instruction_seed stringlengths 40 1.39k | answer stringlengths 1 95 | source stringclasses 1
value | domain listlengths 1 3 | llama8b_solve_rate float64 0 0.98 | _source stringclasses 1
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|---|---|---|---|---|---|
Two bags contain marbles of different colors. The first bag contains 2 red marbles and 2 blue marbles. The second bag contains 2 red marbles, 2 blue marbles, and g green marbles, where g > 0. Calculate the value of g such that the probabilities of randomly drawing two marbles of the same color from each bag, without re... | 5 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the points (3, 1) and (-4, 6) are on opposite sides of the line $3x-2y+a=0$, find the range of values for $a$. Express your answer as an interval, including the endpoints if they are included. | (-7, 24) | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\tan (\alpha - \frac{\pi }{4}) = \frac{1}{2}$, calculate the value of $\frac{\sin \alpha + \cos \alpha }{\sin \alpha - \cos \alpha }$. | 2 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the domain of the function $f(x)$ is $(-5,5)$, its derivative is $f'(x)$, and $f(x) + xf'(x) > 2$, find the solution set of the inequality $(2x-3)f(2x-3) - (x-1)f(x-1) > 2x-4$. Express your answer as an interval in standard interval notation. | (2,4) | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
When a plane figure is projected obliquely, which type of quadrilateral retains its defining property of having opposite sides that are parallel?
Express your answer as the name of the shape (e.g., square, rectangle, rhombus, parallelogram). | parallelogram | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Given $a= \sqrt {3}+ \sqrt {2},b= \sqrt {3}- \sqrt {2}$, find the arithmetic mean of $a$ and $b$. Express your answer as a simplified radical expression. | \sqrt{3} | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Express 876,000 in scientific notation in the form $a\times10^n$, where $1\leq|a|<10$ and $n$ is an integer. Provide your answer in the format $\boxed{a\times10^n}$. | 8.76\times10^5 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Convert the decimal number 38 into binary form. Express your answer in binary with a subscript 2 to denote the base, e.g., 100110_{(2)}. | 100110_{(2)} | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Given 1987 sets, each containing exactly 45 elements, and the union of any two sets contains exactly 89 elements, calculate the number of elements in the union of these 1987 sets. Express your answer as a single integer. | 87429 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Find the angle between 0° and 360° that has the same terminal side as the -415° angle. Express your answer in degrees, as a whole number. | 305 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions -> Other"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\sqrt[3]{2.37}\approx 1.333$ and $\sqrt[3]{23.7}\approx 2.872$, use the properties of cube roots to find the approximate value of $\sqrt[3]{2370}$. Express your answer as a decimal number rounded to two decimal places. | 13.33 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Given the piecewise function $f(x)= \begin{cases} x+3 & (x > 10) \\ f(f(x+5)) & (x\leqslant 10) \end{cases}$, find the value of $f(5)$. Express your answer as a single number. | 24 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given the plane vectors $\overrightarrow {a} = (1,2)$ and $\overrightarrow {b} = (-3,x)$, if $\overrightarrow {a} \parallel \overrightarrow {b}$, find the value of $x$. Express your answer as a single integer. | -6 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
The total cost of 4 one-litre cartons of milk is $4.88. Find the cost of 1 one-litre carton of milk. Express your answer in dollars and cents. | 1.22 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Find the smallest positive value of m such that when the graph of the function y = sin^2(x - π/4) is translated rightward along the x-axis by m units, the resulting graph is symmetric about the y-axis. | \frac{\pi}{4} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Given the polynomials 29 * 42 y^2 + my + 2 and y^2 + my + 2, find the value of m such that the remainders when these polynomials are divided by y-1 and y+1 are equal. Express your answer as a single numerical value. | 0 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Find the condition on $x$ that makes the fraction $\frac{1}{{x-4}}$ meaningful. Express your answer as an inequality or equation using mathematical notation. | x \neq 4 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(1, -3, 2)$ and $\overrightarrow{b}=(-2, 1, 1)$, find the magnitude of $2\overrightarrow{a}+\overrightarrow{b}$. Express your answer in simplest radical form if necessary. | 5\sqrt{2} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given two vectors $\overrightarrow{a}=(-2,k)$ and $\overrightarrow{b}=(2,4)$ in a plane, and it's given that $\overrightarrow{a}\perp \overrightarrow{b}$, calculate the magnitude of the difference between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$, i.e., $|\overrightarrow{a}-\overrightarrow{b}|$. Express you... | 5 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Let $i$ be the imaginary unit. If $(1-i)z=2i$, find the modulus of the complex number $z$ ($|z|=$). Express your answer as a single value. | \sqrt{2} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
A house and store were sold for $12,000 each. The house was sold at a loss of 20% of the cost, and the store at a gain of 20% of the cost. Calculate the overall gain or loss from the entire transaction. Express your answer in dollars, and specify if it's a gain or loss. | \$1000 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence $\sqrt{2}, \sqrt{5}, 2\sqrt{2}, \sqrt{11}, \ldots$, with the general formula $a_n = \sqrt{3n-1}$, find the position of the term $2\sqrt{5}$ in the sequence. Express your answer as a whole number. | 7 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Ben lives in a large house with his father, mother, sister, and brother, as well as 2 dogs, 3 cats, 4 parrots, and 5 goldfish. Calculate the total number of legs in the house. | 38 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Five girls ran a race. Fiona started first, followed by Gertrude, then Hannah, then India, and lastly Janice. Whenever a girl overtook another girl, she was awarded a point. Given that India finished first, followed by Gertrude, Fiona, Janice, and lastly Hannah, calculate the minimum total number of points that could h... | 5 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Define the function $g(x)=e^{x}+e^{-x}+|x|$ on $\mathbb{R}$. Find the range of $x$ that satisfies the inequality $g(2x-1) < g(3)$. Express your answer as an interval or a combination of intervals. | (-1,2) | big_math | [
"Mathematics -> Calculus -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = x^3 - ax, where a > 0, and the interval [1, +∞), find the maximum value of a such that the function has no extreme values on the interval. Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\alpha$ is an angle in the second quadrant, simplify the expression $$\frac{2\sin{\alpha}}{\sqrt{1 - \cos^{2}{\alpha}}} + \frac{\sqrt{1 - \sin^{2}{\alpha}}}{\cos{\alpha}}$$ and find its value. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Four girls — Mary, Alina, Tina, and Hanna — sang songs in a concert as trios, with one girl sitting out each time. Hanna sang 7 songs, which was more than any other girl, and Mary sang 4 songs, which was fewer than any other girl. Calculate the total number of songs sung by these trios. | 7 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given that point P(a,b) is in the fourth quadrant, determine the quadrant in which point Q(-a,b-1) lies. Express your answer as the name of the quadrant. | Third quadrant | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the exact value of the expression:
\[
\frac{19}{\left(2^{3}-1\right) \times\left(3^{3}-1\right)}+\frac{37}{\left(3^{3}-1\right) \times\left(4^{3}-1\right)}+\frac{61}{\left(4^{3}-1\right) \times\left(5^{3}-1\right)}+\frac{91}{\left(5^{3}-1\right) \times\left(6^{3}-1\right)}
\]
Express your answer as a simplifi... | \frac{208}{1505} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Consider the sets $P = \{-1, 0, 1\}$ and $Q = \{y | y = \cos(x), x \in \mathbb{R}\}$. Determine the intersection $P \cap Q$. Express your answer as a set. | \{-1, 0, 1\} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Given the quadratic equation \( 3y^2 - 4(x+1) \cdot y + 12(x-2) = 0 \), what type of conic section does it represent? Express your answer as the name of the conic section (e.g., ellipse, hyperbola, circle, etc.). | Hyperbola | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given a complex number $Z$ satisfying $(2+i)Z=3-i$, find the magnitude of $Z$. Express your answer as a numerical value, including the square root symbol if necessary. | \sqrt{2} | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets $M = \{x | x^2 - 2x < 0\}$ and $N = \{x | x > 1\}$, determine the set $M \cap \lnot_R N$. Express your answer in interval notation, using parentheses or brackets as necessary. | (0, 1] | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence $\{a_n\}$ defined by $a_1=1$ and $a_{n+1}=a_n^2-1$ ($n\ge1$), calculate the sum of the first 5 terms $a_1+a_2+a_3+a_4+a_5$. Express your answer as a single integer. | -1 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
In the sequence $\{a_n\}$, $a_1=1$, $a_{n+1}=a_n^2-1$ ($n\ge1$). Calculate the sum of the first 5 terms: $a_1+a_2+a_3+a_4+a_5$. Express your answer as a single integer. | -1 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
You and five friends need to raise $1500$ dollars in donations for a charity, dividing the fundraising equally. How many dollars will each of you need to raise? Express your answer as a whole number in dollars. | 250 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
A forest farm has 30,000 saplings, among which 4,000 are pine saplings. To investigate the growth condition of the saplings, a stratified sampling method is used to draw a sample of 150 saplings. Calculate the number of pine saplings in the sample. | 20 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Two squares, each with side length 5 cm, overlap as shown. The shape of their overlap is a square, which has an area of 4 cm². Find the perimeter, in centimeters, of the shaded figure. Express your answer as a whole number. | 32 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Find the solution set of the inequality (3-x)(1+x) > 0. Express your answer in interval notation, including all necessary details for the solution set. | (-1, 3) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
If positive numbers x and y satisfy the equation 3/x + 1/y = 5, find the minimum value of 3x + 4y. Express your answer as a single number. | 5 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given a set of data $x_{1}$, $x_{2}$, $\ldots$, $x_{n}$ with a variance of $2$, what is the variance of the dataset after applying the transformation $3x_{i} + 4$ to each data point? Provide your answer as a single number. | 18 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Given $\lim_{Δx→0}\frac{{f(4+Δx)-f(4-Δx)}}{{Δx}}=-10$, find the derivative f'(4). Express your answer as a single number. | -5 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
The equation of a circle is given by $(x+1)^2+(y-1)^2=2$. Find the equation of the line that passes through the center of this circle and is perpendicular to the line $x+y=0$. Express your answer in the form $Ax + By + C = 0$. | x-y+2=0 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given that \( b > 1 \), \( \sin x > 0 \), \( \cos x > 0 \), and \( \log_b (\sin x) = a \), find the expression for \( \log_b (\cos x) \) in terms of \( a \) and \( b \). Express your answer as a logarithmic expression using the same base \( b \) as in the original equation. | \frac{1}{2} \log_b (1 - b^{2a}) | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Logarithmic Functions"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Let the sequence $(a_n)$ be defined as $a_1 = \\frac{1}{3}$ and $a_{n+1} = \\frac{a_n}{\\sqrt{1 + 13a_n^2}}$ for every $n \\geq 1$. Find the largest value of $k$ such that $a_k < \\frac{1}{50}$. Express your answer as a single integer value for $k$. | 193 | big_math | [
"Mathematics -> Sequences and Series -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Find the range of the function $y=2^{-x^{2}+2x}$. Express your answer as an interval, specifying the endpoints and whether they are included or excluded. | (0,2] | big_math | [
"Mathematics -> Precalculus -> Functions"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
In the polar coordinate system, given two points A(2, π/6) and B(3, π/6), calculate the length of the line segment AB. | 1 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the right focus of the hyperbola x^2/m - y^2/5 = 1 coincides with the focus of the parabola y^2 = 12x, determine the equation of the asymptotes of this hyperbola. Express your answer in the form y = ±kx, where k is a constant. | y = \pm \frac{\sqrt{5}}{2}x | big_math | [
"Mathematics -> Geometry -> Non-Euclidean Geometry -> Other"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets $M=\{x|0 < x < 3\}$ and $N=\{x|x > 2\}$, find the intersection of $M$ and the complement of $N$, denoted as $M\cap(\complement_{\mathbb{R}}N)$. Express your answer as an interval using interval notation (e.g., (a, b], [a, b), etc.). | (0,2] | big_math | [
"Mathematics -> Set Theory -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = 3x + 1 + 9/(3x - 2) and the constraint x < 2/3, calculate the maximum or minimum value of f(x). Be sure to specify whether the value you find is a maximum or minimum. | -3 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Find the value range of the function $y= \frac{x^{2}-1}{x^{2}+1}$. Express your answer as an interval. | [-1,1) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
A cone has a base radius of 4cm and a slant height of 5cm. Calculate the lateral surface area of the cone. Express your answer in square centimeters (cm²), including π if necessary. | 20\pi | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Given complex numbers $z_1 = 3+4i$ and $z_2 = 3-4i$, calculate the difference $z_1 - z_2$. Express your answer as a complex number in the form $a+bi$ and simplify to its simplest form. | 8i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
The set \( S \) consists of 9 distinct positive integers. The average of the two smallest integers in \( S \) is 5, and the average of the two largest integers in \( S \) is 22. Calculate the maximum possible average of all integers in \( S \). Express your answer as a single number. | 16 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Find the negation of the proposition ∀x ∈ ℝ, x² - x ≥ 0. Express your answer as a logical statement using proper notation for quantifiers (e.g., ∀, ∃) and relations (e.g., ≥, <). | ∃x ∈ ℝ, x² - x < 0 | big_math | [
"Mathematics -> Discrete Mathematics -> Logic -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Use the completing the square method to solve the quadratic equation $x^{2}-6x+8=0$. What is the resulting equation in the form of a squared binomial? | (x-3)^{2} = 1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Quadratic Functions"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\tan \alpha > 0$ and $\sin \alpha + \cos \alpha > 0$, determine the quadrant in which the terminal side of $\alpha$ lies. Express your answer as the name of the quadrant (e.g., first, second, third, fourth). | \text{first} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given x = 2, y = x^2 - 5, and z = y^2 - 5, calculate the value of z. Express your answer as a single integer. | -4 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the terminal side of angle $\alpha$ passes through the point $P\left( \frac {1}{2}, \frac { \sqrt {3}}{2}\right)$, find the value of $\sin\alpha$. Express your answer as a fraction or decimal. | \frac{\sqrt{3}}{2} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
In triangle \( \triangle ABC \), \( \angle A = 80^\circ \). Points \( D, E, F \) are on sides \( BC, AB \), and \( AC \) respectively, with \( BD = BE \) and \( CD = CF \). Find the measure of \( \angle EDF \) in degrees. | 50^\circ | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $(5-\sqrt{y^2-25})^2$ using algebraic methods. | y^2 - 10\sqrt{y^2-25} | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Determine the fixed point (x, y) through which the line mx + y - m - 1 = 0 (where m is a parameter and m ∈ ℝ) always passes. Express your answer as an ordered pair (x, y). | (1, 1) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of $\left(-3x\right)^{2}\cdot 2x$. Express your answer as a simplified algebraic expression. | 18x^{3} | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
The lines $x+ky=0$, $2x+3y+8=0$, and $x-y-1=0$ intersect at a single point. Find the value of $k$. Express your answer as a numerical value. | -\dfrac{1}{2} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
In triangle $\triangle ABC$, $\overrightarrow{AB}=\overrightarrow{c}$, $\overrightarrow{AC}=\overrightarrow{b}$. If point $M$ satisfies $\overrightarrow{MC}=2\overrightarrow{BM}$, express $\overrightarrow{AM}$ in terms of vectors $\overrightarrow{b}$ and $\overrightarrow{c}$. Provide your answer in the format $\overrig... | \frac{1}{3}\overrightarrow{b} + \frac{2}{3}\overrightarrow{c} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets M and N defined as:
\[ M = \left\{ z \left\lvert\, z = \frac{t}{1+t} + \mathrm{i} \frac{1+t}{t} \right., t \in \mathbf{R}, t \neq -1, t \neq 0 \right\} \]
\[ N = \left\{ z \mid z = \sqrt{2} \cdot [\cos (\arcsin t) + \mathrm{i} \cos (\arccos t)], t \in \mathbf{R}\, \lvert\, t \leqslant 1 \right\} \]
Find ... | 0 | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
We are preparing to distribute 7 identical fitness equipment to 5 different communities, where communities A and B each must receive at least 2 pieces of equipment, and the other communities may receive none. Calculate the total number of different distribution schemes. Express your answer as a single integer. | 35 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
What is the range of values that the probability of any event A must satisfy? Express your answer as a mathematical inequality. | 0 \leq P(A) \leq 1 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Other"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
The positive integers are arranged in rows and columns as shown below.
| Row 1 | 1 |
| Row 2 | 2 | 3 |
| Row 3 | 4 | 5 | 6 |
| Row 4 | 7 | 8 | 9 | 10 |
| Row 5 | 11 | 12 | 13 | 14 | 15 |
| Row 6 | 16 | 17 | 18 | 19 | 20 | 21 |
|... |
More rows continue to list the positive integers in order, with each new row contain... | 16 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given that $|\vec{a}|=1$, $\vec{b}=(\sqrt{3},1)$, and the angle between $\vec{a}$ and $\vec{b}$ is $90^\circ$, find the magnitude of the vector $2\vec{a}+\vec{b}$. Express your answer as a simplified radical expression. | 2\sqrt{2} | big_math | [
"Mathematics -> Linear Algebra -> Vectors"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\overrightarrow{a}=(1,2)$ and $\overrightarrow{b}=(2x,-3)$ are parallel vectors, find the value of $x$. Express your answer as a fraction or decimal. | -\frac{3}{4} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Given the line $l:x-y+a=0$, points $A(-2,0)$, $B(2,0)$. If there exists a point $P$ on line $l$ such that $AP \perp BP$, find the range of values for the real number $a$. Express your answer as an interval. | \left[-2 \sqrt{2}, 2 \sqrt{2}\right] | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given the hyperbola $x^{2}- \frac {y^{2}}{3}=1$, find the angle between its two asymptotes. Express your answer in degrees. | 60^{\circ} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a$, $b$, $c$ are positive real numbers and satisfy $abc = 1$, find the minimum value of the algebraic expression $(a+1)(b+1)(c+1)$. Provide your answer as a single number. | 8 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Given two vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ that satisfy $|\overrightarrow{a}|=2$, and the projection of $\overrightarrow{b}$ onto $\overrightarrow{a}$ is $1$. If there exists a real number $\lambda$ such that $\overrightarrow{a}$ is perpendicular to $\overrightarrow{a}-\lambda\overrightarrow{b}$, f... | 2 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets $M=\{1,2\}$ and $N=\{b\,|\,b=2a-1, a\in M\}$, find the union $M\cup N$. Express your answer as a set with elements enclosed in curly brackets and separated by commas. | \{1,2,3\} | big_math | [
"Mathematics -> Algebra -> Set Theory -> Other"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given a geometric sequence {$a_n$} where the sum of the first n terms is $S_n = (x^2 + 3x)2^n - x + 1$, find the value of $a_3$. Express your answer as a single integer. | -8 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given that \( x, y, z \) are real numbers and \( x^2 + y^2 + z^2 = x + y + z \), calculate the sum of the maximum and minimum values of \( x + y + z \). Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
4 students are competing for championships in three events. Each championship can have one of the 4 students as the winner. Calculate the total number of possible outcomes for the champions in all three events. | 64 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
What is the value of \( 2014 - 4102 \)? Express your answer as an integer. | -2088 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Given $a > b > c$, determine the largest positive integer $k$ such that the inequality $$\frac{1}{a-b} + \frac{1}{b-c} \geq \frac{k}{a-c}$$ always holds. Provide your answer as a single integer value. | 4 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given the plane vectors $\mathbf{a} = (1, -3)$, $\mathbf{b} = (4, -2)$, if $\lambda\mathbf{a} + \mathbf{b}$ is perpendicular to $\mathbf{a}$, then find the value of $\lambda$. Express your answer as a single numerical value. | -1 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Given three numbers $a=\log_{0.3}6$, $b=0.3^{6}$, $c=6^{0.3}$, determine the correct ordering of these numbers from smallest to largest. Express your answer using the variables a, b, and c (e.g., a < b < c). | a < b < c | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
Let $f(x)$ be differentiable at $x=x_0$, and $$\lim_{\Delta x \to 0} \frac{f(x_0+3\Delta x)-f(x_0)}{\Delta x}=1.$$ Calculate $f'(x_0)$ using the given limit. Express your answer in the form $\boxed{[answer]}$. | \frac{1}{3} | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Let $A=\{x\in \mathbb{N} | -2 \lt x\leqslant 1\}$, $B=\{0,1,2\}$. Find the intersection $A \cap B$. Express your answer as a set of elements using proper set notation. | \{0, 1\} | big_math | [
"Mathematics -> Discrete Mathematics -> Other"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Given an odd function $f(x)$ with domain $\mathbf{R}$, where $f(x+2)$ is an even function, and $f(1) = a$, find the sum $f(1) + f(3) + f(5) + \cdots + f(2019)$. Express your answer in terms of $a$. | 2a | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation of the ellipse $x^{2}+2y^{2}=2$, calculate the length of its major axis. Express your answer as a numerical value. | 2\sqrt{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Let $\vec{a} = (1,2)$, $\vec{b} = (2,k)$. If $(2\vec{a} + \vec{b}) \perp \vec{a}$, find the value of the real number $k$. Express your answer as a single real number. | -6 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Put the numbers 0.28, -0.2, 2.8, -8.2 in increasing order. Express your answer as a list of numbers, including all numbers in the correct order. | -8.2, -0.2, 0.28, 2.8 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Decimals"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence {an} with a non-zero common difference d, where a1 = 18, and a1, a4, a8 form a geometric sequence, find the value of the common difference d. | d = 2 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Given the integer division formula \(a \div b = c \cdots r\), where \(a = b \cdot c + r\) with \(0 \leq r < b\), find the new remainder when both \(a\) and \(b\) are multiplied by 3. Express your answer in terms of the original remainder \(r\). | 3r | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Solve the system of equations:
2x + y = 6
x + 2y = 3
Find the value of x - y. Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Systems of Equations"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z=\frac{1+i}{1+ai}$, and knowing that $z$ is a pure imaginary number, find the value of the real number $a$. Express your answer as a single real number. | -1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a} = (-1, 2)$ and $\overrightarrow{b} = (m, 1)$, if the vector $\overrightarrow{a} + 2\overrightarrow{b}$ is parallel to $2\overrightarrow{a} - \overrightarrow{b}$, then what is the value of the dot product $\overrightarrow{a} \cdot \overrightarrow{b}$? Express your answer as a single num... | \frac{5}{2} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, two circles overlap. The area of the overlapped region is \(\frac{3}{5}\) of the area of the small circle and \(\frac{6}{25}\) of the area of the large circle. Calculate the ratio of the area of the small circle to the area of the large circle. Express your answer as a ratio in the form a:b, where a and... | 2:5 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
The sequence {a_n} is defined by the general term a_n = sin(n/3 * pi). Find the value of the sum a_1 + a_2 +... + a_2007 + a_2008. | \frac{\sqrt{3}}{2} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a > 1$, $b > 2$, and $x+b=5$ (which implies $a+b=5$ since $x=a$), find the minimum value of $\frac{1}{a-1} + \frac{9}{b-2}$. Express your answer as a single number. | 8 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
In triangle ABC (as shown in the figure), $\angle ABC = 60^{\circ}$, $\angle ACB = 45^{\circ}$. Also, $AD$ and $CF$ are the heights on the sides of the triangle, intersecting at $P$. The bisector of $\angle ABC$, $BE$, intersects $AD$ and $CF$ at $Q$ and $S$, respectively. Count the number of isosceles triangles in the... | 5 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
A and B independently solve the same math problem. The probability of A solving the problem is 0.8, and the probability of B solving the problem is 0.6. Find the probability that at least one of them solves the problem. Express your answer as a decimal value between 0 and 1, rounded to 2 decimal places. | 0.92 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
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