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 |
|---|---|---|---|---|---|
Given a complex number $z=1-i$, calculate the value of $\frac{1}{{{z^2}}}-\overline{z}$. Express your answer as a complex number in the form $a + bi$, and include the boxed notation around your final answer. | -1 - \frac{1}{2}i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
In the rectangular coordinate system xy, the center of the hyperbola is at the origin, the focus is on the x-axis, and the asymptote equation is 4 ± 3y = 0. Find the eccentricity of the hyperbola. Express your answer as a fraction or decimal value. | 5/3 | big_math | [
"Mathematics -> Geometry -> Analytic Geometry -> Other"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Find the derivative of the function y = 3x^2. Express your answer as a function of x. | y' = 6x | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Let the complex number $x= \frac{2i}{1-i}$ (where $i$ is the imaginary unit). Calculate the value of $C_{2016}^{1}x + C_{2016}^{2}x^2 + C_{2016}^{3}x^3 + \ldots + C_{2016}^{2016}x^{2016}$ and express your answer as a complex number in the form $a+bi$, simplified to its most reduced form. | 0 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions",
"Mathematics -> Algebra -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Let X be a random variable that follows a binomial distribution B(5, 1/3). Calculate the probability P(X ≤ 4). Express your answer as a simplified fraction. | \frac{242}{243} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
What is the opposite of -8? Express your answer as a single integer. | 8 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
A sample of 20 students is to be selected from 140 students using systematic sampling (with equal intervals). The 140 students are numbered from 1 to 140 and divided into 20 groups in order (1~7, 8~14,..., 134~140). If the number drawn from the 17th group is 117, what is the number drawn from the first group? Express y... | 5 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
In a geometric sequence {a_n}, if a_2 + a_4 = 32 and a_6 + a_8 = 16, then find the value of a_10 + a_12 + a_14 + a_16. Express your answer as a single numerical value. | 12 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Find the coefficient of the x term in the expansion of $(\\sqrt{x}-1)^4 \\cdot (x-1)^2$. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation of the ellipse $\frac{x^2}{4} + \frac{y^2}{9} = 1$, find the length of its major axis. Express your answer as a single number. | 6 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression: $\frac{2^1+2^0+2^{-1}}{2^{-2}+2^{-3}+2^{-4}}$. Provide your answer as a single number. | 8 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given a triangle whose three sides are all positive integers, and only one side length is 5, which is not the shortest side, count the number of triangles that meet this condition. Express your answer as a single integer. | 10 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
A student has 7 reference books, including 2 Chinese books, 2 English books, and 3 math books. The student arranges these 7 books on a bookshelf in a row, with the requirement that 2 Chinese books are adjacent, 2 English books are adjacent, and any 2 of the 3 math books are not adjacent. Calculate the total number of d... | 48 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$, where $|\overrightarrow{a}|=2$ and $|\overrightarrow{b}|=1$, and $\overrightarrow{a}$ is perpendicular to $\overrightarrow{b}$, find $|2\overrightarrow{a}-\overrightarrow{b}|$. Express your answer as a single value. | \sqrt{17} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Express the total population of approximately 4.6 billion people in scientific notation. Use the format $a \\times 10^b$, where $a$ is the coefficient and $b$ is the exponent. | 4.6 \\times 10^9 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function y = 2^x and the domain P = {1, 2, 3}, calculate the range of the function. Express your answer as a set of values. | {2, 4, 8} | big_math | [
"Mathematics -> Algebra -> Functions"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Find the product of three different positive integers whose sum is 7. Express your answer as a single integer. | 8 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given the vectors $\overrightarrow{a} = (n, 1)$ and $\overrightarrow{b} = (4, n)$, find the value of $n$ that makes these vectors collinear. Express your answer as a numerical value, including the ± symbol if necessary. | \pm 2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Given the original function y = φ(x), its inverse function y = φ^(-1)(x), and the instruction to find the third function symmetric to the inverse with respect to the line x + y = 0, determine the expression for the third function in terms of φ(x). | y = -φ(-x) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of $(-1)^{2002}$. Express your answer as an integer. | 1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence 1, 3, 7, 15, 31,..., derive a general formula for the nth term of this sequence. Express your answer as a formula in terms of n. | a_n = 2^n - 1 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence {a_n} defined by a_1 = 2 and a_{n+1} = a_n + 2n for n >= 1, calculate the exact value of a_{100}. Express your answer as a single integer. | 9902 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Given the inequality $e^{x}-x\cos x+\cos x\ln \cos x+ax^{2}\geqslant 1$ for $x\in({0,\frac{π}{2}})$, find the smallest integer value of $a$ that satisfies this condition. Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Calculus -> Other",
"Mathematics -> Other"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Solve for the constant $m$ in the equation $\frac{x-6}{x-5}+1=\frac{m}{x-5}$, given that the equation has a repeated root. Express your answer as a single integer. | -1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
A batch of goods is transported uniformly at a speed of $v$ km/h (where $100 \leq v \leq 120$) from location A to location B with 17 trucks. The distance between locations A and B is 600 km. For safety reasons, the distance between two trucks must not be less than $\left(\frac{v}{20}\right)^2$ km (the length of the tru... | 9.8 \text{ hours} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Four students are admitted to three universities. Calculate the probability that each university admits at least one student. Express your answer as a simplified fraction. | \frac{4}{9} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
The curve C is defined by the equation (x - arcsin a)(x - arccos a) + (y - arcsin a)(y + arccos a) = 0. The line x = pi/4 intersects this curve, resulting in a chord of length d. Find the minimum value of d as a varies. Express your answer in terms of pi. | \frac{\pi}{2} | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Let x be a certain positive decimal number. When the decimal point of x is moved four places to the right, the new number is four times the reciprocal of the original number. Calculate the value of x. Express your answer as a decimal number. | 0.02 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $i$ is the imaginary unit, find the value of the real number $m$ such that the expression $\left(1+mi\right)\left(2-i\right)$ is a pure imaginary number. Express your answer as a single real number. | -2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
In a 4x4x3 rectangular parallelepiped, vertices A, B, and C are adjacent to vertex D. Calculate the perpendicular distance from D to the plane containing A, B, and C. Express your answer as a decimal value, rounded to one decimal place. | 2.1 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
The product \(8 \times 48 \times 81\) is divisible by \(6^k\). Calculate the largest possible integer value of \(k\). | 5 | big_math | [
"Mathematics -> Number Theory -> Factorization"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
A straight one-way city street has 8 consecutive traffic lights. Every light remains green for 1.5 minutes, yellow for 3 seconds, and red for 1.5 minutes. The lights are synchronized so that each light turns red 10 seconds after the preceding one turns red. Calculate the longest interval of time, in seconds, during whi... | 20 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
A movie theater has eleven rows of seats. The rows are numbered from 1 to 11. Odd-numbered rows have 15 seats, and even-numbered rows have 16 seats. Calculate the total number of seats in the theater. Express your answer as a whole number. | 170 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Let \( z_{1} \) and \( z_{2} \) be a pair of distinct conjugate complex numbers, with \( \left|z_{1}\right| = \sqrt{3} \) and \( \frac{z_{1}^{2}}{z_{2}} \) being a real number. Determine the value of \( \left|z_{1} - z_{2}\right| \). Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
A high school has 680 students, consisting of 360 male and 320 female students. A lecture on exam psychology was attended in the academic hall, and the school used stratified sampling to draw a sample. It is known that there are 16 female students in this sample. Find the number of male students in the sample. | 18 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
What is the opposite number of $a-b$? Express your answer as a simplified algebraic expression. | b-a | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z= \frac {1-i}{1+i}$, calculate its conjugate. Express your answer as a complex number in the form a+bi. | i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Given the universal set $U=\{-3,-2,-1,0,1\}$, set $A=\{-2,-1\}$, and set $B=\{-3,-1,0\}$, find the result of the operation $(\complement _{U}A)\cap B$. Express your answer as a set of elements enclosed in curly braces {}, separating elements with commas. | \{-3,0\} | big_math | [
"Mathematics -> Other -> Other"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given the random variable ξ follows a binomial distribution B(2,p) and η follows a binomial distribution B(4,p), if P(ξ≥1) = 5/9, find the value of P(η≥2). Express your answer as a simplified fraction. | 11/27 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
The given equations show a pattern where the denominator can be expressed as $x^2 - 1$, where x is the coefficient of the square root. Using this pattern, find the value of n for the equation $8\sqrt{\frac{8}{n}} = \sqrt{8\frac{8}{n}}$. Express your answer as a single integer. | n = 63 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\overrightarrow{AP}= \frac{1}{3} \overrightarrow{PB}$ and $\overrightarrow{AB}=λ \overrightarrow{BP}$, find the value of λ. Express your answer as a single numerical value. | -\frac{4}{3} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given two circles $⊙C_{1}$: $x^{2}+y^{2}+D_{1}x+E_{1}y-3=0$ and $⊙C_{1}$: $x^{2}+y^{2}+D_{2}x+E_{2}y-3=0$ both pass through point $A(2,-1)$, find the equation of the line passing through points $(D_{1},E_{1})$ and $(D_{2},E_{2})$. Express your answer in the standard form of a line equation (Ax + By + C = 0). | 2x-y+2=0 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
In a geometric sequence $\{a_n\}$ where each term is positive, if $a_6a_4+2a_8a_5+a_9a_7=36$, then find the value of $a_5+a_8$. Express your answer as a single numerical value. | 6 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Find the minimum value of the function $$f(x)= \sqrt {x^{2}+4x+5}+ \sqrt {x^{2}-2x+10}$$. Express your answer as a single numerical value. | 5 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation 3m + 3/(n + 1/p) = 17, where m, n, and p are positive integers, find the value of p. Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of five-digit positive integers such that the sum of their digits is 2, given that the first digit cannot be zero. Express your answer as a whole number. | 5 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $a=(1,-1)$, $b=(-1,1)$, and $c=(5,1)$, calculate the sum $c+a+b$. Express your answer as a vector in the form $(x, y)$. | (5,1) | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Given $\overrightarrow{a} = (3, 1)$, $\overrightarrow{b} = (x, x-1)$ and $\overrightarrow{a} \parallel \overrightarrow{b}$, find the value of $x$. Express your answer as a single number. | \frac{3}{2} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(1,m)$, $\overrightarrow{b}=(-1,1)$, $\overrightarrow{c}=(3,0)$, if $\overrightarrow{a}$ is parallel to ($\overrightarrow{b}+\overrightarrow{c}$), find the value of m. Express your answer as a single number. | \frac{1}{2} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x)=2\sin(\omega x+\frac{\pi}{4})$ $(\omega > 0)$, find the range of $\omega$ such that the graph of the function has exactly $3$ highest points in the interval $[0,1]$. Express your answer as an interval. | \left[ \frac{17\pi}{4},\frac{25\pi}{4} \right) | big_math | [
"Mathematics -> Precalculus -> Functions"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence \(\{a_n\}\), \(a_3+a_4=12\) and the common difference \(d=2\). Find the value of \(a_9\). Express your answer as a single integer. | 17 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the number of zeros in the integer equal to (10^100) * (100^10). Express your answer as a whole number. | 120 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(n) = n^2\cos(n\pi)$, and $a_n = f(n) + f(n+1)$, calculate the sum $a_1 + a_2 + a_3 + \ldots + a_{100}$. Express your answer as a single integer. | -100 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given the polynomial $(m+1)x^3y^{|m|}+xy^2+8^5$ is a quartic trinomial, find the value of $m$. Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Algebra -> Polynomial Operations"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given a sequence {a_n} with the sum of its first n terms denoted as S_n. It is known that a_1 = 4/5 and a_n+1 is defined as follows:
$$
a_{n+1} =
\begin{cases}
2a_n, & 0 \leq a_n \leq \frac{1}{2}, \\
2a_n - 1, & \frac{1}{2} < a_n \leq 1.
\end{cases}
$$
Calculate the value of S_2018. Express your answer as a fraction. | \frac{5047}{5} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, the line segment with endpoints \( P(-4,0) \) and \( Q(16,0) \) is the diameter of a semi-circle. If the point \( R(0, t) \) is on the circle with \( t > 0 \), find the value of \( t \). Express your answer as a single integer. | 8 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Given $\tan x = \frac{1}{3}$, calculate the value of $\sin x \cos x + 1$. Express your answer as a fraction. | \frac{13}{10} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence $\left\{ a_n \right\}$, where $a_1=2$, $a_n=1-\frac{1}{a_{n-1}}$ for $n \geqslant 2$, calculate the value of $a_{2017}$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
A termite eats its way along the diagonal from point P to point Q in a 3 x 4 x 5 cuboid consisting of 60 identical small cubes. The diagonal does not intersect the edges of any small cube inside the cuboid. Calculate the number of small cubes the termite passes through on its journey. Express your answer as a whole num... | 10 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the scores 9, 9, x, and 8, where the mode equals the mean, calculate the value of x. Express your answer as a single numerical value. | 10 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
The terms of a geometric sequence {a_n} with a common ratio of 2 are all positive numbers, and a_3 * a_11 = 16. Find log_2 a_10. Express your answer as a single integer. | 5 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Gaussville Elementary School bought 8-slice pizzas for $6.85 each. The school bought 55 pizzas and sold every slice for $1. Calculate the school's profit. Express your answer in dollars, rounded to 2 decimal places. | 63.25 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Given that A(x1, 2023) and B(x2, 2023) are two points on the graph of the quadratic function y = ax^2 + bx + 5, where a!= 0, find the value of the quadratic function when x = x1 + x2. | 5 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Quadratic Functions"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Points B, C, and D lie on a straight line, and the angle BCD is 125 degrees. In triangle ABC, the angle BAC is 50 degrees. Additionally, points E, A, and D lie on a straight line, and the angle DAE is 80 degrees. Find the value of angle x. Express your answer in degrees as a whole number. | 25 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
What is the largest positive integer n that satisfies the inequality n^{200} < 3^{500}? Express your answer as a single integer. | 15 | big_math | [
"Mathematics -> Algebra -> Inequalities -> Other"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
In triangle $\triangle ABC$, if $ac=8$, $a+c=7$, $B=\frac{π}{3}$, calculate the length of side $b$. Express your answer as a single number. | 5 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a \in \mathbb{R}, b \in [0, 2\pi]$, if for any real number $x$, it satisfies $\sin\left(3x - \frac{\pi}{3}\right) = \sin\left(ax + b\right)$, then find the number of ordered real number pairs $(a, b)$ that satisfy the condition. Express your answer as a whole number. | 2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Find the symmetric center of the graph of the function y = sinx - $$\sqrt {3}$$cosx + 1. Express your answer as an ordered pair (x, y), with the exact value of x in terms of π. | \left(\frac{\pi}{3}, 1\right) | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $|\overset{→}{a}|=2, |\overset{→}{b}|=4$, and $(\overset{→}{a}+\overset{→}{b})$ is perpendicular to $\overset{→}{a}$, find the value of $|2\vec{a}-\vec{b}|$. Express your answer as a numerical value, possibly with a square root. | 4\sqrt{3} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
In a geometric sequence, the first term $a_{1} = \frac{1}{2}$, the common ratio $q = \frac{1}{2}$, and the $n$-th term $a_{n} = \frac{1}{32}$. Determine the value of $n$. Express your answer as a whole number. | 5 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Given a geometric sequence {a_n}, the sum of the first n terms is denoted as S_n. If a_{2013} = 2S_{2014} + 6 and 3a_{2014} = 2S_{2015} + 6, find the common ratio q of the sequence {a_n}. Express your answer as a numerical value or values. | \frac{1}{2} \text{ or } 1 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
A regular tetrahedron has two spheres that are inscribed in and circumscribed around it. Additionally, in each face of the regular tetrahedron and between it and the circumscribed sphere, there are four spheres touching the center of the face. Given that point P is inside the circumscribed sphere, calculate the probabi... | 0.19 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Let $A=20072009$, and $n$ is an 800-digit number formed by writing 100 $A$s side by side. Calculate the remainder when $n$ is divided by 11. Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
A number $x$ is randomly chosen from the interval $[0,2]$. Find the probability that the event "$-1 \leqslant \log_{\frac{1}{2}}\left(x+\frac{1}{2}\right) \leqslant 1$" occurs. Express your answer as a fraction. | \frac{3}{4} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
In the Cartesian coordinate system, point B has coordinates (2, -3). Calculate the distance from point B to the x-axis. Express your answer as a single numerical value. | 3 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
In a competition, student A solves the problem with a probability of $\frac{1}{2}$, student B solves it with a probability of $\frac{1}{3}$, and student C solves it with a probability of $\frac{1}{4}$. The events of A, B, and C solving the problem are independent. Calculate the probability that only one of them will so... | \frac{11}{24} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Given sets A = {2, 4, 6, 16, 29} and B = {4, 16, 20, 27, 29, 32}, find the intersection of A and B, denoted as A ∩ B. Express your answer as a set of numbers, separated by commas and enclosed in curly brackets. | {4, 16, 29} | big_math | [
"Mathematics -> Discrete Mathematics -> Set Theory -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given $z=\frac{1}{1+i}$ (where $i$ is the imaginary unit), calculate the conjugate of $z$, denoted as $\overline{z}$. Express your answer in the form $a + bi$, where $a$ and $b$ are real numbers. | \frac{1}{2} + \frac{1}{2}i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
From 6 students, 4 are to be chosen to be assigned four different tasks labeled as A, B, C, and D. If students Jia and Yi cannot be assigned to task A, calculate the total number of different assignment arrangements. Express your answer as a single integer. | 240 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Permutations"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Express 300,670 in scientific notation in the form $a\\times10^{n}$, where $1\\leq|a|<10$, and $n$ is an integer. | 3.0067\times10^{5} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation (1+i)z = 2i, where i is the imaginary unit, calculate the value of z / conjugate(z). Express your answer as a single complex number in standard form. | i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
A rectangular solid with length, width, and height of 1, 2, and 3, respectively, is circumscribed by a sphere. Calculate the volume of this sphere. Express your answer in terms of pi. | \dfrac{7\sqrt{14}}{3}\pi | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Volume"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given $a=e^{0.11}$, $b=1.1^{1.1}$, $c=1.11$, determine the correct ordering of a, b, and c. Express your answer as a sequence of variables using greater-than (>) symbols (e.g., a > b > c). | a > b > c | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
In triangle ABC, the measure of angle ABC is 50 degrees, AD bisects angle BAC, and DC bisects angle BCA. Find the measure of angle ADC. Express your answer in degrees as a whole number or decimal if necessary. | 115^\circ | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given the arithmetic sequence {a_n}, where a_1 = 20, a_n = 54, and S_n = 999, find the value of n. Express your answer as a whole number. | 27 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the derivative of the function $f(x)$ is ${f'}(x)$, and ${f'}(2)=4$, calculate the limit as $\Delta x$ approaches $0$ of the expression $\frac{f(2)-f(2-\Delta x)}{2\Delta x}$. Express your answer as a numerical value. | 2 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Given a non-zero common difference arithmetic sequence {a_{n}}, the sum of its first n terms is S_{n}, and a_{10} = S_{4}. Calculate the value of S_{8} / a_{9}. | 4 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Abby has 23 coins. The coins have a total value of $4.55. If she has only quarters (worth 25 cents each) and nickels (worth 5 cents each), how many quarters does she have? Express your answer as a whole number. | 17 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Xiao Zhao, Xiao Qian, Xiao Sun, and Xiao Li were asked who had been to the Great Wall. Xiao Zhao said: "I haven't been"; Xiao Qian said: "Xiao Li has been"; Xiao Sun said: "Xiao Qian has been"; Xiao Li said: "I haven't been." Assuming that only one of the four students is lying, determine the name of the student who de... | Xiao Qian | big_math | [
"Mathematics -> Discrete Mathematics -> Logic"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Xiao Wang and Xiao Li play a game of "Rock, Paper, Scissors". The rules of the game are: win once to get 3 points, tie to get 1 point, lose to get 0 points. A total of 7 games are played, and the one with the higher score at the end of the games wins. If Xiao Wang scores 10 points at the end of the games, calculate the... | 2 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of $|-3| - 3$. Express your answer as a single integer. | 0 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Positive integers a and b satisfy a * b = 2010. If a > b, find the smallest possible value of a - b. Express your answer as a whole number. | 37 | big_math | [
"Mathematics -> Number Theory -> Factorization"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets $A=\left\{x\in \mathbb{R} \mid \dfrac{1}{x}\leqslant 0\right\}$ and $B=\left\{x\in \mathbb{R} \mid x^{2}-1 \lt 0\right\}$, describe the union $A\cup B$ using interval notation. | (-\infty, 1) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the line $3x+(3^a-3)y=0$ is perpendicular to the line $2x-y-3=0$, find the value of $a$. Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given the plane vectors $\mathbf{a}=(3,5)$ and $\mathbf{b}=(-2,1)$, calculate the result of $\mathbf{a}-2\mathbf{b}$. Express your answer as an ordered pair (x, y). | (7,3) | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
A product's purchase price is 200 yuan, and its marked price is 300 yuan. To achieve a 20% profit on this product, calculate the discount rate as a percentage of the marked price at which the product should be sold. Express your answer as a percentage, including the percentage sign (%). | 20\% | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
In a Huangpu River transportation and sightseeing map with a scale of 1:38000, a certain tunnel is about 7 cm long. Calculate the actual length of the tunnel in kilometers. Express your answer as a decimal value with two places. | 2.66 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
A team has x people, and each pair of people sends each other a message card to cheer each other up. The whole team sent a total of 56 cards. Write an equation that represents the total number of cards sent. | x(x-1) = 56 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the sum of two numbers and their quotient are both 6, calculate the product of these two numbers minus their difference (greater minus smaller). Express your answer as a fraction in simplest form. | \frac{6}{49} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Using only the digits 1, 3, and 5, how many three-digit numbers greater than 200 can be written? Express your answer as a whole number. | 18 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
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