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 |
|---|---|---|---|---|---|
Let $a$, $b$, $c$, $x$, $y$, $z$ be positive numbers, and $a^{2}+b^{2}+c^{2}=10$, $x^{2}+y^{2}+z^{2}=40$, $ax+by+cz=20$. Calculate the ratio $\dfrac {a+b+c}{x+y+z}$. Express your answer as a simplified fraction. | \dfrac{1}{2} | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Applied Mathematics -> Other"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Determine the coefficient of $x^2$ in the expansion of $(1+\\frac{1}{x^2})(1+x)^6$. Express your answer as a single numerical value. | 30 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given $\sin \alpha + \cos \alpha = - \sqrt{2}$, find the value of $\tan \alpha + \frac{1}{\tan \alpha}$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Find the constant term in the expansion of \((2x - \frac{1}{x})^{4}\). Express your answer as a single numerical value. | 24 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Find the maximum value of the function $y= \frac {\ln x}{x}$. Express your answer as a boxed numerical value. | \frac{1}{e} | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given the piecewise function $f(x)= \begin{cases} x-5 &(x\geqslant 7) \\ f(x+3) &(x < 7) \end{cases} (x\in N)$, calculate the value of $f(3)$. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Nine balls numbered 1, 2, …, 9 are placed into a bag. Person A draws a ball from the bag, and its number is $a$. After putting it back, person B draws another ball from the bag, and its number is $b$. Calculate the probability that the event $a-2b+10>0$ occurs. Express your answer as a fraction in simplest form. | \frac {61}{81} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given the circle equation x^{2}+y^{2}-4x+6y-3=0 and the line equation mx+y+m-1=0, find the value of m that maximizes the distance from the center of the circle to the line. Express your answer as a single value. | -\frac{3}{4} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Transform the quadratic function y = x^2 - 2x + 3 into vertex form. Express your answer as a complete analytical expression in vertex form. | y = (x - 1)^2 + 2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
The first AMC 8 was held in 1985 and has been held annually since. Samantha participated in the 7th AMC 8 when she was 12 years old. In which year was Samantha born? Express your answer as a single year in the format YYYY. | 1979 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Given that the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is ${60}^{\circ },\left|\overrightarrow{a}\right|=2,\left|\overrightarrow{b}\right|=5$, calculate the magnitude of $2\overrightarrow{a}-\overrightarrow{b}$. Express your answer as a square root if necessary. | \sqrt{21} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
A bamboo has nine nodes, with the total capacity of the lower three nodes being four liters and the total capacity of the upper four nodes being three liters. If the capacity of each node changes uniformly, forming an arithmetic sequence, what is the capacity of the middle node? Express your answer as a fraction. | \frac{67}{66} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Find the center and radius of the circle given by the equation $x^2+y^2+4x=0$. Express your answer in the format: Center: (x, y), Radius: r. | Center: (-2, 0), Radius: 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence {a_n}, the sum of the first, second, twelfth, and thirteenth terms is 24. Find the value of the seventh term, a_7. Express your answer as a single integer. | 6 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Among the algebraic expressions $\frac{1}{x}$, $2x+y$, $\frac{1}{3}a^{2}b$, $\frac{{x-y}}{π}$, $\frac{{5y}}{{4x}}$, and $0.5$, how many are polynomial expressions? Express your answer as a whole number count. | 4 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x)= \frac{1}{e} \cdot e^{x}+ \frac{a}{2}x^{2}-(a+1)x+a (a > 0)$, where $e$ is the base of the natural logarithm. If the function $y=f(x)$ and $y=f[f(x)]$ have the same range, find the maximum value of the real number $a$. | 2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = x^3 + ax + 2, find the range of values for 'a' that allows f(x) to have 3 zeros. Express your answer as an interval or range of values for 'a'. | (-\infty, -3) | big_math | [
"Mathematics -> Algebra -> Polynomial Operations -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given the measured value of the constant c as 2.43865 with an error range of ±0.00312, what is the most precise value of c that can be announced, with each digit being significant? Express your answer as a decimal value with the maximum possible significant figures. | 2.44 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Given the hyperbola $\frac{x^{2}}{16} - \frac{y^{2}}{9} = 1$, determine the distance between the right focus and the asymptote. Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Determine the total count of three-digit positive integers (a1 a2 a3) that satisfy the condition (a1 < a2 > a3). Express your answer as a single integer. | 240 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
From 10 college graduates, 3 people are to be selected to serve as village assistants. Find the number of different ways to select them such that at least one of A and B is chosen, but C is not chosen. Express your answer as a single numerical value. | 49 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
The Huawei Mate20 phone is equipped with the world's first 7-nanometer process chip, where 7 nanometers is equal to 0.000000007 meters. Express this number in scientific notation, including the coefficient and the exponent. | 7 \times 10^{-9} | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Xiangyang No. 3 Middle School recently held a new member election activity. It is known that the probability of outstanding students in Xiangyang No. 3 Middle School is about 10%. Among all students, 20% are members of the group, and the probability of outstanding students among group members is about 40%. Calculate th... | 2.5% | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a$ is a real number, if $\frac{a-i}{1+i}$ is a pure imaginary number, where $i$ is the imaginary unit, find the value of $a$. Express your answer as a single real number. | 1 | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence ${a_{n}}$, $a_{8}=8$. Calculate the value of $S_{15}$. Express your answer as a single integer. | 120 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Given a daily "progress rate" and "lag rate" of 20%, calculate the number of days it would take for the "progress" to be 10000 times that of "lagging". Express your answer as a whole number of days. | 23 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Let $P=\{x||x|\leqslant 2,x\in \mathbb{Z}\}$, $Q=\{x|-1＜x＜\frac{5}{2}\}$. Find the intersection $P$⋂$Q$ and express your answer as a set of integers using set notation, e.g., {a, b, c}. | \{0,1,2\} | big_math | [
"Mathematics -> Set Theory -> Basic Set Operations"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Select 4 students from 9 to participate in a debate competition, where at least two of the students A, B, and C must be selected. Find the number of different selection methods. Express your answer as a numerical value. | 51 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Combinations"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Let the sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$ be $S_n$, with $S_{m-1}=-2$, $S_m=0$, and $S_{m+1}=3$. Find the value of $m$. Express your answer as a whole number. | 5 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Dana creates a "rectangular spiral" by drawing line segments of lengths, in cm, of \(1,1,2,2,3,3,4,4, \ldots\). Dana's pen runs out of ink after the total of all the lengths he has drawn is \(3000 \, \text{cm}\). Calculate the length of the longest line segment that Dana draws. Express your answer as a whole number in ... | 54 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression cos24°cos36°-sin24°cos54°. Express your answer as a fraction. | \frac{1}{2} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Ten people are each waiting for a 45-minute operation. The first operation starts at 8:00 a.m., and each of the other operations starts at 15-minute intervals thereafter. What is the end time of the last operation? Express your answer in 12-hour clock format (e.g., 11:00 a.m.). | 11:00 a.m. | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
In a geometric sequence {a_n}, a_1 = 2, and if the sequence {a_n + 1} is also a geometric sequence, find the sum of the first n terms of sequence {a_n}, S_n. Express your answer as a function of n. | 2n | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
It is known that 2 kilograms of apples, 1 kilogram of pears, and 4 kilograms of peaches total 6 yuan. Also, 4 kilograms of apples, 2 kilograms of pears, and 2 kilograms of peaches total 4 yuan. Calculate the total cost to buy 4 kilograms of apples, 2 kilograms of pears, and 5 kilograms of peaches. Express your answer i... | 8 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
When $0 < x < \frac{\pi}{2}$, find the minimum value of the function $f(x)= \frac{1+\cos 2x+8\sin ^{2}x}{\sin 2x}$. Express your answer as a single number. | 4 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of intersection points between the graphs of $y_1 = \sin x$ and $y_2 = \lg x$ in the interval between $(1, 0)$ and $(10, 1)$. Express your answer as a whole number. | 3 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Among the expressions $a+2b$; $\frac{a-b}{2}$; $\frac{1}{3}(x^{2}-y^{2})$; $\frac{2}{a}$; $0$, how many are polynomials? Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Polynomials"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Given set $A=\{x|x^{2}-2x \lt 0\}$ and set $B=\{y|y=\log_{2}(2-x^{2})\}$, calculate the union $A\cup B$. Express your answer in interval notation. | (-\infty, 2) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities",
"Mathematics -> Precalculus -> Functions"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given two parallel lines $3x - 4y + 1 = 0$ and $3x - 4y - 4 = 0$, calculate the distance between the two lines using the formula for the distance between parallel lines. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
A point is randomly selected on a plane, where its Cartesian coordinates are integers with absolute values less than or equal to 4, and all such points are equally likely to be chosen. What is the probability that the selected point is at most 2 units away from the origin? Express your answer as a fraction in simplest ... | \frac{13}{81} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Let $\{a_k\}$ be a sequence of integers such that $a_1 = 1$ and $a_{m+n} = a_m + a_n + mn$ for all positive integers $m$ and $n$. Find the value of $a_{12}$. Express your answer as a single integer. | 78 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the sum of all positive irreducible proper fractions with a denominator of 100. Express your answer as a single integer. | 20 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of positive integer factors of 2012. Express your answer as a whole number. | 6 | big_math | [
"Mathematics -> Number Theory -> Factorization"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Given a sequence $\{a_n\}$ that is both an arithmetic sequence and a geometric sequence, find the sum of the first $n$ terms of this sequence. Express your answer in terms of $n$ and $a_1$. | na_1 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given the definition of conjugate points ${P'}(-y+1,x+1)$ and the initial point $A_{1}(2,2)$, find the coordinates of point $A_{2023}$. Express your answer as a coordinate pair $(x, y)$. | (-2,0) | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
What is the sum of the decimal numbers 0.4, 0.02, and 0.006? Express your answer as a decimal number with all necessary decimal places. | 0.426 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Decimals"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Using three out of four line segments with lengths of 13cm, 10cm, 5cm, and 7cm as sides, calculate the number of triangles that can be formed. Express your answer as a whole number. | 3 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given a complex number $z$ that satisfies $i\cdot z=z+a\cdot i$ (where $i$ is the imaginary unit) and $|z|=\sqrt {2}$, find the value of the positive number $a$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Given the line $x - \sqrt{3}y - 2 = 0$, calculate the acute angle $\theta$ that the line makes with the positive $x$-axis. Express your answer in degrees. | 30^{\circ} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Find the largest number by which the expression $n^3 - n$ is divisible for all possible integral values of $n$. Express your answer as a single integer. | 6 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Given an even function $f(x)$ ($x \neq 0$) that is strictly monotonic on the interval $(0, +\infty)$, find the sum of all $x$ that satisfy $f(x^2-2x-1) = f(x+1)$. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the line $x+y+m=0$ is tangent to the circle $x^{2}+y^{2}=m$, find the value of $m$. Express your answer as a single numerical value. | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the line l is perpendicular to the line x-2y-1=0 and passes through the point (1, 1), find the equation of l in the standard form (Ax + By + C = 0). | 2x + y - 3 = 0 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
For the algebraic expression $\frac{x-1}{x+1}$ to be meaningful, what is the condition on the value of $x$? Express your answer using mathematical notation (e.g., ≠, =, >, <). | x \neq -1 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $\sqrt{(-3)^2}$. Provide your answer as a single numerical value. | 3 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. Given that $b^2=ac$ and $c=2a$, find the value of $\cos C$. Express your answer as a fraction or decimal. | -\dfrac{\sqrt{2}}{4} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence {a_n} with the sum of the first n terms denoted as S_n, if S_15 = 45, calculate 2a_12 - a_16. | 3 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Express the number 2370000 in scientific notation. Provide your answer in the form of a decimal part multiplied by 10 raised to a power, e.g., $a \times 10^{b}$. | $2.37 \times 10^{6}$ | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the sum of the first 5 terms of an arithmetic sequence is 6 and the second term is 1, find the common difference. Express your answer as a simplified fraction. | \frac{1}{5} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x-1) = x^2 - 2x$, and $f(a) = 3$, find the value(s) of the real number $a$. Express your answer as a numerical value, including all possible values of $a$. | \pm 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the probability of drawing a first-class product is 0.65, the probability of drawing a second-class product is 0.2, and the probability of drawing a third-class product is 0.1, calculate the probability of drawing a non-first-class product. Express your answer as a decimal value between 0 and 1. | 0.35 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Find a counterexample to the statement "If n is not prime, then n-2 is prime." In other words, identify a composite number n such that n-2 is also composite. Provide your answer as a specific numerical value of n. | 27 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the hyperbola x^2/a^2 - y^2/b^2 = 1 (a > 0, b > 0) with its left and right foci being F1 and F2 respectively. If there exists a point P on the right branch of the hyperbola such that |PF1| = 3|PF2|, calculate the range of the eccentricity e of the hyperbola. Express your answer as an interval. | (1, 2] | big_math | [
"Mathematics -> Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the quadratic function y = ax^2 + 4ax + 3 (a > 0) and points A(m-1, y1) and B(m, y2) on its graph, where y1 < y2, find the range of values for m that satisfies this condition. Express your answer as an inequality for m. | m > -\frac{3}{2} | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Quadratic Functions"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
If the sales tax rate were to increase from 7% to 7.5%, calculate the increase in tax on a $1000 item. Express your answer in dollars, rounded to two decimal places. | 5.00 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Points A(11, 9) and B(2, -3) are vertices of triangle ABC with AB = AC. The altitude from A meets the opposite side at D(-1, 3). Calculate the coordinates of point C. Express your answer as a coordinate pair (x, y). | (-4, 9) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Find the median of the following dataset of scores: 86, 95, 97, 90, 86. Express your answer as a single number. | 90 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number equation iz = 2 - 4i, find the coordinates of the point corresponding to the conjugate of z in the complex plane. Express your answer as an ordered pair of coordinates (x, y). | (-4, 2) | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given the set of numbers {89, 95, 99, 132, 166, 173}, how many subsets containing three different numbers can be selected such that the sum of the three numbers is even? Express your answer as a single integer. | 12 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Combinations"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Given the mapping f: A → B, where A = B = ℝ, and the corresponding rule f: x → y = -x^2 + 2x, find the range of values for k ∈ B such that there exist two different pre-images in set A. Express your answer as an inequality in terms of k. | k < 1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Quadratic Functions"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x)=x^{2}-x+2$, calculate the definite integral $\int_{0}^{1}f(x)dx$. Express your answer as a simplified fraction. | \frac{11}{6} | big_math | [
"Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
It is known that locations A and B are 10 kilometers apart. Two people, person A and person B, start walking towards each other from locations A and B, respectively. Person A has a dog with him. As soon as person A starts, the dog also starts running towards B. Upon reaching B, the dog immediately turns around and runs... | 12 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Find the smallest positive period of the function $y=5\sin (3x+ \frac {\pi}{6})$, where $x\in R$. Express your answer as a fraction of π. | \frac{2\pi}{3} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given the ordinary generating function $A(t)$ of the sequence $\{a_{0}, a_{1}, a_{2}, \cdots \}$, find the ordinary generating function of the sequence $\{a_{0}, a_{0} + a_{1}, a_{0} + a_{1} + a_{2}, \cdots \}$. Express your answer as a function of $A(t)$. | \frac{1}{1-t} A(t) | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Given the circle equation (x+2)^2 + y^2 = 2016 and the line of symmetry x - y + 1 = 0, find the equation of the symmetric circle with respect to the line. Express your answer as a circle equation in standard form. | (x+1)^2 + (y+1)^2 = 2016 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
How many numbers exist such that the number is its own opposite? Provide your answer as a single number representing the count. | 1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Find the minimum value of the function $f(x)= \frac {1}{3}x^{3}-4x+4$ in the interval $[0,3]$. Express your answer as a single number in the form $\boxed{[answer]}$. | -\frac{4}{3} | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a^{m}=2$ and $a^{n}=3$, calculate the value of $a^{3m+2n}$. Express your answer as a single number. | 72 | big_math | [
"Mathematics -> Algebra -> Exponential Functions"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Let the complex numbers \( z_1 \) and \( z_2 \) correspond to the points \( A \) and \( B \) on the complex plane respectively, and suppose \( \left|z_1\right| = 4 \) and \( 4z_1^2 - 2z_1z_2 + z_2^2 = 0 \). Let \( O \) be the origin. Calculate the area of triangle \( \triangle OAB \). Express your answer as a numerical... | 8\sqrt{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other",
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Analyze the logical relationship between the conditions M∪N≠∅ and M∩N≠∅ for two sets M and N. Specifically, determine and explain whether M∪N≠∅ is a sufficient condition, a necessary condition, both, or neither for M∩N≠∅. Provide your answer in a clear logical explanation. | M∪N≠∅ is a necessary but not sufficient condition for M∩N≠∅. | big_math | [
"Mathematics -> Discrete Mathematics -> Logic"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow {a}=(1, n, 2)$ and $\overrightarrow {b}=(-2, 1, 2)$, where $2\overrightarrow {a} - \overrightarrow {b}$ is perpendicular to $\overrightarrow {b}$, find the magnitude of $\overrightarrow {a}$. Express your answer as a single value, including any necessary square roots. | \frac{3\sqrt{5}}{2} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Given the system of linear inequalities in one variable x: 2(x+1) < x+3 and x-a ≤ a+5, where the solution set for x is x < 1, and a is a non-positive integer, calculate the number of values of a that satisfy the conditions. Express your answer as a whole number. | 3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Given the piecewise function $$f(x)= \begin{cases} f(x-5), & x>0 \\ 2^{x}+ \int_{ 0 }^{ \frac {\pi}{6} } \cos3tdt, & x\leq 0 \end{cases}$$, calculate the value of f(2017). Express your answer as a single fraction. | \frac {11}{24} | big_math | [
"Mathematics -> Calculus -> Integral Calculus -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
A woman is good at weaving cloth. She weaves faster each day than the day before, and the amount she increases each day is the same. It is known that she wove 5 chi of cloth on the first day and a total of 390 chi of cloth in 30 days. What is the increase in chi that the woman weaves each day? Express your answer as a ... | \frac{16}{29} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Find the maximum value of the function $f(x) = \lg[x(x - \frac{3}{2}) + 1]$ over the domain $x \in [1, \frac{3}{2}]$. Express your answer as a single number. | 0 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
Let \( a, b \in \mathbf{R}_{+} \) and satisfy the equation:
\[
\sqrt{\lg a} + \sqrt{\lg b} + \lg \sqrt{a} + \lg \sqrt{b} = 100,
\]
where \( \sqrt{\lg a}, \sqrt{\lg b}, \lg \sqrt{a}, \lg \sqrt{b} \in \mathbf{Z}_{+} \). Calculate the value of \( ab \). Express your answer as an exponent of 10, including the base 10. | 10^{164} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Anna walked at a constant rate. She walked 600 metres in 4 minutes. If she continued walking at the same rate, how far did she walk in 6 minutes? Express your answer in meters as a whole number. | 900 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Given the expansion of $(1+x)(x+ \frac {1}{x^{2}})^{n}$ ($n \in \mathbb{N}^*, n < 10$) with no constant term, determine the maximum value of $n$ that satisfies this condition. Express your answer as a single integer. | 7 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence {a_n} defined by:
{ (\frac{1}{3}-a)n+2, & n>8 \\ a^{n-7}, & n\leq8 }
and the condition a_n > a_{n+1} holds for all n, find the range of the real number a that satisfies these conditions. Express your answer as an interval (lower bound, upper bound). | (\frac{1}{2}, 1) | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Real numbers $x$ and $y$ satisfy the equation $x^2+y^2=10x-6y-34$. Find the value of $x+y$. Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
A certain product decreases its price by 25% every three years. If its current price is 640 yuan, calculate the price of this product after 9 years. Express your answer as a whole number in yuan. | 270 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Given the system of inequalities $\left\{\begin{array}{l}1+x \gt a\\2x-4\leqslant 0\end{array}\right.$, find the range of values for $a$ that allows the system to have a solution. Express your answer as an inequality in terms of $a$. | a < 3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the result of the expression $-1÷(-5)÷(-\frac{1}{5})$. Express your answer as a single number. | -1 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the following expression:
\[\frac{\left(3^{2008}\right)^2-\left(3^{2006}\right)^2}{\left(3^{2007}\right)^2-\left(3^{2005}\right)^2}\]
Express your answer as a simplified numerical value. | 9 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the line that is symmetric to the line x - 2y + 3 = 0 with respect to the X-axis. Express your answer as a linear equation in the form Ax + By + C = 0. | x + 2y + 3 = 0 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Given the hyperbola $mx^{2}-ny^{2}=1\,(m > 0,n > 0)$ with an eccentricity of $2$, find the eccentricity of the ellipse $mx^{2}+ny^{2}=1$. Express your answer as a numerical value, including any necessary radicals. | \frac{\sqrt{6}}{3} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Given a sequence {a_n}, where the sum of the first n terms is S_n, and S_n = 2a_n - 2, calculate the value of a_4. Express your answer as a single integer. | 16 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Find the greatest common divisor (GCD) of 459 and 357 using successive division. Express your answer as a single integer. | 51 | big_math | [
"Mathematics -> Number Theory -> Greatest Common Divisors (GCD)"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
In the context of representing quantities with positive and negative numbers, income and expenditure are considered opposite quantities. If an income of $60$ yuan is recorded as $+60$ yuan, how would you represent an expenditure of $40$ yuan? Express your answer as a numerical value with a sign (positive or negative) a... | -40 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Given the parabola equation $y=-\frac{1}{4}x^2+x-4$, find its axis of symmetry. Express your answer as an equation of a line in the form $x=a$. | x=2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Quadratic Functions"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.