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 $a=\sin \frac {2\pi}{7}$, $b=\cos \frac {12\pi}{7}$, $c=\tan \frac {9\pi}{7}$, determine the order of a, b, and c from largest to smallest. Express your answer as an inequality in the format 'x > y > z'. | c > a > b | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
The graph shows the constant rate at which Suzanna rides her bike. If she rides a total of half an hour at the same speed, calculate the total distance she would have ridden. Express your answer in miles. | 6 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Two good friends, Xiaocong and Xiaoming, travel on the same day. Xiaocong goes from Shenzhen to Huangshi, arriving at Wuhan Station at 13:30. He then takes an intercity railway to Huangshi after resting for 1 hour at Wuhan Station. Xiaoming takes a taxi from Chongkou Development Zone to Wuhan Station, arriving at any t... | \frac{1}{2} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Given \\(\\operatorname{tg} x = \\frac{2ab}{a^2 - b^2}\\), where \\(a > b > 0\\) and \\(0^\\circ < x < 90^\\circ\\), calculate \\(\\sin x\\). Express your answer as a fraction or ratio. | \\frac{2ab}{a^2 + b^2} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Find the domain of the function $f(x)=\frac{1}{\sqrt{\ln(5-2x)}}+\sqrt{e^{x}-1}$. Express your answer as an interval or intervals. | [0,2) | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
The digits from 1 to 9 are each written on nine separate pieces of paper and placed in a hat. Jack randomly selects one piece of paper and then returns it to the hat. Jill then randomly selects one piece of paper. Determine the most likely units digit of the sum of the numbers selected by Jack and Jill. Provide an expl... | 0 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $4x^{2}-px+q=0$, which can be transformed by completing the square to $(x-\frac{1}{4})^2=\frac{33}{16}$, calculate the value of $\frac{q}{p}$. Express your answer as a single number. | -4 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
In triangle $\triangle ABC$, if $\overrightarrow {BD}=2 \overrightarrow {DC}$, and $\overrightarrow {AD}=x \overrightarrow {AB}+y \overrightarrow {AC}$, find the value of $\frac {y}{x}$. Express your answer as a single numerical value. | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
When $a > 0$, simplify the expression $\sqrt{a^2}$. Express your answer as a single variable or value. | a | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, \(PRTY\) and \(WRSU\) are squares. Point \(Q\) is on \(PR\) and point \(X\) is on \(TY\) so that \(PQXY\) is a rectangle. Also, point \(T\) is on \(SU\), point \(W\) is on \(QX\), and point \(V\) is the point of intersection of \(UW\) and \(TY\), as shown. If the area of rectangle \(PQXY\) is 30, calcul... | 5.48 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.15625 | 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, find the limiting position o... | \left( \frac{4}{5}, \frac{2}{5} \right) | big_math | [
"Mathematics -> Calculus -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Among all six-digit numbers, how many have digits that sum up to 52? Provide your answer as a single integer count. | 21 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the circle whose center is on the y-axis, is tangent to the x-axis, and passes through the point (3, 1). Express your answer in standard form. | x^2+y^2-10y=0 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
A battery factory tests the lifespan of 10 batteries produced on a certain day, and the data obtained are as follows (in hours): 30, 35, 25, 25, 30, 34, 26, 25, 29, 21. Calculate the estimated average lifespan of these batteries. Express your answer in hours as a whole number. | 28 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
The relationship between the amount of pollutants P (mg/L) in the exhaust gas generated by a factory after filtration and the time t (h) during the filtration process is given by P = P0e^(-kt). If 20% of the pollutants were eliminated in the first 2 hours, calculate the time required to reduce the pollutants by 50%. Ex... | 6 | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Given the arithmetic sequence {a_n}, it is known that a_4 + a_8 = 8. Calculate the sum of the first 11 terms of this sequence, denoted as S_11. Express your answer as a single integer. | 44 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
The sequence 1, 4, 9, 16, 25, … can be rewritten as $1^2, 2^2, 3^2, 4^2, 5^2, …$. Find the general term formula for this sequence. Express your answer in the format $a_n =$ formula. | $a_n = n^2$ | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given that tan(α - π/4) = 1/2, calculate the value of (sin α + cos α)/(sin α - cos α). Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Find the sum of the coefficients of the polynomial obtained by expanding (x - 2y)^18. Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Given $ab=-3$, calculate the value of $a\sqrt{-\frac{b}{a}}+b\sqrt{-\frac{a}{b}}$. Express your answer as a single number. | 0 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
There are 5 math books, 4 physics books, and 5 chemistry books on a bookshelf. How many different ways are there to pick one book? Express your answer as a whole number. | 14 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given \( a, b, c \in \mathbf{R}^{+} \) and \( a + b + c = 1 \), calculate the integer part of \( M = \sqrt{3a + 1} + \sqrt{3b + 1} + \sqrt{3c + 1} \). Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given a set $M \subseteq \{2, 3, 5\}$, and $M$ contains at least one odd number, find the total number of such sets $M$. Express your answer as a whole number. | 6 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Combinations"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Someone deposited an amount of $a$ yuan on New Year's Day of 2008 in a one-year term account, with an annual compound interest rate of $x$ (interest tax not considered). Calculate the amount that can be withdrawn by New Year's Day of 2013. Express your answer as a mathematical expression involving $a$ and $x$. | $a(1+x)^5$ | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Count the number of integers whose absolute value is greater than 2 and less than 7. Express your answer as a whole number. | 8 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $$\frac {2-ai}{1+i}$$, where $i$ is the imaginary unit and $a\in\mathbb{R}$, find the value of $a$ such that the complex number is a pure imaginary number. Express your answer as a single real number. | 2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation \( p + \frac{1}{q + \frac{1}{r}} = \frac{25}{19} \), where p, q, and r are positive integers, calculate the value of q. Express your answer as a single integer. | 3 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given two skew lines $a$ and $b$ that form an angle of $50^\circ$, and a point $P$ in space, determine the number of lines passing through point $P$ that form an angle of $30^\circ$ with both lines $a$ and $b$. Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
ABCD is a trapezium inscribed in a circle centered at O. It is given that AB is parallel to CD, angle COD is three times angle AOB, and the ratio of AB to CD is 2:5. Find the ratio of the area of triangle BOC to the area of triangle AOB. Express your answer as a ratio. | \frac{3}{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
If $f(x)= \begin{cases} x^{3}+\sin x, & -1\leqslant x\leqslant 1 \\ 2, & 1 < x\leqslant 2\end{cases}$, then calculate the value of $\int_{-1}^{2}f(x)dx$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
In the Cartesian coordinate system, two points A(3, 1) and B(-1, 3) are known. Point C satisfies $\overrightarrow{OC} = \lambda_1 \overrightarrow{OA} + \lambda_2 \overrightarrow{OB}$ (where O is the origin), with $\lambda_1, \lambda_2 \in \mathbb{R}$ and $\lambda_1 + \lambda_2 = 1$. Find the equation that describes the... | x + 2y = 5 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given that point P is any point on the ellipse $x^2+4y^2=4$, and A $(4,0)$, if M is the midpoint of segment PA, derive the equation of the trajectory of point M in the standard form $(x-h)^2+a(y-k)^2=r^2$. | (x-2)^2+4y^2=1 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Let \( i = \sqrt{-1} \). Consider a sequence of complex numbers \( z_{1} = 0 \), and for \( n \geq 1 \), \( z_{n+1} = z_{n}^{2} + i \). Calculate the modulus of \( z_{111} \). Express your answer as a mathematical expression. | \sqrt{2} | big_math | [
"Mathematics -> Complex Analysis -> Sequences and Series"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Given sets M = { x | -1 < x - a < 2 } and N = { x | x^2 >= x }, if M union N equals the set of all real numbers, calculate the range of the real number a. Express your answer as an interval. | [-1, 1] | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $|\vec{a}|=4$, $|\vec{b}|=8$, and the angle between $\vec{a}$ and $\vec{b}$ is $60^{\circ}$, calculate the magnitude of the vector $2\vec{a}+\vec{b}$. Express your answer as a numerical value, including the square root symbol if necessary. | 8\sqrt{3} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the sum of the first 9 terms of an arithmetic sequence $\{a_n\}$ is 27, and $a_{10} = 8$, find the value of $a_{100}$. Express your answer as a single numerical value. | 98 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Let $A=\{x| -12 \lt 4x \lt 8\}$, $B=\{x| \frac{1}{x}＜1\}$. Calculate the intersection $A\cap B$ and express your answer in interval notation. | (-3,0)\cup (1,2) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Tyesha earns $5 per hour babysitting, and babysits for 7 hours in a particular week. If she starts the week with $20 in her bank account, deposits all she earns into her account, and does not withdraw any money, calculate the amount she has in her account at the end of the week. Express your answer in dollars as a whol... | 55 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Let the arithmetic sequence $\{a_n\}$ have a non-zero constant difference $d$, and $a_1=1$. If $a_1$, $a_3$, and $a_{13}$ form a geometric sequence, calculate the common difference $d$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
In the three-dimensional Cartesian coordinate system Oxyz, find the point symmetric to (-1,2,1) with respect to the plane xOz. Express your answer as a coordinate (x, y, z). | (-1,-2,1) | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
In a circle with a radius of 1m, find the length of the arc corresponding to a central angle of 60°. Express your answer in meters, using a fraction of pi (π) if necessary. | \frac{\pi}{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given that point $A(a,4)$ lies on the hyperbola $y=-\frac{4}{x}$, find the value of $a$. Express your answer as a single integer. | -1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\tan \alpha = 3$, calculate the value of $\frac {4\sin \alpha - 2\cos \alpha}{5\cos \alpha + 3\sin \alpha}$. Express your answer as a simplified fraction. | \frac{5}{7} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
A certain street has traffic lights at locations A, B, and C. The probabilities of cars passing through these locations due to green lights are 1/3, 1/2, and 2/3, respectively. Calculate the probability that a car will stop once at these locations due to encountering a red light. Express your answer as a simplified fra... | \dfrac{7}{18} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the function f(x) is an odd function, and when x > 0, f(x) = x^2 + 1/x, find the value of f(-1). Express your answer as a single number. | -2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
What is the opposite of 3? Express your answer as a single integer. | -3 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Ricardo has 2020 coins, some of which are 1-cent coins and the rest are 5-cent coins, with at least one 1-cent coin and at least one 5-cent coin. Find the difference between the maximum and minimum possible amounts Ricardo can have. Express your answer in cents as a whole number. | 8072 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
The vertex of a parabola is at the origin, and the equation of the directrix is x = 3. Find the equation of the parabola in the standard form y^2 = ax. | y^2 = -12x | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
A school uses systematic sampling to select 50 students from a total of 500 students for a questionnaire survey. The 500 students are numbered from 1 to 500. Within the range of 1 to 10, a number is randomly drawn, and if the number drawn is 3, use the definition of systematic sampling to calculate the number that shou... | 13 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Charles Dickens' 200th anniversary of birth was celebrated on February 7, 2012, which was a Tuesday. A year is a leap year if it is divisible by 400 or divisible by 4 but not by 100. Using this rule, calculate the day of the week on which Charles Dickens was born. Provide your answer as the full day of the week. | Friday | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given $a=5-2\sqrt{6}$, calculate the value of $a^{2}-10a+1$. Express your answer as a single number. | 0 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Find the value of $\cos 180^\circ$. Express your answer as a single number. | -1 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Let the complex numbers $z_1 = 2 - i$ and $z_2 = a + 2i$ (where $i$ is the imaginary unit and $a \in \mathbb{R}$). If $z_1z_2 \in \mathbb{R}$, find the value of $a$. Express your answer as a single real number. | 4 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence {a_n}, a_5 > 0 and a_4 + a_7 < 0. Find the term S_n in the sum of the first n terms of the sequence that corresponds to the maximum sum. Express your answer in the form S_n, specifying the value of n. | S_5 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Find the reciprocal of -2. Express your answer as a fraction. | -\frac{1}{2} | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
A magic box is defined such that when any pair of real numbers $(a, b)$ is entered into it, it produces a new real number: $a^2 + b + 1$. If the real number pair $(-2, 3)$ is put into it to get a real number $m$, and then the real number pair $(m, 1)$ is put into it, calculate the resulting real number. Express your an... | 66 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the circle with center at (1, -1) and tangent to the line x + y - sqrt(6) = 0. Express your answer in the standard form of the equation of a circle, including the center coordinates and the radius squared. | (x-1)^2 + (y+1)^2 = 3 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Given that θ is an acute angle, and $\cos\left(\theta + \frac{\pi}{12}\right) = \frac{\sqrt{3}}{3}$, calculate the value of $\cos\left(\frac{5\pi}{12} - \theta\right)$. Express your answer as a fraction with a radical. | \frac{\sqrt{6}}{3} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z= \frac{1-i}{1+i}$, find the conjugate of $z$. Express your answer as a complex number. | i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z$ that satisfies $z(1-i)=|1-i|+i$, determine the imaginary part of $z$. Express your answer as a fraction, including the square root symbol if necessary. | \frac{\sqrt{2}+1}{2} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
The four cities \( A, B, C, \) and \( D \) were built along a straight highway. The distance between \( A \) and \( C \) is \( 50 \ \text{km} \) and the distance between \( B \) and \( D \) is \( 45 \ \text{km} \). Additionally, it is known that the distance between the first and last city is \( 80 \ \text{km} \). Find... | 15 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the asymptotes of a hyperbola with foci on the x-axis are described by the equations y = ±(2/3)x, find the eccentricity of this hyperbola. Express your answer as a fraction with a square root. | \frac{\sqrt{13}}{3} | big_math | [
"Mathematics -> Geometry -> Non-Euclidean Geometry -> Hyperbolic Geometry"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
If each interior angle of a polygon is $135^{\circ}$, calculate the number of sides of the polygon. Express your answer as a whole number. | 8 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Polygons"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
When person A and person B play chess, the probability of A winning is 0.4, and the probability of A not losing is 0.9. Calculate the probability that the game ends in a draw. Express your answer as a decimal value between 0 and 1. | 0.5 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
A school has a total of 1800 students in grades 10 and 11. A stratified sampling method is used to select 90 students as a sample for a survey. If there are 42 grade 10 students in the sample, find the total number of grade 10 students in the school. Express your answer as a whole number. | 840 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Given an isosceles triangle with one side length of 5 units and another side length of 10 units, calculate the perimeter of the triangle. Express your answer in units as a whole number. | 25 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangulations"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Let \( a = 1 + 3^{1} + 3^{2} + 3^{3} + 3^{4} + \ldots + 3^{999} + 3^{1000} \). Calculate the remainder when \( a \) is divided by 4. Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Find the sum of all the roots of the polynomial $4x^3-8x^2-63x-9=0$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $i$ is the imaginary unit, and the complex number $z= \\frac{m}{1-i}$ ($m\\in\\mathbb{R}$), calculate the value of $m$ such that $|z|= \\int_{0}^{\\pi}(sinx- \\frac{1}{\\pi})dx$. Express your answer as a numerical value, including the ± symbol if necessary. | \\pm \\sqrt{2} | big_math | [
"Mathematics -> Algebra -> Complex Numbers",
"Mathematics -> Calculus -> Integral Calculus -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
A caterpillar starts from its hole and moves across the ground, turning 90 degrees either left or right after each hour. It moves 2m in the first hour, followed by 3m in the second hour, 4m in the third hour, and so on. Calculate the greatest distance it can be from its hole after seven hours. Express your answer in me... | 25 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
A student uses a calculator to find an answer but instead of pressing the \( x^2 \) key presses the \( \sqrt{x} \) key by mistake. The student's answer was 9. What should the answer have been if the student had pressed the correct key? Express your answer as a single integer. | 6561 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the exact value of $\cos 105^{\circ}\cos 45^{\circ}+\sin 105^{\circ}\sin 45^{\circ}$. Provide your answer in the form $\boxed{\text{your answer here}}$. | \frac{1}{2} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
In the geometric sequence {a_n} with a_n > 0 and q > 1, given that a_3 + a_5 = 20 and a_2 * a_6 = 64, calculate the sum S_6 of the first 6 terms. Express your answer as a single integer. | 63 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
A foreign investor plans to invest in 3 different projects across 4 candidate cities, with no more than 2 projects in the same city. Calculate the total number of different investment plans the investor can have. | 60 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a} = (1, 3)$ and $\overrightarrow{b} = (2, 1)$, if $\overrightarrow{a} + 2\overrightarrow{b}$ is parallel to $3\overrightarrow{a} + \lambda\overrightarrow{b}$, calculate the value of $\lambda$. Express your answer as a single numerical value. | 6 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
The geometric sequence {a<sub>n</sub>} satisfies a<sub>1</sub>=3, a<sub>1</sub>+a<sub>3</sub>+a<sub>5</sub>=21. Find the product of a<sub>2</sub> and a<sub>6</sub>. Express your answer as a single integer. | 72 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x) = \frac{1}{2}x^2 + 2xf'(2016) - 2016\ln x$, find the value of $f'(2016)$. Express your answer as a single integer. | -2015 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
A sequence of numbers is formed by raising 7 to consecutive powers: $7^1$, $7^2$, $7^3$, $7^4$, …, $7^{2009}$. How many terms in this sequence have a last digit of 3? Express your answer as a whole number. | 502 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Companies A, B, and C contract for 6 projects, with A contracting for 3 projects, B for 2 projects, and C for 1 project. Find the number of different contracting schemes. Express your answer as a whole number. | 60 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Combinations"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the third term of an arithmetic sequence $\{a_n\}$ is 3, find the sum of the first 5 terms of the sequence. Express your answer as a single integer. | 15 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = x^2 + 2x - 1 with a domain of [-2, 2], calculate the range of f(x). Express your answer as an interval, including the minimum and maximum values. | [-2, 7] | big_math | [
"Mathematics -> Precalculus -> Functions"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
If $\frac {\sin\alpha - \cos\alpha}{\sin\alpha + \cos\alpha} = \frac {1}{2}$, find the value of $\tan 2\alpha$. Express your answer as a fraction. | -\frac {3}{4} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
In an arithmetic sequence $\{a_n\}$, we have $a_9 = \frac{1}{2}a_{12} + 6$ and $a_2 = 4$. Let the sum of the first $n$ terms of $\{a_n\}$ be $S_n$. Then, calculate the sum of the first 10 terms of the sequence $\left\{\frac{1}{S_{n}}\right\}$. Express your answer as a fraction in simplest form. | \frac{10}{11} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
A regular octagon has a shaded trapezium within it. Find the ratio of the area of the shaded trapezium to the area of the whole octagon. Express your answer as a ratio in the format a:b, where a and b are integers. | 1:4 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Polygons"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Evaluate the expression: \\(\\frac{4 \\times 4 + 4}{2 \\times 2 - 2}\\). Provide your answer as a single number. | 10 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given $\log_{10}2=a$ and $\log_{10}3=b$, express $\log_{5}12$ in terms of $a$ and $b$. Provide your answer as a fraction. | \frac{2a+b}{1-a} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Logarithmic Functions"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Given set $A=\left\{\left(x,y\right)\left|\right.y=x+1,0\leqslant x\leqslant 1\right\}$ and set $B=\left\{\left(x,y\right)\left|\right.y=2x,0\leqslant x\leqslant 10\right\}$, find the intersection $A\cap B$. Express your answer as a set of ordered pairs (x, y). | \left\{\left(1,2\right)\right\} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, rectangles ABCD and EFGH are similar. DK:KC = 3:2. Find the ratio of the areas of rectangle ABCD to rectangle EFGH. Express your answer as a ratio in the form a:b or a fraction. | \frac{9}{4} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Given that point P (x, y) lies on the line 2x+y+5=0, find the minimum value of x^2+y^2. Express your answer as a single numerical value. | 5 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given the polynomial $x^{2}+mx+4$, use the complete square formula to derive the possible values of $m$ that allow the polynomial to be factored as a perfect square. Express your answer as specific numerical values or an algebraic expression for $m$. | \pm 4 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Given a line segment AB with a length of p units and its midpoint M, a perpendicular from M to AB intersects at R, and the length of MR is q units. A semicircle with a radius equal to 1/2 AB intersects AB at T. Formulate the quadratic equation whose roots are the lengths AT and TB. Express your answer in the form ax^2 ... | x^2 - px + q^2 = 0 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other",
"Mathematics -> Algebra -> Equations and Inequalities -> Quadratic Functions"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f satisfies f(x/(x-1)) = 1/x, and 0 < θ < π/2, calculate f(sec^2 θ). Express your answer as a trigonometric function of θ. | sin^2 θ | big_math | [
"Mathematics -> Algebra -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given \(f(x) = x^{2} + 2xf''(1) - 6\), calculate \(f''(1)\). Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Two identical, uniform tetrahedral dice are rolled simultaneously, each with the numbers 1, 2, 3, 5 on their faces. Let x be the sum of the numbers on the two bottom faces. Calculate the probability that x is not less than 6. Express your answer as a simplified fraction. | \frac{1}{2} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
What is the arithmetic square root of 4? Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given triangle ABC with internal angles ∠A, ∠B, and ∠C corresponding to sides a, b, and c respectively, where a, b, and c are rational numbers that satisfy the equation
(a + √2)^2 = (b + √2)(c + √2)
Find the measure of angle A in degrees. | 60^{\circ} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
There are 5 buses, 6 trains, and 2 ferries from place A to place B in one day. Calculate the total number of different ways a person can travel from place A to place B. Express your answer as a whole number. | 13 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
What is one-half of \(1.2 \times 10^{30}\)? Express your answer in scientific notation. | 6.0 \times 10^{29} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Given the numbers \(101, 148, 200, 512, 621\), how many of these cannot be expressed as the sum of two or more consecutive positive integers? Express your answer as a whole number. | 1 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Nicholas is counting the sheep in a flock as they cross a road. The sheep begin to cross the road at 2:00 p.m. and cross at a constant rate of three sheep per minute. After counting 42 sheep, Nicholas falls asleep. He wakes up an hour and a half later, at which point exactly half of the total flock has crossed the road... | 624 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
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