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 |
|---|---|---|---|---|---|
Calculate the difference between the sum of the first 100 odd natural numbers and the sum of the first 100 even natural numbers. Express your answer as a single integer. | 100 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\overrightarrow{a}$ and $\overrightarrow{b}$ are both unit vectors with an angle of $\frac{\pi}{3}$ between them, find the magnitude of $\overrightarrow{a} + \overrightarrow{b}$. Express your answer as a numerical value. | \sqrt{3} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the arithmetic square root of 0.09. Express your answer as a decimal number. | 0.3 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Decimals"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Let the complex number z satisfy the equation z * (i - 1) = 2i, where i is the imaginary unit. Calculate the value of z. Express your answer as a complex number in the form a + bi, where a and b are real numbers. | 1 - i | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the function f(x) is an odd function, and for x < 0, f(x) = 2x^2 - 1, evaluate f(1). Express your answer as a single integer. | -1 | big_math | [
"Mathematics -> Algebra -> Functions"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $\frac{x^2}{1+k} - \frac{y^2}{1-k} = 1$, find the range of values for $k$ that make the equation represent a hyperbola. Express your answer as an interval or inequality. | (-1, 1) | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Given the set $A=\{(x,y)|y=x^{2}\}$ and set $B=\{(x,y)|y=1-|x|\}$, find the number of proper subsets of the intersection of sets A and B, $A \cap B$. Express your answer as a whole number. | 3 | big_math | [
"Mathematics -> Set Theory -> Other"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\overrightarrow{a}=(-3,2,5)$ and $\overrightarrow{b}=(1,x,-1)$, and $\overrightarrow{a} \cdot \overrightarrow{b}=2$, find the value of $x$. Express your answer as a single integer. | 5 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Let \( f(x) = x + \sqrt{x^{2} + 1} + \frac{1}{x - \sqrt{x^{2} + 1}} \). Calculate the value of \( f(2015) \). Express your answer as a single number. | 0 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Jack plans to ride from his home to Jill's home. Jill's home is located three blocks east and two blocks north of Jack's home. At each intersection, Jack can choose to go east or north, but he must avoid the intersection that is one block east and one block north of his home due to frequent traffic accidents. Find the ... | 4 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
If $5^{\lg x} = 25$, then what is the value of $x$? Express your answer as a single numerical value. | 100 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Cicely had her 21st birthday in 1939. In what year did she have her 100th birthday? Provide your answer in the format YYYY. | 2018 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
In the sequence {a_n}, where a_n > 0, a_1 = 1, and (a_{n+1}^2 + a_n^2) / (a_{n+1}^2 - a_n^2) = 2n, calculate the value of a_{113}. Express your answer as a single number. | 15 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Consider the set $M=\{x|x^2-x\leq 0\}$ and the function $f(x)=\log|x|$, whose domain is denoted by $N$. Calculate the intersection $M\cap N$ and express your answer in standard interval notation. | (0, 1] | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities",
"Mathematics -> Calculus -> Other"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Let $f(x)$ be an even function defined on $\mathbb{R}$ and decreasing on $(-\infty, 0)$. If $x_{1} < 0$ and $x_{1}+x_{2} > 0$, compare the values of $f(x_{1})$ and $f(x_{2})$. Express your answer as a comparison of function values (e.g., $f(x_{1}) < f(x_{2})$). | f(x_{1}) < f(x_{2}) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given a hyperbola $C: \\dfrac{y^{2}}{a^{2}}-\\dfrac{x^{2}}{b^{2}}=1\\left(a > 0,b > 0\\right)$, where the length of the real axis is $\\sqrt{2}$ times the length of the imaginary axis, find the equation of the asymptotes of hyperbola $C$. Express your answer in the form $y = \\pm kx$. | y = \\pm \\sqrt{2}x | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a} = (1, \sqrt{1+\sin 20^\circ})$ and $\overrightarrow{b} = \left(\frac{1}{\sin 55^\circ}, x\right)$ are collinear, find the value of the real number $x$. Express your answer as a real number. | \sqrt{2} | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Geometry -> Other"
] | 0.09375 | 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"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
The perimeter of a rectangle is 48 cm, and the length is known to be 2 times the width. Find the length of the rectangle. Express your answer in centimeters (cm) as a whole number. | 16 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Find the sum of the real part and the imaginary part of the complex number $3-4i$. Express your answer as a single numerical value. | -1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Eugene swam on Sunday, Monday, and Tuesday. On Monday, he swam for 30 minutes. On Tuesday, he swam for 45 minutes. His average swim time over the three days was 34 minutes. Find the number of minutes Eugene swam on Sunday. Express your answer in minutes as a whole number. | 27 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(-3,2,5)$ and $\overrightarrow{b}=(1,x,-1)$, and $\overrightarrow{a}\bot \overrightarrow{b}$, find 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 |
The sum of three natural numbers \(A\), \(B\), and \(C\) is 111. It is known that the average of \(A\) and \(B\) is 31, and the average of \(A\) and \(C\) is 37. Calculate the average of \(B\) and \(C\). Express your answer as a whole number. | 43 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Convert the Cartesian equation $x=1$ into polar coordinates using the relationship $x = \rho \cos \theta$. Express your answer as a polar coordinate equation. | \rho \cos \theta = 1 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = e^x - e^(1-x) - ax, if x1 and x2 are two zeros of the function g(x) = e^x + e^(1-x) - a, and f(x1) + f(x2) = -4, calculate the real number a. | 4 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given a complex number z satisfies the equation z(1+3i) = 4+i, solve for z in standard complex number format (a+bi). | \frac{7}{10} - \frac{11}{10}i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = cos(x) + 2sin(x), find f'(π/4). Express your answer as a boxed value. | \frac{\sqrt{2}}{2} | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
To make a regular dodecagon coincide with itself after rotation, what is the minimum degree it should be rotated around its center? Express your answer in degrees as a whole number. | 30 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
A factory has three workshops: A, B, and C, which produced the same kind of product in quantities of 600, 400, and 300 pieces, respectively. Using stratified sampling to draw a sample of size n, if 6 pieces are drawn from workshop C, find the value of n. Express your answer as a whole number. | 26 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Let $i$ be the imaginary unit. For the complex number $z= \frac {a}{1-2i}+|i|$, where $a$ is a real number, the real part and the imaginary part are opposite numbers. Find the value of $a$. Express your answer as a single real number. | -\frac{5}{3} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Consider a hyperbola given by the equation $\frac{y^2}{64} - \frac{x^2}{16} = 1$. If the distance from a point $P$ on this hyperbola to one of its foci is $4$, find the distance from point $P$ to the other focus. Express your answer as a whole number. | 20 | big_math | [
"Mathematics -> Geometry -> Non-Euclidean Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence {an} with a common difference of 2, if a1, a3, and a4 form a geometric sequence, then find the value of a2. Express your answer as a single integer. | -6 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Four points \( B, A, E, L \) are on a straight line. \( G \) is a point off the line such that \(\angle B A G = 120^\circ\) and \(\angle G E L = 80^\circ\). Calculate the reflex angle at \( G \) in degrees. | 340 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, the points (2,1), (4,1), and (2,5) are three vertices of a rectangle. Find the coordinates of the fourth vertex of the rectangle. Express your answer as coordinates (x, y). | (4,5) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the random variable $ξ∼N(-1,σ^{2})$, and $P(-3≤ξ≤-1)=0.4$, calculate the probability $P(ξ≥1)$. Express your answer as a decimal value rounded to 1 decimal place. | 0.1 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
A group of people divide $90$ dollars equally, with each person receiving a certain amount. If another $6$ people are added and $120$ dollars are divided equally, then each person receives the same amount as the first time. Let the number of people who divide the money the second time be $x$. Write an equation that rep... | $\frac{90}{x-6}=\frac{120}{x}$ | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of the complex number expression $i^3(1+i)^2$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Determine the sign of the expression $\sin 2\cos 3$. Is it positive, negative, or zero? Provide your answer in the form $\boxed{\text{positive}}$, $\boxed{\text{negative}}$, or $\boxed{\text{zero}}$. | \text{negative} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
If both $x$ and $y$ in the fraction $\frac{2xy}{x+y}$ are increased by a factor of $2$, by what factor does the value of the fraction change? Express your answer as a single numerical factor. | 2 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
A travel company leads tourists to visit four out of five pavilions, A, B, C, D, and E, in the Expo Park. Pavilion A must be visited and must be the first one visited. If Pavilion C is visited, it cannot be the last one visited. Calculate the total number of different visiting sequences. | 18 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
If $\alpha$ is in the third quadrant, calculate the value of $$\frac {\cos\alpha}{ \sqrt {1-\sin^{2}\alpha}}+ \frac {2\sin\alpha}{ \sqrt {1-\cos^{2}\alpha}}.$$ Express your answer as a single numerical value. | -3 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the terminal side of angle α passes through point P(-5,12), determine the value of cos α. Express your answer as a fraction. | -\dfrac{5}{13} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(1,3)$ and $\overrightarrow{b}=(3,\lambda)$, if $\overrightarrow{a}⊥\overrightarrow{b}$, calculate the value of the real number $\lambda$. Express your answer as a single real number. | -1 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Let $S_{n}$ be the sum of the first $n$ terms of an arithmetic sequence $\{a_{n}\}$. Given $a_{3}=3$ and $S_{7}=14$, find the common difference $d$. Express your answer as a single number. | -1 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Xiao Ming has already played 20 matches and has a winning rate of 95%. If he does not lose any more matches, how many more matches does he need to win to achieve a winning rate of 96%? Express your answer as a whole number. | 5 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the result of the expression $-1-3$. Express your answer as a single integer. | -4 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number z that satisfies z(3-4i)=1+2i, calculate the conjugate of z. Express your answer as a complex number in the form a+bi. | -\dfrac{1}{5}-\dfrac{2}{5}i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function \( f(x) = \frac{x^5}{5x^4 - 10x^3 + 10x^2 - 5x + 1} \), calculate the sum \( \sum_{i=1}^{2009} f\left( \frac{i}{2009} \right) \). Express your answer as a single integer. | 1005 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
How many arithmetic sequences satisfy the conditions that the first term is 1783, the last term is 1993, the number of terms is at least 3, and the common difference is a natural number greater than 2? Express your answer as a whole number. | 13 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(\tan x) = \cos (2x + \frac{\pi}{3}) - 1$, find the value of $f(\sqrt{3})$. Express your answer as a single numerical value. | -2 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Find the total number of different six-digit even numbers that can be formed by rearranging the digits of the number "124057". Express your answer as a single integer. | 312 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Permutations"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
The ages of three contestants in the Pascal Contest are 14 years, 9 months; 15 years, 1 month; and 14 years, 8 months. Calculate the average (mean) age of the contestants. Express your answer in years and months. | 14 years, 10 months | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given $0 < a < 1$, and $a^{2\log_{3}a} = 81\sqrt{3}$, calculate the value of $\frac{1}{{a}^{2}}+\log_{9}a$. Express your answer as a single fraction or decimal value. | \frac{105}{4} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Logarithmic Functions"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given a complex number $z = \frac {a}{2+i}+ \frac {2+i}{5}$, where the sum of its real part and imaginary part is $1$, find the value of the real number $a$. Express your answer as a single real number. | 2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given the system of inequalities:
\[
\begin{cases}
5x + 10y \leq 30 \\
2x - y \leq 3 \\
x, y \in \mathbb{N}
\end{cases}
\]
Find the maximum value of \( x + y \). Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Determine the number of times the loop body is executed in the following statements:
i = 1
Do
i = i + 1
i = 5 * i
LOOP UNTIL i > 15
Express your answer as a whole number. | 2 | big_math | [
"Mathematics -> Discrete Mathematics -> Algorithms -> Other"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Given the parametric equations of the curve C:
$$
\left\{ \begin{matrix}
x=3+\cos \theta, \\
y=\sin \theta
\end{matrix} \right.
$$
where θ is the parameter, find the maximum value of the distance |OM| from the origin O to any point M on the curve C. Express your answer as a single numerical value. | 4 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Solve the system of linear equations $\left\{{\begin{array}{l}{x+y=6}\\{x=2y}\end{array}}\right.$ and express your answer as an ordered pair (x, y). | (4,2) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Systems of Equations"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
In a geometric sequence with a common ratio of 2, given that the product of the second term (a2) and the fifth term (a5) equals 32, find the product of the fourth term (a4) and the seventh term (a7). Express your answer as a single numerical value. | 512 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Given the inequality $\frac{1}{x} + \frac{a}{y} \geqslant \frac{16}{x+y}$, which holds true for any positive real numbers $x$ and $y$, find the minimum value of the positive real number $a$. Express your answer as a single number. | 9 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z= \dfrac {1+2i}{1-i}$, find the quadrant in which its conjugate complex number $\overset{.}{z}$ lies. Express your answer as the name of the quadrant (e.g. first quadrant, second quadrant, etc.). | third quadrant | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
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