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 |
|---|---|---|---|---|---|
In right triangle $ABC$, where $\angle B = 90^\circ$, $BC = 15$, and $AC = 17$, a semicircle is drawn with $AB$ as its diameter. Find the area of this semicircle. Express your answer in terms of $\pi$. | 8\pi | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
The equation of the circle is $x^{2}+y^{2}-10x+6y+25=0$. Find the coordinates of the center of the circle. Express your answer as a coordinate pair (x, y). | (5,-3) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Four students participate in three different competitions, and each student must participate in one of the competitions. Using the principle of step-by-step counting, calculate the total number of different outcomes. Express your answer as a single numerical value. | 81 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Given the advertising costs $x_i$ (in thousands of yuan, where $i=1,2,3,4$) and sales $y_i$ (in ten thousand yuan, where $i=1,2,3,4$) of a product over four consecutive months, the following data information is obtained after processing:
1. $x_1+x_2+x_3+x_4=18$, $y_1+y_2+y_3+y_4=14$;
2. There is a strong linear correla... | 4.7 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
From four students, A, B, C, and D, three students need to be selected to participate in competitions in three different subjects. Student A must participate. Calculate the total number of different participation plans possible, considering the selection of students and the assignment of subjects. Provide your answer a... | 18 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation of the hyperbola \(\dfrac{x^{2}}{3}-y^{2}=1\), find the coordinates of the foci. Express your answer as a pair of coordinates (x, y), including the ± symbol to indicate both foci. | (\pm2,0) | big_math | [
"Mathematics -> Geometry -> Non-Euclidean Geometry -> Other"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
If $m$ men can complete a job in $d$ days, how many days will it take for $m+r$ men to complete the same job? Express your answer in days, as a fraction or decimal if necessary. | \frac{md}{m+r} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z=1-2i$, what is the imaginary part? Express your answer as a numerical value. | -2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the vector $\overrightarrow{n} = (2, 0, 1)$ is a normal vector of the plane $\alpha$, and point $A(-1, 2, 1)$ lies on $\alpha$, find the distance from point $P(1, 2, -2)$ to the plane $\alpha$. Express your answer as a fraction with a radical in the numerator. | \frac{\sqrt{5}}{5} | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Let vectors $\overline{a}$ and $\overline{b}$ be non-collinear. If $\overline{AB}=2\overline{a}+p\overline{b}$, $\overline{BC}=\overline{a}+\overline{b}$, and $\overline{CD}=\overline{a}-2\overline{b}$, and points A, B, and D are collinear, calculate the value of the real number $p$. Express your answer as a single rea... | -1 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
In $\Delta ABC$, given that $a^2 - c^2 + b^2 = -\sqrt{3}ab$, calculate the measure of angle $C$. Express your answer in degrees as a whole number. | 150 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(1,0)$ and $\overrightarrow{b}$ with magnitude 2, and an angle of $120^{\circ}$ between them, calculate the magnitude of $2\overrightarrow{a}+\overrightarrow{b}$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
A semicircle with radius R is rolled into a cone with the largest possible base. Calculate the height of the resulting cone. Express your answer as a mathematical expression involving R. | \frac{\sqrt{3}R}{2} | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
The sequence {a_n} satisfies a_n = n^2 + 3n + 2. Find the sum of the first 10 terms of the sequence {1/a_n}. Express your answer as a simplified fraction. | \frac{5}{12} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a} = (8, \frac{1}{2}x)$ and $\overrightarrow{b} = (x, 1)$, with $x > 0$, find the value of $x$ such that $\overrightarrow{a} - 2\overrightarrow{b}$ is collinear with $2\overrightarrow{a} + \overrightarrow{b}$. Express your answer as a single numerical value. | 4 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given a sequence ${a_n}$ where $a_1=1$, and for any $m$, $n \in \mathbb{N}^*$, $a_{m+n}=a_m+a_n+mn$, find the value of $\sum_{i=1}^{2017} \frac{1}{a_i}$. Express your answer as a single fraction in simplest form. | \frac{2017}{1009} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
A stationery store sold two scientific calculators for 120 yuan each. One was sold at a 20% profit, and the other at a 20% loss. Calculate the overall profit or loss incurred by the stationery store. Express your answer in yuan, indicating clearly whether it is a profit or loss. | -10 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
A survey shows that the scores of senior high school students in a certain school (total score 750) ξ approximately follow a normal distribution, with an average score of 500. Given that P(400<ξ<450) = 0.3, calculate the probability P(550<ξ<600). Express your answer as a decimal value between 0 and 1. | 0.3 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given the two circles with equations $x^{2}+y^{2}-4x+2y+1=0$ and $x^{2}+y^{2}+4x-4y-1=0$, determine the number of common tangent lines they have. Express your answer as a single numerical value. | 3 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
In a certain grade, there are 560 male students and 420 female students. If stratified sampling is used to draw a sample of 280 students from all the students in this grade, calculate the number of male students in this sample. Express your answer as a whole number. | 160 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the sequence {a_n} is an arithmetic sequence, and the sequence {b_n} is a geometric sequence with all positive terms and a common ratio q!= 1. If a_2 = b_2 and a_10 = b_10, compare a_6 and b_6. Express your answer as an inequality (e.g., a_6 > b_6, a_6 < b_6, etc.). | a_6 > b_6 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
In the plane rectangular coordinate system $xOy$, the coordinates of the point $P$ are $(-3,5)$. What are the coordinates of point $P$ after reflecting it across the $y$-axis? Express your answer in the format $(x, y)$. | (3,5) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
A solid is made up of cubes with a side length of 3 cm, arranged in multiple layers. The base layer forms a 4x4 grid of cubes, and the second layer forms a 2x2 grid of cubes. Calculate the maximum volume of the solid in cubic centimeters (cm³). | 540 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Volume"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Given $ \cos\left( \frac{\pi}{2} - \alpha \right) = \frac{1}{3} $ and $ \frac{\pi}{2} < \alpha < \pi $, calculate the value of $ \sin 2\alpha $. Express your answer as a fraction or decimal, including the negative sign if applicable. | -\frac{4\sqrt{2}}{9} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
The radius of the base of a cone is 1 cm, and the slant height is 3 cm. Calculate the central angle of the unfolded side surface of the cone in degrees. | 120^{\circ} | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
A rectangular sign with dimensions 9 m by 16 m has a square advertisement painted on it. The border around the square must be at least 1.5 m wide. Find the area of the largest square advertisement that can be painted on the sign. Express your answer in square meters (m^2). | 36 \text{ m}^2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Let $\overrightarrow{a}=(1,2)$, $\overrightarrow{b}=(1,1)$, and $\overrightarrow{c}= \overrightarrow{a}+k \overrightarrow{b}$. If $\overrightarrow{b} \perp \overrightarrow{c}$, find the value of the real number $k$. Express your answer as a single real number. | -\dfrac{3}{2} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Let {a_n} be an arithmetic sequence with a positive common difference. If a_1 + a_2 + a_3 = 15 and a_1a_2a_3 = 80, then find the value of a_{11} + a_{12} + a_{13}. Express your answer as a single integer. | 105 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Trapezoid \(ABCD\) has one base \(AB\) that is twice as long as the other base \(DC\). \(E\) is the intersection point of the two diagonals. If the length of the diagonal \(AC\) is 11 units, calculate the length of \(EC\). Express your answer as a mixed number or decimal, including units. | 3 \frac{2}{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given that function f(x) is an odd function defined on ℝ and is monotonically decreasing on the interval (-∞, 0], and f(1) = -1. Let g(x) = log2(x + 3). Find the range of x that satisfies f(x) ≥ g(x). Express your answer as an interval. | (-3, -1] | big_math | [
"Mathematics -> Applied Mathematics -> Other",
"Mathematics -> Calculus -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Find the direction vector of the line $x+1=0$. Express your answer as an ordered pair $(x, y)$ representing the direction vector's components. | (0,1) | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
If the complex number $(1+i)(1+ai)$ is a real number, find the value of the real number $a$. Express your answer as an integer. | -1 | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
The circumference of the circle defined by $x^2+y^2-2x+6y+8=0$ is to be calculated. Express your answer in terms of $\pi$, including the appropriate numerical coefficient. | 2\sqrt{2}\pi | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Given $t > 0$, solve for $t$ in the equation $\int_{0}^{t} (2x - 2) \, dx = 8$. Express your answer as a single positive number. | 4 | big_math | [
"Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Let $S_n$ be the sum of the first $n$ terms of an arithmetic sequence $\\{a_n\\}$. If $\\dfrac{S_3}{S_6} = \\dfrac{1}{3}$, find $\\dfrac{S_6}{S_{12}}$. Express your answer as a simplified fraction. | \\dfrac{3}{10} | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
A safe has five locks, and the keys are cloned and distributed among eight people such that any five of the eight people can open the safe. What is the least total number of keys required to satisfy this condition? | 20 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given \(x > 0\), \(y > 0\), and \(x+2y+2xy=8\), find the minimum value of \(x+2y\). Express your answer as a single number. | 4 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the sequence $\{a_n\}$ is an arithmetic sequence and $a_1 - a_9 + a_{17} = 7$, find the value of $a_3 + a_{15}$. Express your answer as a single number. | 14 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
The National Stadium "Bird's Nest" has a construction area of 258,000 square meters. Express 258,000 square meters in scientific notation with 2 significant figures. Use the format $a\\times10^{n}$. | 2.6\times10^{5} | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Given real numbers $a$ and $b$ satisfy $2^a=3$, $3^b=2$, find the number of zeros of the function $f(x)=a^x+x-b$. Express your answer as a whole number. | 1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
There are four numbers, and their sum is 45. If you add 2 to the first number, subtract 2 from the second number, multiply the third number by 2, and divide the fourth number by 2, the results are all the same. Find the exact values of the original four numbers in order, separated by commas. | 8, 12, 5, 20 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the derivative of the function $f(x)$ is $f'(x)$, and it satisfies $f(x) = 2xf'(2) + x^3$, calculate the value of $f'(2)$. Express your answer as an integer. | -12 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\vec{a}=(3,2)$ and $\vec{b}=(x,1-y)$, where $\vec{a}$ is parallel to $\vec{b}$ and both $x$ and $y$ are positive numbers, find the minimum value of $\frac{3}{x}+\frac{2}{y}$. Express your answer as a single number. | 8 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the asymptotes of the hyperbola $\frac{{x}^{2}}{4}-\frac{{y}^{2}}{9}=1$ in the format $y = \pm \frac{b}{a}x$. | y = \pm \frac{3}{2}x | big_math | [
"Mathematics -> Geometry -> Non-Euclidean Geometry -> Other"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Two couples each bring one child to visit the zoo. After purchasing tickets, they line up to enter the zoo one by one. For safety reasons, the two fathers must be positioned at the beginning and the end of the line. Moreover, the two children must be positioned together. Calculate the total number of different ways the... | 24 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Permutations"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Mary had ten pieces of paper. Some of them were squares, and the rest were triangles. She cut three squares diagonally from corner to corner. She then found that the total number of vertices of the 13 pieces of paper was 42. Calculate the initial number of triangles Mary had before making the cuts. Express your answer ... | 4 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence $\{a_n\}$ with general term $a_n = \frac{2}{n(n+1)}$, calculate the sum of the first $n$ terms, denoted as $S_n$. Express your answer as a fraction in simplest form, using the boxed notation around the final answer. | \frac{2n}{n+1} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
In a high school's senior year with 300 students numbered from 0, 1,..., 299, a sample of 60 students is to be drawn using systematic sampling. If the number drawn in the 60th segment is 298, calculate the number drawn in the 1st segment. Express your answer as a whole number. | 3 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Let S_n = 1 - 2 + 3 - 4 +... + (-1)^n-1 * n, where n = 1, 2,.... Calculate S_17 + S_33 + S_50. Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Find the range of values for the real number $a$ such that the inequality $2_x \ln x \geqslant -x^2 + ax - 3$ holds for all $x \in (0, +\infty)$. Express your answer in interval notation. | (-\infty, 4] | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Evaluate the definite integral $\int_{0}^{1} (e^{x}+2x) d x$. Express your answer as a single mathematical expression. | e | big_math | [
"Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $\dfrac{1}{a}-\dfrac{1}{b}=\dfrac{1}{3}$, calculate the value of $\dfrac{ab}{a-b}$. Express your answer as a single number, including the negative sign if applicable. | -3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Six swimmers line up from left to right for a group photo. The leftmost position can only be occupied by swimmer A or B, and swimmer A cannot be in the rightmost position. Calculate the total number of different arrangements. | 216 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Negate the proposition "$\exists x\in \mathbb{R}$, $x^{2}+2x+2\leqslant 0$". Express your answer as a logical statement using proper notation and quantifiers. | \forall x\in \mathbb{R}, x^{2}+2x+2 > 0 | big_math | [
"Mathematics -> Discrete Mathematics -> Logic -> Other"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence \{a_n\} with the first term being a positive number, and S_n representing the sum of the first n terms, find the value of n for which S_n is at its maximum, given that S_3 = S_11. Express your answer as a single integer. | 7 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the vector $\overrightarrow {a} = (3\cos\alpha, 2)$ is parallel to the vector $\overrightarrow {b} = (3, 4\sin\alpha)$, find the value of the acute angle $\alpha$. Express your answer in radians. | \frac{\pi}{4} | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Geometry -> Other",
"Mathematics -> Linear Algebra -> Vectors"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
In a unit square (a square with side length 1), five arbitrary points are placed on or inside the square. Find the smallest number a such that among these five points, there always exists a pair of points with a distance no greater than a. Express your answer as a mathematical expression or a numerical value. | \frac{\sqrt{2}}{2} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Given a complex number $z=\frac{-1-2i}{(1+i)^{2}}$, simplify $z$ and express your answer in standard complex number format, including both real and imaginary parts. | -1 + \frac{1}{2}i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = sin(ωx) - √3cos(ωx) (ω > 0), with f(x1) = 2 and f(x2) = 0, and the minimum value of |x1 - x2| is 3π, calculate the value of ω. Express your answer as a fraction. | ω = 1/6 | big_math | [
"Mathematics -> Precalculus -> Functions",
"Mathematics -> Calculus -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Mark has a bag that contains 3 black marbles, 6 gold marbles, 2 purple marbles, and 6 red marbles. Mark adds a number of white marbles to the bag such that if a marble is now drawn at random from the bag, the probability of it being black or gold is \(\frac{3}{7}\). Calculate the number of white marbles that Mark adds ... | 4 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $i$ is the imaginary unit, find the coordinates of the point corresponding to the complex number $\frac{2i}{1+i}$ in the complex plane. Express your answer as an ordered pair (a,b). | (1,1) | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
The operation \( \diamond \) is defined by \( a \diamond b = a^{2} b - a b^{2} \). Compute the value of \( 2 \diamond 7 \). Express your answer as an integer. | -70 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
A certain model of mobile phone has undergone two price reductions, with its unit price dropping from 2000 yuan to 1280 yuan. Calculate the average percentage decrease in price per reduction. Express your answer as a percentage. | 18\% | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
The polar equation of a circle is $\rho=1$, and the polar equation of a line is $\rho\cos\theta= \frac {1}{2}$. Calculate the length of the chord cut by the circle from the line. Express your answer as a numerical value, including the square root symbol if necessary. | \sqrt{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Let p be a prime number. The sum of all positive divisors of p^4 is a perfect square. Find the number of prime numbers p that satisfy this condition. Express your answer as a whole number. | 1 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given the functions ${f_1(x) = x}$, ${f_2(x) = \sin x}$, ${f_3(x) = \cos x}$, and ${f_4(x) = \frac{1}{x}}$, calculate the probability that the product of two randomly selected functions is an odd function. Express your answer as a simplified fraction. | \frac{1}{2} | big_math | [
"Mathematics -> Applied Mathematics -> Probability"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
In a certain singles table tennis tournament, it was originally planned that each pair of players would play one match, but three players quit after each playing two matches. As a result, a total of 50 matches were played. How many matches were played among the three players? Express your answer as a whole number. | 1 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow {a}=(3,4), \overrightarrow {b}=(4,3), \overrightarrow {c}=\lambda \overrightarrow {a}- \overrightarrow {b} (\lambda\in\mathbb{R})$, and the angle between vector $\overrightarrow {c}$ and $\overrightarrow {a}$ is equal to the angle between $\overrightarrow {c}$ and $\overrightarrow {b}$, f... | -1 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Other"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Given $$\overrightarrow {OA}=(2, 1)$$ and $$\overrightarrow {OB}=(-3, 4)$$, calculate $$\overrightarrow {AB}$$. Express your answer as an ordered pair (x, y). | (-5, 3) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Given the line equation ax + y + a + 1 = 0, find the fixed point through which this line always passes. Express your answer as an ordered pair (x, y). | (-1, -1) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
If a two-digit integer is equal to k times the sum of its digits, what multiple of the sum of the digits is the new number formed by reversing its digits? Express your answer as an algebraic expression in terms of k. | 11 - k | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Let max{p, q} represent the larger of p and q. If the function f(x) = max{1-x, 2^x}, find the set of x for which f(x) > 4. Express your answer as an interval or union of intervals in standard notation (e.g., (-∞, -3) ∪ (2, +∞)). | (-∞, -3) ∪ (2, +∞) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given $\tan \alpha = -\frac{3}{4}$, and $\tan (\alpha + \beta) = 1$, calculate the value of $\tan \beta$. Express your answer as a single number. | 7 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Express the number 50300 in scientific notation in the form $a \times 10^{b}$, where $a$ is the coefficient and $b$ is the exponent. | 5.03 \times 10^{4} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
The function f(x) = x^2 + 2(a-1)x + 2 is increasing on the interval [4, +∞). Calculate the range of 'a' for which this condition holds. Express your answer in standard interval notation. | [-3, +∞) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the piecewise function $f(x) = \begin{cases} \sin \frac{\pi}{2}x, & x < 0 \\ f(x-1)+2, & x \geq 0 \end{cases}$, find the value of $f(2)$. Express your answer as a single number. | 5 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Given that f(x) is an odd function on R, and for x >= 0, f(x) = x^2 + 2x. Find the expression for f(x) when x < 0. Express your answer as a function of x. | f(x) = -x^2 + 2x | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given the ratios $\frac{b}{a} = 2$ and $\frac{c}{b} = 3$, find the ratio of $a + b$ to $b + c$. Express your answer as a simplified fraction. | \frac{3}{8} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = 2^x - a^2 - a, find the range of values for the positive real number a such that the function has a zero in the interval (-∞, 1]. Express your answer as an interval in standard interval notation. | (0, 1] | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Calculus -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given the curve with parametric equations \(\begin{cases} x=2\cos θ, \\ y=\sqrt{3}\sin θ \\end{cases}\) and its intersection points with the $x$-axis, $M$ and $N$. If $P$ is a point on the curve, find the product of the slopes of the lines containing $PM$ and $PN$. Express your answer as a numerical value. | -\frac{3}{4} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
A rhombus is formed by two radii and two chords of a circle whose radius is 16 feet. Calculate the area of the rhombus in square feet. Express your answer as a numerical value with or without a square root, and include the appropriate units. | 128\sqrt{3} | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given lines $l_1: (a - 1)x + y - 1 = 0$ and $l_2: 3x + ay + 2 = 0$, find the value of the real number $a$ if the two lines are perpendicular. Express your answer as a single real number. | \frac{3}{4} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Find the distance from the point $(3,0)$ to one of the asymptotes of the hyperbola $\frac{{x}^{2}}{16}-\frac{{y}^{2}}{9}=1$. Express your answer as a fraction or decimal. | \frac{9}{5} | big_math | [
"Mathematics -> Geometry -> Analytic Geometry -> Conic Sections"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Solve the inequality $\log_{3}|x-2| < 2$ to find the range of x values. Then, count the number of integers within this range. Express your answer as a single integer. | 17 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Logarithmic Functions"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Throw a fair dice twice. Let $A=\left\{both numbers are odd\right\}$, $B=\left\{the sum of the two numbers is 4\right\}$. Calculate the conditional probability $P(B|A)$. Express your answer as a fraction. | \dfrac{2}{9} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Find the derivative of the function sin(2x) with respect to x. Express your answer as a function of x. | 2\cos(2x) | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
How many pairs of real numbers $(x, y)$ satisfy the equation $x^{4} - 2^{-y^{2}} x^{2} - \\left\\| x^{2} \\right\\| + 1 = 0$, where $\\left\\| a \\right\\|$ denotes the greatest integer not exceeding $a$? Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence $\{a_n\}$ with the general term formula $a_n = (-1)^n(n+1)$, calculate the sum of the first 10 terms: $a_1+a_2+a_3+\ldots+a_{10}$. Express your answer as a single integer. | 5 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Given a sequence of positive integers {a_n} that satisfies a_{p+q} = a_p + a_q for any p, q in N^*, and a_2 = 4, calculate the value of a_9. Express your answer as a single integer. | 18 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow {a} = (1, 2)$ and $\overrightarrow {b} = (x, -2)$, and the scalar product $\overrightarrow {a} \cdot \overrightarrow {b} = -3$, find the magnitude of $\overrightarrow {a} + \overrightarrow {b}$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Back in 1930, Tillie had to memorize her multiplication facts from $0 \times 0$ to $12 \times 12$. The multiplication table she was given had rows and columns labeled with the factors, and the products formed the body of the table. Calculate the fraction of the numbers in the body of the table that are odd. Express you... | 0.21 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
The equation $x^{2}+y^{2}-x+y+m=0$ represents a circle. Find the condition on $m$ for this equation to hold true. Express your answer as an inequality. | m < \dfrac {1}{2} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
A man is good at walking, and he increases his walking distance by the same amount every day. In nine days, he walks a total of 1,260 li. The sum of the distances he walked on the first, fourth, and seventh days is 390 li. What is the distance he walked on the sixth day? Express your answer as a whole number in li. | 150 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets $M=\{x\in\mathbb{Z}|-1\leq x\leq 1\}$ and $N=\{x|x^2=x\}$, find the intersection of sets M and N, $M\cap N$. Express your answer as a set of numbers, including all elements of the intersection. | \{0,1\} | big_math | [
"Mathematics -> Set Theory -> Other"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the fraction \(\frac{20+15}{30-25}\) and express your answer as a single number. | 7 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Given the alternating series \( S_n = 1 - 2 + 3 - 4 + \cdots + (-1)^{n-1}n \), calculate the sum \( S_{17} + S_{33} + S_{50} \). Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given a geometric sequence $\{a_n\}$ with all positive terms and $\lg(a_3 \cdot a_8 \cdot a_{13}) = 6$, calculate the value of $a_1 \cdot a_{15}$. Express your answer as a power of 10. | 10^4 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Find the coefficient of $x^7$ in the expansion of $\left(\frac{x^2}{2}-\frac{2}{x}\right)^8$. Express your answer as a single integer. | -14 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Given that the arithmetic sequence {a_n} has a common difference of 5, the sum of the first n terms is S_n, and a_1, a_2, a_5 form a geometric sequence, find the exact value of S_6. | 90 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the coordinates of point $A$ are $A(5,4)$, if point $A$ is moved $3$ units down and then $4$ units to the left, what are the coordinates of the new point $A'$? Express your answer as an ordered pair $(x, y)$. | (1, 1) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
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