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 |
|---|---|---|---|---|---|
Calculate the result of $\sqrt{8}+|-2| imes \cos 45^{\circ}$. Express your answer in simplest radical form if necessary. | 3\sqrt{2} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
A line passing through the focus of the parabola $y^2 = 2px$ ($p > 0$) with a slope of 1 intersects the parabola at points $A$ and $B$. If $|AB| = 4$, find the equation of the parabola in the form $y^2 = 2px$. Express your answer as the equation of the parabola with the value of $p$. | y^2 = 2x | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function y = x^4 - 8x^2 + c, find the value of c such that the function has a minimum value of -14 on the interval [-1, 3]. Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
A row of 7 seats is available for two people, A and B, to take seats. It is required that there is at least one empty seat between A and B. Calculate the number of different seating arrangements that meet this condition. Express your answer as a whole number. | 30 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given the conditions:
1. $|\overrightarrow{a}|=1$
2. $(\overrightarrow{a}+\overrightarrow{b})\bot \overrightarrow{a}$
3. $(2\overrightarrow{a}+\overrightarrow{b})\bot \overrightarrow{b}$
Find the magnitude of vector $\overrightarrow{b}$. Express your answer as a numerical value. | \sqrt{2} | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Geometry -> Other"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
If 50% of \( P \) equals 20% of \( Q \), then find \( P \) as a percent of \( Q \). Express your answer as a percentage, including the percentage sign (%). | 40% | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Find the sum of the arithmetic series given by $(-300) + (-297) + (-294) + \ldots + 306 + 309$. Express your answer as a single numerical value. | 918 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
When tossing a uniform coin four times, what is the probability of getting at least two heads? Express your answer as a fraction in simplest form. | \frac{11}{16} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = x / sqrt(1 - x), calculate the value of f(-3). Express your answer as a single value. | -3/2 | big_math | [
"Mathematics -> Precalculus -> Functions"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = x^2 - 4ax + a^2, where a > 0, find the minimum value of the expression x_1 + x_2 + a/(x_1*x_2), where x_1 and x_2 are the zeros of the function. | 4 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Given the plane point sets
\[
\begin{array}{l}
A=\left\{(x, y)|(y-x) y-\frac{1}{x} \geqslant 0\right\}, \\
B=\left\{(x, y) \mid (x-1)^{2}+(y-1)^{2} \leqslant 1\right\},
\end{array}
\]
calculate the area of the plane figure represented by $A \cap B$. Express your answer as a single value. | \frac{\pi}{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
A square is inscribed in a unit circle, then a circle is inscribed in this square, followed by inscribing another square in this circle. This process is repeated infinitely. Let \( C \) represent the total area of all circles and \( S \) represent the total area of all squares. Calculate the ratio \( \frac{C}{S} \). Ex... | \frac{\pi}{2} | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
In the given $\triangle ABC$, $AB=4$ and $AC=8$. If $M$ is the midpoint of $BC$ and $AM=3$, what is the length of $BC$? Express your answer as a numerical value, including the square root symbol if necessary. | 2 \sqrt{31} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
The product of three positive integers is 42, and the sum of two of these integers is 9. Find the third integer. Express your answer as a single positive integer. | 3 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Suppose there is a special key on a calculator that replaces the number x currently displayed with the number given by the formula 1/(1-x). If the calculator is initially displaying 5, what is the result after pressing the special key 100 times? | -\frac{1}{4} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
The line \((1-2a)x-2y+3=0\) is perpendicular to the line \(3x+y+2a=0\). Determine the value of the real number \(a\). Express your answer as a single numerical value. | \dfrac{1}{6} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Let $M=\{x \mid x^2+x-6 < 0\}$ and $N=\{x \mid 1 \leq x \leq 3\}$. Calculate the intersection $M \cap N$ and express your answer in interval notation, including any relevant brackets or parentheses to denote inclusion or exclusion of endpoints. | [1, 2) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
What is the largest possible prime value of the expression \( n^2 - 12n + 27 \), where \( n \) ranges over all positive integers? Express your answer as a single integer. | 7 | big_math | [
"Mathematics -> Number Theory -> Prime Numbers"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation x^2 + ax + 1 = 0, and knowing that x = 1 is a solution, find the value of a. Express your answer as a single integer. | -2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Quadratic Functions"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
What is the value of \(2021 - 2223 + 2425\)? Express your answer as a single integer. | 2223 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Find the smallest possible integer side length of a square that can be divided into 10 smaller squares, all of which have integer side lengths, and at least 8 of which have an area of 1. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Find the x-intercept of the line y = 2x + 1. Express your answer as a single fraction or decimal. | -\dfrac{1}{2} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Linear Equations"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets $M=\{x|x^{2}-5x\leqslant 0\}$ and $N=\{x|p < x < 6\}$, and $M\cap N=\{x|2 < x\leqslant q\}$, calculate the sum of parameters p and q. | 7 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Given $a=-\dfrac{1}{4}$, $b=-2$, and $c=-2\dfrac{3}{4}$, calculate the value of $|a|+|b|-|c|$. Express your answer as a simplified fraction. | -\dfrac{1}{2} | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number expression $a+bi= \\frac {5}{1+2i}$, calculate the product $ab$. Express your answer as a single integer. | -2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=({m+1,m-1})$, $\overrightarrow{b}=({-1,m})$, $\overrightarrow{c}=({-1,1})$, find the value of $m$ such that $({2\overrightarrow{a}+\overrightarrow{b}})⊥\overrightarrow{c}$. Express your answer as a single numerical value. | 3 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the sum of the coefficients of all terms in the expansion of ${(3{x}^{2}+\frac{1}{x})}^{n}$ is $256$, find the value of $n$. Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
In the Cartesian coordinate system, the coordinates of point P are (3,-2). Find the distance from point P to the x-axis. Express your answer as a whole number. | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number z satisfies |z + i| + |z - i| = 2, find the minimum value of |z + i + 1|. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Find the minimum value of the function f(x) = (5 - 4x + x^2) / (2 - x) on the interval (-∞, 2). Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Determine the number of positive integer triplets (a, b, c) that satisfy the system of equations:
\[
\begin{cases}
ab + bc = 44 \\
ac + bc = 23
\end{cases}
\]
Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
In triangle $\triangle ABC$, given $a=4\sqrt{3}$, $c=12$, and $C=\frac{\pi}{3}$, calculate the measure of angle $A$. Express your answer in radians as a fraction of $\pi$. | \frac{\pi}{6} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
In the Cartesian coordinate system, the coordinates of a point are initially (-1, 3). If the point is moved 5 units to the right, what are its new coordinates? Express your answer as a coordinate pair (x, y). | (4, 3) | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Find the polar coordinates of the center of the circle given by the polar equation $\rho=\cos \left(\theta+ \frac{\pi}{3}\right)$. Express your answer as an ordered pair of polar coordinates. | \left( \frac{1}{2},- \frac{\pi}{3}\right) | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = sin(ωx + π/6) (ω > 0), find the range of values for ω such that the function is monotonic on the interval (0, π/3). Express your answer as an interval. | (0, 1] | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(1,3)$ and $\overrightarrow{b}=(m,-2)$, and the condition that $(\overrightarrow{a}+\overrightarrow{b})$ is perpendicular to $\overrightarrow{a}$, find the value of $m$. Provide your answer as a single integer. | -4 | big_math | [
"Mathematics -> Linear Algebra -> Vectors"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Under the given mapping $f: (x, y) \rightarrow (xy, x+y)$, calculate the image of $(4, -2)$. Express your answer as an ordered pair. | (-8, 2) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = x^3 - 3x + m, find the value of the real number m such that the maximum and minimum values of f(x) in the interval [-3, 0] sum up to -14. Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors a = (1, -cosθ) and b = (1, 2cosθ), and a ⊥ b, find the value of cos2θ. | 0 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Trigonometric Functions"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Given line l1: x*sin(alpha) + y - 1 = 0, and line l2: x - 3y*cos(alpha) + 1 = 0. If l1 is perpendicular to l2, then find the value of sin(2*alpha). Express your answer as a numerical value. | \frac{3}{5} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Let the polynomial x^3 - ax^2 + bx - a (where a, b are real numbers) have all its roots as real numbers. If a is the smallest positive real number satisfying this condition, and for this real number a, the value of b is uniquely determined, then what is the value of b? Express your answer as a single integer. | 9 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
In the Cartesian coordinate system $xOy$, it is known that quadrilateral $ABCD$ is a parallelogram, $\overrightarrow{AB}=(1,-2)$, $\overrightarrow{AD}=(2,1)$. Calculate the value of $\overrightarrow{AD} \cdot \overrightarrow{AC}$. Express your answer as a single integer. | 5 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
A cylinder is formed by rotating a rectangle with a length of 4 and a width of 2 around one of its sides. Find the lateral surface area of this cylinder. Express your answer in terms of pi. | 16\pi | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Surface Area"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
A rectangle has a width of \( w \), a length of 8, and a perimeter of 24. Find the ratio of its width to its length. Express your answer as a simplified ratio, using a colon (:) to separate the values. | 1:2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the function $f(x)$ is differentiable, and $f'(1)=1$, find the value of $$\lim_{\Delta x \to 0} \frac{f(1-\Delta x) - f(1)}{-\Delta x}.$$ Express your answer as a single numerical value. | 1 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
The radioactive isotope Holmium-166 (^166Ho) has 67 protons in its nucleus. Given that the mass number of this isotope is 166, calculate the difference between the number of neutrons and the number of protons in the nucleus of ^166Ho. Express your answer as a whole number. | 32 | big_math | [
"Mathematics -> Other -> Other"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
On a 5x5 chessboard, place 3 black pieces and 2 white pieces such that no two pieces are in the same row or column. Calculate the total number of different arrangements. | 1200 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets $M=\left\{ (x,y) \mid y=3x^{2} \right\}$ and $N=\left\{ (x,y) \mid y=5x \right\}$, determine the number of elements in the intersection of $M$ and $N$. Express your answer as a whole number. | 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Max sold glasses of lemonade for 25 cents each. He sold 41 glasses on Saturday and 53 glasses on Sunday. What were his total sales for these two days? Express your answer in dollars and cents, rounding to two decimal places if necessary. | 23.50 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
The sequence $\{a_n\}$ satisfies: $$a_n=\begin{cases} (4-a)n-10,(n\leqslant 7) \\ a^{n-6},(n > 7) \end{cases}$$ and $\{a_n\}$ is an increasing sequence. Determine the range of the real number $a$ that satisfies these conditions. Express your answer as an interval or range of values for $a$. | (2, 4) | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
The distance between the centers of two circles with polar equations $\rho = 4\cos \theta$ and $\rho = 4\sin \theta$ is to be calculated. Find the distance between the centers. Express your answer as a numerical value. | 2\sqrt{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
A piece of rope is randomly cut into two parts. Find the probability that the longer segment is at least x times the shorter segment. Express your answer as a fraction or ratio. | \frac{2}{x+1} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given points A(-3,0), B(0,2), and O as the origin, point C is inside ∠AOB with |OC|=2√2 and ∠AOC=π/4. If OC=λOA+OB (λ∈R), calculate the value of λ. Express your answer as a single numerical value. | \frac{2}{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other",
"Mathematics -> Algebra -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Suppose hops, skips and jumps are specific units of length. If $b$ hops equals $c$ skips, $d$ jumps equals $e$ hops, and $f$ jumps equals $g$ meters, then calculate the number of skips equivalent to one meter. Express your answer as a fraction. | \frac{cef}{bdg} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence {a_n}, a_3 and a_9 are the two real roots of the quadratic equation x^2 - 16x + c = 0 (c < 64). Calculate the sum of the first 11 terms of this sequence, S_11. Express your answer as a single integer. | 88 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the algebraic expression $\sqrt{5-2 \sqrt{6}}+\sqrt{7-4 \sqrt{3}}$. Express your answer as a simplified algebraic expression. | 2 - \sqrt{2} | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given that $14.97\times 46=688.62$, calculate the result of $1.497\times 4.6$ after scaling down both numbers by a factor of 10. Express your answer as a decimal number with all decimal places. | 6.8862 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Two poles, one 20 inches high and the other 80 inches high, are 100 inches apart. Find the height of the intersection of the lines joining the top of each pole to the foot of the opposite pole. Express your answer in inches as a whole number. | 16 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
A shopping mall noticed that its sales in February increased by 10% compared to January, and then decreased by 10% in March compared to February. Calculate the percentage change in sales in March compared to January. Express your answer as a percentage and include the direction of change (increase or decrease). | -1\% | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $$\frac {3+4i}{i} = \frac {z}{1+i},$$ calculate the value of z. Express your answer as a complex number in the form a+bi. | 7+i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ such that $\overrightarrow{a} \perp \overrightarrow{b}$, $|\overrightarrow{a}| = 2$, and $|\overrightarrow{b}| = 1$, calculate the magnitude of $\overrightarrow{a} - 2\overrightarrow{b}$. Express your answer as a simplified radical expression. | 2\sqrt{2} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
The line y = 3x is rotated counterclockwise around the origin by 90 degrees, then translated to the right by 1 unit. Calculate the resulting line equation in slope-intercept form (y = mx + b). | y = -\frac{1}{3}x + \frac{1}{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the following product of fractions:
\[
\left(\frac{4}{5}\right)\left(\frac{5}{6}\right)\left(\frac{6}{7}\right)\left(\frac{7}{8}\right)\left(\frac{8}{9}\right)
\]
Express your answer as a simplified fraction in the format \boxed{fraction}. | \frac{4}{9} | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Fractions"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
In a triangle with two sides measuring $3cm$ and $7cm$ respectively, use the triangle inequality theorem to determine the possible range of lengths for the third side. Express your answer as a range in the format (lower bound, upper bound) in centimeters. | (4, 10) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the function y=f(x) has an inverse function y=f^(-1)(x), and the graph of the function y=2x-f(x) passes through the point (2,1), calculate the point on the graph of the function y=f^(-1)(x)-2x. Express your answer as an ordered pair (x,y), including negative signs if necessary. | (3,-4) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Let $P$ be a point on the circle $(x-5)^{2}+(y-3)^{2}=9$. Find the maximum distance from point $P$ to the line $3x+4y-2=0$. Express your answer as a whole number. | 8 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
In the expansion of $\left( \sqrt {x}+ \frac{1}{ \sqrt[3]{x}} \right)^{24}$, how many terms have an exponent of $x$ that is an integer? Provide your answer as a single integer. | 5 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Squares of side length 1 are arranged to form the figure shown. Calculate the perimeter of the figure. Express your answer as a whole number. | 26 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $\frac{a^{2}}{\sqrt{a} \cdot \sqrt[3]{a^{2}}}$. Express your answer as a simplified algebraic expression with fractional exponents. | a^{\frac{5}{6}} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Given a random variable $X \sim B(n, 0.8)$, and $D(X) = 1.6$, calculate the value of $n$. Express your answer as a whole number. | 10 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z=i(4-3i)$, where $i$ is the imaginary unit, find the conjugate of $z$. Express your answer as a complex number in the form $a + bi$. | 3-4i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function \( f(x) = 2x + 3 \), find the inverse of the function \( f^{-1}(x+1) \). Express your answer as a function of x in the form y = f(x). | y = 2x + 2 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Let the universal set $U=\{4,5,6,7,8\}$, and the sets $M=\{5,8\}$, $N=\{1,3,5,7,9\}$. Find the intersection of $N$ and the complement of $M$ with respect to $U$, denoted $N \cap (\complement_{U}M)$. Express your answer as a set using standard set notation. | \{7\} | big_math | [
"Mathematics -> Discrete Mathematics -> Logic"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the line that passes through the point (1,6) and is parallel to the line x-2y=0. Express your answer in the standard form of a line equation (Ax + By + C = 0). | x-2y+11=0 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Other"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
A man travels m feet due north at 2 minutes per mile. He returns due south to his starting point at 2 miles per minute. Calculate the average rate in miles per hour for the entire trip. Express your answer as a whole number. | 48 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Players A and B play a game. Player A thinks of a number denoted as $a$, and player B guesses the number thought by player A, denoting the guessed number as $b$. Both $a$ and $b$ belong to the set $\{1,2,3\}$. If $|a-b| \leqslant 1$, then it is said that A and B are "telepathically linked". Calculate the probability th... | \dfrac {7}{9} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
In the expansion of $(x^2-x+2y)^5$, find the coefficient of $x^4y^2$. Express your answer as a single integer. | 120 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\sqrt{18n}$ is an integer, find the smallest positive integer value of $n$. Express your answer as a single positive integer. | 2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Given sets \( A = \{a, b, c\} \) and \( B = \{-1, 0, 1\} \), a mapping \( f \) from \( A \) to \( B \) satisfies \( f(a) = f(b) + f(c) \). How many such mappings \( f \) are there? Express your answer as a single integer. | 7 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
The function defined on the set of real numbers, \(f(x)\), satisfies \(f(x-1) = \frac{1 + f(x+1)}{1 - f(x+1)}\). Calculate the value of \(f(1) \cdot f(2) \cdot f(3) \cdots f(2008) + 2008\). Express your answer as a single integer. | 2009 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given $x$, $y \in \mathbb{R}$, and $i$ is the imaginary unit, if $x+yi$ and $\frac{2+i}{1+i}$ are conjugate complex numbers, find the value of $x+y$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Given the universal set U = {x | x is a positive integer less than 9}, set M = {1, 2, 3}, and set N = {3, 4, 5, 6}, calculate the intersection of the complement of M in U and N. Express your answer as a set of integers. | {4, 5, 6} | big_math | [
"Mathematics -> Discrete Mathematics -> Logic -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Four line segments a, b, c, d are in proportion, where a = 3cm, d = 4cm, c = 6cm. Find the value of b. Express your answer in centimeters (cm). | 2 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Given $\tan \left(\alpha +\dfrac{\pi }{4}\right)=\dfrac{3}{4}$, calculate $\cos ^{2}\left(\dfrac{\pi }{4}-\alpha \right)$. Express your answer as a simplified fraction. | \dfrac{9}{25} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Translate the line y = -2x downwards by 2 units. Express the resulting line equation in slope-intercept form (y = mx + b). | y = -2x - 2 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Other"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
A, B, C, D, and E are five people working on a project. If A, B, C, and D work together, it takes 6 days to complete the project; if B, C, D, and E work together, it takes 8 days to complete the project; if A and E work together, it takes 12 days to complete the project. How many days will it take E to complete the pro... | 48 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $f(x)$ is an odd function defined on $\mathbb{R}$, and for any $x \in \mathbb{R}$, the equation $f(x+2) = f(x) + f(1)$ holds, calculate the value of $f(2011)$. Express your answer as a single integer. | 0 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
The number \( x \) has the following property: subtracting \(\frac{1}{10}\) from \( x \) gives the same result as multiplying \( x \) by \(\frac{1}{10}\). What is the number \( x \)? Express your answer as a fraction. | \frac{1}{9} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Four students randomly select one of two universities, A and B, to visit. Calculate the probability that both universities A and B are visited by the students. Express your answer as a simplified fraction. | \frac{7}{8} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Find the coordinates of the focus of the parabola given by the equation $y^{2}=4x$. Express your answer as a coordinate pair (x, y). | (1,0) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(\cos 45^\circ, \sin 45^\circ)$ and $\overrightarrow{b}=(\cos 15^\circ, \sin 15^\circ)$, calculate the dot product $\overrightarrow{a} \cdot \overrightarrow{b}$. Express your answer in the form $\boxed{[answer]}$. | \frac{\sqrt{3}}{2} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
A mixture is prepared by adding 50.0 mL of 0.200 M NaOH to 75.0 mL of 0.100 M NaOH. Calculate the [OH^-] in the mixture. Express your answer in M (molarity) with appropriate precision. | 0.140 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = ax^2 + bx with a > 0, b > 0, find the minimum value of (8a + b) / ab, where the tangent line at the point (1, f(1)) has a slope of 2. Express your answer as a single number. | 9 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given the set S = {A0, A1, A2, A3} and the custom operation ⊕ defined as Ai ⊕ Aj = Ak, where k is the remainder when i + j is divided by 4, find the number of elements x that satisfy the equation (x ⊕ x) ⊕ A2 = A0. Express your answer as a whole number. | 2 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Saba, Rayan, and Derin are working together to complete a task at constant rates. When they all work together, it takes 5 minutes to finish the task. When Saba and Derin work together, it takes 7 minutes. When Rayan and Derin work together, it takes 15 minutes. How many minutes does it take for Derin to complete the ta... | 105 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $2x+3y+4z=1$, find the minimum value of $x^{2}+y^{2}+z^{2}$. Express your answer as a fraction in the form $\boxed{\frac{a}{b}}$, where $a$ and $b$ are integers. | \frac{1}{29} | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Solve the inequality 2x - 3 > 7 - x. Express your answer as an inequality using x as the variable. | x > \frac{10}{3} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
A regular octagon has sides of length 2 cm. When rectangles PQTU and RSVW are shaded, four small triangles inside the octagon remain unshaded. Find the total area of these four triangles. Express your answer in square centimeters (cm²) as a whole number. | 4 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
A square piece of paper is folded along its vertical centerline, and then the folded paper is cut along a dotted line, resulting in three rectangular pieces of paper: one large and two small. Calculate the ratio of the perimeter of each small rectangle to the perimeter of the large rectangle. Express your answer as a s... | \frac{5}{6} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the parametric equations $\begin{cases} x=a+t\cos\theta \\ y=b+t\sin\theta \end{cases}$, where $t$ is the parameter, and the parameter values $t_1$ and $t_2$ for points B and C, find the parameter value corresponding to the midpoint M of the line segment BC. Express your answer as a single mathematical expression... | \frac{t_1 + t_2}{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
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