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 |
|---|---|---|---|---|---|
A convex polyhedron has all its faces as quadrilaterals and 16 vertices. Using Euler's formula, calculate the number of faces of this polyhedron. Express your answer as a whole number. | 14 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the remainder of $91^{91}$ divided by 100. Express your answer as a single integer. | 91 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the angle between two skew lines a and b is $50^{\circ}$, and P is a point in space, determine the number of unique lines passing through point P that make an angle of $30^{\circ}$ with both lines a and b. | 2 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Natascha cycles 3 times as fast as she runs. She spends 4 hours cycling and 1 hour running. Find the ratio of the distance she cycles to the distance she runs. Express your answer as a simplified ratio in the form a:b. | 12:1 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of positive integer solutions to the equation 4x + 5y = 98. | 5 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
In a certain population, the ratio of the number of females to the number of males is 11:10. If the average age of the females is 34 years, and the average age of the males is 32 years, calculate the average age in this population. Express your answer as a mixed number or decimal. | 33 \frac{1}{21} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the derivative of the function $f(x)=x^{m}+nx$ is $f′(x)=2x+1$, calculate the definite integral $\int_{1}^{2}f\left(-x\right)dx$. Express your answer as a simplified fraction. | \frac{5}{6} | big_math | [
"Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Let $\mathbf{i} = (1, 0)$, $\mathbf{j} = (0, 1)$, $\mathbf{a} = 2\mathbf{i} + 3\mathbf{j}$, $\mathbf{b} = k\mathbf{i} - 4\mathbf{j}$. Find the value of the real number $k$ such that $\mathbf{a}$ is perpendicular to $\mathbf{b}$. Express your answer as a single number. | 6 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the domain of the function $y=f\left(x\right)$ is $\left[-8,1\right]$, determine the domain of the function $g\left(x\right)=\frac{f(2x+1)}{x+2}$. Express your answer as an interval or union of intervals. | \left[-\frac{9}{2}, -2\right) \cup \left(-2, 0\right] | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Functions"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Using the operation defined as $a\triangle b=a\cdot b-a-b+1$, calculate the result of $(-3)\triangle 4$. Provide your answer as a single integer. | -12 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Solve for $x$ in the equation $x^2 + b^2 = (a - x)^2$. Express your answer as a single algebraic expression involving variables $a$ and $b$. | \frac{a^2 - b^2}{2a} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Given a complex number $z=2-i$, calculate the value of $\frac{1}{{\overline{z}+i}}$. Express your answer as a complex number in the form $a+bi$. | \frac{1}{4} - \frac{1}{4}i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
In an arithmetic sequence, the sum of the first three terms is 34, the sum of the last three terms is 146, and the sum of all terms is 390. Find the number of terms in the sequence. Express your answer as a whole number. | 13 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given a real number \( a \) such that there is only one real number \( x \) satisfying the inequality \( \left|x^{2}+2 a x+3 a\right| \leqslant 2 \), determine the number of distinct real numbers \( a \) that meet this condition. Express your answer as a whole number. | 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.25 | 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 in the form $y = c$. | y=- \dfrac {1}{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Driving between two towns at 110 km/h instead of 100 km/h saves 9 minutes. Calculate the distance in kilometers between the two towns. | 165 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Let the sum of the first $n$ terms of a geometric sequence be $S_n$. If $\frac{S_{10}}{S_5} = 1:2$, then calculate the ratio $\frac{S_{15}}{S_5}$. Express your answer as a ratio in simplest form. | \frac{3}{4} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Let the function $f(x)=\log_{a}x$ ($a > 0$ and $a \neq 1$). If $f(x_{1}x_{2}\cdots x_{2017})=8$, calculate the value of $f(x_{1}^{2})+f(x_{2}^{2})+\cdots +f(x_{2017}^{2})$. Express your answer as a single numerical value. | 16 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Logarithmic Functions"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(1,1)$, $\overrightarrow{b}=(2,x)$, if $\overrightarrow{a}+\overrightarrow{b}$ is parallel to $\overrightarrow{a}-\overrightarrow{b}$, find the value of the real number $x$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Let A be an even number but not divisible by 10. Find the last two digits of A^20. Express your answer as a two-digit number. | 76 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Given the angle $\alpha = -\frac{35}{6}\pi$, calculate the value of $$\frac{2\sin(\pi + \alpha)\cos(\pi - \alpha) - \cos(\pi + \alpha)}{1 + \sin^2\alpha + \sin(\pi - \alpha) - \cos^2(\pi + \alpha)}.$$ Express your answer as a single numerical value. | \sqrt{3} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given a complex number $z= \frac {1+2i^{3}}{2+i}$ (where $i$ is the imaginary unit), find the coordinates (a, b) of the corresponding point of $z$ in the complex plane. Express your answer in the format $\boxed{(a, b)}$. | (0, -1) | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Two circles have equations x^2 + y^2 + 4x - 4y + 7 = 0 and x^2 + y^2 - 4x - 10y + 13 = 0. How many lines are tangent to both circles? Express your answer as a whole number. | 3 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
If $x > 0$, $y > 0$, and $x + y = 1$, find the minimum value of $xy + \frac{2}{xy}$. Express your answer as a single numerical value. | \frac{33}{4} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
The edge length of a cube is 2, and its 8 vertices all lie on the same spherical surface. Find the surface area of this sphere. Express your answer in terms of pi. | 12\pi | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
There are 12 items displayed on a shelf, arranged with 4 items on the upper layer and 8 items on the lower layer. Now, if 2 items are to be selected from the lower layer to be moved to the upper layer, and the relative order of the other items remains unchanged, calculate the number of different ways to adjust the item... | 840 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Combinations"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Among the real numbers $\sqrt{4}$, $\sqrt{3}$, $0$, $\frac{22}{7}$, $\sqrt[3]{0.125}$, $0.1010010001\ldots$, $\frac{\pi}{2}$, how many are irrational? Express your answer as a whole number. | 3 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Find the solution set of the inequality $3 \leqslant |2x-5| < 9$ in interval notation, including the union of both cases. | (-2,1] \cup [4,7) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the following expression:
$$
\left(\left((2+1)^{-1}+1\right)^{-1}+1\right)^{-1}+1
$$
Express your answer as a simplified fraction. | \frac{11}{7} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Find the slope of the tangent line to the graph of the function $f(x)= \frac {2x-5}{x^{2}+1}$ at the point $(0,f(0))$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
One pen costs $2. Calculate the total cost of 10 pens. Express your answer in dollars. | $20 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given the universal set $U=\{1,2,3,4,5\}$, set $A=\{1,3,4\}$, and set $B=\{2,4\}$, find the result of the set operation $(\complement_{U}A)\cup B$. Express your answer as a set of elements enclosed in curly brackets, with elements separated by commas. | \{2,4,5\} | big_math | [
"Mathematics -> Discrete Mathematics -> Logic -> Other"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $3^{x}=a$ and $5^{x}=b$, calculate the value of $45^{x}$ in terms of $a$ and $b$. Express your answer as an algebraic expression. | a^{2}b | big_math | [
"Mathematics -> Algebra -> Exponential Functions"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the result of $x^{3}\cdot x^{3}$ using the rule of exponents for multiplication. Express your answer as a single algebraic expression. | $x^{6}$ | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
The sum of the interior angles of a convex polygon minus the measure of one interior angle is 2570 degrees. Find the measure of this interior angle. Express your answer in degrees as a whole number. | 130 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
A, B, and C are running at uniform speeds over a distance d. A can beat B by 20 yards, B can beat C by 10 yards, and A can beat C by 28 yards. Find the distance d, in yards, that satisfies all these conditions. | 100 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Find the limiting sum S of the infinite series $1-\frac12-\frac14+\frac18-\frac{1}{16}-\frac{1}{32}+\frac{1}{64}-\frac{1}{128}-\cdots$. Express your answer as a simplified fraction. | \frac{2}{7} | big_math | [
"Mathematics -> Calculus -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence ${a_n}$ with its first $n$ terms sum $S_n$, where $a_1=1$, $a_2=2$, and $a_{n+2} - 2a_{n+1} + a_n = 0 (n \in \mathbb{N}^*)$. Let $T_n = \frac{1}{S_1} + \frac{1}{S_2} + \cdots + \frac{1}{S_n}$. Calculate the value of $T_{2018}$. Express your answer as a fraction in simplest form. | \frac{4036}{2019} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Convert the number 116,000,000 into scientific notation. Express your answer in the form $a \times 10^{n}$, where $1 \leq a < 10$ and $n$ is an integer. | 1.16 \times 10^{8} | big_math | [
"Mathematics -> Other -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
A gumball machine that randomly dispenses one gumball at a time contains 13 red, 5 blue, 1 white, and 9 green gumballs. What is the least number of gumballs that Wally must buy to guarantee that he receives 3 gumballs of the same color? Express your answer as a whole number. | 8 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets $M=\\{x\\mid-\\frac{1}{2}<x<\\frac{1}{2}\\}$ and $N=\\{x\\mid x(x-1)\\leqslant 0\\}$, determine the interval representing $M\\cup N$. Express your answer in interval notation, using brackets or parentheses as necessary to denote inclusivity or exclusivity. | (-\\frac{1}{2},1] | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Let S<sub>n</sub> be the sum of the first n terms of the sequence {a<sub>n</sub>}. If S<sub>n</sub> = $$\frac {1}{2}$$a<sub>n</sub> + 1, n ∈ N<sub>+</sub>, calculate the value of a<sub>5</sub>. Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
There are 3 different movie tickets to be distributed among 10 people, with each person receiving at most one ticket. Calculate the total number of different distribution methods. Express your answer as a single integer. | 720 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Permutations"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Given an element $a \in \{0, 1, 2, 3\}$, and $a \notin \{0, 1, 2\}$, find the value of $a$. Express your answer as a single integer. | 3 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
In the addition shown, a digit, either the same or different, can be placed in each of the two boxes. Find the sum of the two missing digits.
$7 \square 8$
$\overline{2182}$
863
$\square 91$
Express your answer as a single integer. | 7 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation 45a + b = 2021, where a and b are positive integers, calculate the minimum value of a + b. Express your answer as a single integer. | 85 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number z = 1 - i, calculate the magnitude of the vector OZ corresponding to the complex number 2/z + z^2. Express your answer as a single value. | \sqrt{2} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Given the regression equation $$\overset{\hat }{y} = 0.85x - 85.7$$, calculate the residual at the sample point (165, 57). Express your answer as a single number. | 2.45 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
What is the smallest number of plastic snap-together cubes that can be snapped together such that only receptacle holes are showing, with all protruding snaps hidden? Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number z that satisfies the equation i * z = 3 - 4i, calculate the magnitude of z, denoted as |z|. Express your answer as a single numerical value. | 5 | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given a geometric sequence $\{a_n\}$ with all positive terms, where $a_2=1$ and $a_3a_7-a_5=56$, find the sum of the first 5 terms, denoted as $S_5$. Express your answer as a fraction or decimal, if necessary. | \frac{31}{2} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{v} = \left(a_{n+1} - \frac{a_n}{2}, \frac{a_{n+1}^2}{2a_n}\right)$ and $\overrightarrow{\mu} = (3, 3)$, and $\overrightarrow{v}$ is parallel to $\overrightarrow{\mu}$, if $a_1 = 5$, find the sum of the first 10 terms of the sequence $\{a_n\}$. Express your answer as a single integer. | 50 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Six straight lines lie in the same plane, no two of which are parallel, and no three of which are concurrent. Determine the number of regions these six lines divide the plane into. Express your answer as a whole number. | 22 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence $\{a_n\}$, it is known that $a_3=5$, and $a_7=-7$. Find the value of $a_{10}$. Express your answer as a single integer. | -16 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Xiaocong and Xiaoming travel on the same day. Xiaocong goes from Shenzhen to Huangshi, arriving at Wuhan Station at 13:30. He then takes an intercity railway to Huangshi after resting for 1 hour at Wuhan Station. Xiaoming takes a taxi from Chongkou Development Zone to Wuhan Station, arriving at any time between 14:00 a... | \frac{1}{2} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Given a sequence ${a_n}$ with a sum of its first $n$ terms denoted as $S_n$, where $a_{n+1}=2S_{n}+1$ for all positive integers $n$, and $S_3=13$, find the smallest value of $n$ such that $S_{n} > a_{5}$. Express your answer as a single integer. | 5 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.484375 | 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; and it is given that $a=1$, $b=2$, and $C=150^\circ$. Calculate the area of $\triangle ABC$. Express your answer as a numerical value. | \frac{1}{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
If the line $ax+2y+1=0$ is perpendicular to the line $x+3y-2=0$, then what is the value of $a$? Express your answer as a single number. | -6 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x) = \begin{cases} \log_{\frac{1}{3}}x, &x > 0 \\ (\frac{1}{3})^x, &x \leq 0 \end{cases}$, find the value of $f(f(5))$. Express your answer as a single value. | 5 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Functions"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $(a-2)^2+\sqrt{b+3}=0$, find the value of $ab$. Express your answer as a single integer. | -6 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
A sequence of numbers is formed by raising 7 to consecutive powers: $7^1$, $7^2$, $7^3$, $7^4$, …, $7^{2009}$. How many terms in this sequence have a last digit of 3? Express your answer as a whole number. | 502 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the tangent line to the function f(x) = x^3 at the point x = 1. Express your answer in the form y = mx + b, where m and b are the slope and y-intercept of the tangent line, respectively. | y = 3x - 2 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Xiao Wang places some equilateral triangle paper pieces on the table. The first time he places 1 piece; the second time he places three more pieces around the first triangle; the third time he places more pieces around the shape formed in the second placement, and so on. The requirement is: each piece placed in each su... | 571 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the lines $x = \\frac{\\pi}{4}$ and $x = \\frac{5\\pi}{4}$ are adjacent axes of symmetry for the graph of the function $f(x) = \\sin(\\omega x + \\phi)$, calculate the value of $\\phi$. Express your answer in radians. | \\frac{\\pi}{4} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = x^2 + 2xf'(-1), find f'(0). Express your answer as a single number. | 4 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
A gold hammer is 5 feet long, with the first foot weighing 4 pounds and the last foot weighing 2 pounds. If the weight of each foot of the gold hammer forms an arithmetic sequence, calculate the total weight of the gold hammer in pounds. | 15 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Find the minimum value of the quadratic expression $5x^2 + 4y^2 - 8xy + 2x + 4$. Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Quadratic Functions"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Find the maximum value of the function y = -x - 9/x + 18 for x > 0. Express your answer as a single number. | 12 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the normal vector of plane α is (1,2,-2) and the normal vector of plane β is (-2,-4,k), and that α and β are parallel planes, find the value of k. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Find the range of values for a such that the function y = x + a*ln(x) has an extreme value point in the interval (1/e, e). Express your answer as an interval. | (-e,-\frac{1}{e}) | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Given the definitions of r! = r \cdot (r-1) \cdots \cdot 1 and C_{j}^{k} = \binom{j}{k} = \frac{j!}{k!(j-k)!}, where r, j, k are positive integers and k < j, find the value of n such that C_{n}^{1}, C_{n}^{2}, and C_{n}^{3} form an arithmetic sequence, where n is a positive integer greater than 3. Express your answer a... | 7 | big_math | [
"Mathematics -> Combinatorics -> Counting Methods -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Jurgen is traveling to Waterloo by bus. He packs for 25 minutes. He then walks to the bus station, which takes 35 minutes. He arrives 60 minutes before his bus leaves. His bus leaves at 6:45 p.m. At what time did he start packing? Express your answer in p.m. time format (e.g., 4:45 p.m.). | 4:45 p.m. | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
If \( x \) and \( y \) are real numbers, find the minimum possible value of the expression \( (x+3)^{2}+2(y-2)^{2}+4(x-7)^{2}+(y+4)^{2} \). Express your answer as a single integer. | 104 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
The sequence $\{a_n\}$ satisfies the recursive relation $a_n = 4a_{n-1} + 3$ and $a_1 = 0$. Calculate the fifth term of this sequence. Express your answer as a single integer. | 255 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Given that the average monthly growth rate of a factory's total output is $p$, calculate the average annual growth rate. Express your answer as a formula in terms of $p$. | (1+p)^{12} - 1 | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Given the set $S=\{(x,y)|1\leqslant x\leqslant 10,1\leqslant y\leqslant 10,x\in \mathbb{N},y\in \mathbb{N}\}$, find the maximum number of elements in a subset $A\subseteq S$ such that for any $(a,b)\in A$ and $(s,t)\in A$, the condition $(a-s)(b-t)\leqslant 0$ holds. Express your answer as a single integer. | 19 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Among the shapes of an equilateral triangle, parallelogram, regular pentagon, and circle, how many are symmetric about an axis but not symmetric about the center? Provide your answer as a whole number. | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
The sum of the areas of the squares on the sides of a right-angled isosceles triangle is 72 cm². Find the area of the triangle. Express your answer in square centimeters (cm²) as a whole number. | 9 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Count the number of two-digit decimals between 0.40 and 0.50. Provide your answer as a whole number. | 9 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x)= \begin{cases} \log_{3}(x+2)+a,x\geqslant 1 \\ e^{x}-1,x < 1 \end{cases}$, if $f[f(\ln 2)]=2a$, calculate the value of $f(a)$. Express your answer as a single numerical value. | 2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
In right triangle $ABC$, $\angle C = 90^\circ$, $AC = 3$, and $BC = 4$. Find the length of $AB$. Express your answer as a whole number. | 5 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Suppose that \(x\) and \(y\) are real numbers that satisfy the two equations:
\[
\begin{gathered}
x^{2} + 3xy + y^{2} = 909 \\
3x^{2} + xy + 3y^{2} = 1287
\end{gathered}
\]
Find a possible value for \(x + y\). Express your answer as a single numerical value. | 27 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Solve the system of equations $\left\{\begin{array}{l}x+y=5\\ x-y=3\end{array}\right.$ and find the values of x and y. Express your answer in the format x = [value], y = [value]. | x = 4, y = 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> System of Equations"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Given a geometric sequence \(\{a_n\}\) where each term is a positive number, the sum of the first two terms is 6, and the sum of the first six terms is 126, calculate the sum of the first four terms. Express your answer as a single integer. | 30 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Given the universal set $U=\{x \in \mathbb{Z} | -5 < x < 5\}$, and the set $S=\{-1, 1, 3\}$, find the number of sets $P$ such that the complement of $P$ in $U$, denoted $\complement_{U}P$, is a subset of $S$. Express your answer as a single integer. | 8 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
For each value of x, f(x) is defined to be the minimum value of the three numbers 2x+2, (1/2)x+1, and (-3/4)x+7. What is the maximum value of f(x)? Express your answer as a single numerical value. | \frac{17}{5} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
To estimate the number of fish in a pond, 40 fish were first caught and marked, then released back into the pond. After the marked fish were completely mixed with the rest of the fish in the pond, 100 fish were caught again, and 5 of them were found to be marked. Estimate the approximate total number of fish in the pon... | 800 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Find the monotonic increasing interval of the function y = -x^2 + 1. Express your answer as an interval using standard interval notation. | (-\infty, 0] | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given the expression \( x^{\ln \ln x} - (\ln x)^{\ln x} \) and the condition \( x > 1 \), evaluate the value of this expression. Express your answer as a single numerical value. | 0 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x)= \begin{cases} 3x^{2}-4 & (x > 0) \\ \pi & (x=0) \\ 0 & (x < 0) \end{cases}$, find the value of $f(f(0))$. Express your answer as an exact value, including any necessary mathematical constants. | 3\pi^{2}-4 | big_math | [
"Mathematics -> Algebra -> Functions"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given the parametric equations of a line:
\[
\left\{
\begin{array}{l}
x = -1 + t\sin{40°} \\
y = 3 + t\cos{40°}
\end{array}
\right.
\]
Calculate the slope of the line in degrees. | 50 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
In a $4\times 4\times 3$ rectangular parallelepiped, vertices $A$, $B$, and $C$ are adjacent to vertex $D$. Calculate the perpendicular distance from $D$ to the plane containing $A$, $B$, and $C$. Express your answer as a decimal value, rounded to one decimal place. | 2.1 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given rectangle ABCD with AD = 5 and CD = 3, draw three circles with centers at A, B, and C and radii of 1, 2, and 3, respectively. Calculate the approximate area of the shaded region. Express your answer as a single number with one decimal place. | 4.0 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given
$$
\begin{cases}
x=2 \\
y=1
\end{cases}
$$
is the solution to the system of equations
$$
\begin{cases}
ax+by=5 \\
bx+ay=1
\end{cases}
$$
find the value of $3-a-b$. Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence ${a_{n}}$ satisfies $a_{1}=2$, $a_{n+1}=\frac{1+{a}_{n}}{1-{a}_{n}}$ $(n∈N^{∗})$, find the product of the first 2018 terms of the sequence, i.e., $a_{1}·a_{2}·a_{3}·…·a_{2018}$. | -6 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given that \(a, b, c,\) and \(d\) are real numbers, and assuming that all the roots of the equation \(z^4 + az^3 + bz^2 + cz + d=0\) lie on the circle in the complex plane with center \(0 + 0i\) and radius 1, find the sum of the reciprocals of these roots. Express your answer in the form \(\boxed{[answer]}\). | -a | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Convert 28nm to scientific notation, given that 1nm is equal to $10^{-9}$ meters. Express your answer in scientific notation, including units (meters). | 2.8 \times 10^{-8}\text{m} | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of the expression \(2016 - 2017 + 2018 - 2019 + 2020\). Express your answer as a single integer. | 2018 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Find the range of values for the real number $a$ such that the inequality $|x-a|+|x-1|\leqslant 3$ holds true for some real number $x$. Express your answer as an interval. | [-2,4] | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
A tangent line is drawn from a point on the circle $x^2+y^2=1$ to intersect the positive x-axis and y-axis at points A and B, respectively. Calculate the minimum product of $|\overrightarrow{OA}|$ and $|\overrightarrow{OB}|$. Express your answer as a single numerical value. | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
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