instruction_seed stringlengths 40 1.39k | answer stringlengths 1 95 | source stringclasses 1
value | domain listlengths 1 3 | llama8b_solve_rate float64 0 0.98 | _source stringclasses 1
value |
|---|---|---|---|---|---|
A bag contains 8 red balls, a number of white balls, and no other balls. If \(\frac{5}{6}\) of the balls in the bag are white, then find the number of white balls in the bag. Express your answer as a whole number. | 40 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given that α∈({0,π/2}) and cos(α+π/6) = 1/3, calculate the value of sin(2α+π/3). Express your answer as a single value in the form \boxed{[answer]}. | \frac{4\sqrt{2}}{9} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Find the polar coordinates (r, $\theta$) of the center of the circle given by the polar equation $\rho = -2\cos\theta$. Express your answer in the format (r, $\theta$), where r is a numerical value and $\theta$ is in radians. | (1, \pi) | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Use the "Method of Continued Proportionate Reduction" to find the greatest common divisor (GCD) of 168 and 486. Express your answer as a whole number. | 6 | big_math | [
"Mathematics -> Number Theory -> Greatest Common Divisors (GCD)"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
There are $3$ activity groups, and students A and B each participate in one of them. The probability of each student participating in any group is the same. What is the probability that these two students are not in the same interest group? Express your answer as a simplified fraction. | \dfrac{2}{3} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Distribute 10 identical balls into 3 distinct boxes labeled 1, 2, and 3, such that each box contains at least as many balls as its label number. Calculate the total number of ways to do this. Express your answer as a single integer. | 15 | big_math | [
"Mathematics -> Applied Mathematics -> Combinatorics"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
To fill 100 boxes with cups, where each box contains a different number of cups and no box is empty, what is the minimum number of cups needed? Express your answer as a whole number. | 5050 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
The exterior angle of each side of an n-sided polygon is 24°. Find the number of sides n. Express your answer as a whole number. | 15 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
The line $3x + y + a = 0$ passes through the center of the circle $x^2 + y^2 + 2x - 4y = 0$. Find the value of $a$. Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
The slant height of a cone is 10, and its height is 8. Calculate the surface area of this cone. Express your answer in terms of π. | 96\pi | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of positive two-digit numbers that are divisible by both 3 and 4 and have a tens digit greater than the ones digit. Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the product \(\left(1-\frac{1}{3}\right)\left(1-\frac{1}{4}\right)\left(1-\frac{1}{5}\right)\) and express your answer as a simplified fraction. | \frac{2}{5} | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Fractions"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Find the constant term in the expansion of $(\frac{1}{x^{2}}+1)(x^{2}+\frac{1}{\sqrt{x}})^{6}$. Express your answer as a single integer. | 15 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Divide 2 teachers and 4 students into 2 groups to participate in social practice activities at locations A and B, respectively. Each group consists of 1 teacher and 2 students. Find the number of different arrangement schemes. Express your answer as a whole number. | 12 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Combinations"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Abel took 45 Yuan to the market. After buying 4 bottles of honey and 2 pieces of cake, he had money left. If he uses the remaining money to buy 2 more pieces of cake, he would still be short of 0.2 Yuan; but if he uses the remaining money to buy 2 more bottles of honey, he would have 4.4 Yuan left. Calculate the price ... | 6.8 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given a random variable ξ follows the normal distribution N(2016, σ^2), find the probability P(ξ < 2016). Express your answer as a numerical value. | 0.5 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Other"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Find the domain of the function $$y=\log_{3}(-x^{2}-2x)$$. Express your answer as an interval in standard notation. | (-2, 0) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Two people, A and B, independently shoot at the same target once, with hit probabilities of 0.6 and 0.7, respectively. Given that the target is hit, find the probability that both A and B hit the target simultaneously. Express your answer as a simplified fraction. | \frac{21}{44} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(2,-1)$ and $\overrightarrow{b}=(1,3)$, and $\overrightarrow{a}\perp (\overrightarrow{a}+m\overrightarrow{b})$, calculate the value of $m$. Express your answer as a single integer. | 5 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
A bag contains red, blue, and purple marbles, and does not contain any other marbles. The ratio of the number of red marbles to the number of blue marbles is 4:7. The ratio of the number of blue marbles to the number of purple marbles is 2:3. There are 32 red marbles in the bag. Calculate the total number of marbles in... | 172 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
The 64 whole numbers from 1 through 64 are written, one per square, on a checkerboard (an 8 by 8 array of 64 squares). The first 8 numbers are written in order across the first row, the next 8 across the second row, and so on. Find the sum of the numbers in the four corners of the checkerboard. Express your answer as a... | 130 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Given the hyperbola x^2 - y^2 = 1 and the midpoint of a chord as (2, 1), calculate the equation of the line containing this chord. Express your answer in slope-intercept form (y = mx + b), including all necessary coefficients and constants. | y = 2x - 3 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given a cone with a sector whose central angle is 120 degrees and a radius l, find the ratio of the total surface area to the lateral surface area of the cone. Express your answer as a ratio in the form a:b. | 4:3 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Surface Area"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Find the derivative of the function f(x) = (2πx)^2. Express your answer as a function of x. | f'(x) = 8π^2x | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given the quadratic equation $x^{2}-2x-3=0$, what is the value of $b$ in the discriminant $b^{2}-4ac$? Provide your answer as the numerical value of $b$. | -2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Find the maximum value of the function f(x) = cos 2x + 6cos(π/2 - x). Express your answer as a single number. | 5 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Round the number 98516 to the nearest thousand and express it in scientific notation using the format $a \\times 10^b$ where $a$ and $b$ are numbers. | 9.9 \times 10^4 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
In the three-dimensional Cartesian coordinate system, if $P(3,-2,1)$, find the coordinates of the point symmetric to $P$ with respect to the $xOz$ plane. Express your answer as a set of coordinates $(x, y, z)$. | (3,2,1) | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
In a bag of marbles, 3/5 of the marbles are blue and the rest are red. If the number of red marbles is doubled and the number of blue marbles stays the same, what is the new fraction of red marbles? Express your answer as a simplified fraction. | \frac{4}{7} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
In the geometric sequence {a_n}, a_n > 0, a_2 = 1 - a_1, a_4 = 9 - a_3. Find the value of a_4 + a_5. Express your answer as a single integer. | 27 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
A class of 30 students wrote a history test. Of these students, 25 achieved an average of 75%. The other 5 students achieved an average of 40%. Calculate the overall class average percentage, rounded to the nearest whole number. | 69 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Given a quadratic equation x^2 - 21x + t = 0 with integer coefficients and roots a and b that are prime numbers, find the value of the expression b/a + a/b. Express your answer as a simplified fraction. | \frac{365}{38} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation ${(x-3)}^{2}{(x+1)}^{8}={a}_{0}+{a}_{1}x+{a}_{2}{x}^{2}+⋯+{a}_{10}{x}^{10}$, find the value of $\log _{2}(a_{0}+a_{1}+a_{2}+\cdots +a_{10})$. Express your answer as a single integer. | 10 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Within the plane of triangle $ABC$, point $M$ satisfies the condition $\vec{MA} + \vec{MB} + \vec{MC} = \vec{0}$. If $D$ is the midpoint of $BC$, calculate the value of $\frac{S_{\triangle ABC}}{S_{\triangle MBC}}$. Express your answer as a single numerical value. | 3 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Select 3 students from 4 to participate in an activity. How many different methods are there to make this selection? Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Combinations"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Three sets of parallel lines and another set of five parallel lines intersect on a plane. Calculate the number of parallelograms that can be formed. Express your answer as a single integer. | 30 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence {a_n} with the general term formula $$a_{n}= \begin{cases} \left.\begin{matrix}-n, & n\leq4 \\ \sqrt {n^{2}-4n}-n, & n>4\end{matrix}\right.\end{cases}(n\in\mathbb{N}^*)$$, find the limit of a_n as n approaches infinity. Express your answer as a single numerical value. | -2 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
A square floor is tiled with a large number of regular hexagonal tiles. The tiles are coloured blue or white. Each blue tile is surrounded by 6 white tiles and each white tile is surrounded by 3 white and 3 blue tiles. Ignoring part tiles, calculate the ratio of the number of blue tiles to the number of white tiles. Ex... | \frac{1}{2} | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
A boat's speed in still water is 15 mph, and a creek's current speed is 5 mph. If the boat travels back and forth over a certain distance, what is the ratio of the boat's average speed for the round trip to its speed in still water? Express your answer as a simplified fraction. | \frac{8}{9} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
The general term of the sequence $\{a_n\}$ is given by $a_n = \dfrac{1}{n(n+1)}$. If the sum of the first $n$ terms of the sequence is $\dfrac{10}{11}$, find the value of $n$. Express your answer as a whole number. | 10 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given the curve $f(x)= \frac{2}{3}x^{3}$ at the point $(1,f(1))$, the slope angle of the tangent line is $\alpha$. Calculate $\frac{{\sin }^{2}\alpha-{\cos }^{2}\alpha}{2\sin \alpha\cos \alpha+{\cos }^{2}\alpha}$. Express your answer as a single fraction or number. | \frac{3}{5} | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
In a dairy factory, the production volumes of four types of dairy products, A, B, C, and D, are 2000 boxes, 1250 boxes, 1250 boxes, and 500 boxes, respectively. A sample of size 60 is randomly selected using a stratified sampling method based on production volumes. Calculate the number of type B dairy products in the s... | 15 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Represent the number 2200 in standard scientific notation form, where the coefficient is between 1 and 10, and the power of 10 is an integer. Express your answer in the form a × 10^n. | 2.2 × 10^3 | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given that positive numbers x and y satisfy the equation x+y=xy, find the minimum value of x+2y. Express your answer as a single numerical value. | 3+2\sqrt{2} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $x, y \in \mathbb{R}$ and $x - 2y - 4 = 0$, find the minimum value of $2^x + \frac{1}{4^y}$. Express your answer as a single number. | 8 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
A painting measuring 18 inches by 24 inches is to be placed into a wooden frame with the longer dimension vertical. The wood at the top and bottom is twice as wide as the wood on the sides. If the frame area equals that of the painting itself, calculate the ratio of the smaller to the larger dimension of the framed pai... | 2:3 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Nine cards numbered from 1 to 9 were placed face down on the table. Aleksa, Bart, Clara, and Deindra each picked up two of the cards.
- Aleksa said, "My numbers add up to 6."
- Bart said, "The difference between my numbers is 5."
- Clara said, "The product of my numbers is 18."
- Deindra said, "One of my numbers is tw... | 9 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
A bag contains \( x \) green and \( y \) red sweets. A sweet is selected at random from the bag and its color noted. It is then replaced into the bag together with 10 additional sweets of the same color. A second sweet is next randomly drawn. Find the probability that the second sweet is red. Express your answer as a f... | \frac{y}{x + y} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
If the radius of a circle is reduced to half of its original length, and the arc length is increased to $\frac{3}{2}$ times its original length, then what is the ratio of the new central angle to the original central angle? Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Find the maximum value of the function $f(x)=\cos 2x+6\cos(\frac{\pi}{2}-x)$. Express your answer as a single number. | 5 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Find the simplest common denominator of the fractions $\frac{1}{x+y}$, $\frac{1}{x-y}$, and $\frac{1}{{x^2}-{y^2}}$. Express your answer as an algebraic expression. | (x+y)(x-y) | big_math | [
"Mathematics -> Algebra -> Rational Expressions -> Other"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
The sequence $\{{a_n}\}$ has initial terms ${a_1 = -7, a_2 = 5}$, and satisfies the recurrence relation ${a_{n+2} = a_n + 2}$ for all positive integers $n$. Find the sum $S_{18}$ of the first 18 terms of the sequence. Express your answer as a single integer. | 126 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
For the parabola $x^2=8y$, find the distance from the focus to the directrix. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the domain of the function f(x) is {x|x!=0}, and it satisfies the equation f(x) + 2f(1/x) = 6/x - 3, find an expression for f(2x) in terms of x. | -\frac{1}{x} + 8x - 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given the geometric progression 1 + 2 + 2^2 +... + 2^n, where n is a positive integer, find the minimum value of n such that the sum of the progression is greater than 128. Express your answer as a whole number. | 7 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Find the solution set for the inequality $-x^{2}+3x-2 \geqslant 0$. Express your answer using interval notation (e.g., [a, b], (a, b), etc.). | [1,2] | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Given $a=2^{-\frac{1}{3}}$, $b=(2^{\log_{2}3})^{-\frac{1}{2}}$, $c=\cos50°\cos10°+\cos140°\sin170°$, determine the correct ordering of the real numbers $a$, $b$, and $c$. Express your answer in the format $a \square b \square c$ where $\square$ represents $>$, $<$, or $=$. | a > b > c | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Precalculus -> Trigonometric Functions",
"Mathematics -> Algebra -> Exponential Functions"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Given the quadratic equation $2x^2-2x+3m-1=0$ with two real roots $x_1$ and $x_2$, and the inequality $x_1 \cdot x_2 > x_1 + x_2 - 4$, determine the range of the real number $m$ that satisfies both conditions. Express your answer as an inequality in terms of $m$. | -\frac{5}{3} < m \leq \frac{1}{2} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the line equation mx+2ny-4=0 and the circle equation x^{2}+y^{2}-4x+2y-4=0, find the relationship between m and n such that the line always bisects the circumference of the circle. Express your answer as an equation in the form of m (+/-) n (+/-) number = 0. | m-n-2=0 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Let $f(x) = \cos(x + \theta) + \sqrt{2}\sin(x + \phi)$ be an even function, where $\theta$ and $\phi$ are acute angles, and $\cos\theta = \frac{\sqrt{6}}{3}\sin\phi$. Calculate the sum of $\theta$ and $\phi$ in radians. Express your answer in terms of $\pi$ if necessary. | \frac{7\pi}{12} | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ have an angle of $120^{\circ}$ between them, $|\overrightarrow{a}|=1$, and $|\overrightarrow{b}|=3$, calculate the magnitude of the difference between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$. Express your answer as a numerical value. | \sqrt{13} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Eleven points are marked from left to right on a straight line. The sum of all the distances between the first point and the other points is $2018 \mathrm{~cm}$. The sum of all the distances between the second point and the other points, including the first one, is $2000 \mathrm{~cm}$. What is the distance between the ... | 2 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the function y = f(x), find the fourth function obtained by the following transformations:
- Take the inverse of the function y = f(x) to get the second function.
- Shift the graph of the second function 2 units to the left.
- Shift the resulting graph 1 unit up.
- Reflect the resulting graph concerning the line ... | y = -f(-x-1) + 2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Evaluate the expression \( \frac{x^{2}-4}{x^{2}-2x} \) when \( x = \frac{1}{5} \). Express your answer as a single number. | 11 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given the circle $\rho=8\sin \theta$ and the line $\theta=\frac{\pi}{3}$, calculate the maximum distance from a point on the circle to the line. Express your answer as a single number. | 6 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given that the negative of one of the solutions of x^2 - 3x + c = 0 is a solution of x^2 + 3x - c = 0, find the solutions of x^2 - 3x + c = 0. Express your answer as two numerical values. | 0, 3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Quadratic Functions"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $a=(2,4)$ and $b=(-1,1)$, calculate the result of the vector operation $2a-b$. Express your answer as an ordered pair $(x, y)$. | (5,7) | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Evaluate the expression \((999 - 99 + 9) \div 9\) and provide your answer as a whole number. | 101 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Let $f_{0}(x) = \cos x$, $f_{1}(x) = f'_{0}(x)$, $f_{2}(x) = f'_{1}(x)$, $f_{n+1}(x) = f'_{n}(x)$ $(n \in \mathbb{N})$. Find $f_{2012}(x)$. Express your answer as a trigonometric function of x. | \cos x | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Given a complex number $z=x+yi (x,y∈R)$, and $|z-2|= \sqrt{3}$, find the maximum value of $\frac{y}{x}$. Express your answer as a numerical value. | \sqrt{3} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
There are 5 different textbooks, including 2 Chinese books, 2 math books, and 1 physics book. If these books are placed on the same shelf in a row such that books of the same subject are not adjacent, how many arrangements are possible? Express your answer as a whole number. | 48 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Permutations"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given the standard equation of a parabola is $y^2=6x$, find the exact coordinates of its focus. Express your answer as an ordered pair (x, y). | \left( \frac {3}{2},0 \right) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the complex number z satisfies the equation z(2-i) = 18+11i, find the real part of z. Express your answer as a single integer. | 5 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Given that angle α = 2kπ - π/5 (k ∈ ℤ), if angle θ has the same terminal side as angle α, calculate the value of y = sin θ / |sin θ| + cos θ / |cos θ| + tan θ / |tan θ|. Express your answer as a single integer. | -1 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
A cube, with each face painted, is cut into 27 smaller cubes of equal size. Calculate the probability that a randomly selected small cube has exactly two faces painted. Express your answer as a simplified fraction. | \frac{4}{9} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
Square \( ABCD \) has a side length of 12 inches. A segment \( AE \) is drawn where \( E \) is on side \( DC \) and \( DE \) is 5 inches long. The perpendicular bisector of \( AE \) intersects \( AE, AD, \) and \( BC \) at points \( M, P, \) and \( Q \) respectively. Calculate the ratio of the segments \( PM \) to \( M... | 5:19 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the equations 60^a = 3 and 60^b = 5, find the value of 12^((1-a-b)/2(1-b)). Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Among the letters M, X, S, and E, how many are symmetrical about an axis? Express your answer as a whole number. | 3 | big_math | [
"Mathematics -> Other -> Other"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
The math teacher of the ninth grade teaches an average of 6 tutoring classes per month. If the classes are taught by female teachers, each female teacher should teach 15 classes per month. Let $a$ be the number of classes taught by each male teacher if only male teachers are teaching. Using the information given, calcu... | 10 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ with an angle of $60^\circ$ between them, $|\overrightarrow{a}|=3$, $|\overrightarrow{b}|=2$. If $(3\overrightarrow{a}+5\overrightarrow{b})$ is perpendicular to $(m\overrightarrow{a}-\overrightarrow{b})$, calculate the value of $m$. Express your answer as a si... | \frac{29}{42} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Other"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given $x \in \left(-\frac{\pi}{2}, 0\right)$ and $\cos x = \frac{4}{5}$, calculate the value of $\tan 2x$. Express your answer as a single number. | -\frac{24}{7} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Given the polynomial P(x) = (x - 1^2)(x - 2^2)⋯(x - 100^2), how many integers n satisfy P(n) ≤ 0? Express your answer as a single integer. | 5100 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
21 people sit in a circle and count in sequence from 1 to 3. The person who counts "3" comes out to perform the program, and those who have performed do not participate in the counting again. Calculate the total number of counts made when only two people who have not performed the program are left. | 64 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets $A=\{-1,1,3,5,7\}$ and $B=\{x|2^{x} > 2\sqrt{2}\}$, find the intersection $A \cap B$. Express your answer as a set of integers in the format $\boxed{\{...\}}$. | \{3,5,7\} | big_math | [
"Mathematics -> Discrete Mathematics -> Set Theory -> Other"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Given that \(\{a_n\}\) is a geometric sequence, and \(a_na_{n+4}=9^{n+1} (n\in N^*)\), calculate the common ratio \(q\). Express your answer as a numerical value, including the ± symbol if applicable. | \pm 3 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of intersection points between the curves \( C_{1}: x^{2}-y^{2}+4y-3=0 \) and \( C_{2}: y=ax^{2} \) where \( a > 0 \) is a constant. | 4 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
A certain commodity is purchased at a price of 100 yuan per piece. It is sold at a 25% increase on the purchase price. Later, due to overstock, the price is reduced, and it is sold at a 10% discount. Calculate the profit per piece. Express your answer in yuan. | 12.5 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the angle between the plane vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is $60^{\circ}$, $\overrightarrow{a}=(2,0)$, and $|\overrightarrow{b}|=1$, calculate the magnitude of the vector $\overrightarrow{a}+2\overrightarrow{b}$. Express your answer as a numerical value. | 2\sqrt{3} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Convert the polar coordinate equation $\rho=4\cos\theta$ into a Cartesian coordinate equation in the form of $(x-h)^2 + (y-k)^2 = r^2$. | (x-2)^2+y^2=4 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Find the range of the function y=2tan(x-π/6), where x∈[-π/6, 5π/12]. Express your answer as an interval, including the minimum and maximum values. | [-2√3,2] | big_math | [
"Mathematics -> Precalculus -> Functions"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given an ellipse $C:\\frac{{x}^{2}}{{a}^{2}}+\\frac{{y}^{2}}{{b}^{2}}=1$ where $a>b>0$, and the foci $F_{1}$ and $F_{2}$, with point $P$ on the ellipse such that $PF_{2}\\bot F_{1}F_{2}$. A line is drawn perpendicularly from $P$ to $F_{1}P$, intersecting the $x$-axis at point $A$, and it is given that $|AF_{2}|=\\frac{... | \frac{3-\sqrt{5}}{2} | big_math | [
"Mathematics -> Geometry -> Differential Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence {a_n} satisfying a_1 = 1 and a_{n+1} = na_n + 1, calculate the fifth term a_5. Express your answer as a single integer. | 65 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Let $i$ be the imaginary unit. If $(\sqrt{3} + i)z = (1 - \sqrt{3}i)$, then find the value of $|z|$. | 1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
A subset of the integers $1,2,\cdots,100$ has the property that none of its members is 3 times another. What is the largest number of members such a subset can have? Express your answer as a whole number. | 76 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $|\overrightarrow{a}|=1$, $|\overrightarrow{b}|=\sqrt{2}$, and $\overrightarrow{a} \bot (\overrightarrow{a}-\overrightarrow{b})$, calculate the projection of vector $\overrightarrow{a}$ in the direction of vector $\overrightarrow{b}$. Express your answer as a fraction or decimal value. | \frac{\sqrt{2}}{2} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, $ABCDEFGH$ is a cube with an edge length of $12 \text{ cm}$. An ant sits on the cube at vertex $A$. The ant can only walk along the edges of the cube, and cannot walk along any edge more than once. What is the greatest distance that the ant can walk before it cannot continue? Express your answer in cent... | 108 | big_math | [
"Mathematics -> Discrete Mathematics -> Graph Theory"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
The equation 2x^2 + xy - y^2 = 14 has how many integer solutions? Express your answer as a whole number. | 8 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given a complex number $z=2-i$, calculate the imaginary part of $z(\overline{z}+i)$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the average of the 20 data points \\(a_{1}\\), \\(a_{2}\\), \\(a_{3}\\),..., \\(a_{20}\\) is \\(\bar{x}\\), and the variance is 0.21, calculate the variance of the 21 data points \\(a_{1}\\), \\(a_{2}\\), \\(a_{3}\\),..., \\(a_{20}\\), \\(\bar{x}\\). Express your answer as a decimal value rounded to 2 places... | 0.20 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Given the recursive sequence $\{a_n\}$ defined by $a_{n+1}= \frac {a_n}{2a_n+1}$ and $a_1=1$, calculate the value of $a_6$. Express your answer as a simplified fraction. | \frac{1}{11} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
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