numerical value. | \frac{17}{16} | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Sequences and Series"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Find the domain of the function f(x) = $$\frac{\ln(x+2)}{\sqrt{1-3^x}}$$. Express your answer in interval notation. | (-2, 0) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
In an opaque box, there are 4 small balls with the same shape, size, and texture. These balls are labeled with the numbers -2, 0, 1/4, and 3 respectively. Calculate the probability that a randomly drawn ball from the box has a positive number. Express your answer as a simplified fraction. | \frac{1}{2} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
If 50% of \( P \) equals 20% of \( Q \), then find \( P \) as a percent of \( Q \). Express your answer as a percentage, including the percentage sign (%). | 40% | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the area of a sector is $\dfrac{3\pi}{8}$ and its radius is $1$, find the central angle of the sector. Express your answer in radians as a fraction of $\pi$. | \dfrac{3\pi}{4} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Find the minimum value of the function y = (x^2 + 7x + 10) / (x + 1), given x > -1. Express your answer as a single number. | 9 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the exact value of the expression \( 5 + 4 - 3 + 2 - 1 \). Express your answer as a single integer. | 7 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
If each interior angle of a polygon is $120^{\circ}$, find the number of its sides. Express your answer as a whole number. | 6 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Polygons"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets U = {0,1,2,3,4}, A = {0,1,3}, and B = {2,3}, calculate A⋂(∁_U B). Express your answer as a set of elements enclosed in curly brackets, with elements separated by commas. | {0,1} | big_math | [
"Mathematics -> Discrete Mathematics -> Logic -> Set Theory"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given $\overrightarrow{a}=(1,\sin α) $ and $\overrightarrow{b}=(2,4\cos α) $, if $\overrightarrow{a}//\overrightarrow{b}$, find the value of $\tan α$. Express your answer as a single numerical value. | 2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
A star player gives all 5 identical small gifts to 3 different fans, ensuring each fan receives at least one gift. Find the total number of different ways to distribute the gifts. Express your answer as a whole number. | 6 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Find the smallest natural number that has exactly 20 divisors. Express your answer as a single integer. | 240 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
There are 100 balls in a container, of which 36% are red balls and the rest are blue balls. How many blue balls should be removed so that red balls account for 72% of the remaining balls? Express your answer as a whole number. | 50 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Under normal circumstances, the size of machine parts manufactured by workers follows a normal distribution $N(\mu, \sigma^2)$. Using the $3\sigma$ principle, calculate the most likely number of parts out of $1000$ that fall within the size range $(\mu-3\sigma, \mu+3\sigma)$. Provide your answer as a whole number. | 997 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z=i(2-i)$ (where $i$ is the imaginary unit), find the conjugate of $z$, denoted as $\overline{z}$. Express your answer in the form $a + bi$, where $a$ and $b$ are real numbers. | 1-2i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Find the function that results from reflecting the graph of y = ln(x) in the y-axis and then translating it 2 units to the right, thereby making it symmetric to y = ln(x) with respect to the line x = 1. Express your answer as a function of x in the form y = f(x). | y = ln(2 - x) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
6 athletes are standing on 6 tracks ready to participate in a competition. Among them, athlete A cannot stand on the first or second track, and athlete B must stand on either the fifth or sixth track. Calculate the total number of different arrangements possible for the athletes. Express your answer as a single numeric... | 144 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Find the slope angle of the tangent line to the graph of the function f(x) = e^x * cos(x) at the point (0, f(0)). Express your answer in radians. | \dfrac {\pi}{4} | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Given the table of x and x^2 values, and the inequality a < √13 < b, where a and b represent two adjacent numbers, find the value of a. Express your answer as a decimal.
| x | 3 | 3.1 | 3.2 | 3.3 | 3.4 | 3.5 | 3.6 | 3.7 | 3.8 | 3.9 | 4 |
|-------|-------|-------|-------|-------|-------|-------|-------|-------|--... | 3.6 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(1,2)$ and $\overrightarrow{b}=(x,1)$, and $\overrightarrow{a} \perp \overrightarrow{b}$, find the value of $x$ that satisfies this condition. Express your answer as a single number. | -2 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
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