question stringlengths 40 1.39k | answer stringlengths 1 90 | source stringclasses 1
value | domain listlengths 1 3 | llama8b_solve_rate float64 0 0.98 |
|---|---|---|---|---|
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 \( \left\lfloor \frac{n}{k} \right\rfloor \) (where \( \left\lfloor x \right\rfloor \) denotes the greatest integer less than or equal to the real number \( x \)) and are c... | 14 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.390625 |
From the numbers 1, 2, 3, 4, 5, if we randomly pick any three different numbers, what is the probability that these three numbers form a set of Pythagorean triples? Express your answer as a simplified fraction within the boxed format. | \dfrac{1}{10} | big_math | [
"Mathematics -> Applied Mathematics -> Probability"
] | 0.6875 |
Given two vectors $\overrightarrow{a}=(\cos \alpha, -2)$ and $\overrightarrow{b}=(\sin \alpha, 1)$, and $\overrightarrow{a}$ is parallel to $\overrightarrow{b}$, find the value of $\tan\left(\alpha- \frac{\pi}{4}\right)$. Express your answer as a single number. | -3 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.421875 |
A countryman's son's age is 5 times his daughter's age. His wife's age is 5 times his son's age, and his age is twice his wife's age. The sum of their ages is exactly the age of his grandmother, who is celebrating her 81st birthday today. Calculate the age of the countryman's son. Express your answer as a whole number. | 5 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.859375 |
Given the sequence 2009, 2010, 1, -2009, -2010,..., where starting from the second term, each term is equal to the sum of its adjacent terms, calculate the sum of the first 2010 terms of this sequence. Express your answer as a single integer. | 0 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.015625 |
Given the sequence $\{a_n\}$ with general term $a_n = \frac{2}{n(n+1)}$, find the sum of the first $n$ terms, denoted as $S_n$. Express your answer as a simplified fraction. | \frac{2n}{n+1} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.703125 |
Three real numbers \(a\), \(b\), and \(c\) have a sum of 114 and a product of 46656. If \(b = ar\) and \(c = ar^2\) for some real number \(r\), calculate the value of \(a + c\). Express your answer as a single integer. | 78 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.078125 |
Express the number 2034000 in scientific notation. Provide your answer in the form of a number multiplied by 10 raised to a power, where the number is between 1 and 10. | 2.034 \times 10^6 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.75 |
Find the equation of the line that passes through the intersection point of two lines $l_1$: $2x - 3y + 2 = 0$ and $l_2$: $3x - 4y - 2 = 0$, and is parallel to the line $4x - 2y + 7 = 0$. Express your answer in the standard form of a linear equation (Ax + By + C = 0). | 2x - y - 18 = 0 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Linear Equations"
] | 0.296875 |
Given that the volume of the circumscribed sphere of the prism $ABCD-A_1B_1C_1D_1$ is $\frac{4}{3}\pi$, and the base $ABCD$ is a square with side length 1, calculate the lateral area of the prism $ABCD-A_1B_1C_1D_1$. Express your answer as a numerical value. | 4\sqrt{2} | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Volume"
] | 0.03125 |
Given that the sequence {a_n} is an arithmetic sequence, if a_2 = 3 and a_1 + a_6 = 12, find the value of a_7 + a_8 + a_9. Express your answer as a single numerical value. | 45 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.65625 |
Given that f(x) is an odd function and g(x) is an even function, and f(x) + g(x) = 1/(x-1), calculate the value of f(3). Express your answer as a simplified fraction. | \dfrac {3}{8} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.359375 |
A polygon, after removing one of its interior angles, has the sum of the remaining interior angles equal to 2000°. Find the number of sides of this polygon. Express your answer as a whole number. | 14 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Polygons"
] | 0.15625 |
Given vectors $\overrightarrow {a} = (1, -\frac {3}{2}, \frac {5}{2})$ and $\overrightarrow {b} = (-3, \lambda, -\frac {15}{2})$ such that $\overrightarrow {a}$ is parallel to $\overrightarrow {b}$, find the value of $\lambda$. Express your answer as a single numerical value. | \frac{9}{2} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.625 |
Let $\alpha$ be an angle in the second quadrant, and $P(x,1)$ be a point on its terminal side, such that $\cos \alpha = \frac{1}{3}x$. Calculate $\tan \alpha$ exactly, expressing your answer as a simplified fraction. | -\frac{\sqrt{2}}{4} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.0625 |
Given the arithmetic sequence {a_n} where the sum of the first 11 terms is 22, find the sum of a_3, a_5, and a_{10}. Express your answer as a single integer. | 6 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.609375 |
A person walks on a 400-meter circular track at a speed of 6 kilometers per hour. They start from point A and walk clockwise for 1 minute, then counterclockwise for 3 minutes, and clockwise again for 5 minutes. What is the minimum time they need to return to the starting point A? Express your answer in minutes as a who... | 1 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.0625 |
What is the reciprocal of -3? Express your answer as a fraction. | -\frac{1}{3} | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.921875 |
A number, when its decimal point is moved one place to the right, becomes 34.65 larger than the original number. Find the original number. Express your answer as a numerical value. | 3.85 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 0.875 |
A basketball player has the same free-throw shooting percentage for every attempt, and the probability that they make at most one basket in two attempts is 16/25. Calculate the player's free-throw shooting percentage for each attempt. Express your answer as a fraction. | \frac{3}{5} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.21875 |
A recipe calls for 4 1/2 cups of flour. If you only make half of the recipe, calculate the exact amount of flour needed. Express your answer as a mixed number. | 2 \frac{1}{4} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.921875 |
Given \(f(x)=ax^{5}+bx- \dfrac {c}{x}+2\), and \(f(2)=4\), calculate \(f(-2)\). Express your answer as a single number. | 0 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.21875 |
Given that $a,b$ are real numbers, solve for $a$ and $b$ in the complex number equation $\frac{1+2i}{a+bi}=1+i$. Express your answer as a pair of values $(a, b)$. | (\frac{3}{2}, \frac{1}{2}) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.28125 |
Given the equation of a circle $(x-2)^{2}+y^{2}=3$, find the range of values for the expression $\frac{y}{x}$. Express your answer as an interval. | [-\sqrt{3}, \sqrt{3}] | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.015625 |
When A and B play chess, the probability of A winning is 0.3, and the probability of A not losing is 0.8. Calculate the probability of A and B drawing the game. Express your answer as a decimal value between 0 and 1. | 0.5 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Other"
] | 0.828125 |
Given the function $f_{(0)}(x)=\sin x$, define $f_{(1)}(x)=f''_{(0)}(x)$, $f_{(2)}(x)=f''_{(1)}(x)$,..., $f_{(n)}(x)=f''_{(n-1)}(x)$. Calculate the value of $f_{(1)}(15^{\circ})+f_{(2)}(15^{\circ})+f_{(3)}(15^{\circ})+...+f_{(2017)}(15^{\circ})$. Express your answer as a single value, including any necessary radicals. | \dfrac{\sqrt{6}+\sqrt{2}}{4} | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.015625 |
Given sin(2α) = 1/4, calculate cos²(α - π/4). Express your answer as a simplified fraction. | \frac{5}{8} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.296875 |
A Buddhist pagoda has seven levels, with the number of red lamps on each level being double that of the level below. If the total number of lamps is 381, how many lamps are there at the top level? Express your answer as a whole number. | 3 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.828125 |
Calculate the value of the following expression:
$$ C _{ 3n }^{ 38-n } + C _{ n+21 }^{ 3n } $$
where $n \in \mathbb{N}^*$ satisfies the system of inequalities:
$$
\begin{cases}
0 \leq 38-n \leq 3n \\
n+21 \geq 3n
\end{cases}
$$
Express your answer as a single integer. | 466 | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.59375 |
Given the relationships between the number of days required by teams A, B, and C working alone and together, calculate the value of the expression
\[
\frac{1}{a+1} + \frac{1}{b+1} + \frac{1}{c+1}
\]
where a, b, and c are defined as in the original problem. Express your answer as a single numerical value. | 1 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.03125 |
Given the function f(x) = 2cos^2(πx - π/6) - 1, stretch the horizontal coordinates of all points on the graph to twice their original length, keep the vertical coordinates unchanged, and then shift the resulting graph to the right by φ (φ > 0) units. Determine the minimum value of φ such that the final graph correspond... | \frac{1}{6} | big_math | [
"Mathematics -> Precalculus -> Functions"
] | 0.078125 |
What is the last digit of the number \( 2^2 + 20^{20} + 200^{200} + 2006^{2006} \)? Provide your answer as a single digit. | 0 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.859375 |
Suppose
$$
\frac{1}{x}=\frac{2}{y+z}=\frac{3}{z+x}=\frac{x^{2}-y-z}{x+y+z}.
$$
Find the value of $\frac{z-y}{x}$. Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.3125 |
A powderman set a fuse for a blast to take place in 30 seconds. He ran away at a rate of 8 yards per second. Sound travels at the rate of 1080 feet per second. Calculate the distance the powderman had run when he heard the blast. Express your answer in yards, rounded to the nearest whole number. | 245 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.09375 |
Simplify the expression $\lg 8 + 3\lg 5$. Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Logarithmic Functions"
] | 0.8125 |
Find a unit vector perpendicular to the vector $\overrightarrow{a}=(1,- \sqrt{3})$. Express your answer as an ordered pair (x, y) with exact values. | \left( \dfrac{ \sqrt{3}}{2}, \dfrac{1}{2}\right) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.359375 |
Four students who have passed the transfer test are to be placed into classes A, B, and C. Each class must have at least one student. Calculate the total number of different ways to place these students into the classes. | 36 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Combinations"
] | 0.015625 |
Given the complex number z that satisfies (3+4i)z = 25, find \\overline{z}. Express your answer in the form a + bi, where a and b are real numbers. | 3+4i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.703125 |
Find the equation of the line passing through the point (-1,3) and parallel to the line y = (1/2)x + 1. Express your answer in standard form (Ax + By + C = 0). | x - 2y + 7 = 0 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.28125 |
If z = 1 + 2i + i^3, calculate the magnitude |z| and express your answer as a single value. | \sqrt{2} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.921875 |
A number line has 40 consecutive integers marked on it. If the smallest of these integers is -11, what is the largest integer? Express your answer as a whole number. | 28 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 0.765625 |
Given \\( \\frac{a+2i}{i} = b+i \\), where \\(i\\) is the imaginary unit and \\(a, b \in \mathbb{R}\\), find the sum of \\(a\\) and \\(b\\). Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.421875 |
Given a complex number z satisfying the equation z + (z - 1)i = 3, calculate the magnitude |z| of the complex number z. Express your answer as a numerical value, including the square root symbol if necessary. | \sqrt{5} | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.28125 |
What is the negation of the proposition "$\exists x \in \mathbb{R}, \text{ such that } 2^x \leq 0$"? Express your answer as a logical proposition. | $\forall x \in \mathbb{R}, 2^x > 0$ | big_math | [
"Mathematics -> Discrete Mathematics -> Logic -> Other"
] | 0.046875 |
On the number line, points $A$ and $B$ are on the same side of the origin $O$, representing numbers $a$ and $1$ respectively. Point $A$ is moved $3$ units to the left to point $C$. If the numbers represented by points $C$ and $B$ are opposite, then what is the value of $a$? Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Algebra -> Simple Equations"
] | 0.78125 |
Given the perimeter p of an equilateral triangle inscribed in a circle, calculate the area of the circle in terms of p and π. | \frac{\pi p^2}{27} | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.421875 |
Given that the non-pure imaginary number z satisfies z(1+i) = |z|^2, find the imaginary part of the conjugate of z. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.296875 |
Given the arithmetic sequence {a_n}, if a_2 = 10 and a_5 = 1, find the sum of the first 7 terms. Express your answer as a single integer. | 28 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.828125 |
Given $\overrightarrow{a}=(10,20)$, $\overrightarrow{b}=(5,5)$, $\overrightarrow{c}=\overrightarrow{a}+k\overrightarrow{b}$, find the value of the real number $k$ such that $\overrightarrow{b} \perp \overrightarrow{c}$. Express your answer as a single real number. | -3 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.796875 |
Three people, A, B, and C, each hold some money. A says to B and C: "Each of you give me half of the money you hold, and then I will have 90"; B then says to A and C, "Each of you give me half of the money you hold, and then I will have 70"; C also says to A and B, "Each of you give me half of the money you hold, and t... | 32 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.015625 |
Given the equation (2x-5)(x+m) = 2x^2 - 3x + n, find the values of m and n. Express your answer as a pair of values (m, n). | (1, -5) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.640625 |
In a round-robin tournament with 6 teams, where each team plays one game against each other team and each game results in one team winning and one team losing, what is the maximum number of teams that could be tied for the most wins at the end of the tournament? Provide your answer as a single integer. | 5 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.125 |
The sum of seven consecutive positive integers is 77. What is the largest of these integers? Express your answer as a whole number. | 14 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.875 |
In right triangle $ABC$, the hypotenuse $BC=4$. Find the value of $AB^{2}+AC^{2}+BC^{2}$. Express your answer as a single integer. | 32 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.96875 |
Find the minimum positive period of the function $f(x)=( \sqrt {3}\sin x+\cos x)( \sqrt {3}\cos x-\sin x)$. Express your answer in radians as a fraction of pi if possible. | \pi | big_math | [
"Mathematics -> Precalculus -> Functions"
] | 0.484375 |
Solve the inequality $\frac{1+x}{2-x}\geqslant 0$ and express the solution set in interval notation. | [-1,2) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.171875 |
In the complex plane, the point $Z_1$ corresponds to the complex number $4+i$. If the midpoint of the line segment $Z_1Z_2$ corresponds to the complex number $1+2i$, calculate the complex number corresponding to the point $Z_2$. Express your answer in the form $a+bi$. | -2+3i | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.84375 |
Given that the random variable X follows a binomial distribution B(6, 1/2), determine the value of P(X=3). Express your answer as a fraction in the simplest form. | \frac{5}{16} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.90625 |
Given the universal set U = {x | x is a positive integer less than 9}, set M = {1, 2, 3}, and set N = {3, 4, 5, 6}, find the intersection of the complement of M in U and N. Express your answer as a set of integers. | {4, 5, 6} | big_math | [
"Mathematics -> Discrete Mathematics -> Logic -> Other"
] | 0.015625 |
Given that $\{a_n\}$ is an arithmetic sequence with the first term $a_1 = 1$ and common difference $d=3$, find the position $n$ when $a_n = 2005$. Express your answer as a whole number. | 669 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.96875 |
Given a complex number $z$ satisfying $z(2+i)=3+i$ (where $i$ is the imaginary unit), find the conjugate complex number $\overline{z}$. Express your answer in the form $a + bi$, where $a$ and $b$ are real numbers. | \overline{z} = \frac{7}{5} + \frac{1}{5}i | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.046875 |
Given the inequality $|2a-b|+|a+b| \geq |a|(|x-1|+|x+1|)$ holds for any non-zero real numbers $a$ and $b$, find the range of the real number $x$ that satisfies this inequality for all $a$ and $b$. Express your answer as an interval or describe the range. | [-\frac{3}{2}, \frac{3}{2}] | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.015625 |
Given the binomial expansion $(1-2x)^7 = a_0 + a_1x + a_2x^2 + \ldots + a_7x^7$, find the sum of the absolute values of the coefficients $|a_0| + |a_1| + |a_2| + \ldots + |a_7|$. Express your answer as a single integer. | 2187 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0 |
Let the sum of the first n terms of a geometric sequence {a_n} be S_n, satisfying a_n > 0, q > 1, and a_3 + a_5 = 20, a_2 * a_6 = 64. Find S_6, the sum of the first 6 terms of the sequence. Express your answer as a single numerical value. | 63 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.21875 |
Given that $\overrightarrow{e_{1}}$ and $\overrightarrow{e_{2}}$ are two unit vectors with an angle of $60^{\circ}$ between them, find the real number $k$ such that the statement "$(2\overrightarrow{e_{1}}-k\overrightarrow{e_{2}}) \perp \overrightarrow{e_{1}}$" holds. Express your answer as a single real number. | 4 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.953125 |
In a school with 1000 second-year high school students, 200 of them are arts students. A stratified sampling method by subject (arts and sciences) is employed to draw a sample of 100 students. Find the number of arts students in the sample. Express your answer as a whole number. | 20 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.984375 |
In a $7 \times 7$ grid, some squares are painted black and the rest are painted white, with at least one square of each color. A pattern is called "symmetric" if it looks the same when rotated $90^{\circ}$ and if it can be folded along either diagonal such that the two halves match. Find the total number of symmetric p... | 1022 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0 |
Given the ellipse C: x^2/16 + y^2/7 = 1, find the maximum value of |PF1|, where P is a point on the ellipse and F1 is its left focus. Express your answer as a single number. | 7 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.25 |
Let $P$ and $Q$ be two sets, where $P = \{x \mid \log_2 x < 1\}$ and $Q = \{x \mid x^2 - 4x + 4 < 1\}$. Find the intersection $P \cap Q$ and express your answer in interval or set notation. | \{x \mid 1 < x < 2\} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.046875 |
The minimum positive period of the function $y=\sin^2\left(\frac{\omega}{2}x-\frac{\pi}{4}\right)$ ($\omega>0$) is $\pi$. Find the value of $\omega$. Express your answer as a numerical value. | 2 | big_math | [
"Mathematics -> Precalculus -> Functions"
] | 0.234375 |
Find the remainder when $2^{1000}$ is divided by $13$. Express your answer as a single integer. | 3 | big_math | [
"Mathematics -> Number Theory -> Congruences"
] | 0.625 |
A batch of bagged rice follows a normal distribution N(10, 0.01) (unit: kg). Given Φ(1) = 0.8413 and Φ(2) = 0.9772, calculate the probability that a randomly selected bag of rice weighs between 9.8 kg and 10.2 kg. Express your answer as a decimal value between 0 and 1. | 0.9544 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.625 |
Corina was supposed to add the numbers P and Q, but she subtracted Q from P by mistake. The correct sum would have been 16, and her incorrect answer was 4. Using the equations P + Q = 16 and P - Q = 4, calculate the value of P. Express your answer as a single integer. | 10 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.9375 |
For the inequality $-4x-k\leqslant 0$, if the negative integer solutions are $-1$ and $-2$, calculate the range of values for $k$. Express your answer in interval notation (e.g., $[a, b)$ or $(a, b]$). | [8, 12) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.03125 |
Find the exact number of real roots of the equation sin(x) = lg(x). Express your answer as a whole number. | 3 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.109375 |
Find the sum of all real roots of the equation \(\left|x^{2}-12 x+34\right|=2\). Express your answer as a single numerical value. | 18 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.875 |
If the vertex of the parabola $y = x^2 - 4x + c$ is on the x-axis, find the value of $c$. Express your answer as a single number. | 4 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Quadratic Functions"
] | 0.921875 |
Given the equation for heat generated by a current flowing through a conductor is Q=I^2Rt, where Q is the heat in joules (J), I is the current in amperes (A), R is the resistance in ohms (Ω), and t is the time in seconds (s). If the resistance of the conductor is 5Ω, and the conductor generates 30J of heat in 1s, calcu... | \sqrt{6} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.71875 |
A bank savings card has a 6-digit password. When someone tries to withdraw money from an ATM, they forget the last digit of the password but remember that the last digit is an even number. Calculate the probability that they will enter the correct password in no more than two attempts. Express your answer as a simplifi... | \frac{2}{5} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0 |
Given that $\sum_{i=1}^{n} \cos ^{-1}\left(\alpha_{i}\right)=0,$ find the value of $\sum_{i=1}^{n} \alpha_{i}$. Express your answer as a single numerical value. | n | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.25 |
Given that the line $x+y+m=0$ is tangent to the circle $x^{2}+y^{2}=m$, find the value of $m$. Express your answer as a single numerical value. | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.71875 |
Given the parametric equations of a line $$\begin{cases} x=1+2t \\ y=2-3t \end{cases}$$ (where $t$ is the parameter), calculate the slope of the line. Express your answer as a fraction. | -\frac{3}{2} | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Other"
] | 0.890625 |
Find the maximum value of the function y = 1/4*sin(x - π/6) - cos(x - 2π/3). Express your answer as a single numerical value. | 3/4 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.03125 |
In a school, 4 osmanthus trees are transplanted. It is known that the survival probability of each tree is $\dfrac{4}{5}$. The number of surviving trees is represented by the random variable $\xi$, which follows a binomial distribution. Calculate the variance of $\xi$. Express your answer as a fraction or decimal value... | \dfrac{16}{25} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.921875 |
Point P is a center of symmetry of the graph of the function $f(x) = \cos \omega x$ ($\omega > 0$). If the minimum distance from point P to the axis of symmetry of the graph is $\pi$, find the smallest positive period of the function $f(x)$. Express your answer in terms of $\pi$ as a single numerical value. | 2\pi | big_math | [
"Mathematics -> Precalculus -> Functions"
] | 0.34375 |
Given $a={4}^{0.5}, b=\sqrt{2}, c=0．{5}^{4}$, order the values of $a$, $b$, and $c$ from smallest to largest. Express your answer as a sequence using the variables $a$, $b$, and $c$ (e.g., $a < b < c$). | c < b < a | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions"
] | 0.40625 |
Analyze the proposition "(x-1)^2+(y-2)^2=0" in relation to the condition "(x-1)(y-2)=0". Determine if the proposition is sufficient, necessary, both, or neither for the condition. Provide your answer in terms of sufficiency and necessity. | The proposition is sufficient but not necessary for the condition. | big_math | [
"Mathematics -> Algebra -> Other"
] | 0 |
Let $S$ and $T$ be two non-empty sets, and $S \notin T$, $T \notin S$. Let $X = S \cap T$. What is the union of $S$ and $X$? Express your answer using proper set notation. | S | big_math | [
"Mathematics -> Discrete Mathematics -> Logic"
] | 0.453125 |
Calculate the value of \((\sqrt{1+\sqrt{1+\sqrt{1}}})^{4}\). Express your answer in simplest radical form if necessary. | 3+2\sqrt{2} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.53125 |
Let p(x) = x^4 - 4x^3 + 2x^2 + ax + b. Suppose that for every root λ of p, 1/λ is also a root of p. Calculate the value of a + b. | -3 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.15625 |
Given that the function $f(x)$ is an odd function defined on the interval $[-a, a]$, and $g(x) = f(x) + 2$, calculate the sum of the maximum and minimum values of $g(x)$. Express your answer as a single number. | 4 | big_math | [
"Mathematics -> Precalculus -> Functions"
] | 0.84375 |
If $|x| = 3x + 1$, then calculate the value of $(4x + 2)^{2005}$. Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.765625 |
The "Wuyang" building, if constructed jointly by team A and team B, would need more than 6 months. If constructed by team A and team C, it would need more than 5 months. If constructed by team B and team C, it would need more than 4 months. Calculate the maximum number of months it would take for teams A, B, and C to b... | 4 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.09375 |
Given that the solution set of the quadratic inequality $ax^2 - x + b \geq 0$ about $x$ is $[-2, 1]$, determine the solution set of the quadratic inequality $bx^2 - x + a \leq 0$ about $x$. Express your answer as an interval. | [-\frac{1}{2}, 1] | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Quadratic Functions"
] | 0.078125 |
A number x is randomly chosen from the interval [0,2]. Find the probability that the event "-1 ≤ log_{1/2}(x+1/2) ≤ 1" occurs. Express your answer as a fraction. | 3/4 | big_math | [
"Mathematics -> Applied Mathematics -> Probability"
] | 0.4375 |
Select two numbers from the set {1, 2,..., 37} such that the sum of the remaining 35 numbers is equal to the product of these two numbers. Calculate the difference between the two numbers that satisfy this condition. Express your answer as a whole number. | 10 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.234375 |
Given the equation 1/a + 1/2a + 1/3a = 1/(b^2 - 2b), where a and b are positive integers, calculate the smallest possible value of a + b. Express your answer as a single integer. | 50 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.234375 |
Given that the line $ax+y-2=0$ intersects the circle $C: (x-1)^2+(y-a)^2=4$ at points $A$ and $B$, and the segment $AB$ is the longest chord of circle $C$, find the real number $a$. Express your answer as a single real number. | 1 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.328125 |
If \( x = 2018 \), then what is the value of the expression \( x^{2} + 2x - x(x + 1) \)? Express your answer as a single integer. | 2018 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.703125 |
Calculate the exact value of $(-0.125)^{2007} \times (-8)^{2008}$. Express your answer as an integer. | -8 | big_math | [
"Mathematics -> Algebra -> Exponential Functions"
] | 0.203125 |
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