instruction_seed stringlengths 40 1.39k | answer stringlengths 1 95 | source stringclasses 1
value | domain listlengths 1 3 | llama8b_solve_rate float64 0 0.98 | _source stringclasses 1
value |
|---|---|---|---|---|---|
Find the minimum value of the function $$f(x)=2^{x}+ \frac {1}{4\cdot 2^{x}}$$. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
In a frequency histogram, a group \( [a, b) \) has a frequency of m. If the height h of the frequency histogram on this group is given, find the absolute value of the group width \( |a-b| \). Express your answer as a formula using the variables m and h. | \frac{m}{h} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Given the equations \(60^a = 3\) and \(60^b = 5\), calculate the value of \(12^{\left(\frac{1-a-b}{2(1-b)}\right)}\). Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Exponential Functions"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Given that plane vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $({3\overrightarrow{a}-2\overrightarrow{b}})⊥({5\overrightarrow{a}+\overrightarrow{b}})$, and $\overrightarrow{a}•\overrightarrow{b}=\frac{1}{7}$, if $|{\overrightarrow{a}}|=1$, calculate the magnitude of vector $\overrightarrow{b}$, $|{\ove... | \sqrt{7} | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Linear Algebra -> Vectors"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Among the numbers from 1 to 10, when selecting any 4 numbers, how many ways are there for the second largest number to be 7? Express your answer as a single integer. | 45 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
The equation of the directrix of the parabola $y=ax^{2}$ is $y=2$. Calculate the value of $a$. Express your answer as a fraction, including the negative sign if applicable. | -\dfrac{1}{8} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
You are responsible for making digitally symmetric license plates that start with an 8 or 9 as the first digit following the letter "C" and have five digits in total. A digitally symmetric plate has five digits that are symmetric around the middle digit. Calculate the maximum number of such plates that can be made. Exp... | 200 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
Given the universal set U = {0,1,2,3,4,5}, set A = {0,1,3}, and set B = {2,3,5}, find the intersection of set A with the complement of set B in U, denoted as A ∩ (∁U B). Express your answer as a set enclosed in curly brackets, with elements separated by commas. | {0,1} | big_math | [
"Mathematics -> Discrete Mathematics -> Logic -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Find the third term in the expansion of $(\\frac{a}{\\sqrt{x}}-\\frac{\\sqrt{x}}{a^2})^6$. Express your answer as a simplified fraction, including any necessary variables. | \\frac{15}{x} | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Binomial Theorem"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $a$ and $b$ that satisfy $|a+b|=\sqrt{10}$ and $|a-b|=\sqrt{6}$, calculate the dot product $a \cdot b$. | 1 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Given sets A={0} and B={0,1,2}, find the intersection of sets A and B, denoted as A∩B. Express your answer as a set using curly brackets. | {0} | big_math | [
"Mathematics -> Discrete Mathematics -> Other"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Given a hyperbola with eccentricity $\sqrt{3}$ and equation $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, find the equation of its asymptotes. Express your answer in the form $y = mx$. | y = \pm \sqrt{2}x | big_math | [
"Mathematics -> Geometry -> Non-Euclidean Geometry -> Other"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Given the universal set U = {1, 2, 3, 4} and set A = {1, 2}, determine the number of sets B that satisfy the condition A ∪ B = U. Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
In the production process of a certain chemical plant, two different chemical raw materials need to be added in sequence. It is known that there are five available raw materials, but raw materials A and B cannot be used at the same time. Additionally, if raw material A is used, it must be added first. Find the total nu... | 15 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Four cards with the same texture, size, and back design are randomly placed face down on a table. The front sides of the cards are drawn with a circle, a rectangle, an equilateral triangle, and a regular pentagon, respectively. If one card is drawn at random, what is the probability that the drawn card has a front desi... | \frac{1}{2} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Given the system of equations $\left\{\begin{array}{c}x=m\\ y=n\end{array}\right.$ and the equation $x-y=2$, find the value of the algebraic expression $n-m$. Express your answer as a single integer. | -2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = e^(1-x) + t*ln(x), where t is a real number, find the minimum value of t such that f(x) is monotonically increasing in its domain (0, +inf). Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence {a<sub>n</sub>} with a<sub>1</sub> = -2 and a<sub>n+1</sub> = 1 - 1/a<sub>n</sub>, calculate the value of a<sub>2016</sub>. Express your answer as a simplified fraction. | \frac{1}{3} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
A bar of soap loses 10% of its volume each time it is used. What is the minimum number of times a new bar would have to be used so that less than one-half of its original volume remains? Express your answer as a whole number. | 7 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $f'(x_0)=2$, calculate the value of $\lim_{k\rightarrow 0} \frac{f(x_0-k)-f(x_0)}{2k}$. Express your answer as a single number. | -1 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ that satisfy the equations $\overrightarrow{a} + 2\overrightarrow{b} = \overrightarrow{0}$ and $(\overrightarrow{a} + \overrightarrow{b}) \cdot \overrightarrow{a} = 2$, find the value of $\overrightarrow{a} \cdot \overrightarrow{b}$. Express your answer as a s... | -2 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
A large block, which has dimensions n by 11 by 10, is made up of a number of unit cubes and one 2 by 1 by 1 block. There are exactly 2362 positions in which the 2 by 1 by 1 block can be placed. Calculate the value of n that satisfies this condition. Express your answer as a whole number. | 8 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given real numbers $a > 0$, $b > 0$, and the inequality $(a-1)^{3}+(b-1)^{3}\geqslant 3(2-a-b)$, find the minimum value of $a^{2}+b^{2}$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $F\_1$ and $F\_2$ are the left and right foci of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a > 0, b > 0)$, a line parallel to one asymptote of the hyperbola passes through $F\_2$ and intersects the other asymptote at point $M$. If point $M$ is on the circle with $F\_1F\_2$ as its diameter, calcul... | 2 | big_math | [
"Mathematics -> Geometry -> Non-Euclidean Geometry -> Hyperbolic Geometry"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
The line $2x - y - 2 = 0$ is rotated counterclockwise by $\frac{\pi}{2}$ around its intersection with the $y$-axis. Find the equation of the resulting line in standard form (Ax + By = C). | x + 2y + 4 = 0 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = 4x^3 - ax^2 - 2bx, where a > 0 and b > 0, and an extreme value at x = 1, find the value of a + b. Express your answer as a single integer. | 6 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
The sum of 2023 consecutive integers is 2023. Find the sum of the digits of the largest of these integers. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Xiao Liang starts from point A and moves forward 10 meters, then turns right 60°, moves forward another 10 meters, turns right 60° again, and so on. When he returns to the starting point A for the first time, what is the total distance he has walked? Express your answer in meters as a whole number. | 60 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given two straight lines $ax + 2y - 1 = 0$ and $3x - 6y - 1 = 0$ are perpendicular to each other, find the value of $a$. Express your answer as a single number. | 4 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Given the corresponding values of $x$ and $y$ as shown in the table below, if $y$ is linearly related to $x$ and the regression line equation is $\hat{y}=1.4x+1.4$, calculate the value of $m$.
| $x$ | $0$ | $2$ | $4$ | $6$ | $8$ |
|-----|-----|-----|-----|-----|-----|
| $y$ | $1$ | $m+1$ | $2m+1$ | $3m+3$ | $11$ |
Ex... | 3 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
A function $f(x)$ satisfies $f(m+n) = f(m) \cdot f(n)$ for any positive integers $m, n$, and $f(1) = 2$. Calculate the following expression:
$$
\frac{f(2)}{f(1)} + \frac{f(4)}{f(3)} + \frac{f(6)}{f(5)} + \dots + \frac{f(2016)}{f(2015)}
$$
Express your answer as a single integer. | 2016 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex equation $(3+2i) \cdot z = 5-i$, calculate the modulus $|z|$ of the complex number $z$. Express your answer as a single value, possibly with a square root. | \sqrt{2} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(2,-1)$ and $\overrightarrow{b}=(-3,4)$, calculate the value of $2\overrightarrow{a}+\overrightarrow{b}$. Express your answer as a vector in the form (x, y). | (1,2) | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
A school arranges four classes in the morning, each lasting 40 minutes. The first class starts from 8:00 to 8:40, with a 10-minute break between classes. If a student arrives late, and he arrives at the classroom between 9:10 and 10:00, what is the probability that he listens to the second class for no less than 10 min... | \dfrac{1}{5} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Given the inverse proportion function y = (k-1)/x, and two points A(x1, y1) and B(x2, y2) on the graph, where 0 < x1 < x2 and y1 < y2, what is the range of values for k that satisfies these conditions? Express your answer as an inequality in terms of k. | k < 1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the opposite of the number $a$ is $-2023$, find the value of $a$. Express your answer as a single integer. | 2023 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given the geometric sequence $\{a_n\}$ with the equation $a_n+a_{n+1}=3\cdot2^n$, calculate the expression $S_{k+2}-2S_{k+1}+S_{k}$. Express your answer as a power of 2, including the variable k. | 2^{k+1} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a < \frac{1}{2}$, simplify the expression $\sqrt[4]{(2a - 1)^{2}}$. Express your answer as a simplified mathematical expression. | \sqrt{1 - 2a} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
In a summer camp, there are 100 campers, each of whom can sing, dance, or perform at least one skill. Some campers have more than one skill, but no one has all three skills. If 42 campers do not sing, 65 campers do not dance, and 29 campers do not perform, how many campers have at least two skills? Provide your answer ... | 64 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation |x-8y| + (4y-1)^2 = 0, solve for x and y, and then calculate the value of log_2 y^x. Express your answer as a single numerical value. | -4 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other",
"Mathematics -> Precalculus -> Functions -> Other"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Given the set M = {-1, 0, 1} and N = {x | x^2 - 2x < 0}, find the intersection of M and N. Express your answer as a set using standard set notation. | \{1\} | big_math | [
"Mathematics -> Set Theory -> Operations on Sets"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $x^{m}=5$ and $x^{n}=-2$, calculate the value of $x^{m+2n}$. Express your answer as a single numerical value. | 20 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the tangent line to the parabola y = x^2 that is parallel to the line y = 2x. Express your answer in the general form of a line equation (Ax + By + C = 0), including all necessary coefficients and constants. | 2x - y - 1 = 0 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Suppose f(x) is defined on [0,1]. Find the range of values for a such that the function f(x-a) + f(x+a) is defined. Express your answer as an interval. | \left[-\frac{1}{2}, \frac{1}{2}\right] | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence {a_n}, the condition a_2 + a_8 = 15 - a_5 holds. Find the value of a_5. Express your answer as a single number. | 5 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
A company has 180, 150, and 120 sales points in cities A, B, and C, respectively. The company wants to investigate the product sales situation and needs to sample 90 sales points from these 450 sales points using the stratified sampling method. Calculate the number of sales points that should be sampled from city B. Ex... | 30 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Given the hyperbola $\dfrac {x^{2}}{a^{2}}- \dfrac {y^{2}}{b^{2}}=1$ ($a > 0,b > 0$) where the distance from the foci to the asymptote is $2$, and one of the asymptotes of the hyperbola is parallel to the line $x-2y+3=0$, find the equation of the hyperbola. Express your answer as a standard equation of a hyperbola. | \dfrac {x^{2}}{16}- \dfrac {y^{2}}{4}=1 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
In a certain sports competition, the scoring rules are: winning a match gives 3 points, drawing a match gives 1 point, and losing a match gives 0 points. If a particular athlete plays 15 matches and scores a total of 22 points, calculate the number of possible scenarios. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z=3+ \frac {3+4i}{4-3i}$, calculate the value of $|z|$. Express your answer as a mathematical expression, including any necessary radicals or absolute value signs. | \sqrt{10} | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $(\sqrt{3}+2)^{2023}(\sqrt{3}-2)^{2023}$ and provide your answer as a single numerical value. | -1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Selecting a student at random from a class, the probability that the student's height is less than 160cm is 0.2, and the probability that the student's height is between 160cm and 175cm is 0.5. Find the probability that the student's height exceeds 175cm. Express your answer as a decimal value between 0 and 1. | 0.3 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
The convex pentagon ABCDE has all sides of equal length, and angles A and B are both 90 degrees. What is the degree measure of angle E? Express your answer in degrees as a whole number. | 150 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Polygons"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given that the function f(x) is defined on the interval [0, +∞) and is monotonically increasing on this interval, solve the inequality f(2x-1) < f(1/3) for x. Express your answer as an interval, using brackets or parentheses to denote inclusion or exclusion of endpoints as necessary. | [\frac{1}{2},\frac{2}{3}) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Find the coefficient of $x^2$ in the expansion of $(1+2x)^3$. Express your answer as a numerical value. | 12 | big_math | [
"Mathematics -> Algebra -> Algebra -> Algebraic Expressions"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Three friends have a total of 6 identical pencils, and each one has at least one pencil. Find the number of ways this can happen. Express your answer as a whole number. | 10 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Combinations"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Solve for x in the equation 1/(x-1) = 2/(x-2). Express your answer as a single value or a list of values. | 0 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
Given sets $A=\{0,1,2,3\}$ and $B=\{x|x < 3, x\in \mathbb{N}\}$, find the intersection $A \cap B$. Express your answer as a set of numbers. | \{0,1,2\} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
If 3 ones and 2 zeros are randomly arranged in a row, calculate the probability that the 2 zeros are not adjacent. Express your answer as a decimal value between 0 and 1. | 0.6 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
In the Cartesian coordinate system, a point is randomly chosen within the square formed by the lines x=1, y=1, and the coordinate axes. What is the probability that this point falls within the region below the curve y=x^2? Express your answer as a fraction. | \frac{1}{3} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Given $\sin \left( \frac {\pi}{2}-\alpha \right) = \frac {3}{5}$, calculate $\cos (\pi-2\alpha)$. Express your answer as a fraction. | \frac{7}{25} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $(x-1)^5+5(x-1)^4+10(x-1)^3+10(x-1)^2+5(x-1)$. Express your answer as a simplified algebraic expression. | x^5-1 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Let the set $U = \{x \in \mathbb{N} | 0 < x \leq 8\}$, $S = \{1, 2, 4, 5\}$, and $T = \{3, 4, 5, 7\}$. Determine $S \cap (\complement_U T)$. Express your answer as a set, with elements separated by commas and enclosed in curly brackets {}. | \{1, 2\} | big_math | [
"Mathematics -> Discrete Mathematics -> Set Theory -> Other"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
For any positive integers \( n \) and \( k \) such that \( k \leq n \), let \( f(n, k) \) denote the number of positive integers that do not exceed \( \left\lfloor \frac{n}{k} \right\rfloor \) and are coprime with \( n \). Calculate the value of \( f(100, 3) \). Express your answer as a whole number. | 14 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
A cone has a vertex angle of 90°, and a cross-section is made at a 45° angle with the axis. What type of curve is formed by this cross-section? Provide your answer as the name of the curve. | Parabola | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the quadratic function f(x) = ax^2 + bx + c satisfying 2a + c/2 > b and c < 0, determine an interval containing a zero of the function f(x). Express your answer as an interval (lower bound, upper bound) using parentheses. | (-2, 0) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the hyperbola $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ $(a > 0,b > 0)$, where the asymptotes are tangent to the parabola $y=x^2+\dfrac{1}{16}$, calculate the eccentricity of the hyperbola. Express your answer as a single value, including the square root symbol if necessary. | \dfrac{\sqrt{5}}{2} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given a 6 x 5 grid and points A(1,3), B(5,1), and C(4,4), calculate the fraction of the grid area occupied by the area of triangle ABC. Express your answer as a simplified fraction. | \frac{1}{6} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the positive numbers x and y satisfy the equation 2x + y - 3 = 0, calculate the minimum value of 2/x + 1/y. | 3 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
An 18 inch by 24 inch painting is mounted in a wooden frame where the width of the wood at the top and bottom of the frame is twice the width of the wood at the sides. If the area of the frame is equal to the area of the painting, calculate the ratio of the shorter side to the longer side of this frame. Express your an... | 2:3 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given that a, b, and c are distinct positive integers such that abc = 16, find the largest possible value of a^b - b^c + c^a. Express your answer as a single integer. | 263 | big_math | [
"Mathematics -> Number Theory -> Other",
"Mathematics -> Algebra -> Other"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
In the plane rectangular coordinate system xOy, consider the function y=k/x where x < 0. If the function increases as the independent variable x increases, determine the quadrant where the graph of the function lies. Express your answer as the name of the quadrant (e.g., First, Second, Third, Fourth). | Second | big_math | [
"Mathematics -> Precalculus -> Functions"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Let $a_1, a_2,..., a_n$ be a sequence of positive terms, and let $T_n$ be the product of the first $n$ terms. Define the "relative cumulative product" as $\lg (T_1 \cdot T_2 \cdot...T_n)$. If a sequence of 2013 positive terms $a_1, a_2,..., a_{2013}$ has a "relative cumulative product" of 2013, then find the "relative ... | 4027 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Find the distance from point P (-1, 2) to the line 8x - 6y + 15 = 0. Express your answer as a fraction or decimal value. | \frac{1}{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
If \( n \) is \( \frac{5}{6} \) of 240, calculate the value of \( \frac{2}{5} \) of \( n \). Express your answer as a whole number. | 80 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Fractions"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Let the complex number $z$ satisfy $z(1+i) = 4$, where $i$ is the imaginary unit. Find the imaginary part of the complex number $z$. Express your answer as a single number. | -2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Five girls ran a race. Fiona started first, followed by Gertrude, then Hannah, then India, and lastly Janice. Whenever a girl overtook another girl, she was awarded a point. India was first to finish, followed by Gertrude, Fiona, Janice, and lastly Hannah. Calculate the minimum total number of points that could have be... | 5 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the following expression:
$$\frac {\sin(92^\circ) - \sin(32^\circ) \cos(60^\circ)}{\cos(32^\circ)}$$
Express your answer as a simplified fraction or a boxed numerical value. | \frac{\sqrt{3}}{2} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given samples $a_1$, $a_2$, $…$, $a_{10}$ with mean $\bar{a}$, and samples $b_1$, $b_2$, $…$, $b_{10}$ with mean $\bar{b}$, calculate the mean of the combined sample $a_1$, $b_1$, $a_2$, $b_2$, $…$, $a_{10}$, $b_{10}$. Express your answer as a formula involving $\bar{a}$ and $\bar{b}$. | \frac{1}{2}(\bar{a} + \bar{b}) | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Given the positive numbers x and y (x > y), find the minimum value of t, where t = max{x^2, 4/y(x-y)}. Express your answer as a single number. | 4 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given the arithmetic sequence {a<sub>n</sub>}, where a<sub>3</sub>+a<sub>7</sub>=10 and a<sub>8</sub>=8, find the common difference d. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
If the point $(a, 9)$ is on the graph of the function $y = 3^x$, calculate the value of $\tan\frac{a\pi}{6}$. Express your answer as a single value. | \sqrt{3} | big_math | [
"Mathematics -> Precalculus -> Functions",
"Mathematics -> Trigonometry -> Other"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
A chord of a circle has a length equal to the radius of the circle. Find the measure of the central angle corresponding to this chord in radians. | \frac{\pi}{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the point P(x,y) moves along the line x-y-1=0, find the minimum value of (x-2)^2 + (y-2)^2. Express your answer as a single numerical value in the boxed format. | \frac{1}{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
In the Cartesian coordinate system, point A is translated 2 units to the right to obtain point A'(3, 2). Find the original coordinates of point A. Express your answer as an ordered pair (x, y). | (1, 2) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence {a_n} with S_n being the sum of the first n terms, if a_3 + 7 = 2a_5, calculate the value of S_13. Express your answer as a single numerical value. | 91 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Determine the coefficient of the term containing $x^{4}$ in the expansion of $(\sqrt{x}+1)^{10}$. Express your answer as a numerical value. | 45 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence ${{a_n}}$, where ${a_1=1}$ and $a_{n+1}=a_{n}+3$, find the value of ${a_6}$. Express your answer as a single integer. | 16 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Determine the correct range for the inequality involving the square root of 6. Express your answer as an inequality in the format $\boxed{a < \sqrt{6} < b}$, where a and b are integers. | 2 < \sqrt{6} < 3 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Inequalities"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
The terminal side of angle α intersects the unit circle at point (-√5/5, 2√5/5). Find the value of cos2α. Express your answer as a fraction. | -3/5 | big_math | [
"Mathematics -> Trigonometry -> Functions -> Trigonometric Functions"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Let $f(x) = \begin{cases} 2e^{x-1}, & x<2 \\ \log_{3}(x^{2}-1), & x\geq 2 \end{cases}$. Calculate the value of $f(f(2))$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Functions"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Find the hundreds digit of $(20!-15!)$. Express your answer as a single digit. | 0 | big_math | [
"Mathematics -> Other -> Other"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
A line passes through point M(-2, m) and point N(m, 4) with a slope of 1. Find the value of m. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the geometric sequence $\{a_n\}$ is an increasing sequence, and $a_5^2 = a_{10}$, $2(a_n+a_{n+2})=5a_{n+1}$, find the general formula for the sequence $\{a_n\}$. Express your answer as a formula for $a_n$ using standard mathematical notation. | 2^n | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
For \(x \in [0, 2\pi]\), find the sum of all solutions to the equation \(\operatorname{tg}^{2} x - 9 \operatorname{tg} x + 1 = 0\). Express your answer in terms of π. | 3\pi | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given four non-collinear points on a plane, \(O\), \(A\), \(B\), and \(C\), if \( \overrightarrow{OA} - 4 \overrightarrow{OB} + 3 \overrightarrow{OC} = 0\), find the ratio of the magnitudes of vectors \(\overrightarrow{AB}\) and \(\overrightarrow{BC}\). Express your answer as a numerical value. | 3 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
A circle passes through the three vertices of a triangle whose side lengths are $7 \\frac{1}{2}$, $10$, and $12 \\frac{1}{2}$. Find the radius of this circle. Express your answer as a fraction or decimal. | \\frac{25}{4} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
A quadrilateral with consecutive side lengths of 25, 39, 52, and 60 is inscribed in a circle. Calculate the circumference of the circle. Express your answer in terms of π. | 65\pi | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Create a three-digit number with no repeated digits by selecting one digit from {0, 2, 4} and two digits from {1, 3, 5}. How many different three-digit numbers can be formed? Express your answer as a single integer. | 48 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Four students, named A, B, C, and D, each randomly choose one day from Saturday or Sunday to go on a countryside tour. What is the total number of situations where both Saturday and Sunday have students participating in the tour? Express your answer as a whole number. | 14 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z = 2 - 3i$, find the real and imaginary parts of this complex number. Express your answer as a pair of numbers, with the real part first and the imaginary part second. | (2, -3) | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
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