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 |
|---|---|---|---|---|---|
Find the maximum value of the function $y=\frac{x}{e^x}$ on the interval $[0, 2]$. Express your answer as a single value. | \frac{1}{e} | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
A class has 6 periods in one day, 3 in the morning and 3 in the afternoon. The schedule for one day includes Chinese, Mathematics, English, Physics, Physical Education, and Art. The requirements are that the Mathematics class must be scheduled in the morning and the Art class in the afternoon. Calculate the total numbe... | 216 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Dalia's birthday is on a Wednesday and Bruce's birthday is 60 days after Dalia's. On what day of the week is Bruce's birthday? Provide your answer as a full day name (e.g., Sunday, Monday, etc.). | Sunday | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Let the sequence $\{a_n\}$ be a monotonically increasing arithmetic sequence, with $a_1=2$ and $a_1-1$, $a_3$, $a_5+5$ forming a geometric sequence. Calculate the value of $a_{2017}$. Express your answer as a single integer. | 1010 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.125 | 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 |
Let \( C_1 \) and \( C_2 \) be distinct circles of radius \( 7 \mathrm{~cm} \) that are in the same plane and tangent to each other. Find the number of circles of radius \( 26 \mathrm{~cm} \) in this plane that are tangent to both \( C_1 \) and \( C_2 \). Express your answer as a whole number. | 6 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given that {a_n} is an arithmetic sequence and a_1 + a_{15} = 48, find the value of a_3 + 3a_8 + a_{13}. Express your answer as a single integer. | 120 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
If the radius of a circle is increased by 1 unit, what is the ratio of the new circumference to the new diameter? Express your answer as a mathematical constant or a simple fraction if necessary. | \pi | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Consider the statements:
(1) p ∧ ∼q ∧ r
(2) ∼p ∧ ∼q ∧ r
(3) p ∧ ∼q ∧ ∼r
(4) ∼p ∧ q ∧ r
where p, q, and r are propositions. Determine how many of these statements make (p → q) → r true. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Discrete Mathematics -> Logic"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given points A(-1, 1), B(1, 2), and C(2, 3), and $\overrightarrow {AB} \perp (\overrightarrow {BC} + \lambda \overrightarrow {AC})$, calculate the value of lambda. | -\frac{3}{8} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Find the value of the complex number $\left(1+ \frac{1}{i}\right)^{4}$. Express your answer in standard form (a + bi) if it is a complex number, or as a single number if it is purely real. | -4 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function \( f(x) = \log_{2} \log_{2}(2x + 2) + 2^{2x + 2} \) for \( x \in [0,1] \), determine the number of integer values that \( f(x) \) can take within this interval. Express your answer as a single integer. | 14 | big_math | [
"Mathematics -> Precalculus -> Functions",
"Mathematics -> Calculus -> Other"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $2x + 5 - 3x + 7 = 8$, solve for the value of $x$. Express your answer as an integer. | 4 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of $\sqrt{9}$. Express your answer as a whole number. | 3 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
A gold bar has a length of 5 feet. When one foot is cut from the thicker end, it weighs 4 pounds; when one foot is cut from the thinner end, it weighs 2 pounds. Assuming that the weight of the gold bar decreases uniformly from the thicker to the thinner end, forming an arithmetic sequence, calculate the weight of the s... | \frac{7}{2} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Let the function $f(x)=2\sqrt{3}\sin x\cos x-2\sin^{2}x+1$ $(x\in\mathbb{R})$. Calculate the smallest positive period of $f(x)$ in radians. Express your answer in the form $\boxed{[answer]}$. | \pi | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence $\{ a_{n} \}$, it is given that $a_{6}+a_{7}+a_{8}=12$. Determine the sum of the first 13 terms in this sequence. Express your answer as a single integer. | 52 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Given that sin(-80°) = k, determine the value of tan(100°). Express your answer in terms of k and √(1-k^2). | \frac{k}{\sqrt{1-k^2}} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of positive integers n, with n <= 100, such that n^3 + 5n^2 is the square of an integer. Express your answer as a single integer. | 8 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Two positive integers have a sum less than 100, and one integer is twice the other. Find the maximum possible value of their sum. Express your answer as a single integer. | 99 | big_math | [
"Mathematics -> Algebra -> Simple Equations"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Two ants, one black and one white, start from point A on a unit cube and move along the edges. Each time they complete traversing an edge, it is considered as completing one segment. The white ant's path is AA1 → A1D1 → …, and the black ant's path is AB → BB1 → …. They both follow the rule: the segment they traverse in... | 1 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given non-zero vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ with $|\overrightarrow{b}| = 2$, the minimum value of $|\overrightarrow{b} - t\overrightarrow{a}|$ ($t \in \mathbb{R}$) is $\sqrt{3}$. Determine the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ in degrees. | 60^{\circ} \text{ or } 120^{\circ} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given that July 3, 2030 is a Wednesday, determine the day of the week for July 14, 2030. Provide your answer as the full day of the week. | Sunday | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\overrightarrow{e_{1}}$ and $\overrightarrow{e_{2}}$ are non-collinear vectors, $\overrightarrow{AB}=2\overrightarrow{e_{1}}+\overrightarrow{e_{2}}$, $\overrightarrow{BC}=-\overrightarrow{e_{1}}+3\overrightarrow{e_{2}}$, $\overrightarrow{CD}=\lambda\overrightarrow{e_{1}}-\overrightarrow{e_{2}}$, and points ... | 5 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
If there are 4 different red envelopes in a WeChat group of 3 people, and each red envelope can only be grabbed once, and each person must grab at least one red envelope, calculate the total number of ways the red envelopes can be grabbed. Express your answer as a numerical value. | 36 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
In the expansion of $(x- \\frac{1}{3x})^{24}$, calculate the number of terms with integer exponents of $x$. Express your answer as a whole number. | 9 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the line that passes through the point (-1, 2) and is perpendicular to the line 2x-3y+4=0. Express your answer in the standard form of a line equation (Ax + By + C = 0), including all necessary coefficients and constants. | 3x+2y-1=0 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
In the geometric sequence {a_{n}}, the common ratio q = 2, and the sum of the first 87 terms S_{87} = 140. Calculate the sum of the terms a_{3}, a_{6}, a_{9},..., a_{87}. | 80 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Solve for x in the equation \(\frac{6}{x+1} = \frac{3}{2}\). Express your answer as a single integer. | 3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given the quadratic equation $x^2 + 2x - 1 = 0$ with roots $x_1$ and $x_2$, find the sum of the roots $x_1 + x_2$. Express your answer as a single integer. | -2 | big_math | [
"Mathematics -> Algebra -> Algebra -> Equations and Inequalities"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $\frac{S_{2016}}{2016} = \frac{S_{2015}}{2015} + 2$, where $S_n$ denotes the sum of the first n terms of an arithmetic sequence ${a_n}$, find the common difference d of the sequence. Express your answer as a single number. | 4 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the sequence $\{a_n\}$ is a geometric sequence, and $a_2+a_6=3$, $a_6+a_{10}=12$, find the value of $a_8+a_{12}$. Express your answer as a single numerical value. | 24 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given that ${F_1}$ and ${F_2}$ are the left and right foci of the ellipse $E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, $P$ is a point on the line $x = \frac{3a}{2}$, and $\Delta F_2 P F_1$ is an isosceles triangle with a base angle of $30^\circ$, find the eccentricity of $E$. Express your answer as a single ... | \frac{3}{4} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence $\{a_n\}$ where $a_n = (\sqrt{2}+1)^n - (\sqrt{2}-1)^n \quad (n \in \mathbf{N})$, and let $[x]$ denote the greatest integer not exceeding the real number $x$, compute the units digit of $[a_{2017}]$. Express your answer as a single digit. | 2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given the inequality $x^{2}+ax+1 \geqslant 0$ holds for all $x \in (0, \frac{1}{2}]$, find the minimum value of $a$. Express your answer as a single number, including the negative sign if applicable. | -\frac{5}{2} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence ${a_{n}}$, the sum of the first three terms is $3$, and the sum of the $28^{th}$, $29^{th}$, and $30^{th}$ terms is $165$. Calculate the sum of the first $30$ terms in this sequence. Express your answer as a single integer. | 840 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the solution set of the inequality ax - b > 0 with respect to x is (1/2, +∞), determine the solution set of the inequality (ax - 2b) / (-x + 5) > 0 with respect to x. Express your answer as an interval or union of intervals. | (1, 5) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given a normal distribution N(0, $\frac{4}{9}$), calculate the probability of data falling outside the interval (-2, 2). Express your answer as a decimal value rounded to 4 decimal places. | 0.0026 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Consider the sequence {2n - 1} with n in N*, which is grouped such that the nth group contains 2^n terms. Determine the group number in which the number 2009 appears. Express your answer as a whole number. | 9 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the parabola $x=\frac{1}{2}y^2$ and a point $P$ on the parabola with a horizontal coordinate of $1$, find the distance from point $P$ to the focus $F$ of the parabola. Express your answer as a fraction or decimal. | \frac{3}{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
The surface area of a cube with an edge length of 2 can be calculated using the formula: edge length × edge length × 6. Calculate the surface area of the cube. Express your answer in square centimeters (cm²). | 24 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Surface Area"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given a sequence $\{a_n\}$ whose sum of the first $n$ terms $S_n$ satisfies $S_n = An^2 + Bn$, and $a_1 = 1$, $a_2 = 3$, find the value of $a_{2017}$. Express your answer as a single integer. | 4033 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
A set of 3 red balls, 2 white balls, and 2 blue balls is arranged randomly. What is the probability of having 2 adjacent white balls or 2 adjacent blue balls in this arrangement? Express your answer as a fraction. | \frac{10}{21} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Express the number 50300 in scientific notation in the form $a \times 10^{b}$, where $a$ is the coefficient and $b$ is the exponent. | 5.03 \times 10^{4} | big_math | [
"Mathematics -> Other"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Given the system of equations:
\[x + y = m\]
\[2x - y = n\]
and the solution \(x=2\) and \(y=-3\), calculate the value of \(m-n\). Express your answer as a single integer. | -8 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation \((a-i)^{2}=2i\), where \(i\) is the imaginary unit, find the value of the real number \(a\). Express your answer as a single real number. | -1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
There are 60 products numbered from 01 to 60. Now 5 products are selected for inspection using systematic sampling, where the first sampled number is 5. Calculate the sequence of numbers chosen by this sampling method. Express your answer as a sequence of 5 numbers separated by commas. | 5, 17, 29, 41, 53 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Derive the formula for the sum $1^{2}-2^{2}+3^{2}-4^{2}+\ldots+(2n-1)^{2}-(2n)^{2}$. Express your answer as a formula in terms of n. | -n(2n+1) | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Determine the number of terms in the arithmetic sequence -3, -1,..., 2k-1. Express your answer as a simple algebraic expression in terms of k. | k + 2 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Given that real numbers x and y satisfy the equation x + y = 4, find the minimum value of the expression x^2 + y^2. Express your answer as a single numerical value. | 8 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x) = \begin{cases} x+1, & (x\geqslant 0) \\ f(x+2), & (x < 0) \end{cases}$, calculate the value of $f(-3)$. Provide your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(\frac{1}{2},\; \frac{\sqrt{3}}{2})$ and $\overrightarrow{b}=(-\frac{\sqrt{3}}{2},\; \frac{1}{2})$, calculate $(\overrightarrow{a}+\overrightarrow{b})\cdot \overrightarrow{a}$. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Find the value of $a$ in the power function $y=x^{a}$ given that the slope of its tangent line at $x=1$ is $-4$. Express your answer as a single number. | -4 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
In the expansion of $(1+x)^{2}+\left(1+x
ight)^{3}+\ldots +\left(1+x
ight)^{9}$, find the coefficient of $x^{2}$. Express your answer as a single integer. | 120 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Given $x>0$ and $y>0$, and the equation $xy=2x+8y$, find the values of $x$ and $y$ when $xy$ reaches its minimum value. Express your answer as a pair of values $(x, y)$. | (16, 4) | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
If $x$ and $-9$ are reciprocals of each other, find the value of $x$. Express your answer as a fraction. | -\frac{1}{9} | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
A is traveling at 50 mph and needs to pass B, who is traveling at 40 mph, by gaining 30 feet. Meanwhile, C is 210 feet away from A and is headed towards A at 50 mph. If B and C maintain their speeds, what is the minimum speed increase required for A to pass safely? Express your answer in mph. | 5 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
A square has the same perimeter as an equilateral triangle whose sides all have length 12 cm. Find the length of the sides of the square. Express your answer in centimeters (cm) as a whole number. | 9 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given a hyperbola $C$: $\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1$ where $a>0$ and $b>0$, find its eccentricity if one of its asymptotes is perpendicular to the line $2x-y+1=0$. Express your answer as a simplified radical. | \frac{\sqrt{5}}{2} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Count the total number of three-digit positive integers whose digits multiply to 24. Provide your answer as a single integer. | 21 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
There are two rectangles, A and B. The length and width of rectangle A are $x$ cm and $y$ cm, respectively. The length and width of rectangle B are each 1 cm less than those of rectangle A. Calculate the difference in area between rectangle A and rectangle B. Express your answer as an algebraic expression in terms of $... | x + y - 1 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
In the multiplication problem below, $A$, $B$, $C$, $D$ are different digits. Calculate the sum of $A$ and $B$.
\[\begin{array}{cccc}& A & B & A\\ \times & & C & D\\ \hline C & D & C & D\\ \end{array}\]
Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Determine the coefficient of the term containing $x^4$ in the expansion of the binomial $\left(x^2-\frac{1}{x}\right)^5$. Express your answer as a single integer. | 10 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Given a complex number $z$ satisfying $iz=|3+4i|-i$, calculate the imaginary part of the conjugate of $z$. Express your answer as a single integer. | 5 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Let $a$ be a positive real number and $i$ be the imaginary unit. Given that $|a+i|=2$, calculate the value of $a$. Express your answer as a single value, including the square root symbol if necessary. | \sqrt{3} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x) = \begin{cases} 2x-1 & (x > 0) \\ 1-2x & (x \leqslant 0) \end{cases}$, calculate the value of $f(1) + f(-1)$. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x)=\begin{cases}
& (a-2)x, & x\geqslant 2 \\
& \left( \frac{1}{2} \right)^{x}-1, & x < 2 \\
\end{cases}$, find the range of the real number $a$ such that $f(x)$ is a monotonically decreasing function on $\mathbb{R}$. Express your answer as an interval. | \left( -\infty,\frac{13}{8} \right] | big_math | [
"Mathematics -> Calculus -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given that α ∈ (π/2, π) and sin(π-α) - cos(π+α) = √2/3, find the value of sinα - cosα. Express your answer as a single fraction. | 4/3 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
If the sum of the coefficients in the expansion of $(x+3y)^n$ is equal to the sum of the binomial coefficients in the expansion of $(7a+b)^{10}$, find the value of $n$. Express your answer as a single numerical value. | 5 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence $\{a_n\}$ with the general term formula $a_n = n^2 - 14n + 65$, find the item numbers for which $a_n = 20$. Express your answer as integers, separating multiple item numbers with commas if necessary. | 5, 9 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Quadratic Functions"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Given x ∈ (-3π/4, π/4) and cos(π/4 - x) = -3/5, find the value of cos 2x. Express your answer as a fraction. | -\frac{24}{25} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Determine the number of arrangements of the numbers 0, 1, 2,..., 9 such that the odd numbers form an increasing sequence and the even numbers form an increasing sequence. Provide your answer as a single integer representing the total number of such arrangements. | 252 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Combinations"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets $M=\left{ x\mid y=\sqrt{x-1} \right\}$ and $N=\left{ x\mid y=\log_{2}(2-x) \right}$, determine the value of ${C}_{R}\left(M\cap N\right)$. Express your answer in interval notation. | (-\infty,1)\cup[2,\infty) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
What is the absolute value of 2? Provide your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
The operator "$\oplus$" is defined as $a\oplus b=a+b+ab-1$. Calculate the value of $3\oplus 2$. Express your answer as a single integer. | 10 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given the plane vectors $\overset{→}{a}=(1,-3)$, $\overset{→}{b}=(4,-2)$, if $\lambda \overset{→}{a}+ \overset{→}{b}$ is perpendicular to $\overset{→}{a}$, find the value of $\lambda$. Express your answer as a single numerical value. | -1 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Find the median of the dataset: 2, 6, 4, 4, 1. Arrange the data in ascending order and identify the middle number as the median. Provide your answer as a single number. | 4 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the sequence {a_n} is a geometric sequence and (a_1 + 3a_3) / (a_2 + 3a_4) equals 1/2, find the value of (a_4a_6 + a_6a_8) / (a_6a_8 + a_8a_10). Express your answer as a simplified fraction. | \frac{1}{16} | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given the expansion of $(1-x)^9 = a_0 + a_1x + a_2x^2 + \ldots + a_9x^9$, calculate the sum of the absolute values of the coefficients excluding $a_0$, i.e., $|a_1| + |a_2| + |a_3| + \ldots + |a_9|$. Express your answer as a single integer. | 511 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
For any function \( y = f(x) \), in the same Cartesian coordinate system, the graph of the function \( y = f(x-1) \) and the graph of the function \( y = f(-x+1) \) are always symmetric with respect to a certain line. Find the equation of this line of symmetry. Express your answer in the form \( x = a \). | x = 1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the tangent line to the curve f(x) = x - 2ln(x) at the point A(1, f(1)). Express your answer as a linear equation in the form ax + by + c = 0. | x+y-2=0 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
If the sum of 25 consecutive even numbers is 10,000, what is the largest number among these 25 consecutive even numbers? Express your answer as a whole number. | 424 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
If $z \in \mathbb{C}$ and $|z+2-2i|=1$, calculate the maximum value of $|z-1-2i|$. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = x^3 + 2x, find the equation of the tangent line to the curve y = f(x) at the point (1, f(1)). Express your answer in the standard form of a linear equation (Ax + By + C = 0). | 5x - y - 2 = 0 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Given a function $y=a^x$ where $a$ is a constant, the sum of the maximum and minimum values of $y$ on the interval $[0, 1]$ is 3. Find the value of $a$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Functions -> Exponential Functions"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the measurement results ξ follow a normal distribution N(1, σ^2) (σ > 0), and the probability of ξ taking values in the interval (0, 2) is 0.8, find the probability of ξ taking values in the interval (0, 1). Express your answer as a numerical value between 0 and 1. | 0.4 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
What is the result of adding 2 to -1? Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
The heights of 4 athletes on a team are 135 cm, 160 cm, 170 cm, and 175 cm. Laurissa joins the team, making it a total of 5 athletes. Find Laurissa's height such that the mode height of the players is equal to the median height which is equal to the mean (average) height. Express your answer in centimeters as a whole n... | 160 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Find the extremum point of the function $f(x) = x^2 - 2x$. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given that point P is a moving point on the line y = -x - 1, two tangents are drawn from point P to the parabola C: x^2 = 2y, and the points of tangency are denoted as A and B. Calculate the maximum value of the distance from the origin to the line AB. | \sqrt{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given three points $A$, $B$, and $C$ in a plane such that $|\\overrightarrow{AB}| = 3$, $|\\overrightarrow{BC}| = 5$, and $|\\overrightarrow{CA}| = 6$, calculate the value of $\\overrightarrow{AB} \\cdot \\overrightarrow{BC} + \\overrightarrow{BC} \\cdot \\overrightarrow{CA} + \\overrightarrow{CA} \\cdot \\overrightarr... | -35 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Let $P$ be any point on the curve $x^{2}-y-\ln x=0$. Calculate the minimum distance from point $P$ to the line $y=x-2$. Express your answer as a numerical value, including the square root symbol if necessary. | \sqrt{2} | big_math | [
"Mathematics -> Geometry -> Other",
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Diana and Apollo each roll a standard six-sided die. Calculate the probability that Diana's number is larger than Apollo's number. Express your answer as a simplified fraction. | \frac{5}{12} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Define a function $f(x)$ on $\mathbb{R}$ that is an even function, and satisfies $f(1-x) = f(1+x)$. If $f(x) = x^2$ for $x \in [0, 1]$, then calculate the value of $f(-3)$. Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation of a circle $(x-2)^{2}+(y-3)^{2}=1$, find the coordinates of the center of the circle. Express your answer as a coordinate pair (x, y). | (2,3) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the result of $\left(-1\right)^{2019}$. Express your answer as a single integer. | -1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Given the hyperbola x^2/a^2 - y^2/b^2 = 1 (a > 0, b > 0), the points M and N on the hyperbola are symmetrical to the origin, and point P is a point on the hyperbola. The slopes of lines PM and PN are k1 and k2 respectively. If k1 * k2 = 5/4, then find the eccentricity of the hyperbola. Express your answer as a numerica... | \frac{3}{2} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function y = 1/2 * (x^2 - 100x + 196 + |x^2 - 100x + 196|), find the sum of the function values when the variable x takes on the 100 natural numbers 1, 2, 3,..., 100. Express your answer as a single integer. | 390 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Express the wavelength of $0.000 000 94m$ in scientific notation. Include the unit (m) in your answer. | 9.4 \times 10^{-7}m | big_math | [
"Mathematics -> Other -> Other"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
There are 6 different books placed in a row on a desk, with the requirement that books A and B must be placed at the two ends, and books C and D must be adjacent. Find the number of different arrangements of the books. Express your answer as a single integer. | 24 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Permutations"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
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