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 |
|---|---|---|---|---|---|
Find the integer that is closest in value to \(\frac{35}{4}\). Express your answer as a single integer. | 9 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
In the space coordinate system, point B is the projection of point A(1, 2, 3) on the yOz coordinate plane, and O is the origin of the coordinates. Calculate the distance |OB| and express your answer as a square root if necessary. | \sqrt{13} | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
In a class meeting, it is planned to select 4 students to speak from 7 students including A and B, with the requirement that at least one of A and B must participate. Calculate the number of different speaking orders. | 720 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Combinations"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, the lengths of four of the sides of the figure are shown in terms of x. Assuming that x is not equal to 0, find the perimeter of the figure. Express your answer in terms of x. | 10x | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Find the opposite of $0.1$, which is the number that, when added to $0.1$, results in zero. Express your answer as a decimal number. | -0.1 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
A complex number $z$ satisfies $\overline{\dot{z}}\cdot \left(1+2i
ight)=4+3i$. Calculate the value of $z$ and express your answer as a complex number in the form $a+bi$. | 2+i | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Given that the graph of f(x) always passes through the point (1,1), find the point through which the graph of f(x-4) always passes. Express your answer as an ordered pair (x, y). | (5,1) | big_math | [
"Mathematics -> Algebra -> Functions"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Given a sequence {a_n} satisfying a_1 = 1, a_2 = 1, a_{n+1} = |a_n - a_{n-1}| (for n >= 2), calculate the sum of the first 2013 terms of this sequence. Express your answer as a single integer. | 1342 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given the ellipse $\dfrac {x^{2}}{a^{2}}+ \dfrac {y^{2}}{4}=1(a > 0)$ and the hyperbola $\dfrac {x^{2}}{9}- \dfrac {y^{2}}{3}=1$ have the same foci, find the value of $a$. Express your answer as a numerical value. | 4 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Given the expression $(x^{2}+ax+5)\cdot \left(-2x
ight)-6x^{2}$, find the value of $a$ such that the result does not contain the term $x^{2}$. Express your answer as a single number. | -3 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Let $f(x)$ be a decreasing function on $(-\infty,+\infty)$. Find the solution set of the inequality $f(2) < f(2x+1)$. Express your answer in standard interval notation. | (-\infty, \frac {1}{2}) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Five positive consecutive integers starting with a have average b. Find the average of 5 consecutive integers that start with b. Express your answer as an algebraic expression in terms of a. | a + 4 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation x^2 + x - 1 = x e^(x^2 - 1) + (x^2 - 1) e^x, find the sum of squares of all elements in the solution set A. Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
The focus of the parabola $y^2=2px$ ($p>0$) is $F$, and the chord $AB$ passes through point $F$ with an inclination angle of $60^\circ$, $|AF| > |BF|$. Find the ratio of $|AF|$ to $|BF|$. Express your answer as a whole number. | 3 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given the expansion $(2-x)^{6} = a_{0} + a_{1}x + a_{2}x^{2} + \cdots + a_{6}x^{6}$, determine the value of $|a_{1}| + |a_{2}| + \cdots + |a_{6}|$. Express your answer as a whole number. | 665 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $x > 0$ and $y > 0$, and they satisfy the equation $\frac{4}{y} + \frac{1}{x} = 4$, find the minimum value of $x + y$. Express your answer as a single number. | \frac{9}{4} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(\ln x) = 3x + 4$, find the expression for $f(x)$. Provide your answer as a function expression. | 3e^x + 4 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Find the domain of the function $f(x)=\sqrt{3-x}+\log_{\frac{1}{3}}(x+1)$. Express your answer in interval notation, using round parentheses for excluded endpoints and square brackets for included endpoints. | (-1,3] | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the sequence $\{a_n\}$ is an arithmetic sequence with a common difference of 2, and $a_1, a_2, a_5$ form a geometric sequence, find the value of $a_2$. Express your answer as a single integer. | 3 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
A rugby team scored 24 points, 17 points, and 25 points in the seventh, eighth, and ninth games of their season. Their mean points-per-game was higher after 9 games than it was after their first 6 games. Calculate the smallest number of points that they could score in their 10th game for their mean number of points-per... | 24 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
If $\sqrt{63n}$ is an integer, find the smallest value of the positive integer $n$. Express your answer as a single integer. | 7 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
In a certain unit, the ratio of the number of elderly, middle-aged, and young people is 2:3:5, respectively. Using stratified sampling, a sample is drawn. If the sample contains 12 middle-aged people, find the sample size. Express your answer as a whole number. | 40 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
In a geometric sequence {a_n} with a common ratio not equal to 1, there exist s and t in N* such that a_s * a_t = a_5^2. Find the minimum value of 4/s + 1/(4t). | \frac{5}{8} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Square \(PQRS\) has side length 60 and centre \(C\). Point \(W\) lies on \(PS\) so that \(WS = 53\). Point \(X\) lies on \(SR\) so that \(XR = 40\). The midpoint of \(QR\) is \(Y\). Point \(Z\) lies on \(PQ\). Determine the length of \(ZQ\) so that the total area of the shaded regions is equal to the total area of the ... | 17 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Find the maximum value of the function f(x) = (x^2 + ax + 1)e^x, given that x=3 is an extremum point. Express your answer as a single value with exponential notation (e.g., 6e^{-1}). | 6e^{-1} | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Anna enjoys dinner at a restaurant in Washington, D.C., where the sales tax on meals is 10%. She leaves a 15% tip on the price of her meal before the sales tax is added, and the tax is calculated on the pre-tip amount. If she spends a total of $27.50 for dinner, what is the cost of her dinner without tax or tip in doll... | 22 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $$\overrightarrow {a}=(4,2)$$ and $$\overrightarrow {b}=(6,m)$$, and knowing that $$\overrightarrow {a} \perp \overrightarrow {b}$$, find the value of $m$ that makes the vectors perpendicular. Express your answer as a single number. | -12 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $\sin 43^{\circ}\cos 17^{\circ}+\cos 43^{\circ}\sin 17^{\circ}$. Express your answer as a fraction or a radical. | \frac{\sqrt{3}}{2} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=5,|\overrightarrow{b}|=6,\overrightarrow{a}\cdot\overrightarrow{b}=-6$, calculate the magnitude of the vector $\overrightarrow{a} + \overrightarrow{b}$. Provide your answer as a numerical value. | 7 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfying $|\overrightarrow{a}|=3$, $|\overrightarrow{b}|=2$, and $|\overrightarrow{a} + \overrightarrow{b}|=4$, calculate the magnitude of $|\overrightarrow{a} - \overrightarrow{b}|$. Express your answer as a square root if necessary. | \sqrt{10} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Square \(P Q R S\) has an area of \(900\). \(M\) is the midpoint of \(P Q\) and \(N\) is the midpoint of \(P S\). Calculate the area of triangle \(P M N\). Express your answer as a decimal number. | 112.5 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Five integers have an average of 69. The middle integer (the median) is 83. The most frequently occurring integer (the mode) is 85. The range of the five integers is 70. Calculate the second smallest of the five integers. Express your answer as a whole number. | 77 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
A hyperbola $\frac{x^2}{m} - \frac{y^2}{n} = 1 \ (m > 0, n > 0)$ and an ellipse $\frac{x^2}{a} + \frac{y^2}{b} = 1 \ (a > b > 0)$ share the same foci $F_1$ and $F_2$. Let $M$ be the intersection point of the two curves. Find the product $|MF_1|\cdot|MF_2|$ in terms of $a$ and $m$. | a - m | big_math | [
"Mathematics -> Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
If the complex number $z=(a^2+2a-3)+(a+3)i$ is purely imaginary (where $i$ is the imaginary unit), then find the value of the real number $a$. Express your answer as a single real number. | 1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Quadratic Functions"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Given the universal set U = {0,1,2,3,4,5}, M = {0,3,5}, and N = {1,4,5}, calculate M ∩ (∁U N). Express your answer as a set using standard set notation. | {0,3} | big_math | [
"Mathematics -> Discrete Mathematics -> Logic -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the functions f(x) = x^2 - 2m and h(x) = 6n^2 ln x - 4nx defined on (0, +∞), where n > 0, find the maximum value of m/n if the curves y = f(x) and y = h(x) have a common point and their tangents at the common point are the same. Express your answer using necessary mathematical notation (e.g., e, ln). | 3e^{-\frac{1}{6}} | big_math | [
"Mathematics -> Calculus -> Applications of Derivatives"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overline{m}=(-2,1)$ and $\overline{n}=(1,1)$, find the real number $a$ such that $(\overline{m}-2\overline{n}) \perp (a\overline{m}+\overline{n})$. Express your answer as a single real number in a boxed format. | \frac{5}{7} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
The base-ten representation for $19!$ is $121,6T5,100,40M,832,H00$, where $T$, $M$, and $H$ denote digits that are not given. Calculate the sum of $T$, $M$, and $H$. Express your answer as a single integer. | 12 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
A square base with a side length of 6 cm is surrounded by four identical isosceles triangles that meet at a point to form a pyramid. The height of this pyramid is 4 cm. Calculate the total area of the four triangles and the square. Express your answer in square centimeters (cm²). | 96 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Surface Area"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(3,1), \overrightarrow{b}=(1,3), \overrightarrow{c}=(k,-2)$, if $(\overrightarrow{a}- \overrightarrow{c}) // \overrightarrow{b}$, calculate the cosine value of the angle between vector $\overrightarrow{a}$ and vector $\overrightarrow{c}$. Express your answer as a fraction or decimal, i... | \dfrac{ \sqrt{5}}{5} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence {a_n} with the sum of its first n terms denoted as S_n, if points M, N, and P are collinear, and O is the origin, and the vector equation $$\overrightarrow {ON}=a_{15} \overrightarrow {OM}+a_{6} \overrightarrow {OP}$$ holds (the line MP does not pass through point O), calculate the value o... | 10 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Solve the equation $2^{\log_{3} x} = \frac{1}{4}$. Express your answer as a fraction. | \frac{1}{9} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Given the hyperbola equation x^2/a^2 - y^2/b^2 = 1 (a > 0, b > 0) and the asymptote equations y = ±x, calculate the eccentricity of the hyperbola. Express your answer as a single value. | \sqrt{2} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $\left(a^{3}\right)^{2}$ using the appropriate exponent rules. Express your answer as a simplified expression. | a^{6} | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Find the range of the real number $a$ such that the inequality $x^{2}-4x-2-a > 0$ has solutions in the interval $(1,4)$. Express your answer as an interval. | (-\infty,-2) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\overrightarrow{a}$ and $\overrightarrow{b}$ are unit vectors with an angle of $120^{\circ}$ between them, calculate the value of $(\overrightarrow{a}-2\overrightarrow{b})\cdot\overrightarrow{b}$. Express your answer as a single number. | -\frac{5}{2} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Find the domain of the function f(x) = sqrt(x) / ln(2-x). Express your answer as an interval or union of intervals. | [0,1)\cup(1,2) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
If w is a complex root of the equation x^3 = 1, calculate the product (1 - w + w^2)(1 + w - w^2). Express your answer as a single number. | 4 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given the circle equation x²+(y-1)²=2, find the equation of the tangent line that passes through the point (1,2). Express your answer in the standard form of a linear equation (Ax + By + C = 0). | x+y-3=0 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $f\left(x\right)$ is an odd function, and when $x \lt 0$, $f\left(x\right)=e^{x}$, calculate $f\left(e\right)$ using the property of odd functions. Express your answer as a single mathematical expression. | -e^{-e} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
At a crossroads, if vehicles are not allowed to turn back, calculate the total number of possible driving routes. Express your answer as a single integer. | 12 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
In a class of 80 students, systematic sampling is used to select a number of students for a survey. Each student is randomly assigned a number from 1 to 80. Given that the two smallest numbers among the selected students are 6 and 14, calculate the largest number among the selected students. Express your answer as a wh... | 78 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Given the vector $(\overrightarrow{AB} + \overrightarrow{MB}) + (\overrightarrow{BO} + \overrightarrow{BC}) + \overrightarrow{OM}$, simplify the expression using the rules of vector addition. Express your answer as a simplified vector using standard vector notation. | \overrightarrow{AC} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation of the ellipse $x^{2}+2y^{2}=4$, calculate the coordinates of the foci. Express your answer as a pair of coordinates $(x, y)$. | (\sqrt{2}, 0), (-\sqrt{2}, 0) | big_math | [
"Mathematics -> Geometry -> Analytic Geometry -> Conic Sections"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given the vector $a=(\sqrt{2},-\sqrt{7})$, $a\cdot b=-9$, and the angle between vectors $a$ and $b$ is $120^{\circ}$, calculate the magnitude of vector $b$, $|b|$. Express your answer as a numerical value. | 6 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.859375 | 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 -> Other"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $\log_{8}9 \cdot \log_{27}32$. Express your answer as a single fraction or number. | \frac{10}{9} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Logarithmic Functions"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the square of the complex number (2+i). Express your answer as a complex number in the form a+bi, including both real and imaginary parts. | 3+4i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
The Statistical Bulletin of National Economic and Social Development of Henan Province reported that at the end of 2022, the total resident population of the province included 42.39 million rural residents. Express 42.39 million in scientific notation. Provide your answer in the form $\boxed{a \times 10^b}$, where $a$ ... | 4.239 \times 10^7 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
If \( m \) and \( n \) are consecutive positive integers and \( n^{2} - m^{2} > 20 \), calculate the minimum possible value of \( n^{2} + m^{2} \). Express your answer as a single integer. | 221 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Let $(1-x)^n = a + a_1x + a_2x^2 + \ldots + a_nx^n$ ($n \geq 3$, $n \in \mathbb{Z}$). If $a_3 + 2a_2 = 0$, calculate the value of $n$. Express your answer as a single integer. | 8 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Celyna bought 300 grams of candy A for $5.00, and x grams of candy B for $7.00. She calculated that the average price of all of the candy that she purchased was $1.50 per 100 grams. Find the value of x. Express your answer in grams as a whole number. | 500 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
February 1, 2012, was a Wednesday. Calculate the day of the week for June 1, 2012. Express your answer as a day of the week (e.g., Monday, Tuesday, etc.). | Friday | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
In the plane of $\triangle ABC$, there is a point $P$ such that $\overrightarrow{PA}+ \overrightarrow{PB}+ \overrightarrow{PC}= \overrightarrow{AB}$. Calculate the ratio of the area of $\triangle PBC$ to the area of $\triangle ABC$. Express your answer as a simplified fraction. | \dfrac{2}{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
On the number line, point A represents the number -5, and point B represents the number 2. Find the length of segment AB. Express your answer as a positive integer. | 7 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
In the expansion of $x(1+x)^6$, find the coefficient of the $x^3$ term. Express your answer as a single integer. | 15 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Binomial Theorem"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Classify the function \( f(x)=\log _{\frac{1}{3}}\left(2 x^{2}+2 x \sqrt{x^{2}+1}+1\right)^{x} \) as even, odd, or neither. | even | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Precalculus -> Functions"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
If the graph of the quadratic function y = x^2 is shifted 1 unit to the left, what is the analytical expression of the shifted quadratic function? Express your answer as a function of x, using standard function notation (e.g., y = f(x)). | y = (x + 1)^2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Quadratic Functions"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given a parabola $y=ax^{2}+bx+c\left(a\neq 0\right)$ with its vertex at $\left(-3,2\right)$ and opening downwards, determine the type of extremum (maximum or minimum) and its value. Express your answer in the format "The parabola has a [maximum/minimum] value of [value]." | The parabola has a maximum value of 2. | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Quadratic Functions"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given the following code snippet, calculate the output when a = 3:
```
IF a < 10 THEN
y = 2 * a
ELSE
y = a * a
```
Express your answer as a single numerical value. | 6 | big_math | [
"Mathematics -> Discrete Mathematics -> Algorithms -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Determine the constant term in the expansion of the binomial $(x^{2}- \dfrac {2}{ \sqrt {x}})^{5}$. Express your answer as a single integer. | 80 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Unit A has 5 male volunteers and 7 female volunteers; Unit B has 4 male volunteers and 2 female volunteers. One unit is randomly selected from the two units, and then one volunteer is randomly selected from the chosen unit. Calculate the probability of selecting a male volunteer. Express your answer as a fraction in si... | \frac{13}{24} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Find the solution set of the inequality $(1-x)(3+x) > 0$. Express your answer in interval notation. | (-3, 1) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Given $x, y \\in \\mathbb{R}$, find the minimum value of the expression $(x+y)^{2}+(x- \\frac {2}{y})^{2}$. Provide your answer as a single number. | 4 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\alpha$ is an angle in the fourth quadrant and $\tan \alpha = -\frac{5}{12}$, calculate the value of $\cos \alpha$. Express your answer as a simplified fraction in the form $\boxed{\frac{a}{b}}$. | \frac{12}{13} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Given sets $M=\{x|2x+1 \lt 3\}$ and $N=\{x|x \lt a\}$, describe the range of real number $a$ such that $M\cap N=N$. Express your answer as an interval. | (-\infty, 1] | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given that α∈(0, π/2) and β∈[0, π/2], find the range of the expression 2α - β/3. Express your answer as an interval. | (-π/6, π) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence ${\{a_n\}}$ that satisfies $a_n = \log_{n+1}(n+2) \ (n \in \mathbb{N}^*)$, define $k \ (k \in \mathbb{N}^*)$ that makes the product ${a_1} \cdot {a_2} \cdot {a_3} \cdot \ldots \cdot {a_k}$ a positive integer as a "desired number." Calculate the sum of all "desired numbers" within the interval $[1, 20... | 2026 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the random variable ξ follows a normal distribution N(2,σ^2), and P(ξ<4) = 0.8, calculate the probability P(0<ξ<2). Express your answer as a decimal value between 0 and 1, rounded to 1 decimal place. | 0.3 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Find the range of values for the real number $a$ such that the inequality $2x+ \\frac{2}{x-1}\\geqslant a$ holds true for all $x \\in (1, +\\infty)$. Express your answer as an interval. | (-\\infty, 6] | big_math | [
"Mathematics -> Algebra -> Inequalities"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Two fair coins are tossed at the same time. What is the probability that they will both land as "HEADS"? Express your answer as a fraction. | \frac{1}{4} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given the definition of factorial for any natural number n as n! = 1 × 2 × 3 × ⋯ × n, calculate the last digit of the expression 2014! - 3!. Express your answer as a single digit, resulting from the calculation modulo 10. | 4 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Let $f(x)$ be a differentiable function, and it satisfies $$\lim_{\Delta x \to 0} \frac {f(1+2\Delta x)-f(1)}{\Delta x} = -2,$$ then find the derivative of the function $y=f(x)$ at $x=1$. Express your answer as a single numerical value. | -1 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the tangent line to the curve y = xe^(x-1) at the point (1,1). Express your answer as a linear equation in the form y = mx + b, where m and b are constants. | y = 2x - 1 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
If $n$ non-coincident points on a plane can determine at most 45 lines, what is the value of $n$? Express your answer as a positive integer. | 10 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics -> Other"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
A cube has 27 points consisting of its 8 vertices, the midpoints of its 12 edges, the centers of its 6 faces, and the center of the cube. How many sets of three collinear points are there among these 27 points? Express your answer as a whole number. | 28 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
When the number of sides of a polygon increases by 1, what is the change in the sum of its interior angles? Express your answer in degrees as a whole number. | 180 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation \( 2x^2 - 6x + y^2 = 0 \), find the maximum value of the expression \( x^2 + y^2 + 2x \). Express your answer as a single number. | 15 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the tangent line to the curve f(x) = 2x - e^x at the point (0, f(0)). Express your answer as an equation in standard form. | x - y - 1 = 0 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\\vec{a}$ and $\\vec{b}$ satisfying $|\\vec{a}|=1$, $|\\vec{a}-\\vec{b}|=\\sqrt{3}$, and $\\vec{a}\\cdot(\\vec{a}-\\vec{b})=0$, find the value of $|\\vec{b}-2\\vec{a}|$. Express your answer as a numerical value. | 2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Vectors"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
In the Cartesian coordinate system, point $P$ is symmetric to point $A$ with respect to the $x$-axis, and point $P$ is symmetric to point $B$ with respect to the $y$-axis. Given that point $B$ has coordinates $(1,2)$, find the coordinates of point $A$. Express your answer as a coordinate pair $(x, y)$. | (-1,-2) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
A list of 5 positive integers has the following properties: the only integer that occurs more than once is 8, its median is 9, and its average (mean) is 10. What is the largest possible integer that could appear in the list? Express your answer as a single integer. | 15 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
A flagpole that is 4 meters high casts a shadow that is 6 meters long. At the same time, a large tree next to the flagpole casts a shadow that is 12 meters long. Find the height of the tree. Express your answer in meters as a whole number. | 8 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\overline{z}$ is the complex conjugate of $z$, if $\overline{z} + z = 2$ and $(\overline{z} - z)i = 2$ (where $i$ is the imaginary unit), calculate the value of $z$. Express your answer in the form $a + bi$. | 1 + i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence $\{a_n\}$ defined by $a_1 = 1$ and $a_{n+1} = 2S_n + 1$ (where $n \ge 1$ and $n \in \mathbb{N}$), derive the general term formula for the sequence. Express your answer as a formula for $a_n$ and provide a clear explanation or proof. | 3^{n-1} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
In triangle $\triangle OAB$, let $\overrightarrow{OA}=(2\cos \alpha, 2\sin \alpha)$ and $\overrightarrow{OB}=(5\cos \beta, 5\sin \beta)$. If $\overrightarrow{OA} \cdot \overrightarrow{OB} = -5$, then calculate the area of $\triangle OAB$, $S_{\triangle OAB}$. Express your answer as a numerical value. | \dfrac{5\sqrt{3}}{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Given the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a > 0, b > 0)$ and the parabola $y^2 = 2px (p > 0)$, the asymptotes of the hyperbola intersect the directrix of the parabola at points $A$ and $B$. If the eccentricity of the hyperbola is $2$ and the area of $\triangle AOB$ is $\sqrt{3}$, find the value of $p$... | 2 | big_math | [
"Mathematics -> Geometry -> Other",
"Mathematics -> Algebra -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $(1-2x)^{2016} = a_0 + a_1x + a_2x^2 + \ldots + a_{2016}x^{2016}$, where $x \in \mathbb{R}$, find the value of $(a_0 + a_1) + (a_0 + a_2) + (a_0 + a_3) + \ldots + (a_0 + a_{2016})$. Express your answer as a single integer. | 2016 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given the parabola y = 3 - x^2 and the line of symmetry x + y = 0, find the distance |AB| between two distinct points A and B on the parabola that are symmetric with respect to the line. | 3\sqrt{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the operation definitions: $a \triangle b = a + b + ab - 1$ and $a \otimes b = a^2 - ab + b^2$, calculate the value of $3 \triangle (2 \otimes 4)$. Express your answer as a single integer. | 50 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
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