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 an arithmetic sequence $\{a_n\}$ with the sum of the first $n$ terms $S_n$, if $a_2 + a_3 + a_{10} = 9$, then find the value of $S_9$. Express your answer as a single numerical value. | 27 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation |a+2| + (b-1)^2 = 0, calculate the value of the algebraic expression (a+b)^2005. Express your answer as an integer in the format \boxed{[answer]}. | -1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = 2x + 3, and g(x+2) = f(x), determine the expression for g(x). Express your answer as a function of x. | g(x) = 2x - 1 | big_math | [
"Mathematics -> Algebra -> Functions -> Other"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Given 4 "0"s and 4 "1"s arranged in a row, calculate the number of unique physical signals that can be represented by these 8 digits arranged in a row. | 70 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $x_{1}$ is a solution to the equation $xe^{x}=e^{2}$, and $x_{2}$ is a solution to the equation $x\ln x=e^{2}$, find the product $x_{1}x_{2}$. Express your answer as a single expression involving $e$. | e^{2} | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
A cube measures 10 cm x 10 cm x 10 cm. Three cuts are made parallel to the faces of the cube, creating eight separate solids which are then separated. Calculate the increase in the total surface area. Express your answer in square centimeters (cm²) as a whole number. | 600 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Surface Area"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of pairs of positive integers (m, n) that satisfy the equation 4/m + 2/n = 1. | 4 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
Ali is arranging the books on his bookshelves. He puts half his books on the bottom shelf and two-thirds of what remains on the second shelf. Finally, he splits the rest of his books over the other two shelves so that the third shelf contains four more books than the top shelf. There are three books on the top shelf. C... | 30 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Consider the ellipse $C: \\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1$ $(a > b > 0)$, and let line $l$ intersect the ellipse at points $P$ and $Q$. Let $M$ be the midpoint of segment $PQ$, and let $O$ be the coordinate origin. Denote the slope of line $l$ as $k_1$, and the slope of line $OM$ as $k_2$, where ${k_1}{k_2}=-\\f... | \\frac{\\sqrt{3}}{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Let the set $M = \{x | x^2 - x \leq 0\}$, and the function $f(x) = \log_2(1-|x|)$ have a domain of $N$. Calculate the intersection of $M$ and $N$, expressing your answer as an interval in standard interval notation. | [0, 1) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Count the number of prime numbers less than 100 that can be represented as the sum of squares of consecutive positive integers. Express your answer as a single integer. | 5 | big_math | [
"Mathematics -> Number Theory -> Prime Numbers"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given the flowchart, calculate the output number. The flowchart initializes s = 0 and iterates with a step of 2 starting from 2, adding the iterating value to s at each iteration. The loop terminates before adding i to s when i = 102. Express your answer as a single integer. | 2550 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the line that passes through point P(4, -1) and is perpendicular to the line $3x-4y+6=0$. Express your answer in the standard form of a linear equation (Ax + By + C = 0). | 4x+3y-13=0 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Given two non-empty sets $M$ and $N$, define $M \otimes N = \{(a, b) \mid a \in M, b \in N\}$. If $P = \{0, 1, 2\}$ and $Q = \{1, 2\}$, calculate the number of elements in $P \otimes Q$. Express your answer as a numerical value. | 6 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a+b=3$, simplify the expression $2a^2+4ab+2b^2-4$ to find its numerical value. | 14 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence $\{a_n\}$ with the general term $a_n= \dfrac{1}{\sqrt{n+1}+\sqrt{n}}$ and $S_n= \sqrt{101}-1$, find the value of $n$. Express your answer as a whole number. | 100 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
The line $ax+by=1$ (where $a,b$ are not $0$) intersects the two coordinate axes, forming a triangle. Calculate the area of this triangle. Express your answer as a fraction or decimal, including absolute value signs if necessary. | \dfrac{1}{2|ab|} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the arc length is 4 and the central angle corresponding to this arc is 2 radians, find the radius of the circle containing this arc. Express your answer as a whole number. | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Two poles, one 20 inches high and the other 80 inches high, are 100 inches apart. Find the height of the intersection of the lines joining the top of each pole to the foot of the opposite pole. Express your answer in inches as a whole number. | 16 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given sets A = {1, a-1} and B = {-1, 2a-3, 1-2a}, find the value of a that satisfies the condition A ⊆ B. Express your answer as a single numerical value. | 0 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
How many quadratic functions \( f(x) \) satisfy the conditions \( f\left(x^{2}\right) = f(f(x)) = (f(x))^{2} \)? Express your answer as a whole number. | 1 | big_math | [
"Mathematics -> Algebra -> Algebra -> Quadratic Functions"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
A teacher teaches one class for each of three grades in a day, with a total of 8 periods in a day, 5 in the morning and 3 in the afternoon. The teacher cannot teach three consecutive classes (periods 5 and 6 do not count as consecutive). Calculate the total number of possible schedules for the teacher in a day. Express... | 312 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{AB}=(2,3)$ and $\overrightarrow{AC}=(-1,2)$, find the vector $\overrightarrow{CB}$. Express your answer as a vector in the form (x, y). | (3, 1) | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
A block of wood with a weight of 30 N is completely submerged under water, resulting in a buoyant force of 50 N. When the block is released, it floats at the surface. Calculate the fraction of the block that will be visible above the surface of the water when the block is floating. Express your answer as a simplified f... | \frac{2}{5} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given $|a| = 6$, $|b| = 3$, and $a \cdot b = -12$, calculate the projection of vector $a$ onto vector $b$. Express your answer as a single integer. | -4 | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Round the number 1.647 to the nearest hundredth. Express your answer as a decimal number. | 1.65 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Decimals"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
In triangle $\triangle ABC$, $\angle C=90^{\circ}$, $BC=3$, and point $D$ is the trisection point of segment $AB$ closer to point $B$. Calculate the dot product $\overrightarrow{CB} \cdot \overrightarrow{CD}$. Express your answer as a single numerical value. | 6 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $a$ and $b$ that satisfy $|a|=2$, $|b|=1$, and $a \cdot b = 1$, find the angle between vector $a$ and vector $(a-b)$. Express your answer in radians. | \dfrac{\pi}{6} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the first term of the sequence $\{a_n\}$ is $-1$ and $a_{n+1} = 2a_n + 2$, derive the general term formula of the sequence $\{a_n\}$. Express your answer as a formula in terms of $n$, using standard mathematical notation for sequences. | 2^{n-1} - 2 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets P={2, 3} and Q={4, 5, 6, 7}, and the definition P⊗Q={(a, b) | a∈P, b∈Q}, determine the number of elements in P⊗Q. Express your answer as a whole number. | 8 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given two complex numbers $z_1$ and $z_2$ whose corresponding points in the complex plane are symmetric with respect to the line $y = x$, and $z_1 = 1 + 2i$, calculate the value of $\frac{z_1}{z_2}$. Express your answer as a complex number in the form $a + bi$, where $a$ and $b$ are real numbers. | \frac{4}{5} + \frac{3}{5}i | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Assume y=f(x) is a linear function. If f(0)=1 and f(1), f(4), f(13) form a geometric sequence, then find the value of f(2)+f(4)+...+f(2n). Express your answer as a function of n in the simplest algebraic form. | n(2n+3) | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given $f(x)= \begin{cases} x-5 & (x\geqslant 6) \\ f(x+2) & (x < 6) \end{cases}$, find $f(1)$. Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Find the y-intercept of the graph of the function defined by y = (4(x+3)(x-2) - 24) / (x+4). Express your answer as a single integer. | -12 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the geometric mean of 1 and 9. Express your answer in the form ±n, where n is a whole number. | ±3 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence with a first term a1 = 1 and a common difference d ≠ 0, where a1, a2, and a5 form a geometric sequence, find the value of d. Express your answer as a single numerical value. | 2 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
In the Land of Mathesis, numbers are written in the base r. Jones purchases an automobile for 440 monetary units (m.u.) and pays with a 1000 m.u. bill. He receives 340 m.u. in change. Calculate the base r in which the numbers are represented. Express your answer as a single integer. | 8 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Find the derivative of the function y = (x+1)^2(x-1) at x=1. Express your answer as a single number. | 4 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Given that for any positive integer n, the factorial n! = 1 × 2 ×... × n. Solve for the positive integer N in the equation 5! × 9! = 12 × N!. Express your answer as a positive integer. | 10 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Given the mapping $f: (x, y) \rightarrow (x+2y, 2x-y)$, find the pre-image of (3, 1) under the mapping $f$. Express your answer as an ordered pair (x, y) in the format \boxed{(x, y)}. | (1, 1) | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
A triangle with side lengths 8, 13, and 17 has an incircle. The side length of 8 is divided by the point of tangency into segments r and s, with r < s. Find the ratio r : s. Express your answer as a ratio. | 1:3 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Let the function $f(x) = \begin{cases} \sqrt{x-1} & (x\geq 1) \\ x & (x<1) \end{cases}$. Calculate the value of $f(f(2))$. Express your answer as a single numerical value. | 0 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
A set of data is arranged in ascending order as follows: $9$, $10$, $12$, $15$, $x$, $17$, $y$, $22$, $26$. The median of this data set is $16$, and the $75\%$ percentile is $20$. Find the value of $x + y$. Express your answer as a single numerical value. | 36 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of the expression A_4^2 - C_4^3, where A_4^2 represents the number of arrangements of 4 items taken 2 at a time, and C_4^3 represents the number of combinations of 4 items taken 3 at a time. Express your answer as a single integer. | 8 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Permutations",
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Jupiter's volume is about 30 $\sqrt{30}$ times that of Earth's volume. Calculate the approximate ratio of their surface areas. | 30 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Volume"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Judy had 35 hits during the softball season, consisting of 1 home run, 1 triple, and 5 doubles, with the rest being singles. Calculate the percentage of her hits that were singles. Express your answer as a percentage, including the percentage sign (%). | 80% | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $3 \sqrt {12} \div 3 \sqrt {\frac {1}{3}} - 2 \sqrt {3}$ and express your answer in simplest radical form. | 6 - 2 \sqrt {3} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Given $(x-1)(x+1)^9 = a + a_1x + a_2x^2 + \ldots + a_9x^9 + a_{10}x^{10}$, calculate the value of $a_2 + a_4 + a_6 + a_8 + a_{10}$. Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the focus of the parabola $x^2 = 2y$ coincides with one focus of the ellipse $\frac{y^2}{m} + \frac{x^2}{2} = 1$, calculate the value of $m$. Express your answer as a fraction. | \frac{9}{4} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
The front, side, and bottom face areas of a rectangular prism measure 12 square inches, 8 square inches, and 6 square inches respectively. Find the volume of the prism. Express your answer in cubic inches as a whole number. | 24 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Volume"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Let the polynomial \(x^3 - ax^2 + bx - a\) (where \(a, b \in \mathbb{R}\)) have all its roots as real numbers. If \(a\) is the smallest positive real number satisfying this condition, and for this real number \(a\), the value of \(b\) is uniquely determined, then what is the value of \(b\)? Express your answer as a sin... | 9 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
The equation 2x^2 + xy - y^2 = 14 has how many integer solutions? Express your answer as a single integer. | 8 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
If a rise in water level by 1 meter is recorded as +1 meter, how should a drop in water level by 2 meters be recorded? Express your answer as a numerical value with a sign, including the unit'meters'. | -2 meters | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
What is the correct factorization of the expression $x^2 - y^2$? Express your answer as a product of two binomials. | (x+y)(x-y) | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
A student typed out several circles on a computer following this pattern:
●○●○○●○○○●○○○○●○○○○○●......
If this pattern continues indefinitely, how many filled circles (●) will there be in the first 2009 circles? Express your answer as a single integer. | 63 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
The shelf life of milk varies with the storage temperature. Assuming the relationship between shelf life and storage temperature is an exponential function y = ka^x, where milk has a shelf life of about 100 hours in a refrigerator at 0°C, and about 80 hours in a refrigerator at 5°C, calculate the shelf life of milk in ... | 64 \text{ h} | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given $\sin\left(\alpha + \frac{\pi}{6}\right) = \frac{2\sqrt{5}}{5}$, find the value of $\cos\left(\frac{\pi}{3} - \alpha\right)$. Express your answer as a fraction with a radical. | \frac{2\sqrt{5}}{5} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the result of $\left(-3\right)-\left(-5\right)$. Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence {a_n}, a_1 = 11 and d = -2. Find the value of n for which the sum S_n is the greatest. Express your answer as a single integer. | 6 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z$ that satisfies the equation $z + |z| = 2 + i$, determine the value of $z$. Express your answer in the form $a + bi$, where $a$ and $b$ are real numbers. | \dfrac{3}{4} + i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $i$ is the imaginary unit, find the minimum value of $n$ ($n \in \mathbb{N}^*$) such that $i^n = -i$. Express your answer as a whole number. | 3 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Given the average of a set of data $a_{1}$, $a_{2}$, $a_{3}$ is $4$, and the variance is $3$, find the variance of the data after applying the transformation $3a_{i}-2$ to each data point. Express your answer as a numerical value. | 27 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Given $0＜θ＜\frac{π}{2}$, if ${(sinθ+cosθ)}^{2}+\sqrt{3}cos2θ=3$, then calculate $ an heta$. Express your answer in exact form. | 2 - \sqrt{3} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
A number $b$ is randomly taken from the interval $[-2, 2]$. If the probability that the line $y = x + b$ intersects with the circle $x^2 + y^2 = a$ is $\frac{1}{2}$, then calculate the value of $a$. Express your answer as a common fraction, simplified if possible. | \frac{1}{2} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
A large cube with an edge length of 10 units is formed by combining unit cubes. After being painted, the large cube is then separated back into the original unit cubes. Calculate the number of unit cubes that have at least one face painted. Express your answer as a whole number. | 488 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Given a parabola $y=ax^2+bx+c$ that opens downward and has its vertex at $(2, -3)$, determine the type of value (maximum or minimum) this point represents for the parabola and its numerical value. Express your answer in the format: 'The parabola has a [maximum/minimum] value of [numerical value]'. | The parabola has a maximum value of -3. | big_math | [
"Mathematics -> Algebra -> Quadratic Functions"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Let f(x) = log2(2^x - 1). The inverse function is f^(-1)(x) = log2(2^x + 1). Solve the equation f(2x) = f^(-1)(x) for x. Express your answer as a numerical value. | 1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Logarithmic Functions"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given a set of data: -8, -1, 4, x, 10, 13, and the median of this set is 7, find the mode of the data. Express your answer as a single numerical value. | 10 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
A pyramid has a right-angled triangle as its base. What is the maximum number of right-angled triangles that can be among its three lateral faces? | 2 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence {u_n} defined by u_{n+1} - u_n = 3 + 4(n-1) for n = 1, 2, 3,..., and the initial condition u_1 = 5, find the polynomial form of u_n and calculate the sum of its coefficients. Express your answer as a single numerical value. | 5 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
If $(3x-1)^7 = a_7x^7 + a_6x^6 + \ldots + a_1x + a$, find the value of $a_7 + a_6 + \ldots + a_1 + a$ by substituting $x=1$ into the equation. Express your answer as a single integer. | 128 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations -> Other"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Solve for □ in the equation 44 × 25 = □ × 100. Express your answer as a single number. | 11 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function y = |x - 1| + |x - 2| + |x - 3| and the domain |x| ≤ 4, calculate the difference between the maximum and minimum values of y. Express your answer as a single number. | 16 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Evaluate $(x^x)^{(x^x)}$ at $x = 2$. Express your answer as a single integer. | 256 | big_math | [
"Mathematics -> Algebra -> Exponential Functions"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Evaluate the expression (2+3)^2 - (2^2 + 3^2) and provide your answer as a single integer. | 12 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Determine the interval(s) where the function f(x) = x^2 - 2ln(x) is strictly decreasing. Express your answer as interval(s) in the format (a, b), including all intervals where the function is strictly decreasing. | (0, 1) | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
A shop sign says, "T-shirts. Three for the price of two. Equivalent to a saving of £5.50 on each T-shirt." Using this special offer, calculate the total cost of three T-shirts. Express your answer in pounds (£). | £33 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
If $(1+2x)^5 = a_0 + a_1x + a_2x^2 + a_3x^3 + a_4x^4 + a_5x^5$, then find the sum of the coefficients $a_0 + a_1 + a_3 + a_5$. Express your answer as a single integer. | 123 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Binomial Theorem"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Determine the binomial coefficient of the third term in the expansion of \\((x-2)^{5}\\) using the formula:
\[C_n^r = \frac{n!}{r!(n-r)!}.\]
Express your answer as a single integer. | 10 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Among the following real numbers: $-\sqrt{16}$, $-\frac{π}{3}$, $3.14$, $\sqrt[3]{9}$, $\frac{3}{7}$, $0$, how many are irrational? Provide your answer as a whole number. | 2 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Find the sum of the coefficients in the expansion of $(1-2x)^{10}$. Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
The letters of the word 'GAUSS' and the digits in the number '1998' are each cycled separately. If the pattern continues in this way, how many cycles will it take for both 'GAUSS' and '1998' to return to their original configurations? Express your answer as a whole number. | 20 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
The coefficient of $x^2$ in the expansion of $(x+1)^n$ $(n\in \mathbb{N}_+)$ is $15$. Find the value of $n$. Express your answer as a single integer. | 6 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
A batch of goods is transported uniformly at a speed of $v$ km/h (where $100 \leq v \leq 120$) from location A to location B with 17 trucks. The distance between locations A and B is 600 km. For safety reasons, the distance between two trucks must not be less than $\left(\frac{v}{20}\right)^2 km$ (the length of the tru... | 9.8 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
What is the mean of a normal distribution that makes its corresponding normal curve symmetric about the y-axis? Express your answer as a single number. | 0 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression \(\frac{8+4}{8-4}\) to obtain a single numerical value. | 3 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence ${a_n}$, $a_2=2008$, $a_{2008}=a_{2004}-16$. Find the value(s) of $n$ for which the sum of the first $n$ terms, $S_n$, is maximized. Provide your answer as a numerical value(s). | 503 \text{ or } 504 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
In the sequence \(2, 2^2, \cdots, 2^{2015}\), count the numbers that give the same remainder when divided by 9 and have the last digit as 2. Express your answer as a single integer. | 168 | big_math | [
"Mathematics -> Number Theory -> Modular Arithmetic"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation tan(alpha) + tan(beta) - tan(alpha)tan(beta) + 1 = 0, where alpha and beta are in the interval (pi/2, pi), find the value of alpha + beta in radians. | \frac{7\pi}{4} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the complex conjugate of a complex number $z$ is $\overline{z}$, and $z = 1 + i$ (where $i$ is the imaginary unit), calculate the imaginary part of the complex number $\frac{4}{z} - \overline{z}$. Express your answer as a single number. | -1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
A bug starts from the origin of the coordinate plane and first moves one unit to the right to (1,0). Then, it rotates 90° counterclockwise and moves 1/2 unit to (1, 1/2). If it continues this pattern, rotating 90° counterclockwise and moving half the distance of the previous move each time, what is the limiting positio... | \left( \frac{4}{5}, \frac{2}{5} \right) | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $a+i=(1+2i)\cdot ti$, where $i$ is the imaginary unit and $a$, $t$ are real numbers, calculate the value of $t+a$. Express your answer as a single integer. | -1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
A school has 40 classes, each with 50 students. From these, 150 students are selected to participate in the "Student Congress". What is the sample size in this problem? Express your answer as a whole number. | 150 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x)= \begin{cases} \sqrt{x},x\geqslant 4 \\ 2^{x},x < 4 \end{cases}$, find $f[f(2)]$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Functions"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
The price of a favorite soft drink has increased over the past ten years. In four of those years, the price increased by 5 p each year. In the other six years, the price increased by 2 p each year. The drink cost 70 p in 2002. Calculate the price of the drink now, expressing your answer in pounds, rounded to 2 decimal ... | 1.02 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
A class of 6 students recorded the number of plastic bags discarded at their homes over a week, with the results as follows (unit: pieces): 33, 29, 34, 26, 27, 31. If there are 50 students in the class, estimate the total number of plastic bags discarded by all the students' families in the class this week. Provide you... | 1500 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a > b$, what is the correct relationship between $-6a$ and $-6b$? Express your answer as an inequality using the variables $a$ and $b$. | -6a < -6b | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Find the value of a such that in the expansion of (a/x - x^(2/3))^7, the coefficient of the x^3 term is 14. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Anna and Bonnie are racing on a 400-meter elliptical track. They start at the same time, but Anna is 25% faster than Bonnie. Calculate the number of laps it takes for Anna to first overtake Bonnie. Express your answer as a numerical value. | 5 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.203125 | 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_{2n+1} - S_{2n-1} + S_2 = 24, find the value of a_{n+1}. Express your answer as a single numerical value. | 6 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
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