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 |
|---|---|---|---|---|---|
Given the function y = |x - b|, when x = 1 or x = 3, the corresponding two function values are equal. Find the value of the real number b. Express your answer as a single real number. | 2 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Amoebas reproduce by splitting and their numbers double every three minutes. If one amoeba is placed in a bottle, then after 1 hour, the bottle is filled with amoebas. If initially, two amoebas are placed in the bottle, calculate the time it takes for the bottle to be filled with amoebas. Express your answer in minutes... | 57 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Given two independent events A and B, the probability that neither of them occurs is 1/9. If the probability that event A occurs and event B does not occur is equal to the probability that event B occurs and event A does not occur, what is the probability P(A) that event A occurs? Express your answer as a fraction. | \frac{2}{3} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
In a certain number system, the square of 24 is 554. Find the base of this number system. Express your answer as a single integer. | 12 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Alice and Bob play a game involving a circle whose circumference is divided by 12 equally-spaced points. The points are numbered clockwise, from 1 to 12. Both start on point 12. Alice moves clockwise and Bob, counterclockwise. In a turn of the game, Alice moves 5 points clockwise and Bob moves 9 points counterclockwise... | 6 | big_math | [
"Mathematics -> Discrete Mathematics -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given that \(\sin \left(\frac{\pi}{4} - x\right) = \frac{3}{5}\), calculate the value of \(\sin 2x\). Express your answer as a fraction. | \frac{7}{25} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given that α and β are acute angles, and sin(α) = 4/5, cos(β) = 5/13, calculate sin(β - α). Express your answer as a simplified fraction. | 16/65 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
In the geometric sequence $\{a_n\}$, $a_1 + a_3 = 10$ and $a_2 + a_4 = 30$. Calculate the sum of the first five terms of the sequence $S_5$. Express your answer as a single integer. | 121 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
A company currently has 160 staff members, 30 middle managers, and 10 senior managers. To conduct a health check, a sample of m individuals will be drawn using stratified sampling. If the proportion of senior managers in the sample is to be the same as the overall proportion of senior managers in the company, and only ... | 20 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation 5x + 12y = 60, calculate the minimum value of sqrt(x^2 + y^2), which represents the shortest distance from the origin to the line. Express your answer as a fraction in the form of a/b, where a and b are integers. | \frac{60}{13} | big_math | [
"Mathematics -> Geometry -> Other",
"Mathematics -> Calculus -> Other"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
In the geometric sequence \\(\\{a_n\\}\\), where \\(a_1= \\frac{1}{8}\\) and \\(q=2\\), find the geometric mean of \\(a_4\\) and \\(a_8\\). Express your answer as a single positive number. | 4 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a_n$ is the coefficient of the term containing $x^n$ in the expansion of $(1+x)^{n+2}$, find the sum of the first $n$ terms of the sequence $\{\frac{1}{a_n}\}$. Express your answer as a fraction in simplest form. | \frac{n}{n+2} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
A circular mirror has a radius of 5 cm. If its radius is increased by 5 cm, by how many square centimeters does the area of the resulting larger circle increase compared to the original circular area? Express your answer in terms of π and include units. | 75\pi | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Find the domain of the function $y=\log _{(x-2)}(5-x)$. Express your answer in interval notation. | (2,3)\cup(3,5) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
The growth rate of a certain mold doubles every day. If the mold can cover the entire surface of a tank after 14 days, how many days will it take to cover half of the tank's surface? Express your answer as a whole number of days. | 13 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
A cube with a volume of $8 \text{ cm}^3$ has all its vertices on the surface of a sphere. Calculate the surface area of the sphere. Express your answer in square centimeters ($\text{cm}^2$) and include $\pi$ if necessary. | 12\pi | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Let the sequence $a_n$ be an arithmetic sequence, and its sum of the first $n$ terms be $S_n$. Given that $a_1+a_3+a_8=99$ and $a_5=31$, find the value of $k$ such that for all $n \in \mathbb{N}^*$, $S_n \leq S_k$ always holds. Express your answer as a single integer. | 20 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Sequences and Series"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence {a<sub>n</sub>}, a<sub>3</sub> + a<sub>10</sub> = 5, a<sub>7</sub> = 1, and S<sub>n</sub> represents the sum of the first n terms of the sequence {a<sub>n</sub>}. Determine the maximum value of S<sub>n</sub>. Express your answer as a single integer. | 70 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Let the function $y=f(x)$ be defined on $\mathbb{R}$ and be an even function, and let $f(x)$ be increasing on $(-\infty, 0)$. If $x_1 < 0 < x_2$, and $x_1 + x_2 > 0$, compare the values of $f(-x_1)$ and $f(x_2)$. Provide your answer as a direct comparison of the function values, using the format $\boxed{f(-x_1) > f(x_2... | f(-x_1) > f(x_2) | big_math | [
"Mathematics -> Algebra -> Functions"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $2x^{3}\cdot x^{3}$. Express your answer as a simplified algebraic expression. | 2x^{6} | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Find the standard equation of a circle with center at (0, -2) and radius 4. Express your answer in the standard form (x-h)^2 + (y-k)^2 = r^2. | x^2 + (y+2)^2 = 16 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Given the coordinate of the focus of a parabola is (0, 1/2), determine its standard equation. Express your answer in the standard form of a parabola equation. | x^2 = 2y | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}$ with a magnitude of 3 and $\overrightarrow{b}$ with a magnitude of 2, and $(\overrightarrow{a} + 2\overrightarrow{b}) \perp (\overrightarrow{b} - 2\overrightarrow{a})$, find the dot product of vectors $\overrightarrow{a}$ and $\overrightarrow{b}$. Express your answer as a single numbe... | -\frac{10}{3} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
The area of square \(ABCD\) is 64 and \(AX = BW = CZ = DY = 2\). Find the area of square \(WXYZ\). Express your answer as a whole number. | 40 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given that person A and person B have shooting accuracies of $0.6$ and $0.8$ respectively, and their shots do not affect each other, calculate the probability that both players score the same number of points after each shooting once. Express your answer as a decimal value between 0 and 1. | 0.56 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Three switches, A, B, and C, have probabilities of functioning properly over a given period of time of 0.9, 0.8, and 0.7 respectively. Calculate the reliability of this system. Express your answer as a decimal value to three decimal places. | 0.504 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Given $x < -1$, find the maximum value of the function $y = x + \frac{1}{x+1}$. Express your answer as a single number. | -3 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
A parking lot has 25% more cars than trucks. What is the ratio of cars to trucks? Express your answer as a ratio in simplest form, using a colon (:) to separate the parts of the ratio. | 5:4 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
For the function y = x^2 - 2x, find the interval where the function is decreasing. Express your answer in standard interval notation (e.g., (-∞, 1)). | (-∞, 1) | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
The angle between the bisector of an angle and the bisector of its adjacent supplementary angle can be determined using geometric properties. Given that the sum of an angle and its adjacent supplementary angle is 180°, find the angle formed by the bisectors. Express your answer in degrees as a whole number. | 90 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
What is the exact value of $(625^{\log_5 2015})^{\frac{1}{4}}$? Express your answer as a single integer. | 2015 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the line that passes through point A(1, 2) and is at the maximum distance from the origin. Express your answer in the standard form of a linear equation (Ax + By + C = 0). | x+2y-5=0 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Given the expressions \( n+3, n-9, n-4, n+6, n-1 \), arrange them in ascending order and identify the middle expression. Express your answer as an algebraic expression. | n-1 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Let $z=\frac{{1+{i^3}}}{{2-i}}$. Calculate the imaginary part of $z$. Express your answer as a single number. | -\frac{1}{5} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
A bank plans to introduce a new fixed deposit service. It is predicted that the deposit amount is directly proportional to the square of the deposit interest rate, with a proportionality constant of $k$ ($k>0$). The loan interest rate is $0.048$. Assuming that the bank can lend out all the deposits it receives, what de... | 0.032 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $A$ and $B$ are two vertices of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > 0$, $b > 0$), if $P$ is a point on the hyperbola and the symmetric point of $P$ with respect to the $x$-axis is $Q$, and if the slopes of lines $AP$ and $BQ$ are $K_1$ and $K_2$ respectively, and $K_1K_2 = -\frac{4}{9... | \frac{\sqrt{13}}{3} | big_math | [
"Mathematics -> Geometry -> Non-Euclidean Geometry -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
In the geometric sequence {a_n}, a_1 + a_2 = 3 and a_3 + a_4 = 12. Find the value of a_5 + a_6. Express your answer as a single integer. | 48 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
An even function f(x) is monotonic on the interval [0, a] (a > 0) and satisfies f(0) * f(a) < 0. Find the number of zeros of the function f(x) in the interval [-a, a]. Express your answer as a whole number. | 2 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Express 25 billion yuan in scientific notation. Provide your answer in the form of $a \times 10^{b}$, where $a$ is the coefficient and $b$ is the exponent. | 2.5 \times 10^{10} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, the horizontal distance between adjacent dots in the same row is 1. Also, the vertical distance between adjacent dots in the same column is 1. Find the perimeter of quadrilateral PQRS. Express your answer as a whole number. | 14 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
A radioactive substance continuously decays into other substances, with 3/4 of its mass decaying every year. Calculate the minimum number of years required for the remaining mass to be no more than 1% of the original mass. Express your answer as a whole number of years. | 4 | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given the arithmetic sequence {a_n} with specific terms a_2 = 14 and a_5 = 5, find the common difference d. Express your answer as an integer. | -3 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Given the two circles with equations x^2+y^2-2x-5=0 and x^2+y^2+2x-4y-4=0, find the equation of the perpendicular bisector of the line segment connecting their intersection points A and B. Express your answer as a linear equation in the form ax + by + c = 0. | x+y-1=0 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given 4 different styles of shirts and 3 different colors of trousers, calculate the total number of unique combinations when one pair of trousers and one shirt are matched to form a set. | 12 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence {a_n}, given a_3 + a_{17} = 10, calculate the sum of the first 19 terms, S_{19}. Express your answer as a single integer. | 95 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
The photolithography machine uses a technique similar to photo printing to transfer fine graphics on the mask plate to the silicon wafer through the exposure of light, which is the core equipment for chip manufacturing. The $ArF$ excimer laser is one of the commonly used light sources in photolithography machines, with... | 1.93 \times 10^{-7} \text{ meters} | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
A semi-circular iron sheet is used to form the lateral surface of a cone-shaped workpiece with a base radius of $4cm$. Calculate the length of the generatrix of the cone. Express your answer in centimeters (cm) as a whole number. | 8 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given the hyperbola equation $\dfrac{x^2}{8} - y^2 = 1$, find the distance from its foci to its asymptotes. Express your answer as a single numerical value. | 1 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Given a sequence $\{a_n\}$ that satisfies $3a_{n+1} + a_n = 4$ (for $n \geq 1$), and $a_1 = 9$, the sum of the first $n$ terms is denoted as $S_n$. Find the smallest integer $n$ that satisfies the inequality $|S_n - n - 6| < 1$. | 7 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Two straight lines have equations \( y = px + 4 \) and \( py = qx - 7 \), where \( p \) and \( q \) are constants. The two lines meet at the point \( (3, 1) \). Calculate the value of \( q \). Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given a geometric sequence {a_n} with a common ratio of 2, find the value of a_4/a_2. Express your answer as a single number. | 4 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Solve the inequality $15-2x \gt 7$ and find the number of positive integer solutions. Express your answer as a whole number. | 3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given that both $a$ and $b$ are positive numbers, and $a + b = 1$, find the minimum value of $\frac{4}{a} + \frac{9}{b}$. Express your answer as a single number. | 25 | big_math | [
"Mathematics -> Applied Mathematics -> Optimization"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Given cos(π/4 - α) = 4/5, calculate the value of sin(2α). Express your answer as a fraction. | \frac{7}{25} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Given complex numbers $z_{1}=2t+i$ and $z_{2}=1-2i$ (where $i$ is the imaginary unit), if $\frac{z_{1}}{z_{2}}$ is a real number, find the value of the real number $t$. Express your answer as a single real number. | -\frac{1}{4} | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Given the recurrence relation $2a_{n+1}=1+2a_{n}$ and the initial condition $a_1=-2$, calculate the sum of the first 10 terms of the sequence $\{a_n\}$. Express your answer as a single number. | \frac{5}{2} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
The sum of the first $n$ terms of an arithmetic sequence is 48, and the sum of the first $2n$ terms is 60. Calculate the sum of the first $3n$ terms. Express your answer as a single integer. | 36 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Four students, A, B, C, and D, stand in a row to take a photo at their high school graduation. Given that A and B are next to each other, calculate the probability that A and C are also next to each other. Express your answer as a simplified fraction. | \dfrac{1}{3} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, \(\triangle PQS\) is equilateral. Also, \(\triangle PQR\) and \(\triangle PSR\) are isosceles with \(PQ = PR = PS\). If \(\angle RPQ = \angle RPS\), find the measure of \(\angle QRS\). Express your answer in degrees as a whole number. | 30 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
What is the fixed point that the line $kx-y-3k+3=0$ passes through, regardless of the value of $k$? Express your answer as an ordered pair $(x, y)$. | (3,3) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
If in space, the distances between any two of n distinct points are all equal, what is the maximum possible value of the positive integer n? Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Given $a=\log_{3}4$, $b=(\frac{1}{5})^{0}$, $c=\log_{\frac{1}{3}}10$, determine the correct order of the values of a, b, and c. Express your answer as a sequence of inequalities (e.g., a > b > c). | a > b > c | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Logarithmic Functions"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
A five-legged Martian has a drawer full of socks, each of which is red, white or blue, and there are at least five socks of each color. The Martian pulls out one sock at a time without looking. What is the minimum number of socks the Martian must remove from the drawer to be certain there will be 5 socks of the same co... | 13 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
In the Cartesian coordinate system, point P is located in the second quadrant. The distance from point P to the x-axis is 3 units, and the distance from point P to the y-axis is 7 units. Find the coordinates of point P. Express your answer as an ordered pair (x, y), including negative signs if applicable. | (-7, 3) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x) = \log_2{x}$, determine the derivative $f'(x)$. Express your answer as a mathematical expression, including all necessary variables and constants. | \frac{1}{x\ln{2}} | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
If the area of triangle $ABC$ is 64 square inches, and the geometric mean of sides $AB$ and $AC$ is 12 inches, calculate the value of $\sin A$. Express your answer as a simplified fraction or exact decimal if necessary. | \frac{8}{9} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
A sequence of figures is formed by adding 4 unshaded squares to the previous figure. The first figure has 8 unshaded squares. Find the number of unshaded squares in the 10th figure. Express your answer as a whole number. | 44 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the product of $(3-4i)$ and $(3+4i)$. Express your answer as a single integer. | 25 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
In an arithmetic sequence {a_n}, S_n is the sum of the first n terms. If S_5 = 20, find the value of a_2 + a_3 + a_4. Express your answer as a single integer. | 12 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
In the geometric sequence $\{a_{n}\}$, the product of $a_{5}$ and $a_{14}$ is given as $a_{5}a_{14}=5$. Using the property that the product of any four consecutive terms in a geometric sequence is the square of the product of their middle two terms, find the value of $a_{8}a_{9}a_{10}a_{11}$. Express your answer as a s... | 25 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
The line with equation y = 2x - 6 is reflected in the y-axis. Find the x-intercept of the resulting line. Express your answer as a single number. | -3 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Estimate the value range of the expression $({2\sqrt{2}+\sqrt{3}})×\sqrt{2}$. Express your answer as an interval. | (6, 7) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function y = (x+1)(x-a), find the value of 'a' such that the function is even. Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Point P(m+2,3) is moved 3 units to the right to point P', and P' lies on the y-axis. Find the value of m. Express your answer as a single integer. | -5 | big_math | [
"Mathematics -> Algebra -> Simple Equations"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
There are 6 paths from Wolfburg to the Green Meadows, and 20 paths from the Green Meadows to Sheep Village. Wolfburg and Sheep Village are separated by the Green Meadows. Find the total number of different ways a wolf can travel from Wolfburg to Sheep Village. Express your answer as a whole number. | 120 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Angie, Bridget, Carlos, and Diego are seated at random around a square table, one person to a side. Find the probability that Angie and Carlos are seated opposite each other. Express your answer as a fraction or decimal. | \frac{1}{3} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
If \( k=2 \), calculate the value of the expression \(\left(k^{3}-8\right)(k+1)\). Provide your answer as a single number. | 0 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Polynomial Operations"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z= \frac{m+2i}{3-4i}$, if $z$ is a real number, calculate the value of the real number $m$. Express your answer as a real number. | -\frac{3}{2} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a$ is a real number, if $\frac{a-i}{1+i}$ is a pure imaginary number, where $i$ is the imaginary unit, find the value of $a$. Express your answer as a single real number. | 1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
What number must be added to 8 to give the result -5? Express your answer as an integer. | -13 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
In right triangle ABC, AX = AD and CY = CD. Find the measure of angle XDY. Express your answer in degrees as a whole number. | 45 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
A circle with radius $r$ is tangent to sides $AB, AD$, and $CD$ of rectangle $ABCD$ and passes through the midpoint of diagonal $AC$. Calculate the area of the rectangle in terms of $r$. Express your answer as an expression involving $r$. | 8r^2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Xiaoming's family bought many bottles of juice. On New Year's Eve, they drank half of the total amount minus 1 bottle. On the first day of the New Year, they drank half of the remaining amount again. On the second day of the New Year, they drank half of the remaining amount plus 1 bottle, leaving them with 2 bottles. C... | 22 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
A certain grade is holding a basketball friendly match where each pair of teams must play one game, and a total of 28 games are played. Calculate the number of teams participating in this match. Express your answer as a whole number. | 8 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Given the hyperbola $C: \frac{x^2}{4} - \frac{y^2}{b^2} = 1$ ($b > 0$), the distance from its foci to the asymptote is 3. Calculate the length of the imaginary axis of the hyperbola $C$. Express your answer as a single number. | 6 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given a geometric sequence $\{a_n\}$ where all terms are positive, and $a_1=3$, $S_3=21$, calculate the sum of the terms $a_3$, $a_4$, and $a_5$. Express your answer as a single numerical value. | 84 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Given the circle x^2 + y^2 - 2x + a = 0 and the line 8x - 6y - 3 = 0, calculate the value of a such that the length of the chord cut by the line on the circle is sqrt(3). Express your answer as a single real number. | 0 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Determine the domain of the function $f(x) = \frac{\ln(2 + x - x^2)}{|x| - x}$. Express your answer as an interval or a union of intervals in standard interval notation. | (-1, 0) | big_math | [
"Mathematics -> Precalculus -> Functions"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number z = (1-2i)/(1+i), calculate the product of z and its conjugate. Express your answer as a simplified numerical value or fraction. | 5/2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Given x = √23 - 1, calculate the value of the algebraic expression x^2 + 2x + 2. Express your answer as a single integer. | 24 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Express $1.5 trillion yuan in scientific notation. Provide your answer in the form $a \times 10^b$, where $a$ is the coefficient and $b$ is the exponent. | 1.5 \times 10^{12} | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
If the volume of a sphere is expanded to be 8 times its original size, then its surface area is expanded to be how many times its original size? Express your answer as a numerical factor. | 4 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Volume"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
If saving $30m^{2}$ of water is denoted as $+30m^{2}$, then what is the notation for wasting $10m^{2}$ of water? Express your answer in the same notation format, including units. | -10m^{2} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Susan had 50 dollars to spend at the carnival. She spent 12 dollars on food and twice as much on rides. Calculate the amount of money she has left to spend. Express your answer in dollars as a whole number. | 14 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
The PE teacher places 9 identical soccer balls into three boxes numbered 1, 2, and 3, with the requirement that the number of balls in each box is not less than its number. Calculate the number of different ways to place the balls, ensuring that you provide an exact integer answer. | 10 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\log_{3}m + \log_{3}n \geqslant 4$, find the minimum value of $m + n$. Express your answer as a single number. | 18 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Three individuals, A, B, and C, want to sign up for "The Voice of China 2023" by either calling the hotline or logging onto the official website. Calculate the total number of different ways they can sign up. | 8 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
A certain department conducted a sampling test on a batch of products from a table tennis ball manufacturing company, and the results are as follows:
| Number of Balls Sampled | $50$ | $100$ | $200$ | $500$ | $1000$ | $2000$ |
|-------------------------|------|-------|-------|-------|--------|--------|
| Number of Hig... | 0.95 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Count the number of different shapes of isosceles triangles with all sides being positive numbers and their perimeter not exceeding 10. Express your answer as a whole number. | 10 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
When we unfold the side view of a cone, we are essentially spreading out the curved surface into a flat shape. What geometric shape is formed by this unfolded side view? Express your answer as the name of the shape. | sector | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
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