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
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|---|---|---|---|---|---|
Given the pattern in the equations $2\sqrt{\frac{2}{3}} = \sqrt{2\frac{2}{3}}$, $3\sqrt{\frac{3}{8}} = \sqrt{3\frac{3}{8}}$, $4\sqrt{\frac{4}{15}} = \sqrt{4\frac{4}{15}}$, $5\sqrt{\frac{5}{24}} = \sqrt{5\frac{5}{24}}$, find the value of n for the equation $8\sqrt{\frac{8}{n}} = \sqrt{8\frac{8}{n}}$. Express your answer... | 63 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence $\{a_n\}$ with the first term being a positive number and the sum of the first $n$ terms denoted as $S_n$, if $a_{1006}$ and $a_{1007}$ are the two roots of the equation $x^2 - 2012x - 2011 = 0$, find the maximum value of the positive integer $n$ for which $S_n > 0$ holds. Express your answ... | 1007 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given $a=\left(\frac{1}{5}\right)^2$, $b=2^{\frac{1}{5}}$, and $c=\log_{2}\frac{1}{5}$, determine the order of these three values from smallest to largest. Express your answer as a sequence of variables, using the less than (<) symbol to denote the order. | c < a < b | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
A group of people jointly buy goods. If each person contributes 8 coins, there will be 3 coins left over; if each person contributes 7 coins, there will be 4 coins short. Let x be the number of people and y be the price of the goods. Derive the system of equations that represents this situation. Express your answer as ... | \end{array}\right.\] | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
A student has 4 online classes scheduled in the morning (mathematics, Chinese, politics, and geography) and 3 classes scheduled in the afternoon (English, history, and physical education). The student plans to check in for one class in the morning and one class in the afternoon. What is the probability that at least on... | \frac{2}{3} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
If the ratio of the radii of three spheres is 1:2:3, then how many times the sum of the volumes of the other two spheres is the volume of the sphere with the largest radius? Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
A company currently has 160 staff members, 30 middle managers, and 10 senior managers. To conduct a health check, a sample of m individuals will be drawn using stratified sampling. If only 1 senior manager is selected, find the value of m. Express your answer as a whole number. | 20 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Find the coordinates of the intersection point of the lines $y=x$ and $y=-x+2$. Express your answer as an ordered pair $(x, y)$. | (1, 1) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the domain of the function y=f(x) is [1,2], find the domain of the function y=f(sqrt(x)). Express your answer in interval notation. | [1,4] | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
There are 6 contestants speaking in sequence. Contestant A does not speak first or last. Calculate the total number of different speaking orders. Express your answer as a single integer. | 480 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $|a|=2$ and $|b|=a$, find all possible values for the sum $a+b$. Express your answer as a list of integers or simple expressions if necessary. | 0, 4, -4 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number z = -1/2 + sqrt(3)/2 * i, calculate the value of (z^2 + z + 1). Express your answer as a single number in the boxed notation. | 0 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
The equation of line l is x = 2. Find the slope angle of this line in degrees. | 90 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Let i be the imaginary unit, and the complex number z = (3-i)/i. Find the conjugate of z, denoted by \\bar{z}. Express your answer in the form a + bi. | \\bar{z} = -1 + 3i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given the functions f(x) = 3sinx + 4cosx and g(x) = 3sinx - 4cosx, if the graph of function f(x) is shifted to the right by φ units to obtain the graph of function g(x), calculate the value of sinφ. Express your answer as a simplified fraction. | \frac {24}{25} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Find the zeros of the function $y=x^2-x-2$. Express your answer as a pair of exact values. | -1, 2 | big_math | [
"Mathematics -> Algebra -> Algebra -> Equations and Inequalities"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = cos(ωx + π/4), find the minimum value of the real number ω (ω > 0) such that shifting the graph to the left by π/3 units results in an odd function. Express your answer as a single real number. | 3/4 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
If starting from a vertex of a regular polygon, at most 5 diagonals can be drawn, calculate the measure of one of its interior angles. Express your answer in degrees as a whole number. | 135 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Polygons"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
In a certain community, there are currently 480 households, among which there are 200 middle-income families, 160 low-income families, and the rest are high-income families. In a stratified sampling survey conducted for the construction of a happy Guangdong, 6 high-income families were selected. Calculate the total num... | 24 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x)= \begin{cases}x-3,x\geqslant 10 \\ f[f(x+5)],x < 10\end{cases}$, where $x\in N$, find $f(8)$. Express your answer as a single integer. | 7 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
What is the result of subtracting 2 from -1? Express your answer as an integer. | -3 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
The manufacturing of a certain mechanical part consists of two operations. The scrap rate of the first operation is $a$, and the scrap rate of the second operation is $b$. Assuming that the production of scrap from these two operations is independent of each other, find the qualifying rate of the product. Express your ... | 1 - a - b + ab | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
The area of a rectangle is 12 square metres, and the lengths of its sides are whole numbers. Calculate the greatest possible perimeter of this rectangle in metres, ensuring your answer is a whole number. | 26 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Given $$\overrightarrow {a}=(\cos x，-2)， \overrightarrow {b}=(\sin x，1)$$ and $$\overrightarrow {a}$$ is parallel to $$\overrightarrow {b}$$, calculate the value of $\sin 2x$. Express your answer as a single value. | - \frac {4}{5} | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Geometry -> Other"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\overrightarrow{e_{1}}$ and $\overrightarrow{e_{2}}$ are unit vectors satisfying $\overrightarrow{e_{1}} \cdot \overrightarrow{e_{2}} = \frac{1}{2}$, and $\overrightarrow{a} = \lambda_{1} \overrightarrow{e_{1}} + \lambda_{2} \overrightarrow{e_{2}}$ is a non-zero vector where $\lambda_{1}, \lambda_{2} \in \m... | \frac{2\sqrt{3}}{3} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given the solution set of the inequality $(ax-1)(x+1) < 0$ with respect to $x$ is $(-\infty,-1)\cup\left( - \frac{1}{2},+\infty \right)$, calculate the value of $a$. Express your answer as a single integer or fraction. | -2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
A figure is composed of three isosceles right-angled triangles, each with legs of length 4 units. Calculate the total area of the figure. Express your answer in square units as a whole number. | 24 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given the parametric equations of a line $\left\{\begin{array}{l}x=\sqrt{3}-\frac{1}{2}t\\ y=2+\frac{\sqrt{3}}{2}t\end{array}\right.$ and the equation of another line $kx+y+1=0$, calculate the real number $k$ such that the two lines are parallel. Express your answer as a single real number. | \sqrt{3} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Given $0 < \alpha < \pi$, satisfying $3\sin 2\alpha = \sin \alpha$, calculate the value of $\cos (\pi - \alpha)$. Express your answer as a numerical value. | -\frac{1}{6} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given the universal set A = {0, 1, 2}, find the total number of proper subsets of set A. Express your answer as a whole number. | 7 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics -> Other"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the harmonic mean of 1 and 2016 using the formula: $\frac{2ab}{a+b}$. Round your answer to the nearest integer. | 2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Find the constant term in the expansion of (x + 1/x)(2x - 1/x)^5. Express your answer as a single integer. | 40 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
In our school netball league, a team gains a certain whole number of points if it wins a game, a lower whole number of points if it draws a game, and no points if it loses a game. After 10 games, my team has won 7 games, drawn 3, and gained 44 points. My sister's team has won 5 games, drawn 2, and lost 3. Calculate the... | 31 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Suppose f(x) is defined on [0,1]. Determine the range of values for a such that the function f(x-a) + f(x+a) is defined. Express your answer as an interval using standard interval notation. | \left[-\frac{1}{2}, \frac{1}{2}\right] | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
In the geometric sequence {a_n}, the common ratio q = -2, and the sum of the first 5 terms S5 = 44. Calculate the value of the first term a1. Express your answer as a single numerical value. | 4 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
A school library bought 140 new books, ending up with $\frac{27}{25}$ of the number of books it had before the purchase. Find the original number of books in the library. Express your answer as a whole number. | 1750 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence {a_n}, the sum of four consecutive terms is given as a_{1003}+a_{1004}+a_{1005}+a_{1006}=18. Find the sum of the first 2008 terms of this sequence. Express your answer as a single numerical value. | 9036 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x) = x + \log_{2}x$, calculate the value of $f(2) + f(4)$. Express your answer as a single integer. | 9 | big_math | [
"Mathematics -> Algebra -> Functions -> Logarithmic Functions"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
In the geometric sequence {a_n}, a_2 and a_10 are the two roots of the equation x^2-8x+4=0. Find the value of a_6, given that it is a positive value. | 2 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Lines PS, QT, and RU intersect at a common point O. P is joined to Q, R to S, and T to U, to form triangles. Find the sum of angles P, Q, R, S, T, and U. Express your answer in degrees as a whole number. | 360^\circ | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Mateo's 300 km trip from Edmonton to Calgary included a 40 minute break in Red Deer. He started in Edmonton at 7 a.m. and arrived in Calgary at 11 a.m. Not including the break, what was his average speed for the trip? Express your answer in km/h. | 90 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Given positive numbers $a$ and $b$ satisfy $a+b-ab+3=0$, find the minimum value of $ab$. Express your answer as a single number. | 9 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Determine the angle of inclination of the tangent to the curve y = (1/3)x^3 - 2 at the point (1, -5/3). Express your answer in degrees. | 45° | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
The tangent line to the curve y = x^3 + 11 at point P(1,12) intersects the y-axis at a certain y-coordinate. Find this y-coordinate. Express your answer as a single number. | 9 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given the planar vectors $\overset{→}{a} =(1,2)$ and $\overset{→}{b} =(1,-1)$, calculate the vector $\dfrac{1}{3} \overset{→}{a}- \dfrac{4}{3} \overset{→}{b}$. Express your answer as a vector in the form (x, y). | (-1, 2) | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
A box contains 3 red balls, 2 white balls, and 5 black balls, all of the same size and shape. If a red ball is picked from the box without replacement, calculate the probability of picking a white ball on the second draw. Express your answer as a fraction or decimal. | \frac{2}{9} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Other"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence {an}, the first term a1 = 0, and the common difference d ≠ 0. If ak = a1 + a2 + a3 +... + a7, find the value of k. Express your answer as a whole number. | 22 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $\bar{z} \cdot (1+2i)=4+3i$, determine the value of $z$. Express your answer as a complex number in the form $a+bi$. | 2+i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
A circle graph shows the results of asking 200 students to choose pizza, Thai food, or Greek food, with 50% of the students choosing Greek food. Calculate the number of students who chose Greek food. Express your answer as a whole number. | 100 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given points D and E on sides AB and AC of ∠CAB respectively, the angle bisectors of ∠BDE and ∠CED intersect at F. If ∠BAC = 38°, find the measure of ∠DFE. Express your answer in degrees as a whole number. | 71^\circ | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the derivative of the function f(x) defined on R is f'(x), and it satisfies f'(x) - f(x) > 0, f(2023) = e^2023, find the solution set of the inequality f(ln x) < x. Express your answer as an interval. | (0, e^2023) | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(2,0)$ and $\overrightarrow{b}=(\frac{1}{2},\frac{{\sqrt{3}}}{2})$, calculate the value of $\overrightarrow{b}⋅(\overrightarrow{a}-\overrightarrow{b})$. Express your answer as a single number. | 0 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the coordinates of vector $\overrightarrow{p}$ in a unit orthogonal basis $\overrightarrow{a}$, $\overrightarrow{b}$, $\overrightarrow{c}$ in space are $\left(1,3,2
ight)$, find the coordinates of $\overrightarrow{p}$ in the basis $\overrightarrow{a}+\overrightarrow{b}$, $\overrightarrow{a}-\overrightarrow{b... | (2, -1, 2) | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
How many distinct trees with exactly 7 vertices are there? A tree is defined in the language of graph theory as a connected graph with no cycles, simply understood as a graph that connects \( n \) vertices with \( n-1 \) edges. Express your answer as a whole number. | 11 | big_math | [
"Mathematics -> Discrete Mathematics -> Graph Theory"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given two lines $l_1: 3x + 4y - 3 = 0$ and $l_2: 3x + 4y + 7 = 0$, calculate the distance between these two lines using the formula $\text{Distance} = \frac{|C_2 - C_1|}{\sqrt{A^2 + B^2}}$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Let set $A = \{1, 2\}$. Find the number of proper subsets of set $A$. Express your answer as a whole number. | 3 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics -> Other"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $i$ is the imaginary unit and the complex number $z=a+i$ satisfies $z^{2}+z=1-3i$, find the value of $a$. Express your answer as a single real number. | -2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Line $l_{1}$ is perpendicular to line $l_{2}$. If the inclination angle of $l_{1}$ is $30^{\circ}$, find the slope of $l_{2}$. Express your answer as a numerical value, including any necessary mathematical notation. | -\sqrt{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequences 3, 7, 11,..., 139 and 2, 9, 16,..., 142, find the number of common terms between them. Express your answer as a whole number. | 5 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
An elevator starts from rest at the top of a building and travels to the bottom. The elevator accelerates from rest between 2 seconds and 4 seconds, then travels at a constant speed, and finally decelerates to a stop between 22 seconds and 24 seconds. Determine the time interval during which the elevator has its maximu... | [4, 22] | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence $\{a_n\}$ defined by $a_1=1$ and $a_n=a_{n-1}+2n$ for $n\geq2$, derive a formula for $a_n$ and use it to calculate the value of $a_7$. Express your answer as a single integer. | 55 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given the set $A = \{1, 2\}$, find the number of sets $B$ that satisfy $A \cup B = \{1, 2, 3\}$. Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
In a salad dressing, the ratio of oil to vinegar is 3:1. To make the dressing more acidic, the amount of vinegar is doubled. Find the new ratio of oil to vinegar. Express your answer as a ratio in simplest form. | 3:2 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\angle A = 42^{\circ}$, find the degree of the complement of $\angle A$. Express your answer in degrees as a whole number. | 48 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
A school has 120, 180, and 150 boarding students in the first, second, and third grades, respectively. The school wants to conduct a questionnaire survey using stratified sampling based on the proportion of the number of students. If 90 students are to be selected for the survey, calculate the number of boarding studen... | 24, 36, 30 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Let real numbers $a$, $b$, $c$, $d$, $e$ satisfy the relations: $a+b+c+d+e=8$, $a^{2}+b^{2}+c^{2}+d^{2}+e^{2}=16$. Find the maximum value of the real number $e$. Express your answer as a real number. | \dfrac {16}{5} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
A triangle has an inscribed circle with a radius of r inches. If the perimeter of the triangle is p inches, and the area is k square inches, determine the ratio p/k. Express your answer as a fraction involving the variable r. | \frac{2}{r} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
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 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of distinct real roots of the equation $|x^2 - 6x| = 9$. Express your answer as a whole number. | 3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Given two straight lines $ax + 2y - 1 = 0$ and $3x - 6y - 1 = 0$ are perpendicular to each other, find the value of $a$. Express your answer as a single number. | 4 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function y=(a\cos x+b\sin x)\cos x has a maximum value of 2 and a minimum value of -1, calculate the value of (ab)^2. Express your answer as a single number. | 8 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
A certain brand of computer is reduced by 40%, resulting in a selling price of a yuan per unit. Calculate the original price of each unit of this brand of computer in terms of a. | \frac{5}{3}a | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the exact value of the expression: \(\left[\left(0.8+\frac{1}{5}\right) \times 24+6.6\right] \div \frac{9}{14}-7.6\). Provide your answer as a single number. | 40 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
In the figure, ∆ABC is a right-angled triangle with ∠ACB = 90°, AC = AE, and BC = BF. Find the measure of angle ∠ECF in degrees. | 45 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z=m\left(m-2\right)+(m^{2}-4)i$, find the real number $m$ for which $z$ is purely imaginary. Express your answer as a real number in the format $\boxed{[answer]}$. | 0 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\cos (\frac{\pi}{2}+\phi)=\frac{2}{3}$ and $|\phi| < \frac{\pi}{2}$, calculate the value of $\tan \phi$. Express your answer as a fraction or decimal, including the negative sign if applicable. | -\frac{2\sqrt{5}}{5} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
The points \( P(3, -2) \), \( Q(3, 1) \), and \( R(7, 1) \) form a rectangle with the point \( S \). Find the coordinates of point \( S \) such that \( P Q R S \) is a rectangle. Express your answer as a coordinate pair \( (x, y) \). | (7, -2) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Kirra has 7 greeting cards but no envelopes. She can buy packages that contain 8 greeting cards and 10 envelopes each. What is the smallest number of packages Kirra needs to buy to have more envelopes than cards? Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Alice, Bob, and Carl take turns rolling a die. Alice rolls first, Bob always rolls after Alice, and Carl always rolls after Bob. Calculate the probability that Carl is the first to roll a 6, given that the probability of rolling a 6 on any single roll is 1/6 and each roll is independent of the others. Express your answ... | \frac{25}{91} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
In right triangle ABC, AX = AD and CY = CD. Find the measure of angle XDY. Express your answer in degrees as a whole number. | 45 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Rotate the parabola y = x^2 - 4x + 3 clockwise by 90 degrees around its vertex. Express the new equation of the parabola in the format (y ± a)^2 = x ± b. | (y+1)^2=x-2 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Let the universal set $U=\{x\in\mathbb{N}|x\geq2\}$, and the set $A=\{x\in\mathbb{N}|x^2\geq5\}$. Find the complement of set A within the universal set U, denoted as $\complement_U A$. Express your answer as a set of natural numbers using standard set notation. | \{2\} | big_math | [
"Mathematics -> Discrete Mathematics -> Set Theory -> Other"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Given a function f(x) that satisfies f(x+2) = 1/f(x) for any real number x, and f(1) = -5, find the value of f(f(5)). Express your answer as a single value. | -\frac{1}{5} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Given the quadratic equation x^2 + x - 3 = 0 with roots x1 and x2, evaluate the expression x1^3 - 4x2^2 + 19. Express your answer as a single integer. | 0 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
The average of four test marks is 60. The first three marks were 30, 55, and 65. What is the value of the fourth mark? Express your answer as a whole number. | 90 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Find the sum of the squares of all real numbers that satisfy the equation \( x^{256} - 256^{32} = 0 \). Express your answer as a single number. | 8 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Polynomial Equations"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Find the last two digits of the sum \(1 + 2 + 3 + 4 + \cdots + 2003 + 2004 + 2005\). Express your answer as a two-digit number, including a leading zero if necessary. | 15 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
If the generatrix of a cylinder is 5cm long, and the radius of the base is 2cm, calculate the lateral surface area of this cylinder. Express your answer in square centimeters (cm²), including π if necessary. | 20\pi | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
The numbers x, y, p, and q are all integers. x and y are variable, and p and q are constant and positive. The four integers are related by the equation xy = px + qy. When y takes its maximum possible value, calculate the expression equal to y - x in terms of p and q. | (p-1)(q+1) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Determine the number of trailing zeros in the product \(25 \times 25 \times 25 \times 25 \times 25 \times 25 \times 25 \times 8 \times 8 \times 8\). Express your answer as a single integer. | 9 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Given the circle $C: x^{2}+(y+5)^{2}=3$, how many lines are there that are tangent to the circle and have equal y-intercepts and x-intercepts? Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given the expressions for P, Q, and R:
P = -1/(201603 * 201604)
Q = -1/(201602 * 201604)
R = -1/(201602 * 201603)
Determine the exact relationship among P, Q, and R. Express your answer as a string describing the relationship (e.g., P>Q>R). | P>Q>R | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
In a positive geometric sequence, the sum of the first 2 terms (S_2) is 7, and the sum of the first 6 terms (S_6) is 91. Find the sum of the first 4 terms (S_4). Express your answer as a single numerical value. | 28 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
If $M \subseteq \{1, 2\}$, calculate the total number of sets $M$ that satisfy this condition. Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Combinations"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Let vector \(a = (x, 1)\) and \(b = (4, x)\), and vectors \(a\) and \(b\) are collinear. Find the value(s) of \(x\). Express your answer as a numerical value(s). | \pm 2 | big_math | [
"Mathematics -> Linear Algebra -> Vectors"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets $M=\\{x\\mid-\\frac{1}{2}<x<\\frac{1}{2}\\}$ and $N=\\{x\\mid x(x-1)\\leqslant 0\\}$, determine the interval representing $M\\cup N$. Express your answer in interval notation, specifying whether the endpoints are included or excluded. | (-\\frac{1}{2},1] | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Find the slope angle of the line passing through the points (-2,1) and (1,4). Express your answer in degrees, including the degree symbol (\u00B0). | 45\u00B0 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Convert the binary number $11100_2$ into a quaternary (base-4) number. Express your answer in quaternary format, including the base-4 subscript (_4). | 130_4 | big_math | [
"Mathematics -> Other -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
A circle is inscribed in a semicircle with center O and diameter AB. The center of the circle is the point P, where PA = PO. Calculate the ratio of the radius of the circle to the radius of the semicircle. Express your answer as a ratio in the format a:b. | 3:8 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
A regular tetrahedron $ABCD$ has its vertices on a sphere of radius $R$. Calculate the length of edge $AB$ in terms of $R$. | \frac{2 \sqrt{6}}{3} R | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
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