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 |
|---|---|---|---|---|---|
Steve plants ten trees every three minutes. If he continues planting at the same rate, how long will it take him to plant 2500 trees? Express your answer in hours, and include units in your answer. | 12.5 \text{ hours} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence $\{a_n\}$, where $a_1=2, a_{n+1}=a_n+2 (n \in \mathbb{N}^+)$, determine the value of $a_4$. Express your answer as a single integer. | 8 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation of the circle $(x-1)^2 + y^2 = 1$, identify the center $(a, b)$ and radius $r$. Express your answer as an ordered pair for the center and a single value for the radius. | (1, 0), 1 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given the linear function y = (m-1)x - 4 passes through the point (2,4), find the value of m. Express your answer as a single number. | 5 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = x^3 + (a - 2)x^2 + 2x, if f(x) is an odd function, find the equation of the tangent line to the curve y = f(x) at the point (1, 3). Express your answer in slope-intercept form (y = mx + b). | y = 5x - 2 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the tangent line to the curve f(x) = x^2 that is perpendicular to the line x + 4y - 8 = 0. Express your answer in the standard form of a linear equation (Ax + By + C = 0). | 4x - y - 4 = 0 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x) = x^{\frac{1}{4}}$, find the solution set of the inequality $f(x) > f(8x-16)$. Express your answer in interval notation, including all necessary endpoints and parentheses/brackets. | [2, \frac{16}{7}) | big_math | [
"Mathematics -> Algebra -> Functions"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Using three different weights of 1 gram, 3 grams, and 9 grams, various weights of objects can be measured. Assuming the objects to be measured and the known weights can be placed on either side of the balance scale, calculate the total number of different weights of objects that can be measured. Express your answer as ... | 13 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(1,3)$ and $\overrightarrow{b}=(-2,x)$, if $\overrightarrow{a}$ is parallel to $\overrightarrow{b}$, find the magnitude $|\overrightarrow{b}|$. Express your answer with a radical if necessary. | 2\sqrt{10} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Given the piecewise function f(x) defined as:
\[ f(x) =
\begin{cases}
x + \frac{1}{2} & \text{if } x \in \left[0, \frac{1}{2}\right) \\
2(1 - x) & \text{if } x \in \left[\frac{1}{2}, 1\right]
\end{cases}
\]
and the iterative application of f(x) denoted as f_n(x), find the result of f_27(1/5). Express your answer as ... | \frac{4}{5} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Find the domain of the function $f(x) = \lg(4 - x) + \frac{1}{\sqrt{x - 1}}$. Express your answer as an interval. | (1, 4) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Given point P(2,-1), find its coordinates after reflecting across the origin. Express your answer as an ordered pair (x, y). | (-2,1) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given $a$ is a real number, the set $M$ is defined as $M=\{x| x^2-3x-a^2+2=0, x\in \mathbb{R}\}$. Calculate the number of subsets of the set $M$. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, the circle has center O and square OPQR has vertex Q on the circle. If the area of the circle is 72π, calculate the area of the square. Express your answer as a whole number. | 36 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation \(\frac{2+3+4}{3} = \frac{1990+1991+1992}{N}\), calculate the value of \(N\). Express your answer as a whole number. | 1991 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Fractions"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
If the product of four consecutive positive integers is 358800, then find the sum of these four integers. Express your answer as a whole number. | 98 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given a geometric sequence $\{a_n\}$, where $a_4 + a_8 = -2$, find the value of $a_6(a_2 + 2a_6 + a_{10})$. Express your answer as a single numerical value. | 4 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Let the numbers x and y satisfy the conditions $\begin{cases} x^2 + y^2 - xy = 2 \\ x^4 + y^4 + x^2y^2 = 8 \end{cases}$ Calculate the value of $P = x^8 + y^8 + x^{2014}y^{2014}$. Express your answer as a single integer. | 48 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given a non-constant sequence {a_i} satisfying a_{i+1}^2 - a_i a_{i+1} + a_i^2 = 0, with boundary conditions a_1 = a_{n+1} = 1, find the value of the sum ∑_{i=0}^{n-1} a_i. Express your answer as a single integer. | 0 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the sequence $\{a_n\}$ is an arithmetic sequence, and if $a_2=3$, $a_1+a_6=12$, find the value of $a_7+a_8+a_9$. Express your answer as a single numerical value. | 45 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Sequences and Series"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence {a_n}, it is known that a_4 + a_8 = 26. Find the sum of the first 11 terms S_{11}. Express your answer as a single integer. | 143 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a-1+(a-2)i$ is a real number, find the value of $a$ that satisfies this condition. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation of the circle $x^2 + y^2 - 2x - 3 = 0$, find the coordinates of the center in the format (h, k) and the radius r. Express your answer for the center as an ordered pair and the radius as a single number. | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
If the result of (x^2 + ax + 2)(2x - 4) does not contain the term x^2, then what is the value of a? Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
A school cafeteria sells 5 types of dishes at one window. Three students, A, B, and C, each choose one type of dish from the window. Assuming each type of dish is abundant, calculate the total number of different ways the three students can choose their dishes. | 5^3 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Let $f(x)$ be an odd function defined on $\mathbb{R}$. When $x \geqslant 0$, $f(x) = 2^{x} + 2x + b$ (where $b$ is a constant). Calculate the value of $f(-1)$. Express your answer as a single numerical value. | -3 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence $\{a_n\}$ defined by the general term $a_n = 5 \left(\frac {2}{5}\right)^{2n-2} - 4 \left(\frac {2}{5}\right)^{n-1}$ where $n\in \mathbb{N^{*}}$, find the difference between the positions of the minimum and maximum terms of the sequence. Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
What is the opposite (additive inverse) of 2023? Express your answer as a single integer. | -2023 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Express the number 239,000,000 in scientific notation in the form a × 10^n, where a is a number between 1 and 10, and n is an integer. | 2.39 × 10^8 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Among 9 pearls, one is fake and lighter than the real ones. Using a balance scale, what is the minimum number of times you need to weigh the pearls to definitely find out the fake pearl? Express your answer as a whole number. | 2 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Find the minimum value of ω > 0 such that the graph of the function y = sin(ωx + π/3) + 2 coincides with its original graph after being shifted to the right by 4π/3 units. | ω = 3/2 | big_math | [
"Mathematics -> Precalculus -> Functions"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence \( 1, (1+2), (1+2+2^2), \cdots, (1+2+2^2+\cdots+2^n) \), find the sum of the first \( n \) terms of this sequence in terms of \( n \). Express your answer as a formula in terms of \( n \), including all necessary mathematical operations and constants. | 2^{n+1} - n - 2 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Distribute 10 identical one-yuan coins into 3 identical lucky bags, with at least 1 coin in each bag. Ignoring the order of the bags, calculate the number of different ways to distribute the coins. Express your answer as a single integer. | 8 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Combinations"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Find the coefficient of the xy^2 term in the expansion of (1+2x)^6(1+y)^4. | 72 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
The three numbers 2, x, and 10 have an average of x. Find the value of x. Express your answer as a single number. | 6 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Find the range of the function y=(1/2)^x, given that x≥8. Express your answer as an interval in standard interval notation. | (0, 1/256] | big_math | [
"Mathematics -> Algebra -> Functions"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow {AB}=(2, -1)$ and $\overrightarrow {AC}=(-4, 1)$, calculate the coordinates of vector $\overrightarrow {BC}$. Express your answer as an ordered pair (x, y). | (-6, 2) | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $\frac{{a+1}}{{a+2}}+\frac{1}{{a+2}}$. Express your answer as a simplified algebraic expression. | 1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
A rectangular piece of paper measures 17 cm by 8 cm. It is folded so that a right angle is formed between the two segments of the original bottom edge. Calculate the area of the new figure formed after folding the paper. Express your answer in square centimeters (cm²) as a whole number. | 104 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = (x-1)^2(x-a), find the range of values for a such that x = 1 is a local minimum point of the function. Express your answer as an interval using standard interval notation. | (-\infty, 1) | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets $S=\{x\mid (x-2)^{2} > 9\}$ and $T=\{x\mid a < x < a+8\}$, with $S\cup T=\mathbb{R}$, find the range of values for the real number $a$. Express your answer as an interval. | (-3,-1) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
If $f(x)=ax^2+bx$ is an even function defined on the interval $[a-1,2a]$, calculate the value of $a+b$. Express your answer as a single fraction or decimal. | \dfrac{1}{3} | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
The minimum positive period of the function $y=\sin^2\left(\frac{\omega}{2}x-\frac{\pi}{4}\right)$ ($\omega>0$) is $\pi$. Find the value of $\omega$. Express your answer as a numerical value. | 2 | big_math | [
"Mathematics -> Precalculus -> Functions"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
For each vertex of a solid cube, consider the tetrahedron determined by the vertex and the midpoints of the three edges that meet at that vertex. The portion of the cube that remains when these eight tetrahedra are cut away is called a *cuboctahedron*. Calculate the ratio of the volume of the cuboctahedron to the volum... | 0.8333 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Volume"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
A circular disk is divided by 2n equally spaced radii (n > 0) and one chord. Calculate the maximum number of non-overlapping regions into which the disk can be divided. Express your answer as a formula in terms of n. | 3n + 1 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given that vectors $\overrightarrow{a} = (x+1, 2)$ and $\overrightarrow{b} = (1, -1)$ are parallel, find the magnitude of the sum of vectors $\overrightarrow{a}$ and $\overrightarrow{b}$. Express your answer as a numerical value, including the square root symbol if necessary. | \sqrt{2} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
The sides of a triangle are three consecutive integers, and the largest angle is twice the smallest angle. Find the cosine of the smallest angle. Express your answer as a fraction or decimal. | \frac{3}{4} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other",
"Mathematics -> Trigonometry -> Applications of Trigonometric Functions -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Find the coordinates of the point symmetric to point M(3, -2, 1) with respect to the plane yoz. Express your answer as a 3D coordinate (x, y, z). | (-3, -2, 1) | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z= \\frac{3+i}{1-i}$ where $i$ is the imaginary unit, find its complex conjugate $\\overline{z}$. Express your answer as a complex number in the form $a+bi$, including the imaginary unit $i$. | 1-2i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Given a complex number $z=i\left(1+bi\right)+2+3bi\left(b\in R\right)$, if $z$ is a purely imaginary number, find the value of $z$. Express your answer as a purely imaginary number in the form $ai$. | 7i | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
A bamboo stalk has nine sections, with the bottom three sections having a combined capacity of 4 liters and the top four sections having a combined capacity of 3 liters. If the capacity of each section varies uniformly (i.e., forms an arithmetic sequence), what is the total capacity of the bamboo stalk? Express your an... | \frac{201}{22} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the line that passes through the point (-1, 0) and is perpendicular to the line x+y=0. Express your answer in the standard form Ax + By + C = 0. | x-y+1=0 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
The manager of a company planned to distribute a $50 bonus to each employee from the company fund, but the fund contained $5 less than what was needed. Instead, the manager gave each employee a $45 bonus and kept the remaining $95 in the company fund. Calculate the amount of money in the company fund before any bonuses... | 995 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Let $X \sim B(5, \frac{1}{3})$. Calculate the probability $P(X \leq 4)$. Express your answer as a simplified fraction. | \frac{242}{243} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
For the relation {1,2} ⊆ x ⊆ {1,2,3,4,5} (where x is a subset of {1,2,3,4,5}), calculate the number of possible sets x. | 8 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = sinx - sin3x, where x belongs to the interval [0, 2π], find the sum of all the zeros of the function f(x). Express your answer in terms of π. | 7\pi | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Let the arithmetic sequence {a_n} satisfy a_2 = 7 and a_4 = 3. Let S_n be the sum of the first n terms of the sequence {a_n}. Calculate the largest natural number n for which S_n > 0. Express your answer as a single natural number. | 9 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of all sets A that satisfy the condition A ∪ {0, 1, 2} = {0, 1, 2, 3}. Express your answer as a whole number. | 8 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
A coin that is 8 cm in diameter is tossed onto a 5 by 5 grid of squares each having side length 10 cm. A coin is in a winning position if no part of it touches or crosses a grid line, otherwise it is in a losing position. Given that the coin lands in a random position so that no part of it is off the grid, what is the ... | \frac{25}{441} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
In the Cartesian coordinate system, the coordinates of the point $(-2,3)$ after moving $4$ units to the right are to be determined. Find the new coordinates of the point. Express your answer as a coordinate pair $(x, y)$. | (2,3) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
In the sequence $\{a_n\}$, $a_1 = -60$, and $a_{n+1} = a_n + 3$. Calculate the sum of the absolute values of the first 30 terms. | 765 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given a geometric sequence $\{a_n\}$, where the sum of the first 7 terms is 48, and the sum of the first 14 terms is 60, find the sum of the first 21 terms. Express your answer as a single integer. | 63 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given the imaginary unit $i$ and the equation $|{\frac{{a+i}}{i}}|=2$, find the positive real number $a$. Express your answer as a positive real number. | \sqrt{3} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
A club forms four committees according to the following two rules:
1. Each member belongs to exactly two committees.
2. Each pair of committees shares exactly one member.
Find the total number of members in the club. Express your answer as a single integer. | 6 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
A wire 52 cm long is exactly enough to solder into a rectangular solid teaching aid with a length of 6 cm and a width of 4 cm. Calculate the height of the rectangular solid. Express your answer in centimeters (cm) as a whole number. | 3 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
The initial values of two variables are set as S=0 and i=1. Then, the following two statements are executed consecutively three times: S=S+i, i=i+2. What is the value of S after these executions? Express your answer as a single integer. | 9 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given the parabola y = 2x^2, find its equation after shifting 1 unit to the right and 5 units down. Express your answer as an algebraic equation. | y = 2(x - 1)^2 - 5 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = ax^2 - 4x + c with a range of [1, +∞), find the minimum value of 1/(c-1) + 9/a. Express your answer as a single numerical value. | 3 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
A cone has a base area of 18 square centimeters and a height of 4 centimeters. Find the volume of the cone. Express your answer in cubic centimeters as a whole number. | 24 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Volume"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
For any real numbers $x, y$, define the operation $x*y = ax + by + cxy$, where $a, b, c$ are constants. It is known that $1*2 = 4$ and $2*3 = 6$. Find the non-zero real number $m$ such that for any real number $x$, $x*m = x$. Express your answer as a single real number. | 5 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of all possible sets B that satisfy the condition {3, 5} ∪ B = {3, 5, 7}. Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
A company has 840 employees. Using systematic sampling, 42 people are selected for a questionnaire survey. The 840 people are randomly numbered from 1 to 840. How many people among the 42 selected have numbers that fall within the interval [481, 720]? Express your answer as a whole number. | 12 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Other"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Given z1 = (2-2i)/(1+i) and z2 = -i-2, the points corresponding to the complex numbers z1 and the conjugate of z2 in the complex plane are A and B, respectively. Calculate the length of line segment AB. Express your answer as a numerical value, including the square root symbol if necessary. | \sqrt{13} | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $3x-2y-5=0$ and the constraint $1 \leqslant x \leqslant 2$, calculate the range of $\frac{y}{x}$. Express your answer as an interval, using square brackets or parentheses as necessary. | [-1,\frac{1}{4}] | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
The line \(\frac{x}{4} + \frac{y}{3} = 1\) intersects the ellipse \(\frac{x^2}{16} + \frac{y^2}{9} = 1\) at points \(A\) and \(B\). Find the number of points \(P\) on the ellipse such that the area of \(\triangle PAB\) is equal to 3. Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
In triangle \( \triangle ABC \), the sides opposite to angles \( A \), \( B \), and \( C \) are \( a \), \( b \), and \( c \) respectively. If
\[
\frac{a}{\cos A}=\frac{b}{2 \cos B}=\frac{c}{3 \cos C},
\]
find the measure of angle \( A \) in radians. Express your answer as a fraction of \( \pi \). | \frac{\pi}{4} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
When two fair dice are rolled, what is the probability of rolling two fives? Express your answer as a fraction. | \frac{1}{36} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Find the remainder when \( 7^{2008} + 9^{2008} \) is divided by 64. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
In the sequence shown, each figure after the first is formed by adding 4 squares to the previous figure. Find the number of squares that form Figure 2010. Express your answer as a whole number. | 8037 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $\sqrt{(2-3|x|)^{2}}=2+3 x$, determine the range of values for $x$. Express your answer as an interval or a combination of intervals. | \left[-\frac{2}{3}, 0\right] | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
A sequence of figures is formed by joining two rectangles to the bottom of the previous figure. Each individual rectangle has dimensions 10 cm by 5 cm. The perimeter of the first figure is 60 cm, and each subsequent figure adds 10 cm to the perimeter. If the perimeter of Figure n is 710 cm, calculate the value of n. | 66 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Given rectangle ABCD with AD = a and CD = b, points E, F, and G lie on sides BC, AD, and DC respectively, such that angles DAG, CGE, and DGF are equal. Calculate the ratio (EG + FG) / AG. Express your answer as a ratio of a and b. | \frac{2b}{a} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the function $f(x)$ is an even function on $\mathbb{R}$, $g(x)$ is an odd function on $\mathbb{R}$, and $g(x)=f(x-1)$. If $f(2)=2$, calculate the value of $f(2014)$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Determine the greatest common divisor (GCD) of the three numbers 390, 455, and 546. Express your answer as a single integer. | 13 | big_math | [
"Mathematics -> Number Theory -> Greatest Common Divisors (GCD)"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the imaginary part of the complex number $\left(\frac{1+i}{1-i}\right)^{2023}$. Express your answer as a single value, including any necessary mathematical notation (e.g., i for imaginary unit). | - i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given two circles \(C_{1}\): \(x^{2}+y^{2}=1\) and \(C_{2}\): \((x-3)^{2}+(y-4)^{2}=16\), find the number of common tangents between them. Express your answer as a whole number. | 3 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Given \(a > b > 0\) and \(m > 0\), determine the relationship between the fractions \(\frac{b}{a}\) and \(\frac{b+m}{a+m}\). Express your answer using the symbols <, >, or =. | < | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of \(\frac{550+50}{5^{2}+5}\). Express your answer as a single number. | 20 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Find the solution set of the inequality $|4-3x|-5\leqslant 0$. Express your answer in interval notation. | [- \dfrac{1}{3}, 3] | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
The graph of the function $y=kx-2$ passes through the point $P(-1,3)$. Find the value of $k$. Express your answer as a single number. | -5 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Linear Equations"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
A certain store sells washing machines of a certain brand. Each washing machine costs $a$ yuan to purchase. The store increases the cost by $30\%$ for selling, and then offers a $20\%$ discount. Calculate the retail price of the washing machine in terms of $a$. | 1.04a | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Let t_k(n) denote the sum of the k-th powers of the digits of a positive integer n. Find the value of k such that the condition t_k(n) is a multiple of 3 does not imply that n is a multiple of 3. | 6 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
The eccentricity of the ellipse $mx^{2}+y^{2}=1$ is $\dfrac{\sqrt{3}}{2}$. Find the length(s) of its major axis. If there are multiple possible values, list all of them. | 2, 4 | big_math | [
"Mathematics -> Geometry -> Coordinate Geometry -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Two numbers have a sum of 32. If one of the numbers is -36, what is the other number? Express your answer as a single integer. | 68 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
The mean age of the members of a jazz band is 21. The saxophonist, singer, and trumpeter are 19, 20, and 21 years old respectively. The other three musicians are all the same age. Calculate the age of the other three musicians. Provide your answer as a whole number. | 22 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the value of $x^{2}+3x$ is $3$, calculate the value of the expression $-3x^{2}-9x-2$. Express your answer as a single integer, including the negative sign if applicable. | -11 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the line that passes through the intersection point of the two lines $2x - 3y + 2 = 0$ and $3x - 4y - 2 = 0$, and is parallel to the line $4x - 2y + 7 = 0$. Express your answer in the standard form of a linear equation (Ax + By + C = 0). | 2x - y - 18 = 0 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Linear Equations"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Find the value of k such that the pair of equations y = x^2 and y = 3x + k has two identical solutions. Express your answer as a single fraction. | -\frac{9}{4} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Polynomial Equations"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Three congruent isosceles trapeziums are assembled to form an equilateral triangle with a hole in the middle. Calculate the perimeter of the hole in terms of variables a and b. | 6a - 3b | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given a box containing 3 screw base bulbs and 7 bayonet base bulbs, all of which have the same shape and power and are placed with their bases down. An electrician needs a bayonet base bulb and takes one bulb at a time without returning it. What is the probability that he only gets a bayonet base bulb on his third try?... | \frac{7}{120} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
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