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 |
|---|---|---|---|---|---|
Let $a$ and $b$ be two unit vectors on a plane, with $a \cdot b = \frac{3}{5}$. If $m \in \mathbb{R}$, find the minimum value of $|a + mb|$. Express your answer as a single fraction. | \frac{4}{5} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. Given that $B= \frac{\pi}{3}$, $b=6$, and $\sin A - 2\sin C = 0$, calculate the value of $a$. Express your answer as a numerical value. | 4\sqrt{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Let the domain of the function y=f(x) be R and satisfy y=f(x+2) is an odd function. Find the value of f(2). | 0 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $f(x)$ and $g(x)$ are both odd functions, and $F(x) = f(x) + g(x) + 2$ has a maximum value of 8 on $(0, +\infty)$, determine the minimum value of $F(x)$ on $(-\infty, 0)$. Express your answer as a single numerical value. | -4 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Find the opposite of 3, which is defined as the number that, when added to 3, results in zero. Express your answer as an integer. | -3 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given the quadratic equation $x^{2}-8x+5=0$ with roots $m$ and $n$, calculate the value of $\frac{1}{m-1}+\frac{1}{n-1}$. Express your answer as a single number. | -3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Quadratic Functions"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
In the expansion of $(1-x)^6+(1+x)^5$, what is the coefficient of the term containing $x^3$? Express your answer as an integer. | -10 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Given events A and B, with P(B) = 1/3, P(\\overline{B}|A) = 3/4, and P(\\overline{B}|\\overline{A}) = 1/2, calculate the probability of event A, P(A). Express your answer as a fraction. | \\frac{2}{3} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence \\{ a_{n} \\}, a_{7} = 9 and a_{13} = -3. Find a_{10}. Express your answer as a single integer. | 3 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given that two positive numbers a and b have an arithmetic mean of 7/2 and a geometric mean of 2*sqrt(3), with a < b, find the eccentricity e of the hyperbola x^2/a^2 - y^2/b^2 = 1. Express your answer as a fraction. | \frac{5}{3} | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Geometry -> Other"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
A school's labor union is planting trees. If each person plants 2 trees, there are 21 trees left over. If each person plants 3 trees, they are 24 trees short. Write an algebraic equation that represents this situation, where x is the number of students in the labor union. Express your answer as a valid equation. | 2x + 21 = 3x - 24 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given two sound waves represented by y1 = 3√2 sin(100πt) and y2 = 3 sin(100πt - π/4), find the amplitude of the combined sound wave y = y1 + y2. Express your answer as a single numerical value, including any necessary square roots. | 3√5 | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, the line segment with endpoints \( P(-4,0) \) and \( Q(16,0) \) is the diameter of a semi-circle. If the point \( R(0, t) \) is on the circle with \( t > 0 \), find the value of \( t \). Express your answer as a single integer. | 8 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
For variables x and y that have a linear relationship, there is a set of observation data (x_i, y_i) (i=1, 2, 3,..., n), and all observation data are on the regression line equation y = (1/3)x + 2. Calculate the sum of squared residuals for this set of data. | 0 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
The equation of the line 2x - y - 4 = 0 is rotated counterclockwise by π/4 about its intersection point with the x-axis. Calculate the resulting line equation in standard form Ax + By + C = 0. | 3x + y - 6 = 0 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given \( a^2 + b^2 = 4 \), calculate the value of the expression \( \sqrt[3]{a(b-4)} + \sqrt{ab - 3a + 2b - 6} \). Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given that point C is the golden section point of segment AB, and AC > BC, with AB = 200, find the length of AC. Express your answer in exact form. | 100(\sqrt{5}-1) | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
How many moles of oxygen gas are produced by the decomposition of 245 g of potassium chlorate, given the balanced chemical equation: 2KClO3(s) → 2KCl(s) + 3O2(g) and the molar mass of KClO3: 122.6 g/mol? Express your answer in moles, rounded to 2 decimal places. | 3.00 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given sets A = {x ∈ ℕ | 1 < x ≤ 4} and B = {x | x^2 - x - 6 ≤ 0}, find the intersection A ∩ B. Express your answer as a set of numbers that satisfy both conditions. | \{2, 3\} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Among the real numbers $3\pi$, $-\frac{7}{8}$, $0$, $\sqrt{2}$, $-3.15$, $\sqrt{9}$, $\frac{\sqrt{3}}{3}$, how many are irrational? Express your answer as a whole number. | 3 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Five children (Tian Tian, Shi Tou, Kimi, Cindy, Angela) need to switch fathers, with each child choosing any one of the other four fathers except their own. What is the total number of different combinations of choices for the five children? Express your answer as a single integer. | 44 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets M={x|x≤4} and N={x|y=log2x}, find the intersection of sets M and N, denoted as M∩N. Express your answer in interval notation, including the endpoints of the interval if they are part of the solution. | (0,4] | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $\sin 2t = -\int_{0}^{t}{\cos xdx}$, where $t \in (0, \pi)$, calculate the exact value of $t$. Express your answer as a fraction of $\pi$ in radians. | \frac{2\pi}{3} | big_math | [
"Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Find the coefficient $a_2$ of the $x^2$ term in the expansion of $(3x+1)^5$. Express your answer as a numerical value. | 90 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Binomial Expansion"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Evaluate the trigonometric expression $\sin 20^\circ \cos 10^\circ - \cos 160^\circ \sin 170^\circ$. Express your answer as a single fraction or decimal value. | \frac{1}{2} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Given set $A=\{x||x-1| \lt 1\}$ and set $B=\left\{{x|{y=\sqrt{2-x}}}\right\}$, find the intersection $A \cap B$ and express your answer in interval notation. | (0, 2) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given that {a_n} is an arithmetic sequence and a_1 + a_{15} = 48, find the value of a_3 + 3a_8 + a_{13}. Express your answer as a single integer. | 120 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
In the three-dimensional Cartesian coordinate system, find the symmetric point of (2,1,3) with respect to the plane xOz. Express your answer as a coordinate (x, y, z). | (2,-1,3) | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Given a sequence $\{a_n\}$ that satisfies $a_{n+2} = a_{n+1} + a_{n}$, and $a_1 = 1$, $a_5 = 8$, calculate the value of $a_3$. Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Given an equation $x^2 - ax + b = 0$ with integer solutions $n$ and $m$, where $n > 8$. The coefficient $a$ in base-$n$ is represented as $18_n$. Find the base-$n$ representation of $b$. Express your answer in base-$n$ format. | 80_n | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Suppose a point $P$ has coordinates $(m, n)$, where $m$ and $n$ are the numbers obtained by rolling two dice consecutively. Calculate the probability that point $P$ falls within the circle $x^2 + y^2 = 9$. Express your answer as a simplified fraction in the form $\boxed{\frac{a}{b}}$. | \frac{1}{9} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given $\cos \left( \frac{\pi}{2} + \alpha \right) = \frac{3}{5}$, calculate the value of $\cos 2\alpha$. Express your answer as a fraction. | \frac{7}{25} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
When the expressions $x^5$, $x + \frac{1}{x}$, and $1 + \frac{2}{x} + \frac{3}{x^2}$ are multiplied, what is the degree of the resulting polynomial? Express your answer as a single numerical value. | 6 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given the functional equation f(x-1) = (1 + f(x+1))/(1 - f(x+1)), find the value of f(1) * f(2) * f(3) *... * f(2008) + 2008. Express your answer as a single integer. | 2009 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
There are two cards; one is red on both sides and the other is red on one side and blue on the other. The cards have the same probability (1/2) of being chosen, and one is chosen and placed on the table. If the upper side of the card on the table is red, calculate the probability that the under-side is also red. Expres... | \frac{2}{3} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Given $a=\left(\frac{1}{2}\right)^{0.1}$, $b=3^{0.1}$, and $c=\left(-\frac{1}{2}\right)^3$, determine the ordering of $a$, $b$, and $c$ from greatest to least. Express your answer as a sequence of variables (e.g., b > a > c). | b > a > c | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the two real roots of the equation $(\lg x)^{2}-\lg x^{2}-3=0$ are $a$ and $b$, calculate the value of $\log_{a}b+\log_{b}a$. Express your answer as a single number. | -\dfrac{10}{3} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
Three dice, each showing numbers 1 to 6, are colored red, blue, and yellow, respectively. Each of the dice is rolled once. The total of the numbers rolled is 10. Calculate the number of different ways this can happen. Express your answer as a single integer. | 27 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
How many types of positional relationships can a line have with a plane? Provide your answer as a single number. | 3 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the sum of the first 10 terms and the sum of the first 100 terms of an arithmetic sequence are 100 and 10 respectively, what is the sum of the first 110 terms? Express your answer as a single numerical value. | -110 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given the arithmetic sequence {a<sub>n</sub>}, where a<sub>3</sub>+a<sub>7</sub>=10 and a<sub>8</sub>=8, find the common difference d. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(2, -1, 3)$, $\overrightarrow{b}=(-1, 4, -2)$, $\overrightarrow{c}=(7, 5, \lambda)$, if vectors $\overrightarrow{a}$, $\overrightarrow{b}$, and $\overrightarrow{c}$ are coplanar, find the value of the real number $\lambda$. Express your answer as a fraction. | \frac{65}{7} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Find the negation of the proposition "∀x∈R, e<sup>x</sup> > x". Express your answer as a logical proposition using the same notation and formatting as the original proposition. | ∃x∈R, e<sup>x</sup> ≤ x | big_math | [
"Mathematics -> Discrete Mathematics -> Logic -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the three-view diagrams of a solid, determine the volume of the solid. The side view is an isosceles right triangle with leg length 1. The top view is an equilateral triangle with a base length of 1 and a height of 1. The front view is an isosceles triangle with a base length of 1 and a height of 1. Calculate the... | \frac{1}{6} | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Volume"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Represent the following geometric relationships in mathematical notation: point A lies on line a, line a lies in plane α, and point B lies in plane α. Use ∈ for "is an element of" and ⊂ for "is a subset of". Express your answer using these mathematical symbols. | A ∈ a, a ⊂ α, B ∈ α | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
In $\triangle ABC$, the sides opposite to $\angle A$, $\angle B$, and $\angle C$ are denoted as $a$, $b$, and $c$ respectively. Given that $a=\frac{\sqrt{5}}{2}b$ and $\angle A=2\angle B$, calculate the value of $\cos B$. Express your answer as a single value, including the square root symbol if necessary. | \frac{\sqrt{5}}{4} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Given the planar vectors $a=(1,2)$ and $b=(-2,m)$, where $a$ is parallel to $b$, find the value of $2a+3b$. Express your answer as a coordinate pair (x, y). | (-4,-8) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Let set $A = \{-1, 1\}$, and set $B = \{-1, 3\}$. Find the intersection of sets $A$ and $B$, denoted as $A \cap B$. Express your answer as a set using proper set notation. | \{-1\} | big_math | [
"Mathematics -> Discrete Mathematics -> Logic"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Professor Rosseforp runs to work every day. On Thursday, he ran 10% faster than his usual average speed, resulting in a reduction of x minutes in his journey time. Calculate the time it took him to run to work on Wednesday. Express your answer in terms of x. | 11x | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given a geometric sequence {an} with all real numbers, the sum of the first n terms is Sn. If S10 = 10 and S30 = 70, calculate the sum of the first 40 terms, S40. Express your answer as a single integer. | 150 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ with left and right foci $F_1$ and $F_2$, respectively, and a point $P$ on the ellipse such that $|PF_1| = 3|PF_2|$, calculate the range of the eccentricity of this ellipse. Express your answer as an interval. | [\frac{1}{2}, 1) | big_math | [
"Mathematics -> Geometry -> Differential Geometry -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Find the range of real number k such that the inequality kx^2 + 2kx - (k+2) < 0 always holds true. Express your answer in interval notation, including any relevant mathematical notation (e.g., parentheses, brackets). | (-1, 0) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence {a_n} with a_{n+1} - a_n = 2, and S_n representing the sum of the first n terms of {a_n}, if S_{10} = 50, calculate the sum of the first 10 terms of the sequence {a_n + a_{n+1}}. | 120 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
The figure consists of five squares and two right-angled triangles. The areas of three of the squares are 5, 8, and 32, as shown. Find the area of the shaded square. Express your answer as a whole number. | 45 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given an isosceles triangle with a perimeter of 20, find the function y=f(x) that represents the relationship between the length of the base y and the length of one of the equal sides x, including any restrictions on the domain of x. Express your answer in the format y = f(x) (domain of x). | y = 20 - 2x (5 < x < 10) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given that both the "progress rate" and "regress rate" are 1%, calculate the ratio of the "progress value" to the "regress value" after 300 days. Use the reference data: lg 101≈2.0043, lg 99≈1.9956, 10^0.87≈7.41. Express your answer as a single number, rounded if necessary. | 407 | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
The arc length and the area of a sector are both 4. Find the radian measure of the central angle of this sector. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence {a_n}, if a_2 + 2a_6 + a_{10} = 120, then find a_3 + a_9. Express your answer as a single integer. | 60 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Given the set $U=\{x|-3 < x < 3,x∈Z\}$, $A=\{1,2\}$, $B=\{-2,-1,2\}$, find the union of $A$ and the complement of $B$ in $U$. Express your answer as a set of integers, including all elements of the resulting set. | \{0,1,2\} | big_math | [
"Mathematics -> Discrete Mathematics -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\frac{m}{n} = \frac{3}{7}$, find the value of $\frac{m+n}{n}$. Express your answer as a fraction in simplest form. | \frac{10}{7} | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Fractions"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
A truck can carry 2.5 tons of coal in one trip. To transport 48 tons of coal in one trip, what is the minimum number of such trucks needed? Express your answer as a whole number. | 20 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $f(x+1)+f(-x+1)=2$, identify the symmetry type (odd or even) of the transformed function $f(x+1)-1$. Provide your answer in the format "The transformed function $f(x+1)-1$ is [odd/even]". | The transformed function $f(x+1)-1$ is odd. | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence {a_n} that satisfies a_1 = 1 and a_n - a_{n-1} = 2 for n >= 2, find the general term a_n. Express your answer as a formula in terms of n. | a_n = 2n - 1 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
A granary received 1512 shi of rice. Upon inspection, it was found that the rice contained an admixture of wheat. A sample of rice was taken, and out of 216 grains, 27 were wheat. Calculate the approximate amount of wheat mixed in this batch of rice. Express your answer in shi. | 189 \text{ shi} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Given the linear function y = -1/2x + 1, determine the quadrant of the coordinate plane that the graph of this function does not pass through. Express your answer as the name of the quadrant (First, Second, Third, or Fourth). | Third | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
A two-digit number has a tens digit y and a units digit x, and the sum of these digits is 8. When 36 is subtracted from this number, the result is exactly the two-digit number formed by swapping the tens and units digits. Find the original two-digit number. Express your answer as a two-digit number. | 62 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Given that f(x) is an odd function and is increasing on (0, +∞), and f(2) = 0, find the solution set of the inequality (f(x) - f(-x))/x > 0. Express your answer as an interval or union of intervals. | (-∞, -2) ∪ (2, +∞) | big_math | [
"Mathematics -> Calculus -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation of the hyperbola $\frac{x^{2}}{20} - \frac{y^{2}}{5} = 1$, determine its semi-focal distance. Express your answer as a single numerical value. | 5 | big_math | [
"Mathematics -> Geometry -> Non-Euclidean Geometry -> Other"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x)=\begin{cases} x^{2}-x, & x\geqslant 0 \\ g(x), & x < 0 \end{cases}$ is an odd function, calculate the value of $g(f(-2))$. Express your answer as a single integer. | -2 | big_math | [
"Mathematics -> Algebra -> Functions"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy the equation $(\overrightarrow{a} + \overrightarrow{b})(2\overrightarrow{a} - \overrightarrow{b}) = -4$, with $|\overrightarrow{a}| = 2$ and $|\overrightarrow{b}| = 4$. Calculate the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ in radians, usi... | \frac{\pi}{3} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence {a_n} where a_5 = 11 and the sum of the first 12 terms S_{12} = 186, find the value of a_8. Express your answer as a single integer. | 20 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Determine the number of positive integer triplets (a, b, c) that satisfy the system of equations:
\[
\begin{cases}
ab + bc = 44 \\
ac + bc = 23
\end{cases}
\]
Express your answer as a single integer representing the count of solutions. | 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Given a set of points on a plane, it is known that the distances between some pairs of points are 1, 2, 4, 8, 16, 32. What is the smallest possible value of n (the number of points) that allows each of these distances to occur between some pair of points? Express your answer as a single integer. | 7 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the line symmetric to the line 2x-y+3=0 with respect to the fixed point M(-1,2). Express your answer as a linear equation in the form Ax + By + C = 0. | 2x-y+5=0 | big_math | [
"Mathematics -> Geometry -> Analytic Geometry -> Lines"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given a function f(x) that satisfies f(x+1) = x^2 - 1, determine the expression for f(x). Express your answer as a function of x. | f(x) = x^2 - 2x | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = a^(x-1) + 2 (where a > 0 and a!= 1), find a point that the graph of the function definitely passes through. Express your answer as an ordered pair (x, y). | (1,3) | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression:
$$
\left(1+\frac{1}{2}\right)\left(1+\frac{1}{3}\right)\left(1+\frac{1}{4}\right)\left(1+\frac{1}{5}\right)\left(1+\frac{1}{6}\right)\left(1+\frac{1}{7}\right)\left(1+\frac{1}{8}\right)\left(1+\frac{1}{9}\right)
$$
Express your answer as a single number. | 5 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
The projection of the origin on the line L is P(-2,1). Find the equation of line l in standard form (Ax + By + C = 0). | 2x - y + 5 = 0 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given the set $A=\{x| \frac {2x-6}{x+1}\leq0\}$, and $B=\{-2, -1, 0, 3, 4\}$, calculate the intersection of sets A and B, $A\cap B$. Express your answer as a set of numbers, including all elements of the intersection. | \{0, 3\} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
For the quadratic equation $x^{2}-4x+k=0$ to have no real solutions, find the range of values for $k$. Express your answer as an inequality involving $k$. | k > 4 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
In an arithmetic sequence {a_n}, it is known that a_1 = 2 and a_3 + a_5 = 10. Find the value of a_7. Express your answer as a single integer. | 8 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $x$ and $y$ are positive real numbers, and $\dfrac{1}{x+2} + \dfrac{1}{y+2} = \dfrac{1}{6}$, find the minimum value of $x+y$. Express your answer as a single number. | 20 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
A certain type of seed has a germination probability of 0.9. Now, 1000 seeds have been sown. For each seed that does not germinate, 2 more seeds will be sown as replacements. Let the number of replacement seeds be $X$. Calculate the expected value of $X$. Express your answer as a whole number. | 200 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $a = (2\sin 35^{\circ},2\cos 35^{\circ})$ and $b = (\cos 5^{\circ}, -\sin 5^{\circ})$, calculate the dot product $a\cdot b$. Express your answer as a single numerical value. | 1 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Given that cos(π/2 - α) = 1/3, calculate cos(π - 2α) using the double angle identity. Express your answer as a fraction, including the negative sign if applicable. | -\frac{7}{9} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
The sequence 1, -5, 9, -13, 17, -21, …, $(-1)^{n-1}(4n-3)$, …, has a sum of the first n terms denoted as $S_n$. Calculate the value of $S_{15}$. Express your answer as a single integer. | 29 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the curve $\begin{cases}x=3\cos \phi \\ y= \sqrt{5}\sin \phi \end{cases}$ (where $\phi$ is the parameter), calculate the eccentricity of the ellipse. Express your answer as a simplified fraction. | \frac{2}{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x)= \begin{cases} 3^{x}-2,(x\leqslant 1) \\ \log _{3}(x-1),(x > 1)\end{cases}$, find the value of $f(f( \dfrac {5}{3}))$. Express your answer as a single value. | -\dfrac {4}{3} | big_math | [
"Mathematics -> Algebra -> Functions"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Given: $a + b = m$, $ab = -4$, simplify the result of $(a - 2)(b - 2)$. Express your answer as a simplified algebraic expression. | -2m | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
In the figure, \(AB \parallel EF \parallel CD\), given \(AB = 20\), \(CD = 80\), and \(BC = 100\). Find the value of \(EF\). Express your answer as a whole number. | 16 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
What is the integer closest in value to \(\frac{35}{4}\)? Express your answer as a single integer. | 9 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x) = \begin{cases} 1-2^{x} & (x \leqslant 2) \\ x^{2}-x-5 & (x > 2) \end{cases}$, find the value of $f[f(3)]$. Express your answer as a single integer. | -1 | big_math | [
"Mathematics -> Algebra -> Functions"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Let $\overrightarrow{e_1}$ and $\overrightarrow{e_2}$ be two non-collinear vectors in a plane, $\overrightarrow{AB}=(a-1) \overrightarrow{e_1}+ \overrightarrow{e_2}$, $\overrightarrow{AC}=b \overrightarrow{e_1}-2 \overrightarrow{e_2}$ (where $a>0$, $b>0$). If points A, B, and C are collinear, calculate the maximum valu... | \frac {1}{2} | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Linear Algebra -> Vectors"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Determine the number of solutions to the equation x + 2^x + log_2 x = 0. Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Find the smallest positive integer n that satisfies the inequality \\( \\sqrt{n} - \\sqrt{n-1} < 0.01 \\). Express your answer as a single integer. | 2501 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given the lines $l: 4ax + y - 1 = 0$ and $l_{2}: (a-1)x + y + 1 = 0$, if $l_{1} \perp l_{2}$, find the value of $a$. Express your answer as a single value. | \frac{1}{2} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
In triangle ABC, the lengths of the sides opposite to angles A, B, and C are denoted as a, b, and c respectively, and ∠A = 80°, $a^2=b(b+c)$. Find the size of ∠C in degrees. | 60 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given a sequence $\{a_n\}$ that satisfies $a_{n+2}=a_{n+1}-a_{n}$, and $a_{1}=2$, $a_{2}=3$, let $S_{n}$ be the sum of the first $n$ terms of the sequence $\{a_n\}$. Calculate the value of $S_{2016}$. | 0 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the exact value of the expression ${3}^{1+\log_{3}2}+\log5+\log_{3}2 \times \log_{2}3 \times \log2$. Express your answer as a single number. | 7 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Logarithmic Functions"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Find the constant term in the expansion of the algebraic expression $( \sqrt {x}+2)( \frac {1}{ \sqrt {x}}-1)^{5}$. Express your answer as a single integer. | 3 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.