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 vectors $\overrightarrow{m} = (1, 2)$ and $\overrightarrow{n} = (x, 1)$ satisfy $\overrightarrow{m} \perp \overrightarrow{n}$, calculate the magnitude of $\overrightarrow{n}$, $|\overrightarrow{n}|$, using the dot product relationship. Express your answer as an exact value. | \sqrt{5} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given the hyperbola $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1 \ (a > 0, b > 0)$ has an asymptote equation of $y = \sqrt{3}x$, and one of its foci is $(2,0)$, derive the equation of the hyperbola in the standard form. | x^2 - \dfrac{y^2}{3} = 1 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
In the sequence $\{a_n\}$, $a_n = \sin \frac{2n\pi}{3} + \sqrt{3}\cos \frac{2n\pi}{3}$, and $S_n$ is the sum of the first $n$ terms of the sequence. Calculate the value of $S_{2018}$. Express your answer as a single value, including any necessary radicals. | -\sqrt{3} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
In the binomial expansion of $(2x^{2}- \frac {1}{x})^{5}$, calculate the coefficient of the $x$ term. Express your answer as a single integer, including the negative sign if applicable. | -40 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
In the expansion of (x + 3/sqrt(x))^n, the ratio of the sum of the coefficients of all terms to the sum of the binomial coefficients is 32. Find the coefficient of x^2. | 90 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
A population increases by i% from time t=0 to time t=1, and then increases by j% from time t=1 to time t=2. Calculate the overall percentage increase in the population from time t=0 to time t=2. Express your answer as a percentage, using the format \boxed{[answer]%}. | i + j + \frac{ij}{100}% | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given that the sum of the first n terms of a geometric sequence {a_n} is S_n, and it satisfies S_6/S_3 = 9, find the common ratio q. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Given $\sqrt{12.34}=3.512$ and $\sqrt{123.4}=11.108$, calculate $\sqrt{123400}$. Express your answer as a decimal number, rounded to one decimal place if necessary. | 351.2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Logan is constructing a scaled model of his town. The city's water tower stands 40 meters high, and the top portion is a sphere that holds 100,000 liters of water. Logan's miniature water tower holds 0.1 liters. Calculate the height, in meters, that Logan should make his miniature water tower. | 0.4 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Express the number 21,600 in scientific notation in the form $a imes 10^n$, where $1 \leq |a| < 10$ and $n$ is an integer. | 2.16 \times 10^4 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Solve the system of equations $\begin{cases} x+y=2 \\ x-y=0 \end{cases}$ and represent the solution set using listing notation. Express your answer as a set in the form $\{(\quad\quad\quad)\}$. | \{(1, 1)\} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $\frac{3+i}{1-3i}$, find its value. Express your answer as a complex number. | i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
The sequence \(\{a_{n}\}\) satisfies \(a_{1}=3\) and \(a_{n+1}=-\frac{1}{a_{n}+1}\). Calculate the value of \(a_{2008}\). Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the line that passes through point P(1, 3) and forms a triangle with the positive x-axis and y-axis with an area of 6. Express your answer in the standard form of an equation of a line (Ax + By + C = 0). | 3x+y-6=0 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given the ratios x:y = 4:7 and z:x = 3:5, find the ratio (x + y) : (z + x). Express your answer as a ratio in simplest form, using the format a:b. | 55:32 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Given $x > 0$, find the minimum value of the expression $x + \frac{1}{2x}$. Express your answer as a single value. | \sqrt{2} | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Find the domain of the function $f(x)= \sqrt {x-1}+\lg (3-x)$. Express your answer in interval notation, using brackets or parentheses to denote the type of interval. | [1,3) | big_math | [
"Mathematics -> Precalculus -> Functions"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Given that one vertex of the ellipse $\dfrac{{x}^{2}}{{a}^{2}}+ \dfrac{{y}^{2}}{{b}^{2}}=1$ ($a > b > 0$) and its two foci form an equilateral triangle, find its eccentricity. Express your answer as a single value. | \dfrac{1}{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of $\frac{(.2)^3}{(.02)^2}$. Express your answer as a single number. | 20 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a} = (1, -2)$ and $\overrightarrow{b} = (2, x)$, if $\overrightarrow{a} \parallel \overrightarrow{b}$, find the value of $x$. Express your answer as a single integer. | -4 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Solve the equation $x(x-1)=x$ for $x$. If there are multiple solutions, express your answer using subscripts to denote each solution (e.g., $x_{1}=a$, $x_{2}=b$). | x_{1}=0, x_{2}=2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Given a geometric sequence composed of positive numbers with a common ratio q = 2, and the product of the first 30 terms a_1a_2...a_{30} = 2^{30}, find the product of the terms a_3a_6a_9...a_{30}. Express your answer as a power of 2 in the format 2^x. | 2^{20} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Air quality monitoring data from a certain region indicates that the probability of having good air quality on any given day is 0.75. The probability of having good air quality for two consecutive days is 0.6. Given that the air quality is good on a certain day, find the probability that the air quality will also be go... | 0.8 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given the initial value of a = -12 and the operation defined as |a+4|-10, calculate the result after performing this operation 2022 times. Express your answer as a single integer value. | -8 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Jones's four cousins have ages that are four different single-digit positive integers. The product of the ages of two of them is 24, and the product of the ages of the other two is 30. What is the sum of the ages of the four cousins? Express your answer as a single integer. | 22 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given $sin(\alpha+\frac{\pi}{3})-sin\alpha=\frac{4}{5}$, find the value of $sin(2\alpha-\frac{\pi}{6})$. Express your answer as a fraction. | -\frac{7}{25} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Let X be a random variable that follows the normal distribution N(6, 1/2). Given that P(X < a-2) = P(X > 5), find the value of a. Express your answer as a numerical value. | 9 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
In the Cartesian coordinate system, the vertex of angle α coincides with the origin, its initial side coincides with the non-negative half of the x-axis, and its terminal side passes through point P(-√3, -1). Calculate the exact value of sin α. Express your answer as a simplified fraction. | -\frac{1}{2} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Find the sum of the numerical coefficients in the expansion of the binomial $(a+b)^6$. Express your answer as a whole number. | 64 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Three fair dice are tossed at random. What is the probability that the three numbers turned up can be arranged to form an arithmetic progression with common difference one? Express your answer as a fraction. | \frac{1}{9} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given a sequence {a_n} that satisfies a_{p+q} = a_p * a_q for any p, q in N*, and a_2 = 2, calculate the value of a_8. Express your answer as a single integer. | 16 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors a = (1,2), b = (1,-1), and c = (4,5), find the value of the real number λ such that a is perpendicular to b + λc. Express your answer as a real number. | \frac{1}{14} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Find the domain of the function $y=\frac{\sqrt{-x^{2}-3x+4}}{\log(x+1)}$. Express your answer in interval notation. | (-1,0) \cup (0,1] | big_math | [
"Mathematics -> Algebra -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given that one base angle of an isosceles triangle is 30 degrees greater than the vertex angle, find the degree of the vertex angle. Express your answer in degrees as a whole number. | 40 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation of motion s = 2t^2, determine the increment in displacement during the time interval from 2 seconds to (2+d) seconds. Express your answer as an algebraic expression in terms of d. | 8d + 2d^2 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Let \( S = \left\{\left(s_{1}, s_{2}, \cdots, s_{6}\right) \mid s_{i} \in \{0, 1\}\right\} \). For any \( x, y \in S \) where \( x = \left(x_{1}, x_{2}, \cdots, x_{6}\right) \) and \( y = \left(y_{1}, y_{2}, \cdots, y_{6}\right) \), define:
(1) \( x = y \) if and only if \( \sum_{i=1}^{6}\left(x_{i} - y_{i}\right)^{2} ... | 32 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the function f(x) is monotonically decreasing on (-∞,+∞) and is an odd function, and given f(1) = -1, find the range of x such that -1 ≤ f(x-2) ≤ 1. Express your answer as an interval. | [1,3] | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
If $1-\frac{4}{x}+\frac{4}{x^2}=0$, calculate the value of $\frac{2}{x}$. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Determine the range of the real number $a$ for which the proposition "$p$: $\forall x \in \mathbb{R}$, $ax^{2}+2x+1 > 0$" is false. Express your answer using interval notation if applicable. | (-\infty, 1] | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation of the parabola y = -1/8x^2, calculate the coordinates of its focus. Express your answer as an ordered pair (x, y), including any negative signs if applicable. | (0, -2) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Given the mapping f such that (x, y) maps to (x+y, x-y), find the preimage of (1, 2) under f. Express your answer as an ordered pair (x, y) with exact values for x and y. | \left(\frac{3}{2}, -\frac{1}{2}\right) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Find the maximum value of the real number k for which the inequality (1 + sin(x)) / (2 + cos(x)) >= k has a solution. Express your answer as a single real number. | \frac{4}{3} | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z= \frac{4-2i}{1+i}$, find the value of $|z|$. Express your answer as a square root. | \sqrt{10} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets $P=\{x\in\mathbb{R}\mid 0\leqslant x\leqslant 3\}$ and $Q=\{x\in\mathbb{R}\mid x^{2}\geqslant 4\}$, find $P\cap(\neg_{R}Q)$. Express your answer in interval notation. | [0,2) | big_math | [
"Mathematics -> Discrete Mathematics -> Logic"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the directrix of the parabola $x^{2}=2y$. Express your answer as an equation of a line (e.g., $y = c$). | y=- \dfrac {1}{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Given that | \overrightarrow{a}| = 2, | \overrightarrow{b}| = 3, and | \overrightarrow{a} + \overrightarrow{b}| = \sqrt{19}, calculate the value of | \overrightarrow{a} - \overrightarrow{b}|. Express your answer as the magnitude of a vector. | \sqrt{7} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
If sin(α/2) = √3/3, calculate the value of cos(α) using the double angle formula cos(α) = 1 - 2sin^2(α/2). Express your answer as a simplified fraction. | 1/3 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
If a convex polygon has exactly three obtuse angles, what is the maximum number of sides this polygon can have? Express your answer as a whole number. | 6 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
A game starts with the number 15 written on a blackboard. A move consists of erasing the current number x and replacing it with x + y, where y is a randomly chosen number between 1 and 5 (inclusive). The game ends when the number on the blackboard exceeds 51. What is the expected number on the blackboard at the end of ... | 54 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Find the number(s) x that satisfy both x^3 = x and sqrt(x) = x. Express your answer as a list of integers. | 0, 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Let $i$ be the imaginary unit, and let the complex number $z$ satisfy $1+zi=2i$. Find the magnitude $|z|$ of the complex number $z$. Express your answer as a numerical value, including the square root symbol if necessary. | \sqrt{5} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
In triangle $ABC$, with sides $a$, $b$, and $c$ opposite angles $A$, $B$, and $C$ respectively, given that $a=3$, $C=120^{\circ}$, and the area of triangle $ABC$, $S= \frac {15 \sqrt {3}}{4}$, find the value of $c$. Express your answer as a single number. | 7 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Given the expressions $a=\log _{5}3$, $b=e^{-1}$, and $c=\log _{16}9\cdot \log _{27}8$, compare the values of $a$, $b$, and $c$ and specify their order from least to greatest using inequality notation. | b < c < a | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Logarithmic Functions"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Find the range of the piecewise function $f(x)=\left\{{\begin{array}{l}{{x^2}+2x-1,}&{-3≤x≤0,}\\{x-1,}&{0＜x≤5,}\end{array}}\right.$ over its entire domain. Express your answer as an interval, e.g., [a, b]. | [-2, 4] | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Each of the positive integers \(a, b\), and \(c\) has three digits, and for each of these integers the first digit is the same as its last digit. Also \(b = 2a + 1\) and \(c = 2b + 1\). Find the number of possibilities for the integer \(a\). Express your answer as a whole number. | 2 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given the standard equation of the parabola $y^2 = -12x$, calculate the coordinates of its focus. Express your answer as a coordinate pair (x, y). | (-3, 0) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Given points A(a,2), B(b,6), and C(c,d) all lie on the parabola y = (x-1)^2 - 2, and d < 1, determine the correct relationship between a, b, and c (in the format a? c? b, where? represents <, >, or =) when a < 0 and b > 0. Provide your answer based on the analysis of the parabola's properties and the given conditions. | a < c < b | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
What is the exact value of the sum \(9^{20}+9^{20}+9^{20}\)? Express your answer as a single value with an exponent if necessary. | 3^{41} | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Among the following numbers: $-1$, $\frac{\pi}{2}$, $4.11213415$, $0$, $\frac{22}{7}$, $3.14$, how many are rational numbers? Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given $a = \log_{3} \frac{1}{2}$, $b = \left(\frac{1}{2}\right)^{3}$, $c = 3^{\frac{1}{2}}$, determine the correct ordering of the expressions from smallest to largest. Express your answer as a sequence of the variables a, b, and c, separated by less-than signs (<). | a < b < c | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Logarithmic Functions"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
If the monomials $3x^{m}y$ and $-5x^{3}y^{n}$ are like terms, calculate the value of $m + n$. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
An integer is defined to be upright if the sum of its first two digits equals its third digit. For example, 145 is an upright integer since \(1+4=5\). Count the number of positive 3-digit integers that are upright. Express your answer as a whole number. | 45 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Square ABCD has sides of length 3 cm. The points M and N lie on AD and AB so that CM and CN split the square into three pieces of the same area. Find the length of DM. Express your answer in centimeters (cm). | 2 \text{ cm} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression: $(14a^{3}b^{2}-7ab^{2})\div 7ab^{2}$. Express your answer as a simplified algebraic expression. | 2a^{2}-1 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Given a set of data $a$, $b$, $9$, $10$, $11$ with an average of $10$ and a variance of $2$, calculate the absolute difference between $a$ and $b$, $|a-b|$. Express your answer as a single number. | 4 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Let the set $M=\{x|-2<x<3\}$, and $N=\{x|2^{x+1}\leq 1\}$. Calculate the intersection of $M$ and the complement of $N$ in the real numbers, $M\cap (\complement_{\mathbb{R}}N)$. Express your answer as an interval, using parentheses or brackets as necessary. | (-1,3) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence $\left\{a_{n}
ight\}$ where $\sqrt{a_{n+1}}=\sqrt{a_{n}}+\sqrt{2}$ and $a_{1}=8$, derive the general formula for $a_{n}$. Express your answer as a formula in terms of $n$. | 2(n+1)^2 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Find the coefficient of $x^2$ in the expansion of $(1-x)^4$. Express your answer as an integer. | 6 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Among the four functions y = 2^x, y = log_2 x, y = x^2, and y = cos 2x, how many satisfy the inequality f((x1 + x2)/2) > (f(x1) + f(x2))/2 for all 0 < x1 < x2 < 1? Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Five people, namely Dafu, Bugeng, Zanbao, Shangzao, and Gongshe, together hunted 5 deer. They wanted to divide the deer meat, which totals 500 pounds, among themselves according to their ranks in descending order by the same amount. Calculate the total amount of deer meat shared by Bugeng, Zanbao, and Shangzao. Express... | 300 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Given \(X \sim B(8, \frac{1}{2})\), find the value of \(k\) (\(k \in \mathbb{N}, 0 \leqslant k \leqslant 8\)) that maximizes \(P(X=k)\). Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given the quadratic equation $3x^{2}=2x-3$, rearrange it into the standard form $ax^2 + bx + c = 0$ and identify the values of $a$, $b$, and $c$. Express your answer in the format $a=$, $b=$, $c=$. | a=3, b=-2, c=3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Find the solution set of the inequality $\ln^2x + \ln x < 0$. Express your answer as an interval. | (e^{-1}, 1) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Determine the number of unique functions f(x) that are defined on the set of real numbers, are odd functions, and satisfy the condition f(x) = x for x in [0, 2], given that g(x) = f(x+2) is an even function. | 1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Let $a\in\mathbb{Z}$, and $0\leq a<13$. If $12^{20}+a$ is divisible by 13, find the value of $a$. Express your answer as a single integer. | 12 | big_math | [
"Mathematics -> Number Theory -> Congruences"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
A student chooses 2 courses out of 6 elective courses, among which 2 courses have conflicting schedules. Calculate the total number of ways the student can choose the courses. Express your answer as a numerical value. | 14 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Let a, b, c, d be rational numbers with a > 0. If for every integer n >= 0, the number an^3 + bn^2 + cn + d is also an integer, then find the minimal value of a. Express your answer as a rational number. | \frac{1}{6} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
A county-level middle school has 3 math teachers and 6 language teachers participating in "support teaching" activities in 3 township-level middle schools. Each township-level middle school is allocated 1 math teacher and 2 language teachers. Calculate the number of different allocation schemes. | 540 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Combinations"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
A random variable ξ follows a binomial distribution ξ ~ B(n, p), with E(ξ) = 2 and D(ξ) = 1. Calculate the value of p. Express your answer as a numerical value. | 0.5 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Xiao Cong randomly throws a fair die. As the number of throws increases, what is the expected frequency of the face with the number "6" facing up? Express your answer as a decimal value, rounded to two decimal places if necessary. | 0.17 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Other"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Given the parametric equations $\begin{cases} & x=2+3t \\ & y=1-t \end{cases}$, calculate the slope of the line. Express your answer as a single fraction or integer. | -\dfrac{1}{3} | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the algebraic expression $3(x^2 - 2xy + y^2) - 3(x^2 - 2xy + y^2 - 1)$. Express your answer as a single value or simplified expression. | 3 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the following expression:
$$
\frac{\cos ^{4} 75^{\circ}+\sin ^{4} 75^{\circ}+3 \sin ^{2} 75^{\circ} \cos ^{2} 75^{\circ}}{\cos ^{6} 75^{\circ}+\sin ^{6} 75^{\circ}+4 \sin ^{2} 75^{\circ} \cos ^{2} 75^{\circ}}.
$$
Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Analyze the proposition "(x-1)^2+(y-2)^2=0" in relation to the condition "(x-1)(y-2)=0". Determine if the proposition is sufficient, necessary, both, or neither for the condition. Provide your answer in terms of sufficiency and necessity. | The proposition is sufficient but not necessary for the condition. | big_math | [
"Mathematics -> Algebra -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given complex numbers $Z_1=\frac{1-i}{1+i}$ and $Z_2=(3-i)\cdot i$, calculate the sum $Z_1+Z_2$. Express your answer in the form a+bi, where a and b are integers. | 1+2i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z$ satisfies $(3-z)i=1-3i$, calculate the value of $z$. Express your answer as a complex number in the form $a+bi$. | 6+i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given the lengths of the three sides of a triangle are $3$, $1-2a$, and $8$, find the range of values for $a$ that satisfy the triangle inequality. Express your answer as an interval in the form $(\text{lower bound}, \text{upper bound})$. | (-5, -2) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Consider the sets M = {x | x^2 ≤ 2x} and N = {x | y = ln(2 - |x|)}. Calculate the intersection M ∩ N. Express your answer as an interval, using interval notation. | [0,2) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given set A = {x | -1 ≤ x < 3} and set B = {y | 2 ≤ y ≤ 5}, find the intersection A ∩ B. Express your answer in standard interval notation. | [2, 3) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
In the spatial rectangular coordinate system, given point P (0, 0, $\sqrt{3}$) and point C (-1, 2, 0), find the coordinates of point M on the y-axis that is equidistant from P and C. Express your answer as a coordinate (0, y, 0). | (0, \frac{1}{2}, 0) | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Given a point M(m,n,p) in the three-dimensional Cartesian coordinate system O-xyz, find the coordinates of the point symmetric to M with respect to the plane xOy. Express your answer as a coordinate (x, y, z). | (m,n,-p) | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Given the set A={x|y= $\sqrt {(x-1)(5-x)}$, x∈Z}, calculate the number of elements in set A. Express your answer as a single integer. | 5 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Given the ellipse $\dfrac {x^{2}}{25}+ \dfrac {y^{2}}{m^{2}}=1(m > 0)$ and the hyperbola $\dfrac {x^{2}}{7}- \dfrac {y^{2}}{n^{2}}=1(n > 0)$ have the same foci, calculate the maximum value of $m+n$. Express your answer as a single numerical value. | 6 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Translate the parabola y = x^2 - 6x + 5 two units upward and then one unit to the right. Express the resulting parabola equation in the form y = (x-h)^2 + k. | y = (x-4)^2 - 2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Quadratic Functions"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Find the length of the imaginary axis of the hyperbola $\frac{{x}^{2}}{3}-{y}^{2}=1$. Express your answer as a numerical value. | 2 | big_math | [
"Mathematics -> Geometry -> Non-Euclidean Geometry -> Other"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Convert the decimal number 23 into binary. Express your answer in binary format, including the subscript 2 to denote binary (e.g., 10111_{(2)}). | 10111_{(2)} | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
If $f(x)$ is an even function and its derivative exists, find the value of $f'(0)$. Express your answer as a single number. | 0 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given 12 points in a plane, no three of which are collinear, find the number of lines they determine. Express your answer as a whole number. | 66 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics -> Counting Methods"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
A sphere is tangent to all three side surfaces and both bases of a right triangular prism (whose base is an equilateral triangle and whose side edges are perpendicular to the base). Given that the volume of this sphere is $\frac{9\pi}{2}$, calculate the volume of the triangular prism. Express your answer in terms of $\... | \frac{81\sqrt{3}}{4} | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Volume"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given a positive geometric sequence $\{a_{n}\}$ where $a_{3}-a_{1}=2$, find the minimum value of $a_{4}+a_{3}$. Express your answer as a single numerical value. | 8 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
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