instruction_seed stringlengths 40 1.39k | answer stringlengths 1 95 | source stringclasses 1
value | domain listlengths 1 3 | llama8b_solve_rate float64 0 0.98 | _source stringclasses 1
value |
|---|---|---|---|---|---|
A uniform spherical shell is wrapped around the outside of a sphere with radius R. If the volume of the "outer shell" is equal to the volume of the sphere, what is the thickness of the spherical shell? Express your answer in terms of R. | (\sqrt[3]{2} - 1) R | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Volume"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given points A=(1,0) and B=(3,2), determine the vector $\overrightarrow {AB}$. Express your answer as an ordered pair (x, y). | (2,2) | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
An ellipse satisfies the following optical property: a ray of light emitted from one focus of the ellipse, after being reflected by the ellipse, passes through the other focus. Now, consider a horizontally placed elliptical billiard table satisfying the equation: $\dfrac{x^2}{16} + \dfrac{y^2}{9} = 1$, where points $A$... | 16 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
For the binomial $(2\sqrt{x}+\frac{1}{\sqrt[4]{x}})^n$ where $n \in \mathbb{N}$, if the coefficients of the first three terms form an arithmetic sequence, calculate the number of rational terms in the expansion. Express your answer as a whole number. | 3 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Three different prime numbers m, n, p satisfy the equation m + n = p. Find the minimum value of the product mnp. | 30 | big_math | [
"Mathematics -> Number Theory -> Prime Numbers"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
In the geometric sequence {a_n}, it is known that a_3 = 6. The sum of the terms a_3, a_5, and a_7 is 78. Calculate the value of a_5. Provide your answer as a single integer. | 18 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
In a high school grade with 560 male students and 420 female students, a stratified sampling method is used to select a sample of 280 students. Calculate the number of male students in this sample. Express your answer as a whole number. | 160 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Given the parabola y = (x-1)^2 - 4, translate it 3 units to the right and then 2 units up. What is the analytical expression of the parabola after these translations? Express your answer in the standard form y = a(x-h)^2 + k. | y = (x - 4)^2 - 2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Quadratic Functions"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation ${x^2}+6x+9+\sqrt{y-3}=0$, find the value of $x-y$. Express your answer as a single integer. | -6 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression \\frac{P+Q}{P-Q} - \\frac{P-Q}{P+Q}, where P = x + y and Q = x - y. Express your answer as a simplified algebraic expression. | \\frac{x^2 - y^2}{xy} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the central angle of sector OAB is $\frac{5\pi}{7}$, and its perimeter is $5\pi + 14$, find the radius of sector OAB. Express your answer as a whole number. | 7 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Given the equations \(2^{x}=15\) and \(15^{y}=32\), calculate the value of \(xy\). Express your answer as a single integer. | 5 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Exponential Functions"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Given line $l_1: 2x+y+1=0$, and line $l_2: x+ay+3=0$, if $l_1 \perp l_2$, find the value of the real number $a$. Express your answer as a single number. | -2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence ${a_n}$ with the sum of its first $n$ terms denoted as $S_n$, and it is known that $a_1=1$, $2S_n=a_{n+1}a_n$, find the value of $S_{20}$. Express your answer as a single integer. | 210 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given that \( n \) is a positive integer, find the minimum value of \( n \) such that the binomial expansion of \( \left(x^{2}+\frac{1}{x^{3}}\right)^{n} \) contains a term with \( x^{7} \). Express your answer as a single integer. | 6 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of ordered pairs (x, y) that satisfy the equation 3x + y = 100, where x and y are positive integers. | 33 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
The base three representation of $x$ is
\[12112211122211112222\]
Find the first digit of the base nine representation of $x$. Express your answer as a single digit. | 5 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
From 5000 math papers randomly selected in the city, 400 papers were randomly selected, of which 360 papers passed. Use this information to estimate the approximate number of students in the city who passed the math exam. Express your answer as a whole number. | 4500 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
For any x > 1, find the maximum value of a such that (x^2 + 3)/(x - 1) >= a. Express your answer as a single numerical value. | 6 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
A bag contains two balls of the same size and shape, one red and one black. Drawing a ball randomly with replacement 3 times, each time drawing one ball, scores points as follows: a red ball scores 2 points, and a black ball scores 1 point. Calculate the probability of scoring a total of 5 points from 3 draws. Express ... | \frac {3}{8} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Combinations"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $S_{n}$ is the sum of the first $n$ terms of the sequence $\{a_{n}\}$, $a_{1}=1$, and $a_{n}\cdot a_{n+1}+\cos \left(n\pi \right)=\sin (n\pi -\frac{π}{2})$, calculate the value of $S_{2022}$. Express your answer as a single integer. | 3 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
In the complex plane, the points corresponding to the complex numbers ${z_1}$ and ${z_2}$ are $A(2,1)$ and $B(0,1)$, respectively. Calculate the value of $\frac{z_1}{z_2}$ and express your answer in the form $a + bi$. | 1 - 2i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
A football team participates in 15 matches and loses 2 of them. The scoring rules are as follows: 3 points for a win, 1 point for a draw, and 0 points for a loss. If the team needs to accumulate at least 33 points to advance, calculate the minimum number of matches the team must win. Express your answer as a whole numb... | 10 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Solve for $x$ in the equation $\frac{x-2}{1-x}=0$. Express your answer as a single numerical value. | 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
A document has 8 pages, with an average of 605 words per page. Calculate the total number of words in the document. Express your answer as a whole number, including the unit (words). | 4840 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Xiao Li is currently $a$ years old. Teacher Zhang is $2$ years younger than three times Xiao Li's age. Calculate Teacher Zhang's age in $5$ years. Express your answer as an algebraic expression in terms of $a$. | 3a + 3 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function y = x^2 * sin(x), find its derivative y'. Express your answer as a mathematical expression. | y' = 2x*sin(x) + x^2*cos(x) | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given two parallel planes, alpha and beta, with 4 points selected on plane alpha and 5 points selected on plane beta, calculate the maximum number of planes that can be determined by these points. Express your answer as a whole number. | 72 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the ellipse $\frac {x^{2}}{16}+ \frac {y^{2}}{25}=1$, if a point $P$ on the ellipse has a distance of $6$ to one of its foci $F_{1}$, then what is the distance from point $P$ to the other focus $F_{2}$? Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
In triangle $\triangle ABC$, given $AC=\sqrt{3}$, $AB=3$, $A=30^{\circ}$, calculate the length of $BC$. Express your answer as a single value. | \sqrt{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
The results of asking 50 students if they participate in music or sports are shown in the Venn diagram. Calculate the percentage of the 50 students who do not participate in music and do not participate in sports. Express your answer as a percentage. | 20\% | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $p$ and $q$ satisfy $p + 2q - 1 = 0$, find the fixed point (x, y) through which the line $px + 3y + q = 0$ must pass. Express your answer as an ordered pair (x, y) with exact fractions. | \left( \frac {1}{2}, - \frac {1}{6} \right) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Linear Equations"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Determine the type of curve described by the trajectory of a moving point C, given that its distance to the point M(0, 3) is always equal to its distance to the line y = -3. Provide your answer as the name of the curve (e.g., Parabola, Hyperbola, Ellipse, Circle). | Parabola | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Given the hyperbola equation $\frac{x^2}{4} + \frac{y^2}{k} = 1$ with eccentricity $e$ in the interval $(1, 2)$, find the range of $k$. Express your answer in interval notation. | (-12,0) | big_math | [
"Mathematics -> Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Let $a=(\frac{1}{3})^{\frac{2}{5}}$, $b={2}^{\frac{4}{3}}$, $c=\log _{2}\frac{1}{3}$. Determine the correct order of the values of a, b, and c from smallest to largest. Express your answer in the format "x < y < z", where x, y, and z are the values in the correct order. | c < a < b | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
In the sequence {a_n}, a_1 = 1, and a_n - a_{n-1} = n (where n is a positive integer and n >= 2). Calculate the value of a_{10}. Express your answer as a single integer. | 55 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Given four non-coplanar points A, B, C, and D in space, how many planes are equidistant to these four points? Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Given the parabola equation y = x^2, find the equation of its directrix. Express your answer as an equation of a line. | y = -\dfrac{1}{4} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence {a_n}, the sum of the first n terms is S_n, and it is known that S_2 = 10 and S_5 = 55. Calculate the slope of the line passing through points P(n, a_n) and Q(n+2, a_{n+2}) (n in natural numbers). Express your answer as a single numerical value. | 4 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
There are 2 boys and 2 girls who take a vehicle to the Anti-Japanese War Memorial to participate in volunteer service. They board the vehicle one by one. Calculate the probability that the second person to board is a girl. Express your answer as a simplified fraction. | \frac{1}{2} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Real numbers \(a, b, c\) are positioned on the number line as shown:
Evaluate the algebraic expression \( |a| - |a+b| + |c-a| + |b-c| \) in terms of \(a, b,\) and/or \(c\). Express your answer as an algebraic expression. | 2c - a | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
The Pythagoras High School band has 100 female and 80 male members. The Pythagoras High School orchestra has 80 female and 100 male members. There are 60 females who are members in both band and orchestra. Altogether, there are 230 students who are in either band or orchestra or both. Calculate the number of males in t... | 10 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given the ellipse C: $$\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{4}=1(a＞2)$$ and the line l: y=x-2 passes through one focus of C, calculate the eccentricity of the ellipse. Express your answer as a numerical value. | \frac{\sqrt{2}}{2} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation sin(2t) = -∫[0,π] cos(x) dx, where t ∈ (0, π), find the value of t. Express your answer in terms of π. | \frac{\pi}{2} | big_math | [
"Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the complex fraction (3-4i)/i. Express your answer in the form a + bi. | 4 - 3i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function y = 4/x, points A(-2,a), B(-1,b), and C(3,c) lie on the graph of this function. Determine the relationship between a, b, and c, using '<' to denote the relationship. Express your answer in the format "a < b < c" or a similar relationship. | b < a < c | big_math | [
"Mathematics -> Precalculus -> Functions"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given $(x-1)^4 = a + a_1x + a_2x^2 + a_3x^3 + a_4x^4$, calculate the value of $a + a_2 + a_4$. Express your answer as a single integer. | 8 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Count the integers that are greater than \( \frac{5}{7} \) and less than \( \frac{28}{3} \). Express your answer as a whole number. | 9 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Given the ellipse $\frac{x^{2}}{4}+y^{2}=1$ with two foci $F\_1$ and $F\_2$, let $P$ be any point on the ellipse. Find the range of values for $\overrightarrow{PF\_1} \cdot \overrightarrow{PF\_2}$. Express your answer as an interval, e.g., $[a, b]$. | [-2, 1] | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Athlete A randomly chooses a restaurant to dine on the first day. If they go to restaurant A on the first day, the probability of going to restaurant A on the second day is 0.7; if they go to restaurant B on the first day, the probability of going to restaurant A on the second day is 0.8. Find the overall probability t... | 0.75 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.75 | 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 -> Polygons"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given $$\overrightarrow{a} = (4, 8)$$ and $$\overrightarrow{b} = (x, 4)$$, and $$\overrightarrow{a} \parallel \overrightarrow{b}$$, find the value of $x$. Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $$\frac {1}{i}+ \frac {1}{i^{3}}+ \frac {1}{i^{5}}+ \frac {1}{i^{7}}$$ and express your answer as a simplified complex number, if necessary, in the form a + bi, where a and b are real numbers. | 0 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
What is the last digit of \( 7^{(7^7)} \)? Express your answer as a single digit. | 3 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Given sets $A=\{x\left|\right.-3 \lt x \lt 2\}$ and $B=\{x|x^{2}+4x-5\leqslant 0\}$, find the intersection $A \cap B$. Express your answer as an interval, including any relevant brackets or parentheses to denote inclusion or exclusion of endpoints. | (-3, 1] | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
In triangle $\triangle ABC$, it is known that the sides opposite angles $A$ and $B$ are $a$, $b$, $A=\frac{π}{4}$, $B=\frac{π}{6}$, and $a=2$. Find the value of $b$. Express your answer as a numerical value, including the square root symbol if necessary. | \sqrt{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Evaluate the expression $\lg 8+3\lg 5$. Provide your answer as a single number. | 3 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Logarithmic Functions"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Two wheels rotate, and each rotation results in two pointers pointing to two different numbers. Wheel A has the numbers 3, 4, 5, and 8, while Wheel B has the numbers 6, 7, and 9. What is the probability that the sum of the two numbers indicated by the arrows is even? Express your answer as a fraction. | \frac{1}{2} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
A journalist wants to take a photo of 4 volunteers and 1 elderly person they are helping, requiring them to line up with the elderly person in the middle. How many different arrangements are there? Express your answer as a single integer. | 24 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Permutations"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Two fair dice are thrown. Given that the first die shows 6 points, what is the probability that the sum of the points on both dice is not less than 10? Express your answer as a simplified fraction. | 1/2 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $i$ is the imaginary unit and $\overline{z}$ is the conjugate of a complex number $z$, if $\left(1-i\right)z=2$, calculate the value of $\overline{z}i$. Express your answer as a complex number in the form $a+bi$. | 1+i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Choose four different digits from 0 to 9 and fill them in the brackets of the box. How many ways are there to make the statement in the box correct, given the constraints on the counts of numbers greater than 1, 2, 3, and 4? Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Put 4 different colored balls into 3 differently labeled boxes. Calculate the total number of different ways to do this. Express your answer as a single integer. | 81 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics -> Other"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Find the minimum value of the quadratic function y = x^2 - 4x + 5. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Quadratic Functions"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Given the line $2x - ay + 2 = 0$ and the line $x + y = 0$, find the inequality solution for $a$ such that the y-coordinate of the intersection point is less than $0$. Express your answer as an inequality. | a < -2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x)=\left\{\begin{array}{l}\cos(\pi x), x\geq 0\\ \frac{2}{x}, x<0\end{array}\right.$, calculate the value of $f[f(\frac{4}{3})]$. Express your answer as a single number. | -4 | big_math | [
"Mathematics -> Precalculus -> Functions"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
A cuboid with dimensions $77 \times 81 \times 100$ is cut into many small cubes with a side length of $1$. Each face of the small cubes is parallel to the corresponding face of the original cuboid. How many small cubes does a diagonal inside the cuboid pass through? Express your answer as a whole number. | 256 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Use Horner's method to compute the value of $f(x)=x^{6}-12x^{5}+60x^{4}-160x^{3}+240x^{2}-192x+64$. When $x=2$, what is the value of $v_{4}$? Express your answer as a single integer. | 80 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Count the number of positive integers less than 400 that can be created using only the digits 1, 2, or 3, with repetition of digits allowed. Express your answer as a single integer. | 39 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
A bag contains 3 red balls, 4 black balls, and 5 white balls. If 2 balls are randomly drawn from the bag, find the probability that exactly one of the drawn balls is red. Express your answer as a fraction in its simplest form. | \frac{9}{22} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Combinations"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x)=\left\{\begin{array}{c}3x,&x≥0\\ f(x+3),&x＜0\end{array}\right.$, calculate $f(-4)$. Provide your answer as a single number. | 6 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the sequence $\{a_n\}$ is a geometric sequence, and $a_2+a_6=3$, $a_6+a_{10}=12$, find the value of $a_8+a_{12}$. Express your answer as a single numerical value. | 24 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Let \(x > 0\) and \(y > 0\), and \(2x+y=6\). Calculate the minimum value of \(9^{x}+3^{y}\) and specify the necessary conditions for equality to hold. Express your answer as a single numerical value. | 54 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Given sets M = {x|x^2 - x - 2 > 0} and N = {x|y = lg(x+2) + lg(1-x)}, calculate the union of set N and the complement of set M in the real numbers. Express your answer as an interval. | (-2,2] | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
A person orders 4 pairs of black socks and some pairs of blue socks. The price of each pair of black socks is twice the price of each pair of blue socks. However, the colors were reversed on the order form, causing his expenditure to increase by 50%. Find the original ratio of the number of pairs of black socks to the ... | 1:4 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
A two-digit number has the property that the sum of its unit digit and tens digit, added to their product, equals the two-digit number itself. Find the unit digit of this number. Express your answer as a single digit. | 9 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
If you are responsible for making digitally symmetric license plates that start with an 8 or 9 as the first digit following the letter "C" and have five digits in total, calculate the maximum number of such plates that can be made. Express your answer as a single integer. | 200 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z=1+2i$, find its conjugate $\overline{z}$ in the standard form $a - bi$. | 1-2i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Two students, A and B, independently answer the same question. The probabilities that they can correctly answer the question are 3/5 and 1/3, respectively. Given that the question has been answered correctly, calculate the probability that both students A and B can answer the question correctly. Express your answer as ... | 3/11 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Fill the nine numbers $1, 2, \cdots, 9$ into a $3 \times 3$ grid, placing one number in each cell, such that the numbers in each row increase from left to right and the numbers in each column decrease from top to bottom. How many different valid arrangements are there? | 42 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given the inequality $x^{2}+ax+1\geqslant 0$ that holds for all $x\in(0, \frac {1}{2}]$, determine the minimum value of the real number $a$. Express your answer as a single real number. | -\frac{5}{2} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Let {a_n} be an arithmetic sequence with a common difference less than zero, and let S_n denote the sum of its first n terms. If S_8 = S_12, then for what value of n is S_n maximized? Express your answer as a single integer. | 10 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Let the increasing positive sequence $a_{1}, a_{2}, \cdots, a_{n}$ be the simplest proper fractions with a denominator of 60. Evaluate the summation $\sum_{i=1}^{n} \cos a_{i} \pi$. | 0 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems",
"Mathematics -> Trigonometry -> Trigonometric Functions -> Other"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Express the number 0.000000102 in scientific notation. Provide your answer in the form of a number multiplied by 10 raised to a power, where the power reflects the original number's decimal placement. | 1.02 \times 10^{-7} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Define a function $f(x)$ on $\mathbb{R}$ that is both an odd function and a periodic function with a period of $2$. Find the value of $f(1)+f(4)+f(7)$. Express your answer as a single integer. | 0 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Given that point (2, 1) is the midpoint of the line segment intercepted by line l on the ellipse $\frac {x^{2}}{12}+ \frac {y^{2}}{4}=1$, determine the equation of line l in the standard form Ax + By + C = 0. | 2x + 3y - 7 = 0 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
In the sequence of numbers arranged according to a certain rule: $2$, $5$, $11$, $23$, $47$, $x$,..., calculate the value of $x$ based on the pattern in the sequence. Express your answer as a single integer. | 95 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Given a geometric sequence $\{a_n\}$ with a common ratio of 2, and the sum of its first 4 terms is 1, find the sum of its first 8 terms. Express your answer as a whole number. | 17 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
The number 315 can be written as the product of two odd integers each greater than 1. How many distinct pairs of such factors exist? Provide your answer as a single integer. | 5 | big_math | [
"Mathematics -> Number Theory -> Factorization"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the domain of f(x) is R and for any x in R, f(x-2) = f(x), and given the expression for f(x) in the interval [-1,1] as f(x) = x^2 - 2x + 4, find the expression for f(x) in the interval [-3,-1]. Express your answer as an algebraic expression in terms of x. | x^2 + 2x + 4 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
A circular spinner is divided into 8 equal sections. An arrow is attached to the center of the spinner. The arrow is spun once. What is the probability that the arrow stops in a section containing a prime number that is odd? Express your answer as a fraction. | \frac{3}{8} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Find the absolute value of -8. Express your answer as a non-negative integer. | 8 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Eight distinct points are placed on a circle. Each pair of points is joined with a line segment. Calculate the total number of line segments formed by these points. | 28 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the angle between the unit vectors $\overrightarrow{e_{1}}$ and $\overrightarrow{e_{2}}$ is $\frac{\pi}{3}$, calculate the angle between the vector $\overrightarrow{e_{1}}-2\overrightarrow{e_{2}}$ and the vector $\overrightarrow{e_{1}}$. Express your answer in radians as a fraction of $\pi$. | \frac{\pi}{2} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
There are four bus routes in a town:
- Route 1 goes C-D-E-F-G-H-C and is 17 km long.
- Route 2 goes A-B-C-F-G-H-A and is 12 km long.
- Route 3 goes A-B-C-D-E-F-G-H-A and is 20 km long.
- Route 4 goes C-F-G-H-C.
Using the information provided about the lengths of the bus routes, calculate the length of Route 4. Express... | 9 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given the circle equation $2x^2+2y^2+6x-4y-3=0$, find the coordinates of the center and the radius. Express the center coordinates as an ordered pair and the radius as a simplified radical. | \left(-\frac{3}{2}, 1\right), \frac{\sqrt{19}}{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Veronica has 6 marks on her report card. The mean of the 6 marks is 74. The mode of the 6 marks is 76. The median of the 6 marks is 76. The lowest mark is 50, and the highest mark is 94. Only one mark appears twice, and no mark appears more than twice. Assuming all of her marks are integers, find the number of possibil... | 17 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets U = {1,2,3,4,5}, A = {1,2,3}, and B = {2,5}, determine the result of the operation A ∩ (C_UB). Express your answer as a set of elements, separated by commas if necessary. | {1,3} | big_math | [
"Mathematics -> Discrete Mathematics -> Logic"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = lg [sin (πx)⋅sin (2πx)⋅sin (3πx)⋅sin (4πx)], find the number of open intervals in the domain of f(x) that intersect with the interval [0,1]. Express your answer as an integer. | 4 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
If the line $ax+2y+1=0$ is perpendicular to the line $x+y-2=0$, calculate the value of $a$. Express your answer as a single integer. | -2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
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