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 |
|---|---|---|---|---|---|
A child has a set of 96 different blocks. Each block can be either plastic or wood, come in three sizes (large, medium, small), be one of four colors (blue, green, red, yellow), and have one of four shapes (circle, hexagon, square, triangle). Calculate the number of blocks that differ in exactly two attributes from a b... | 29 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Combinations"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f\left(x\right)=\left\{\begin{array}{l}{f(x-1),}&{x>-2}\\{x^2+2x-3,}&{x\leq-2}\end{array}\right.$, calculate $f\left(f\left(1\right)\right)$. Express your answer as a single numerical value. | 0 | big_math | [
"Mathematics -> Precalculus -> Functions"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given that a direction vector of line l is (2,2,-2), and a normal vector of plane α is (1,1,-1), describe the relationship between line l and plane α using standard geometric notation (e.g., ⊥, ∥, ⊂). | l ⊥ α | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the function
$$
f(x)=
\begin{cases}
\ln x, & \text{if } x > 1 \\
2x+\int_{0}^{m}3t^{2}dt, & \text{if } x\leqslant 1
\end{cases}
$$
and the condition $f(f(e))=29$, find the value of $m$. Express your answer as a single numerical value. | 3 | big_math | [
"Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, the line segment has slope -3/2 and extends from the point (0, b) to the point (8, 0). Calculate the value of b. Express your answer as a single integer. | 12 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Let {a_n} be an arithmetic sequence, and let S_n denote the sum of the first n terms. If S_{11} = 44, find the sum of a_4, a_6, and a_8. Express your answer as a single integer. | 12 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Let \(f(x)=\begin{cases}1, & x > 0 \\ 0, & x=0 \\ -1, & x < 0\end{cases}\) and \(g(x)=\begin{cases}1, & x \text{ is a rational number} \\ 0, & x \text{ is an irrational number}\end{cases}\). Evaluate \(f(g(\pi))\). Express your answer as a single number. | 0 | big_math | [
"Mathematics -> Functions -> Other"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Let $z=\frac{3-i}{1+i}$. Calculate the magnitude of $z$, denoted as $|z|$. Express your answer as a numerical value. | \sqrt{5} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
There are some candies. If you eat 3 candies every day, they will be finished in a little over ten days, and on the last day, you eat only 2 candies. If you eat 4 candies every day, they will be finished in less than ten days, and on the last day, you eat 3 candies. Calculate the original number of candies. Express you... | 35 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Kamile turned her computer off at 5 p.m. Friday, at which point it had been on for exactly 100 hours. At what time and day had Kamile turned her computer on? Express your answer in 12-hour clock format (e.g., 1 p.m.) and include the day of the week. | 1 p.m. Monday | big_math | [
"Applied Mathematics -> Math Word Problems -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given a sequence $\{a_n\}$ satisfying $a_1=1$, $a_2=1$, $a_{n+1}=|a_n-a_{n-1}|$ (for $n\geq2$), find the sum of the first 2013 terms of this sequence. Express your answer as a single integer. | 1342 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
An equilateral triangle and a regular hexagon have perimeters of the same length. Calculate the ratio of the area of the equilateral triangle to the area of the regular hexagon. Express your answer as a simplified ratio. | \frac{2}{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Polygons"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
A function f(x) is defined on the set of real numbers and is odd. It satisfies f(3) = 0, and when x > 0, the inequality f(x) > -xf'(x) holds. Determine the number of zeros of the function g(x) = xf(x) + log|x+1|. Express your answer as a whole number. | 3 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = x^3 - 3x - 1, for any x_1, x_2 within the interval [-3, 2], find the minimum value of the real number t such that |f(x_1) - f(x_2)| ≤ t. Express your answer as a single real number. | 20 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\sin (\pi-\alpha)=\log _{8} \frac{1}{4}$ and $\alpha\in\left(-\frac{\pi}{2},0\right)$, calculate the value of $\cos (\pi+\alpha)$. Express your answer as a single value, including any necessary radicals or fractions. | -\frac{\sqrt{5}}{3} | big_math | [
"Mathematics -> Applied Mathematics -> Other",
"Mathematics -> Trigonometry -> Other"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Two circles with radii 13 and 5 intersect, and the distance between their centers is 12. Find the length of their common chord. Express your answer as a single number. | 10 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Evaluate the integral $\int\_{-1}^{1}{\sqrt{1-{{x}^{2}}}}dx$. Express your answer as a fraction or multiple of pi if necessary. | \frac{\pi}{2} | big_math | [
"Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given $a=\log_{2}0.3$, $b=2^{0.3}$, $c=0.3^{0.2}$, determine the correct order of these values from greatest to least. Express your answer in the format "x > y > z", where x, y, and z are the values of a, b, and c. | b > c > a | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation of the hyperbola $\dfrac{x^2}{4}-y^2=1$, find the equation of its asymptotes. Express your answer in the form $y = \pm \dfrac{x}{a}$. | y = \pm \dfrac{x}{2} | big_math | [
"Mathematics -> Geometry -> Non-Euclidean Geometry -> Other"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Solve the inequality $1-x \geqslant 0$ to find the condition that makes the square root $\sqrt{1-x}$ meaningful. Express your answer as an inequality in terms of $x$. | x \leqslant 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
If the temperature at noon on a certain day is 1℃, and it decreases by 3℃ by evening, calculate the temperature in the evening. Express your answer in degrees Celsius (℃) as an integer, including a negative sign if applicable. | -2 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
An ice ballerina rotates at a constant angular velocity at one particular point. That is, she does not translationally move. Her arms are fully extended as she rotates. Her moment of inertia is $I$. Now, she pulls her arms in and her moment of inertia is now $\frac{7}{10}I$. Calculate the ratio of the new kinetic energ... | \frac{10}{7} | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a} = (1, 2)$ and $\overrightarrow{b} = (2, 1)$, if the vector $\overrightarrow{a} - \lambda \overrightarrow{b}$ is collinear with vector $\overrightarrow{c} = (5, -2)$, calculate the value of λ. Express your answer as a scalar value. | \frac{4}{3} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Given that point P(x,y) is any point on the ellipse x^2/4 + y^2 = 1, and point Q(1,0), find the minimum value of |PQ| (the distance between points P and Q). Express your answer in simplest radical form if necessary. | \frac{\sqrt{6}}{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{m}$ and $\overrightarrow{n}$, where $|\overrightarrow{m}|=|\overrightarrow{n}|=1$ and $|3\overrightarrow{m}-2\overrightarrow{n}|=\sqrt{7}$, calculate the projection vector of $\overrightarrow{m}$ onto the direction of $\overrightarrow{n}$. Express your answer as a vector in terms of $\ove... | \frac{1}{2}\overrightarrow{n} | big_math | [
"Mathematics -> Linear Algebra -> Vectors"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of points common to the graphs of $(x-y+2)(3x+y-4)=0$ and $(x+y-2)(2x-5y+7)=0$. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given the distance r from the origin to point P with coordinates (x, y), and the expressions s = y/r and c = x/r, calculate the range of s^2 - c^2. Express your answer as an interval, including the minimum and maximum values of the range. | [-1, 1] | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Evaluate the definite integral $\int_{0}^{\frac{\pi}{2}} \cos x \, dx$. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Let $S_{n}$ represent the sum of the first $n$ terms of the arithmetic sequence ${a_{n}}$. If $\frac{S_{1}}{S_{4}}=\frac{1}{10}$, then find the ratio $\frac{S_{3}}{S_{5}}$. Express your answer as a simplified fraction. | \frac{2}{5} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the given expression:
$$
\frac{bx(a^{2}x^{2} + 2a^{2}y^{2} + b^{2}y^{2})}{bx + ay} + \frac{ay(a^{2}x^{2} + 2b^{2}x^{2} + b^{2}y^{2})}{bx + ay}
$$
Express your answer as a simplified algebraic expression. | (ax + by)^{2} | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given a sequence $\{a_n\}$ satisfying $a_{1}=1$, $a_{2}=1$, and $a_{n+2}=a_{n}+a_{n+1}$ for all $n\in\mathbb{N}^*$, compute the value of $a_{6}$. Express your answer as a single integer. | 8 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
A number $b$ is randomly taken from the interval $[-2, 2]$. If the probability that the line $y = x + b$ intersects with the circle $x^2 + y^2 = a$ is $\frac{1}{2}$, then calculate the value of $a$. Express your answer as a common fraction, simplified if possible. | \frac{1}{2} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
There are 5 shmacks in 2 shicks, 3 shicks in 5 shures, and 2 shures in 9 shneids. Calculate the number of shmacks in 6 shneids. Express your answer as a whole number. | 2 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
A certain supermarket is offering discounts for the Spring Festival. If a certain item is sold at a price of $20$ yuan after an $80\%$ discount, calculate the original price of the item. Express your answer in yuan as a whole number. | 25 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Find the smallest natural number n such that the equation \(\left\lfloor \frac{10^{n}}{x} \right\rfloor = 2006\) has an integer solution for x. Express your answer as the smallest natural number n. | 7 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of intersection points of the curves $x=y^2$ and $y=x^2$. Express your answer as a whole number. | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Given $A=\{y\in \mathbb{N}|y=-x^{2}+4x,x\in \mathbb{Z}\}$ and $B=\{x|\ln x > 1\}$, find the intersection of $A$ and the complement of $B$ in $\mathbb{R}$. Express your answer as a set of numbers. | \{0\} | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Precalculus -> Functions"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence {a_n}, it is known that a_2 + a_10 = 16. Find the value of a_4 + a_6 + a_8. Express your answer as a single numerical value. | 24 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $f(x) = ax^2 + bx$ is an even function defined on the interval $[a-1, 2a]$, find the value of $a + b$. Express your answer as a single fraction or integer. | \frac{1}{3} | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $(1+bi)i=-1+i$, find the value of $b$. Express your answer as a single numerical value. | 1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Paula the painter had just enough paint for 30 identically sized rooms. Unfortunately, on the way to work, three cans of paint fell off her truck, so she had only enough paint for 25 rooms. How many cans of paint did she use for the 25 rooms? Express your answer as a whole number. | 15 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(\lambda,1,3)$ and $\overrightarrow{b}=(0,-3,3+\lambda)$, if $\overrightarrow{a} \perp \overrightarrow{b}$, calculate the value of the real number $\lambda$. Express your answer as a single real number. | -2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of $(\tan 30^\circ)^2 + (\sin 45^\circ)^2$. Express your answer as a simplified fraction in the form $\boxed{\frac{a}{b}}$, where $a$ and $b$ are integers. | \frac{5}{6} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Find the coordinates of the point on the circle $x^{2}+y^{2}=4$ that is farthest from the line $4x+3y-12=0$. Express your answer as a coordinate pair (x, y), including negative signs if applicable. | (- \frac {8}{5},- \frac {6}{5}) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given a point M in the first quadrant on the hyperbolic function y = 5/x, draw MA perpendicular to the x-axis at point A, and MB perpendicular to the y-axis at point B. If MA and MB intersect the hyperbolic function y = 3/x at points C and D respectively, what is the ratio CD:AB? Express your answer as a ratio, includi... | 2:5 | big_math | [
"Mathematics -> Geometry -> Non-Euclidean Geometry -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given a parallelogram $ABCD$ where $BD=2$ and $2\overrightarrow{AD}•\overrightarrow{AB}=|\overrightarrow{BC}{|^2}$, find the length of $\overrightarrow{AB}$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Given set A = {0, a, 1} and set B = {x | 0 < x ≤ 1}, if there are two elements in A∩B, then find the range of the real number a. Express your answer as an interval in standard notation. | (0, 1) | big_math | [
"Mathematics -> Set Theory -> Other"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Seven boxes contain exactly 10 cookies each. If the cookies are shared equally among 5 people, how many cookies does each person receive? Express your answer as a whole number. | 14 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
A quadrilateral is bounded by the lines with equations \( x=0 \), \( x=4 \), \( y=x-2 \), and \( y=x+3 \). Find the area of this quadrilateral. Express your answer as a whole number. | 20 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Add the decimal numbers 0.9 and 0.09. Express your answer as a decimal number. | 0.99 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Decimals"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{AB}=\left(2,3\right)$ and $\overrightarrow{AC}=\left(3,t\right)$, and $|\overrightarrow{BC}|=1$, calculate the dot product $\overrightarrow{AB}\cdot \overrightarrow{BC}$. Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
What is the largest possible area of a quadrilateral with sides 1, 4, 7, and 8? Express your answer as a single number. | 18 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Find the derivative of y = x^2 at x = 1. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x)= \begin{cases} \log_{2}(x^{2}+x+a),x\geqslant 1 \\ 1-x^{2},x < 1 \end{cases}$, where the range of the function is $\mathbb{R}$, determine the range of the real number $a$. Express your answer as an interval. | (-2,0] | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Calculus -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Find the second term of the binomial expansion of $(x- \\frac {2}{x})^{6}$. Express your answer as a polynomial term, including variables and coefficients. | -12x^4 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Arrange four students at three different locations—A, B, and C—for a practical activity, ensuring that at least one student is assigned to each location. Calculate the total number of distinct arrangement plans. Express your answer as a single numerical value. | 36 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
A cube with edge length n (where n > 2 and n ∈ N) is entirely painted black on its surface. The cube is then divided into n^3 smaller cubes, each with edge length 1. If the number of smaller cubes that have exactly one face painted black is equal to the number of smaller cubes that have no faces painted black, calculat... | 8 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Volume"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Given a positive geometric sequence {a<sub>n</sub>} that satisfies a<sub>1</sub> - a<sub>2</sub> = 8, a<sub>3</sub> - a<sub>4</sub> = 2, and the product a<sub>1</sub>a<sub>2</sub>a<sub>3</sub>...a<sub>n</sub> = 1, find the value of n. Express your answer as a whole number. | 9 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given $x^2 = 4$ and $y^2 = 9$, find the possible values of $(x-y)^2$. Express your answer as a list of numerical values. | 1, 25 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors a = (1, k) and b = (2, 2), and that a + b is collinear with a, find the value of k. Express your answer as a single numerical value. | 1 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
There are 99 positive integers whose sum is 101101. What is the greatest possible value of the greatest common divisor (gcd) of these 99 integers? Express your answer as a single integer. | 101 | big_math | [
"Mathematics -> Number Theory -> Greatest Common Divisors (GCD)"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Three circles with radii 2, 3, and 3 touch each other. Find the area of the triangle formed by joining the centers of these circles. Express your answer as a whole number. | 12 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $x^3 = a_0 + a_1(x-2) + a_2(x-2)^2 + a_3(x-2)^3$, calculate the value of $a_1 + a_2 + a_3$. Express your answer as a single integer. | 19 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
In triangle $ABC$, with angles $A$, $B$, $C$ opposite sides $a$, $b$, $c$ respectively, if $a=3$, $C=120^{\circ}$, and the area of $\triangle ABC$ is $S= \frac{15 \sqrt{3}}{4}$, then find the value of side $c$. Express your answer as a single number. | 7 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = a*sin(πx + α) + b*cos(πx + β) + 4, where a, b, α, β are non-zero real numbers, and f(2011) = 5, calculate the value of f(2012). Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the 5th term of an arithmetic sequence $a_5=8$, and $a_1+a_2+a_3=6$, calculate the common difference $d$. Express your answer as a single numerical value. | 2 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Sequences and Series"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
If the graph of the quadratic function y = x^2 is shifted 1 unit to the left, what is the analytical expression of the shifted quadratic function? Express your answer as a function of x, using standard function notation (e.g., y = f(x)). | y = (x + 1)^2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Quadratic Functions"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
What is the opposite of the number 2022? Provide your answer as a single integer. | -2022 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
What is the value of \(\frac{200+10}{20+10}\)? Express your answer as a single number. | 7 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Given the operation $$\begin{bmatrix} a & b \\ c & d\end{bmatrix}$$• $$\begin{bmatrix} e \\ f\end{bmatrix}$$= $$\begin{bmatrix} ae+bf \\ ce+df\end{bmatrix}$$, and α+β=π and α-β= $$\frac {π}{2}$$, calculate the result of $$\begin{bmatrix} \sinα & \cosα \\ \cosα & \sinα\end{bmatrix}$$• $$\begin{bmatrix} \cosβ \\ \sinβ\en... | \begin{bmatrix} 0 \\ 0\end{bmatrix} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Matrices"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation of motion $s = 2t^2$, find the increment in displacement during the time interval from $2$ seconds to $(2+d)$ seconds. Express your answer as an algebraic expression involving $d$. | 8d + 2d^2 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
The function f(x) satisfies f(x+2)f(x) = 1 for any real number x. If f(2) = 2, calculate the value of f(2016). | \frac{1}{2} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the result of $\left(-40\right)\div 5$. Express your answer as a single integer. | -8 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
The line with equation y = x is translated 3 units to the right and 2 units down. Find the y-intercept of the resulting line. Express your answer as a single integer. | -5 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
If \((2)(3)(4) = 6x\), what is the value of \(x\)? Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Solve for $x$ in the equation $\frac{1}{{2+x}}=\frac{2}{{3x-1}}$. Express your answer as a single value, $x=$. | x=5 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Rational Equations"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Given the inequality $\frac{1}{m}+\frac{2}{{1-2m}}\geqslant k$, where $0 < m < \frac{1}{2}$, find the maximum value of $k$. Express your answer as a single numerical value. | 8 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
In the Maths Premier League, teams get 3 points for a win, 1 point for a draw, and 0 points for a loss. Last year, my team played 38 games and got 80 points. We won more than twice the number of games we drew and more than five times the number of games we lost. Calculate the number of games my team drew. Express your ... | 11 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
A bricklayer is able to lay 8 meters of wall per day. How many meters of wall can this bricklayer lay in 15 days? Express your answer in meters as a whole number. | 120 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the derivative of the function $f(x)$ is ${f'}(x)$, if ${f'}(2)=4$, calculate the limit as $\Delta x$ approaches $0$ of the expression $\frac{f(2)-f(2-\Delta x)}{2\Delta x}$. Express your answer as a single numerical value. | 2 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence {a_n}, it is known that a_2 + a_{10} = 16. Find the sum of a_4, a_6, and a_8. | 24 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $3x^3 - x = 1$, calculate the value of the expression $9x^4 + 12x^3 - 3x^2 - 7x + 2001$. Express your answer as a single integer. | 2005 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Polynomial Operations"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given two lines $l_1$: $y=k_1x+4$ and $l_2$: $y=k_2x-2$ intersect at a point on the x-axis, find the ratio of $k_1$ to $k_2$. Express your answer as a simplified ratio or a single number. | -2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
If the average of 4, 20, and x is equal to the average of y and 16, and x and y are positive numbers, find the ratio x:y. Express your answer as a ratio (e.g., 3:2). | 3:2 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
The school plans to select 3 students from 5 to serve as volunteers for 3 different events at the Asian Games, among which student Zhang cannot serve as a volunteer for the shooting event. Calculate the total number of different arrangements possible under this restriction. Express your answer as a single integer. | 48 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given that a > 1 and b is a positive rational number, and a^b + a^{-b} = 2\\sqrt{2}, find the value of a^b - a^{-b}. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Exponential Functions -> Other"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence $\{a_n\}$ satisfies $a_1=0$, $a_2=-2$, and $a_{n+2}=a_{n+1}-a_n$, calculate the value of $a_{2011}$. Express your answer as a single integer. | 0 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Grandma told Xiaoming: "There were 53 Sundays in the year 2006." Clever Xiaoming immediately told Grandma: "Determine the day of the week for New Year's Day in 2007." Express your answer as the full day of the week. | Monday | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a > 0$, find the minimum value of the expression $a + \frac{a+4}{a}$. Express your answer as a single numerical value. | 5 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors a = (2,1) and b = (sin θ, cos θ), if a ∥ b, find the value of tan(θ + 45°). Express your answer as a single numerical value. | -3 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
A rectangular solid has dimensions of length, width, and height as 2, 2, and 1, respectively. If its vertices lie on the same spherical surface, calculate the surface area of the sphere. Express your answer in terms of π. | 9\pi | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Surface Area"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
In the cube $ABCD-A_1B_1C_1D_1$, it is known that $M$ is the midpoint of $BB_1$. Calculate the cosine value of the dihedral angle between planes $M-CD_1$ and $A$. Express your answer in simplified form. | \frac{\sqrt{3}}{3} | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
In a geometric sequence $\{a_n\}$ where each term is positive, if $a_2a_5a_8=8$, then find the value of $\log_2a_4+\log_2a_6$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Given $x = \\frac{1}{2} \\left( \\sqrt[3]{2 + \\sqrt{5}} + \\sqrt[3]{2 - \\sqrt{5}} \\right)$, evaluate the expression $8x^3 + 6x - 1$ and find the largest integer not exceeding the result. Express your answer as a single integer. | 3 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $x < 0$ and $y < 0$, and the equation $3x + y = -2$, find the maximum value of $xy$. Express your answer as a single fraction. | \frac{1}{3} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $3\left(x+y\right)^{2}-7\left(x+y\right)+8\left(x+y\right)^{2}+6\left(x+y\right)$ by combining like terms, considering $\left(x+y\right)$ as a common factor. Express your answer as a simplified algebraic expression. | 11\left(x+y\right)^{2}-\left(x+y\right) | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Polynomial Operations"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
If a particle A moves according to the rule $s=2t^2$, what is the instantaneous velocity of particle A at $t=1$? Express your answer in meters per second (m/s) as a whole number. | 4 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\tan\alpha=2$, simplify the expression $$\frac {\sin\alpha+2\cos\alpha}{\sin\alpha-\cos\alpha}.$$ Express your answer as a simplified expression or a numerical value. | 4 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
A hospital needs to select 3 doctors to participate in a community health check-up activity from 4 female doctors and 3 male doctors. Calculate the probability that at least 1 male doctor participates. Express your answer as a simplified fraction. | \frac{31}{35} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Combinations"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Five positive integers (not necessarily all different) are written on five cards. The sum of the numbers on every pair of cards is calculated, and only three different totals are obtained: 57, 70, and 83. Find the largest integer on any card. Express your answer as a single integer. | 48 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
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