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 the sets S={x|x²+2x=0, x∈R} and T={x|x²-2x=0, x∈R}, find the intersection S∩T. Express your answer as a set. | \{0\} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation of line $l$ is $ax+2y-3=0$, and $a\in[-5,4]$, calculate the probability that the slope of line $l$ is not less than 1. Express your answer as a simplified fraction. | \frac{1}{3} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Given four wooden sticks with lengths of $5$, $5$, $7$, and $a$, which can be nailed into a parallelogram wooden frame, find the value of $a$ that makes this possible. Express your answer as a single number. | 7 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $\dfrac {1}{2^{2}-1}+ \dfrac {1}{4^{2}-1}+ \dfrac {1}{6^{2}-1}+ \dfrac {1}{8^{2}-1}+ \dfrac {1}{10^{2}-1}$. Express your answer as a simplified fraction. | \dfrac {5}{11} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
In January 2000, the price of a kilogram of chicken was R$1.00, and it started to triple every 6 months. How many years will it take for the price to reach R$81.00? Express your answer in years, rounded to 1 decimal place if necessary. | 2 | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = sin(x) - sqrt(3)*cos(x), find the maximum value of f(x). Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Given a geometric sequence where the product of the first three terms is 2, the product of the last three terms is 4, and the product of all terms is 64, find the number of terms in this sequence. Express your answer as a whole number. | 12 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a=\sin 60^{\circ}$, $b=\cos 60^{\circ}$, $A$ is the arithmetic mean of $a$ and $b$, and $G$ is the geometric mean of $a$ and $b$, determine the increasing order of $a$, $b$, $A$, and $G$. Express your answer as a sequence of variables (e.g., b < G < A < a). | b < G < A < a | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
To m ounces of a m% solution of acid, x ounces of water are added to yield a (m-10)% solution. If m > 25, calculate the value of x in terms of m. Express your answer as an algebraic expression. | \frac{10m}{m-10} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the domain of the function $f(x)$ is $\left[1,+\infty \right)$, determine the domain of the function $y=f(x-1)+f(4-x)$. Express your answer as an interval. | \left[2,3\right] | big_math | [
"Mathematics -> Precalculus -> Functions"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Given non-zero vectors $\overrightarrow{a}, \overrightarrow{b}$ that satisfy $\left|\overrightarrow{a}
ight| = \frac{2\sqrt{2}}{3}\left|\overrightarrow{b}
ight|$, and $(\overrightarrow{a} - \overrightarrow{b}) \perp (3\overrightarrow{a} + 2\overrightarrow{b})$, find the angle between $\overrightarrow{a}, \overrightarro... | \frac{\pi}{4} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
In the complex plane, let $z=\frac{1-3i}{1-i}$. Calculate the magnitude $|z|$. | \sqrt{5} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, square ABCD has side length 2, with M the midpoint of BC and N the midpoint of CD. Calculate the area of the shaded region BMND. Express your answer as a single number. | \frac{3}{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
In the figure, lines CD, AE, and BF are one-third of their respective sides. It follows that the ratio of segments AN2, N2N1, and N1D is 3:3:1, and similarly for lines BE and CF. Calculate the area of triangle N1N2N3 in terms of the area of triangle ABC. Express your answer as a fraction of the area of triangle ABC. | \frac{1}{7} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Let set $A=\{4, 5, 7, 9\}$, $B=\{3, 4, 7, 8, 9\}$, and the universal set $U=A \cup B$. Find the number of non-empty subsets of the set $\complement_U (A \cap B)$. Express your answer as a single integer. | 7 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics -> Other"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given the parametric equations x = sqrt(t) and y = 2*sqrt(1-t), convert them to an ordinary equation and specify the ranges of x and y. Express your answer in the format of x^2 + (y^2)/4 = 1 with the ranges of x and y in parentheses. | x^2 + (y^2)/4 = 1 (0 <= x <= 1, 0 <= y <= 2) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given real numbers $x, y, z$ satisfy $x^2+y^2+z^2=4$, find the maximum value of $(2x-y)^2+(2y-z)^2+(2z-x)^2$. Express your answer as a single numerical value. | 28 | big_math | [
"Mathematics -> Applied Mathematics -> Other",
"Mathematics -> Algebra -> Other",
"Mathematics -> Calculus -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given a quadrilateral ABCD, where AB is parallel to CD, AB equals CD, and the perimeter is 40cm. The ratio of the lengths of two adjacent sides is 3:2. Find the length of the longer side. Express your answer in centimeters (cm) as a whole number. | 12 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the binary number $N$ is $11000$, calculate the integer immediately preceding $N$ in binary. Express your answer in binary. | 10111 | big_math | [
"Mathematics -> Discrete Mathematics -> Other"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Given the inequality $a \le 2x + 3 \le b$, the length of the interval of solutions is $10$. Find the value of $b - a$. Express your answer as a single numerical value. | 20 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
If x men working x hours a day for x days produce x articles, then find a formula for the number of articles produced by y men working y hours a day for y days. Express your answer as a formula involving x and y. | \frac{y^3}{x^2} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
The Knave of Hearts stole some tarts. He ate half of them, and half a tart more. The Knave of Diamonds ate half of what was left, and half a tart more. Then the Knave of Clubs ate half of what remained, and half a tart more. This left just one tart for the Knave of Spades. What was the initial number of tarts the Knave... | 15 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Let $a_{1}$, $a_{2}$, $a_{3}$, $a_{4}$ be a sequence of non-zero terms in arithmetic progression with a common difference $d\neq 0$. If the new sequence obtained by removing $a_{2}$ from this sequence (in the original order) is a geometric progression, calculate the value of $\frac{d}{{a}_{1}}$. Express your answer as ... | -\frac{1}{4} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
A cubical room has edges of length 2 meters. In each corner of the room, there is a mosquito killer that can eliminate mosquitoes within a 1-meter range (the volume of the device can be ignored). If a mosquito randomly appears in the room, calculate the probability that it will be eliminated by the mosquito killer. Exp... | \frac{\pi}{6} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given $\log_{5}45=a$, express $\log_{5}3$ in terms of $a$. Provide your answer as an algebraic expression. | \frac {a-1}{2} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Logarithmic Functions"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
If \(x=4\) and \(x+y=0\), what is the value of \(y\)? Express your answer as a single integer. | -4 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Simple Equations"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
A certain company sends 5 employees, including 2 female employees, to study in 3 different places. Each person must go to one place, and each place must have at least one person. The 2 female employees must study together. Calculate the total number of different allocation schemes that satisfy these conditions. | 36 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Let $a$ and $b$ be real numbers, and $2a + b = 6$. Find the minimum value of $2^a + (\sqrt{2})^b$. Express your answer as a single value, potentially with radicals. | 4\sqrt{2} | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Find the symmetry center of the function $y= \frac {1}{3}\tan (-7x+ \frac {\pi}{3})$. Express your answer as an ordered pair (x, y). | \left( \frac {\pi}{21},0\right) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets $M=\{x|-1 < x < 2\}$ and $N=\{x|x^{2}-mx < 0\}$, if $M\cap N=\{x|0 < x < 1\}$, calculate the value of $m$. Express your answer as a single numerical value. | 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Given $\overrightarrow{a}=(2,3)$, $\overrightarrow{b}=(x,-6)$, if $2\overrightarrow{a} \parallel \overrightarrow{b}$, find the value of $x$. Express your answer as a single numerical value. | -4 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation (x-1)(x^2-2x+m) = 0, find the range of the real number m such that its three roots can form the lengths of the sides of a triangle. Express your answer as an inequality or equality describing the range of m. | \frac{3}{4} < m \leq 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
In a statistical analysis, the reading time of 5000 residents is being studied, and a sample of 200 residents was taken for analysis. What term describes the entire reading time of the 5000 residents? Provide your answer as a single word or term. | Population | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Determine the domain of the function y = sqrt(1-x) + lg(x+2). Express your answer in interval notation. | (-2, 1] | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
A group consists of doctors and lawyers, with an average age of 40. The doctors have an average age of 35, and the lawyers have an average age of 50. Find the ratio of the number of doctors to the number of lawyers. Express your answer as a ratio in the format a:b. | 2:1 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Let \(f(x) = x(2018 + \ln x)\). If \(f'(x_0) = 2019\), calculate the value of \(x_0\). Provide your answer as a single value. | 1 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence ${a_n}$ defined by ${a_1 = 1, a_{n+1} = -S_n S_{n+1}}$, where ${S_n}$ is the sum of the first $n$ terms of the sequence, find the value of $n$ that maximizes the expression $\frac{nS_n^2}{1 + 10S_n^2}$. Express your answer as a single integer. | 3 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Find the slope of the line given by the equation 2x + sqrt(3)y - 1 = 0. Express your answer as a fraction or a decimal. | -\frac{2\sqrt{3}}{3} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Linear Equations"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a = \log_2 \frac{1}{2}$, $b = \log_{\sqrt{2}} \sqrt{3}$, and $c = (\frac{1}{4})^{\frac{2}{3}}$, determine the relationship among $a$, $b$, and $c$. Express your answer as a relationship between $a$, $b$, and $c$ (e.g., $a < b < c$). | a < c < b | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Logarithmic Functions"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Let S<sub>n</sub> represent the sum of the first n terms of the arithmetic sequence {a<sub>n</sub>}. If $$\frac {a_{4}}{a_{3}}$$ = $$\frac {3}{4}$$, then what is the value of $$\frac {3S_{5}}{a_{4}}$$? Express your answer as a single number. | 20 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Given the two sets of solutions $\left\{{\begin{array}{l}{x=1}\\{y=2}\end{array}}\right.$ and $\left\{{\begin{array}{l}{x=2}\\{y=5}\end{array}}\right.$ for the equation $ax+by=2$, calculate the values of $a$ and $b$. Express your answer as the ordered pair $(a, b)$. | (6, -2) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Given two vectors $\vec{a}$ and $\vec{b}$ on a plane with an angle of ${60}^{\circ }$ between them, where $\vec{a}=(2,0)$ and $|\vec{b}|=1$, calculate the magnitude of the vector sum $|\vec{a}+2\vec{b}|$. Express your answer as a numerical value. | 2\sqrt{3} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
In the pie chart shown, 168 students chose bananas as their favourite fruit. The sector representing apples has a right angle. Calculate the number of students who chose apples as their favourite fruit. Express your answer as a whole number. | 56 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Let $M$ and $m$ represent the maximum and minimum values of the function $y=\frac{1}{3}\cos x - 1$, respectively. Calculate the sum of $M$ and $m$. Express your answer as a single number. | -2 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Given \( \sin x = 3 \cos x \), find the value of \( \sin x \cos x \). Express your answer as a simplified fraction. | \frac{3}{10} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Given the hyperbola $C$: $\frac{x^{2}}{3}-y^{2}=1$, with $O$ being the origin and $F$ the right focus of $C$. A line passing through $F$ intersects the two asymptotes of $C$ at points $M$ and $N$, respectively. If $\triangle OMN$ is a right-angled triangle, calculate the length of $|MN|$. | 3 | big_math | [
"Mathematics -> Geometry -> Non-Euclidean Geometry -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
In a sector with a radius of $10cm$ and an area of $100cm^{2}$, find the central angle corresponding to the arc. Express your answer in radians. | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Let the function f(x) be differentiable. Find the limit $\lim_{\triangle x \to 0} \frac{f(1)-f(1+\triangle x)}{3\triangle x}$ in terms of f'(1). | - \frac{1}{3}f'(1) | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Let set $A = \{x \,|\, -1 < x < 1\}$, and $B = \{x \,|\, n-m < x < n+m\}$. If "$m=1$" is a sufficient condition for "$A \cap B \neq \emptyset$", then find the range of values for $n$ in interval notation. | (-2, 2) | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Select 3 TVs from 4 Type A and 5 Type B TVs, with the constraint that the 3 selected TVs must include at least one of each type. How many different selections are possible? Express your answer as a single integer. | 70 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of natural numbers from 1 to 1992 that are multiples of 3 but not multiples of 2 or 5. Express your answer as a whole number. | 266 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Other"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
If $f(x) = \sin(2x + \frac{\pi}{6})$, then what is the value of $f′(\frac{\pi}{6})$? Express your answer as a single number in the form $\boxed{[answer]}$. | 0 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
A line passing through the right focus $F$ of the hyperbola $C$: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > 0, b > 0$) and perpendicular to the $x$-axis intersects the hyperbola $C$ and its asymptote in the first quadrant at points $A$ and $B$, respectively. If $A$ is the midpoint of $BF$, find the eccentricity of $... | \frac{2\sqrt{3}}{3} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
A fair standard six-sided dice is tossed three times. Given that the sum of the first two tosses equals the third, calculate the probability that at least one "2" is tossed. Express your answer as a fraction in simplest form. | \frac{7}{15} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
The sequence ${a_n}$ satisfies ${a_{n+1}}={a_n}+\frac{n}{2}$, where $n \in \mathbb{N}^*$, and $a_1=2$. Calculate the value of $a_{20}$. Express your answer as a single integer. | 97 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Suppose 0 < x < π/6. Order the following expressions from smallest to largest: sin(sin x), cos(cos x), sin(cos x), cos(sin x). Express your answer as a sequence of the four expressions in order from smallest to largest, separated by '<'. | sin(sin x) < cos(cos x) < sin(cos x) < cos(sin x) | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Find the conjugate of the complex number $(3i-1)i$. Express your answer as a complex number in the form $a + bi$. | -3+i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
From three study groups consisting of 3, 3, and 4 members, respectively, select 4 individuals to form an environmental propaganda group such that each study group has at least one member in the propaganda group. Find the number of methods to select such individuals. Express your answer as a whole number. | 126 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Combinations"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression \sqrt{\frac{\sqrt{81} + \sqrt{81}}{2}} and provide your answer as a single number. | 3 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
An arithmetic sequence {a_n} satisfies a_2 = 12 and a_6 = 4. Find the common difference d. Express your answer as an integer. | -2 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
In triangle ABC, angle A = 45 degrees, angle B = 60 degrees, and side a = 10 units. Using the Law of Sines, calculate the length of side b. Express your answer as a simplified radical. | 5 \sqrt{6} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
In the list of numbers $-7$, $0$, $-3$, $\dfrac{4}{3}$, $+9100$, $-0.27$, how many numbers are negative? Express your answer as a whole number. | 3 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given a data set $x_1$, $x_2$,..., $x_n$ with a mean of $6$ and a standard deviation of $2$, find the mean and variance of the data set $2x_1 - 6$, $2x_2 - 6$,..., $2x_n - 6$. Express your answer in the format (mean, variance). | (6, 16) | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.65625 | 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, during the last 30 minutes. | 67 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
When travelling from London to Edinburgh by train, you pass a sign saying 'Edinburgh 200 miles'. Then, \(3 \frac{1}{2}\) miles later, you pass another sign saying 'Half way between London and Edinburgh'. Calculate the total distance from London to Edinburgh in miles. | 393 | big_math | [
"Applied Mathematics -> Math Word Problems"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Using Qin Jiushao's algorithm, calculate the value of v_2 in the function f(x) = 2x^4 + 3x^3 + 5x - 4 when x = 2. Express your answer as a single integer. | 14 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
A regular n-gon is inscribed in a circle with radius R, and the area of this n-gon is 3R^2. Find the value of n. Express your answer as a whole number. | 12 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Polygons"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of solutions (P, Q) of the equation P/Q - Q/P = (P+Q)/PQ, where P and Q are integers from 1 to 9 inclusive. Express your answer as a whole number. | 8 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
Given a semi-circle with radius r, find the area of the largest triangle that can be inscribed in it. Express your answer in terms of r. | r^2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Given a parabola with the equation $x^{2}=2py\left(p > 0
ight)$ and a point $M(m,3)$ on the parabola, the distance from $M$ to the focus is $5p$. Find the value of $p$. Express your answer as a single numerical value. | \frac{2}{3} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Solve for x in the equation $8^x = 32$. Express your answer as a fraction or decimal. | \frac{5}{3} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
There are 6 teachers who will be allocated to two classes (Class 1 and Class 2) to join the New Year's Eve celebration. Each class can have at most 4 teachers. Calculate the number of different arrangements for the teachers across these two classes, considering that the order of teachers within each class does not matt... | 50 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Combinations"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given the hyperbola $C: \\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1 \\;(a > 0, \\;b > 0)$ with an eccentricity of $\\sqrt{5}$, and the distance from point $P(\\sqrt{a^2 + b^2}, 0)$ to its asymptote is 8, find the length of the real axis of $C$. Express your answer as a single number. | 8 | big_math | [
"Mathematics -> Geometry -> Non-Euclidean Geometry -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
In a bag, there are 5 balls of the same size, including 2 red balls and 3 white balls. If 2 balls are randomly drawn from the bag, what is the probability that the two balls include both a red ball and a white ball? Express your answer as a simplified fraction. | \frac{3}{5} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Solve for n in the equation 4^n = 64^2. Express your answer as a whole number. | 6 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Given the hyperbola equation $\frac{x^2}{a^2} - \frac{y^2}{3} = 1$ with eccentricity $2$, find the value of $a$ where $a > 0$. Express your answer as a single numerical value. | 1 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the exact value of $\cos 75^{\circ}-\cos 15^{\circ}$. Express your answer as a simplified fraction with a square root if necessary. | $- \frac{\sqrt{2}}{2}$ | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the normal vector of plane α is (1,2,-2) and the normal vector of plane β is (-2,-4,k), and α and β are parallel planes, find the value of k. | 4 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Randomly select a number x in the interval [0,π]. What is the probability of the event -1 ≤ tan x ≤ √3? Express your answer as a fraction. | \dfrac{7}{12} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given the circle equation $2x^2+2y^2+6x-4y-3=0$, find the coordinates of the center and the radius of the circle. Express the center coordinates in the format (x, y) and the radius as a fraction or simplified radical. | \left(-\frac{3}{2}, 1\right), \frac{\sqrt{19}}{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of $(-1)^{5^{2}}+1^{2^{5}}$. Express your answer as a single integer. | 0 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given a geometric sequence $\{b_{n}\}$ with all terms being positive numbers, and the information that $b_{7} \cdot b_{8} = 3$, calculate the value of $\log_{3}b_{1} + \log_{3}b_{2} + \ldots + \log_{3}b_{14}$. Express your answer as a single integer. | 7 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Identify the real and imaginary parts of the complex number (1+√3)i in the standard form a + bi, where a is the real part and b is the imaginary part. Express your answer as two values separated by a comma. | 0, 1+√3 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
If the sum of the interior angles of a polygon is three times the sum of its exterior angles, find the number of sides of the polygon. Express your answer as a whole number. | 8 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Among the integers between 1000 and 9999, how many four-digit numbers have all even digits and are divisible by 5? Express your answer as a whole number. | 100 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Find the smallest positive period of the function f(x) = sin(x) + cos(x). Express your answer in terms of pi. | 2\pi | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Given a random variable X that follows a normal distribution with mean 4 and variance σ^2, and P(X ≤ 2) = 0.3, calculate the probability P(X < 6). Express your answer as a decimal value between 0 and 1. | 0.7 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given $a > 0$ and $a \neq 1$, the graph of the function $f(x) = \log_a(2x-1) + 1$ always passes through a fixed point $P$. Find the coordinates of $P$. Express your answer as an ordered pair $(x, y)$. | (1, 1) | big_math | [
"Mathematics -> Precalculus -> Functions",
"Mathematics -> Algebra -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
In the geometric sequence \(\{a_n\}\), where \(a_3 = 16\) and \(a_4 = 8\), calculate the value of \(a_1\). Provide your answer as a single number. | 64 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\sin\left(x + \frac{\pi}{6}\right) = \frac{1}{4}$, calculate the value of $\cos^2\left(\frac{\pi}{3} - x\right)$. Express your answer as a simplified fraction. | \frac{1}{16} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
In a convex polygon, the measures of the interior angles form an arithmetic sequence. If the smallest angle is $100^\circ$ and the largest angle is $140^\circ$, how many sides does the polygon have? Express your answer as a whole number. | 6 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Let {a_n} be a geometric sequence with the first term 1 and common ratio q, and let S_n be the sum of the first n terms. Define b_n = S_n + 2. Find the value of q such that {b_n} is also a geometric sequence. | \frac{3}{2} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
A sector of a circle has a radius of 6 and an arc length of 10. Calculate the area of the sector. Express your answer as a whole number. | 30 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
$P Q R S$ is a square with side length $8$. $X$ is the midpoint of side $P Q$, and $Y$ and $Z$ are the midpoints of $X S$ and $X R$, respectively. Calculate the area of trapezoid $Y Z R S$. Express your answer in square units. | 24 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given an isosceles triangle with two sides measuring $3$ and $7$ respectively, calculate the perimeter of this isosceles triangle. Consider all possible configurations of the triangle and apply the triangle inequality theorem to determine the validity of each configuration. Express your answer as a whole number. | 17 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
If \( n \) is \(\frac{5}{6}\) of 240, then calculate \(\frac{2}{5}\) of \( n \). Express your answer as a whole number. | 80 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Fractions"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
A certain school selects outstanding class groups based on assessments in four aspects: "study," "hygiene," "discipline," and "participation in activities." Each aspect has a maximum score of 100, with the following weight distribution:
| Aspect | Study | Hygiene | Discipline | Participation in Activities |
... | 82.5 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Find the smallest value of the function f(x) = |x| + |(1 - 2013x)/(2013 - x)|, where x ∈ [-1, 1]. Express your answer as a fraction in simplest form. | \frac{1}{2013} | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Let the set $I=\{x \mid |x| < 3, x \in \mathbb{Z}\}$, $A=\{1,2\}$, and $B=\{-2,-1,2\}$. Calculate the set $A\cup (C_I B)$, where $C_I B$ denotes the complement of $B$ within $I$. Express your answer as a set of integers, including all elements of the resulting set. | \{0,1,2\} | big_math | [
"Mathematics -> Discrete Mathematics -> Set Theory -> Other"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Find the minimum value of the function f(x) = x^3 - 3x + 1 within the interval [0, 3]. Express your answer as a single integer. | -1 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
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