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 cos α = 1/4, calculate sin(π/2 - 2α). Express your answer as a simplified fraction. | -\dfrac{7}{8} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Given vector $\overrightarrow{a}=(1,1)$, and $4\overrightarrow{a}+\overrightarrow{b}=(4,2)$, calculate the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ in radians. | \frac{3\pi}{4} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = x^2 - 2x + m, find the value of the real number m such that the minimum value of the function on the interval [3, +∞) is 1. Express your answer as a single real number. | -2 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
The temperature at noon one day was -2℃, and the temperature at night dropped by 4℃ compared to noon. What was the temperature at night? Express your answer in degrees Celsius (℃) as an integer, including a negative sign if applicable. | -6 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given $a=2^{-\frac{1}{3}}, b=\log_{\frac{1}{4}}\frac{1}{5}, c=\log_{3}\frac{1}{4}$, determine the order of a, b, and c from largest to smallest. Express your answer as an inequality using > to separate the values. | b > a > c | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions",
"Mathematics -> Algebra -> Intermediate Algebra -> Logarithmic Functions"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence $\{a_n\}$, where $d=1$, and $S_{98}=137$, calculate the sum of $a_2 + a_4 + a_6 + \ldots + a_{98}$. Express your answer as a single numerical value. | 93 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Identify a plane figure that is analogous to a space parallelepiped, considering that opposite faces of a parallelepiped are parallel to each other. Express your answer as the name of the plane figure. | Parallelogram | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets $M={x\in\mathbb{Z}\mid-4 < x < 2}$ and $N={x\mid x^2 < 4}$, find the intersection $M\cap N$. Express your answer as a set. | \{-1,0,1\} | big_math | [
"Mathematics -> Set Theory -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Given sets $A=\{x|1\leqslant x \lt 4\}$ and $B=\{x|-2\leqslant x \lt 2\}$, find the intersection of sets $A$ and $B$, denoted as $A \cap B$, in interval notation as [lower bound, upper bound). | [1,2) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given a geometric sequence $\{a_n\}$, where $$\frac {a_{2}+a_{3}}{a_{1}+a_{2}}=2$$ and $a_{4}=8$, find the value of $a_{6}$. Express your answer as a single integer. | 32 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of $\frac{\sin 70^{\circ} \sin 20^{\circ}}{\cos^2 155^{\circ} - \sin^2 155^{\circ}}$. Express your answer as a simplified fraction. | \frac{1}{2} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Let $t_n = \\frac{n(n+1)}{2}$ be the $n$th triangular number. Calculate the sum
\\[
\\frac{1}{t_1} + \\frac{1}{t_2} + \\frac{1}{t_3} +... + \\frac{1}{t_{2002}}
\\]
and express your answer as a fraction in simplest form. | \\frac{4004}{2003} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Given a circle $C: x^{2}+y^{2}+Dx+Ey+3=0$, with its center lying on the line $x+y-1=0$, and the center in the second quadrant, with a radius of $\sqrt{2}$, find the values of $D$ and $E$. Express your answer as specific integer values for $D$ and $E$. | D=2, E=-4 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given a right prism $ABC-A'B'C'$ with volume $V$, points $P$ and $Q$ are on edges $AA'$ and $OC'$ respectively, and $AP = C'Q$. Calculate the volume of the tetrahedron $B-APQC$ in terms of the original volume $V$. Express your answer as a fraction of $V$. | \frac{1}{3} V | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Volume"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $\sqrt {3}\sin x + \cos x = 4 - m$, calculate the range of the real number $m$. Express your answer as an interval. | [2, 6] | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Handy Aaron helped a neighbor $1 \frac14$ hours on Monday, $50$ minutes on Tuesday, from 8:20 to 10:45 on Wednesday morning, and a half-hour on Friday. He is paid $\$3$ per hour. Calculate his total earnings for the week. Express your answer in dollars as a whole number. | 15 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Given the hyperbola y = (1-m)/x, when x > 0, y decreases as x increases. Find the range of m that satisfies this condition. Express your answer as an inequality involving m. | m < 1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
What is the least whole number n such that any set of n integers contains a subset of six elements whose sum is divisible by 6? | 11 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given vector $\overrightarrow{a}=({-1,\sqrt{7}})$, $|{\overrightarrow{b}}|=1$, and the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\frac{π}{4}$, calculate the magnitude of $\overrightarrow{a}-2\overrightarrow{b}$. Provide your answer as a single numerical value. | 2 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given the ellipse $\frac{x^{2}}{4} + \frac{y^{2}}{b^{2}} = 1$ with an eccentricity of $\frac{\sqrt{3}}{2}$, the line $l$ intersects the ellipse $C$ at points $A$ and $B$, with the midpoint of segment $AB$ being $M(-2, 1)$. Calculate the slope of line $l$. Express your answer as a single fraction or decimal value. | \frac{1}{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
If vector $\overrightarrow{a}=(2,3)$ and vector $\overrightarrow{b}=(-1,2)$, then find the coordinates of $\overrightarrow{a} + \overrightarrow{b}$. Express your answer as an ordered pair (x, y). | (1, 5) | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
There are 3 different movie tickets to be distributed among 10 people, with each person receiving at most one ticket. Calculate the total number of different distribution methods. Express your answer as a single integer. | 720 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Convert the decimal number 24 into binary. Express your answer in binary (base 2) using subscript (2) to denote binary, e.g., 11000_{(2)}. | 11000_{(2)} | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given the expressions a = 2ln 1.01, b = ln 1.02, and c = sqrt(1.04) - 1, determine the correct ordering of a, b, and c. Express your answer as a sequence of variables using standard comparison operators (>, <, =). | a > c > b | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
In how many ways can 9 identical balls be placed into 3 different boxes, with each box containing at least one ball? Express your answer as a whole number. | 28 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
A line $l$ passing through points $M(m, 3)$ and $N(1, m)$ is perpendicular to a line with a slope of $-1$. Find the value of $m$. Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Let $O$ be the origin, and the complex numbers corresponding to $\overrightarrow{OA}$ and $\overrightarrow{OB}$ are $2-3i$ and $-3+2i$, respectively. Calculate the complex number corresponding to $\overrightarrow{BA}$. Express your answer in the form $a+bi$. | 5-5i | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
An arithmetic sequence {a_n} has a common difference d!= 0, and a_1, a_3, a_4 form a geometric sequence. Let S_n denote the sum of the first n terms of {a_n}. Calculate the value of (S_4 - S_2) / (S_5 - S_3). Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Suppose that at the end of one year, one unit of money has depreciated by 10% compared to the beginning of that year. Find the smallest integer n such that after n years, one unit of money has depreciated by at least 90%. The value of log 3 accurate to three decimal places is 0.477. Express your answer as a single inte... | 22 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
In a regular quadrilateral prism $ABCDA'A'B'C'D'$ with vertices on the same sphere, $AB = 1$ and $AA' = \sqrt{2}$. Calculate the spherical distance between points $A$ and $C$. Express your answer in radians. | \frac{\pi}{2} | big_math | [
"Mathematics -> Geometry -> Non-Euclidean Geometry -> Spherical Geometry"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Find the minimal value of integer n that guarantees that among n sets, there exist at least three sets such that any of them does not include any other, or there exist at least three sets such that any two of them include the other. Express your answer as a single integer. | 5 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence $\left\{ a_n \right\}$, where $a_1=1$, $a_2=\dfrac{2}{3}$, and $\dfrac{1}{a_{n-1}}+\dfrac{1}{a_{n+1}}=\dfrac{2}{a_n}$ for $n\geqslant 2$, calculate the value of $a_6$. Express your answer as a simplified fraction. | \dfrac{2}{7} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $\frac{{x^2}}{9-k}+\frac{{y^2}}{k-4}=1$, which represents an ellipse with foci on the y-axis, determine the range of real number k that satisfies the conditions for such an ellipse. Express your answer as an interval or range of values for k. | (\frac{13}{2}, 9) | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
When using the completing the square method to solve the equation $x^{2}-2x=2$, what constant needs to be added to both sides to complete the square? Provide your answer as a single number. | 1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\overrightarrow{AP}= \frac{1}{3} \overrightarrow{PB}$ and $\overrightarrow{AB}=λ \overrightarrow{BP}$, find the value of λ. Express your answer as a single numerical value. | -\frac{4}{3} | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Geometry -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
In the expression $\frac{{ab}}{3}$, $-\frac{2}{3}abc$, $0$, $-5$, $x-y$, $\frac{2}{x}$, how many terms are monomials? Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Given the ellipse $\frac{x^{2}}{16} + \frac{y^{2}}{4} = 1$ and a point $P(1, 1)$ within it, find the equation of the chord with $P$ as its midpoint. Express your answer as a linear equation in the form $Ax + By + C = 0$. | x + 4y - 5 = 0 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Two luncheon scenarios are described with their costs. The first luncheon consists of 3 sandwiches, 7 cups of coffee, and 1 piece of pie, costing $3.15. The second luncheon consists of 4 sandwiches, 10 cups of coffee, and 1 piece of pie, costing $4.20. Find the total cost of a luncheon consisting of 1 sandwich, 1 cup o... | 1.05 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Given a right-angled triangle $\triangle ABC$ at $A$, with side lengths $AB = 8$ and $AC = 15$, find the perimeter of the triangle. Express your answer as a whole number. | 40 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z=(2+i)i$, calculate the product of the real part and the imaginary part of the complex number $z$. Express your answer as a single integer. | -2 | big_math | [
"Mathematics -> Algebra -> Complex Numbers -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Two chess players, A and B, are in the midst of a match. Player A needs to win 2 more games to be the final winner, while player B needs to win 3 more games. If each player has a probability of 1/2 to win any given game, what is the probability that player A will win the match? Express your answer as a fraction. | \frac{11}{16} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given two-dimensional vectors $\vec{a} = (1,2)$ and $\vec{b} = (-2,m)$, and $\vec{a}$ is parallel to $\vec{b}$, calculate the result of the vector operation $2\vec{a} + 3\vec{b}$. Express your answer as a vector in the form (x, y), including negative signs if applicable. | (-4, -8) | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = 2 sin(2x + π/6), if its graph is translated to the right by π/6 units, we obtain the graph of the function g(x). Find the equation of one of the symmetry axes of the graph of g(x). Express your answer in the form x =.... | \frac{\pi}{3} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given the operation $a*b$ defined as $a*b= \begin{cases} a, & \text{if } a\leq b \\ b, & \text{if } a>b \end{cases}$, find the range of the function $f(x)=2^x*2^{-x}$. Express your answer using interval notation. | (0,1] | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the function $f(x)$ is an even function, and shifting its graph one unit to the right results in an odd function, and $f(2)=-1$, calculate the sum of $f(1)+f(2)+...+f(2017)$. Express your answer as a single integer. | 0 | big_math | [
"Mathematics -> Algebra -> Functions",
"Mathematics -> Algebra -> Sequences and Series",
"Mathematics -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
The front view of a cylinder is a rectangle with an area of $6$. Find the lateral area of the cylinder. Express your answer in terms of $\pi$. | 6\pi | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $|\overrightarrow{a}|=10$, $|\overrightarrow{b}|=12$, and $\overrightarrow{a} \cdot \overrightarrow{b} = -60$, calculate the angle between vector $\overrightarrow{a}$ and vector $\overrightarrow{b}$ in degrees. Express your answer in the format $\boxed{[answer]°}$. | 120° | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, $\triangle PQR$ is isosceles. Given that $\angle P = 40^\circ$, calculate the value of $x$ in degrees. | 70 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z$ satisfies $\frac{1+2i}{z}=i$, calculate the imaginary part of $z$. Express your answer as a single number. | -1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
From the numbers 1, 2, 3,..., 7, select several numbers such that the sum of the even numbers is equal to the sum of the odd numbers. How many combinations satisfy this condition? Express your answer as a single integer. | 7 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Combinations"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z(1-2i)=2+i$, calculate the value of $z$. Express your answer as a simplified complex number. | i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
When a 10° angle is viewed under a 5x magnifying glass, what is the resulting angle measure in degrees? Express your answer in degrees, including the degree symbol (°). | 10° | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Among the real numbers $\frac{7}{8}$, $\sqrt{36}$, $-3\pi$, $\sqrt{7}$, and $1.41414141$, how many are rational numbers? Express your answer as a single integer. | 3 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Let $a_n$ be the coefficient of the $x^{n-1}$ term in the expansion of $(1+x)^{n+1}$. Calculate the sum of the first $n$ terms of the sequence $\left\{ \frac{1}{a_n} \right\}$. Express your answer as a single fraction. | \frac{2n}{n+1} | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
A school's wind band consists of 100 boys and 80 girls, and its string band consists of 80 boys and 100 girls. In total, 230 students participated in the two bands, with 60 boys participating in both. Calculate the number of girls who participated in the wind band but not in the string band. | 10 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
For the quadratic equation 2x^2 + x - k = 0 to have no real roots, find the range of values for k. Express your answer as an inequality involving k. | k < -\frac{1}{8} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Points $A$ and $B$ lie on a circle centered at $O$, and $\angle AOB = 60^\circ$. A second circle is internally tangent to the first and tangent to both $\overline{OA}$ and $\overline{OB}$. Calculate the ratio of the area of the smaller circle to that of the larger circle. Express your answer as a simplified fraction. | \frac{1}{9} | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Using the ten digits 0, 1, 2, 3, …, 9 and the imaginary unit $i$, calculate the number of distinct imaginary numbers that can be formed. Express your answer as a single integer. | 90 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
There are 8 products, of which 3 are defective. If X represents the number of defective products drawn when taking 3 products at random, calculate P(X≤1). Express your answer as a simplified fraction and include your calculation steps for clarity. | \frac{5}{7} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Combinations"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Given the universal set $U = \{-1, 0, 1, 2\}$ and the set $P = \{x \in \mathbb{Z} \,|\, -\sqrt{2} < x < \sqrt{2}\}$, calculate the complement of $P$ in $U$, denoted $\complement_U P$. Express your answer as a set using standard set notation. | \{2\} | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence {a_n}, the sum of the first n terms is denoted as s_n. If a_1 + a_9 + a_{11} = 30, find the sum of the first 13 terms, s_{13}. Express your answer as a single integer. | 130 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
A piece of wood has an unknown length. When measuring it with a rope, there are 4.5 feet of rope left. When folding the rope in half and measuring the wood again, there is 1 foot of rope left. Let x be the length of the rope and y be the length of the wood. Derive the set of equations that represents these conditions. ... | \] | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Linear Equations"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence {a_n} satisfies 3a_{n+1} + a_n = 4 for n >= 1, and a_1 = 9, with the sum of the first n terms being S_n, find the smallest integer n such that |S_n - n - 6| < 1/125. Express your answer as a single integer. | 7 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Let $a$, $b \in \mathbb{R}$, $c \in [0,2\pi)$. If for any real number $x$, the equation $2\sin (3x- \frac {\pi}{3})=a\sin (bx+c)$ holds, then how many sets of $(a, b, c)$ satisfy the condition? Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
A sequence of integers has a mode of 32, a mean of 22, a smallest number of 10, and a median of m. If m is replaced by m+10, the new sequence has a mean of 24 and a median of m+10. If m is replaced by m-8, the new sequence has a median of m-4. Find the value of m. Express your answer as a single integer. | 20 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
If the domain of the function $y=f(x)$ is $(0,2)$, find the domain of the function $y=f(-2x)$. Express your answer as an interval in standard interval notation. | (-1,0) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of the expression: $1-\sqrt{9}$. Express your answer as a single integer. | -2 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Let set $A=\{x|x+2=0\}$, and set $B=\{x|x^2-4=0\}$. Calculate the intersection of sets A and B, $A \cap B$. Express your answer as a set, including all elements of the intersection. | \{-2\} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
In a factory, Erika assembles 3 calculators in the same amount of time that Nick assembles 2 calculators. Also, Nick assembles 1 calculator in the same amount of time that Sam assembles 3 calculators. Calculate the total number of calculators that can be assembled by Nick, Erika, and Sam in the same amount of time as E... | 33 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Given the three lines $l_1: ax+2y+6=0$, $l_2: x+y-4=0$, and $l_3: 2x-y+1=0$ intersect at the same point, calculate the real number $a$. Express your answer as a real number. | -12 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
The base radius of a cone is 3, and its height is 4. Calculate the lateral surface area of the cone. Express your answer in terms of π. | 15π | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Surface Area"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Express the decimal number $0.000016$ in scientific notation. Provide your answer in the form $a \times 10^b$, where $a$ is the coefficient and $b$ is the exponent. | 1.6 \times 10^{-5} | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{m}=(-1,2)$ and $\overrightarrow{n}=(1,\lambda)$, if $\overrightarrow{m} ot \overrightarrow{n}$, calculate the angle between $\overrightarrow{m}+2\overrightarrow{n}$ and $\overrightarrow{m}$ in radians. Express your answer using $\pi$. | \frac{\pi}{4} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Given the piecewise function $$f(x)= \begin{cases} x+5, & x>1 \\ 2x^{2}+1, & x\leq 1 \end{cases}$$, find the value of $f[f(1)]$. Express your answer as a single number. | 8 | big_math | [
"Mathematics -> Algebra -> Functions"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
In triangle $\triangle ABC$, $A=105^{\circ}$, $B=45^{\circ}$, and $AC=2$. Calculate the length of side $AB$. Express your answer as a simple radical expression. | \sqrt{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence $\{a_{n}\}$ satisfies the rule:
\[
{a}_{n+1}=\left\{\begin{array}{ll}
2{a}_{n}, & 0\leq{a}_{n}<\frac{1}{2},\\
2{a}_{n}-1, & \frac{1}{2}\leq{a}_{n}<1,
\end{array}\right.
\]
and knowing that ${a}_{1}=\frac{2}{5}$, calculate the value of ${a}_{2023}$. Express your answer as a fraction in simplest form... | \frac{3}{5} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
A renovation project has a labor budget of 2000 yuan. The wage of a carpenter is 50 yuan per person, and the wage of a mason is 40 yuan per person. Let the number of carpenters be x and the number of masons be y. Formulate the linear inequality that represents the relationship between the number of workers hired and th... | 5x + 4y ≤ 200 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Given the hyperbola equation $ rac{x^2}{16} - rac{y^2}{9} = 1$, find the equation of its asymptotes. Express your answer in the form $y = mx$ where $m$ is the slope. | y = \pm \frac{3}{4}x | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
On the highway, speed limit signs are posted every 4 kilometers starting from 3 kilometers, and speed monitoring devices are installed every 9 kilometers starting from 10 kilometers. Given that the first time both types of facilities are encountered simultaneously is at 19 kilometers, calculate the distance in kilomete... | 55 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given that {a_n} is an arithmetic sequence, and the sum a_2 + a_3 + a_{10} + a_{11} = 48, find the value of a_6 + a_7. Express your answer as a single integer. | 24 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Sequences and Series"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence $\{a_n\}$, where $a_1=2$, and $a_5+a_7=2a_4+4$, find the value of $a_3$. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = sin(omega*x - pi/6) + 1/2, where x is a real number, and f(alpha) = -1/2 and f(beta) = 1/2. If the minimum value of |alpha - beta| is 3*pi/4, calculate the value of omega. Express your answer as a fraction. | \frac{2}{3} | big_math | [
"Mathematics -> Precalculus -> Functions",
"Mathematics -> Calculus -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Khelen wants to color each quadrilateral in a tessellation of the plane so that no two quadrilaterals that meet (even at a point) have the same color. What is the smallest number of colors he needs? Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Discrete Mathematics -> Graph Theory -> Other"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
A box contains labels numbered from 1 to 10. Two labels are randomly selected without replacement. What is the probability that the numbers on the two labels are consecutive integers? Express your answer as a simplified fraction. | \frac{1}{5} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Sam drove a total of 96 miles in 90 minutes. His average speed during the first 30 minutes was 60 mph, and his average speed during the second 30 minutes was 65 mph. Calculate Sam's average speed, in miles per hour (mph), during the last 30 minutes. | 67 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
In right triangle $ABC$, $\angle C=90^{\circ}$, $BC=3$, $AB=4$. Calculate the value of $\sin A$. Express your answer as a single ratio. | \frac{3}{4} | big_math | [
"Mathematics -> Trigonometry -> Trigonometric Functions -> Other"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
The function f(n) defined on the set of natural numbers N is given by:
f(n) = {
n - 3 (n >= 1000)
f[f(n + 7)] (n < 1000)
}
Calculate the value of f(90). Express your answer as a single integer. | 999 | big_math | [
"Mathematics -> Discrete Mathematics -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Find the conjugate of the complex number $\frac{2+i}{1-2i}$. Express your answer as a complex number in its simplest form. | -i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence, let S_n be the sum of the first n terms. If S_8 - S_4 = 12, find the value of S_{12}. Express your answer as a single integer. | 36 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
The derivative f'(x) of the function f(x) satisfies the relationship f(x) = x^2 + 2xf'(2) - ln(x). Find the value of f'(2). Express your answer as a single value, which can be a fraction. | -\frac{7}{2} | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence $\{a_n\}$ that satisfies $a_{n+2} = a_{n+1} + a_n$, with initial terms $a_1 = 1$ and $a_2 = 2$, calculate the sum of the first 5 terms of this sequence. | 19 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Given the ellipse $\frac{x^2}{4} + \frac{y^2}{3} = 1$, let $(F_1, F_2, P)$ represent the left and right foci and any point on the ellipse, respectively. Calculate the range of values for $|PF_1||PF_2|$ and express your answer as a closed interval. | [3, 4] | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given that $f(x)$ is an increasing function on $\mathbf{R}$, for real numbers $a$ and $b$, if $a+b > 0$, compare $f(a)+f(b)$ and $f(-a)+f(-b)$. Express your answer using inequality notation. | f(a)+f(b) > f(-a)+f(-b) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function \( f(x) \) satisfies the condition \( f(x+2) = \frac{1}{f(x)} \) for any \( x \), and \( f(1) = -5 \), calculate the value of \( f[f(5)] \). Express your answer as a fraction, including the negative sign if applicable. | -\frac{1}{5} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
A person walks a distance of 378 li, walking briskly on the first day, and from the second day onwards, the distance walked each day is half of the previous day's distance. After walking for 6 days, the destination is reached. Find the distance walked on the second day. Express your answer in li as a whole number. | 96 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
A rectangular prism has a volume of 12 cm³. A new prism is formed by doubling the length, doubling the width, and tripling the height of the original prism. Calculate the volume of this new prism. Express your answer in cubic centimeters (cm³) as a whole number. | 144 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Volume"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Find the coefficient of a^(-1/2) in the expansion of (a - 1/sqrt(a))^7. | -21 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Binomial Expansion"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given a sequence {a_n} (n >= 1) satisfying a_{n+2} = a_{n+1} - a_n, and a_2 = 1. If the sum of the first 2011 terms of the sequence is 2012, calculate the sum of the first 2012 terms. | 2013 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
In the product shown, B is a digit. Find the value of B that makes the equation true:
\[\begin{array}{rr} &\text{B}2 \\ \times& 7\text{B} \\ \hline &6396 \\ \end{array}\]
Express your answer as a single digit. | 8 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Find the derivative of $\sin\left(\frac{\pi}{6}\right)$. Express your answer as a single number. | 0 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
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