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 |
|---|---|---|---|---|---|
Find the number of primes p for which there exist unique integers r and s such that for every integer x, the congruence x^3 - x + 2 ≡ (x - r)^2 (x - s) (mod p) holds. | 0 | big_math | [
"Mathematics -> Number Theory -> Prime Numbers"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
In how many ways can the performance order of 6 singers be arranged if singer B and singer C must appear before or after singer A? Express your answer as a single integer. | 480 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Permutations"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given that i is the imaginary unit and conjugate(z) is the conjugate of a complex number z, if (1-i)z = 2, calculate the value of conjugate(z)*i. Express your answer as a complex number in the form a+bi. | 1+i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Five students made an appointment to visit the National Museum to see the exhibition "The Great Change - Celebrating the 40th Anniversary of Reform and Opening up". After the visit, the five students lined up for a group photo. If students A and B are not next to each other, how many different arrangements are there in... | 72 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the probability of choosing a red candy from a bag is 5/6, determine the smallest possible total number of candies in the bag such that the number of red candies is an integer. Express your answer as a single integer. | 6 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
The function $$y=\sin(2x- \frac {π}{4})$$ is translated to the left by $$\frac {π}{4}$$ units. Determine the maximum value of m such that the corresponding function of the resulting graph has no extreme points in the interval (-m, m). Express your answer as a single value, including the π symbol if necessary. | \frac{π}{8} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Find the coordinates of the focus of the parabola given by the equation $y^2 = 4x$. Express your answer as an ordered pair (x, y). | (1, 0) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets $M=\{x|{{2}^{x-1}} < 1,x\in {R}\}$ and $N=\{x|{lo}{{{g}}\_{2}}x < 1,x\in {R}\}$, find the intersection of $M$ and $N$, denoted as $M\cap N$. Express your answer as an interval. | (0,1) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given sets $A=\{x\left|\right.x\geqslant 0\}$ and $B=\{1,2,3,4,5\}$, determine the correct subset relationship between A and B. Express your answer using standard set notation (e.g. A ⊆ B, B ⊆ A, etc.). | B \subseteq A | big_math | [
"Mathematics -> Discrete Mathematics -> Logic"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Given $z= \frac {5i}{1-2i}$ (where $i$ is the imaginary unit), calculate the conjugate of $z$. Express your answer as a complex number in the form $a + bi$. | -2 - i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
In the rectangular coordinate system, find the standard equation of the hyperbola with asymptotes y=$± \sqrt{2}$x and passing through point P($2\sqrt{2}$, -$\sqrt{2}$). Express your answer in the standard form of a hyperbola equation. | \frac{x^2}{7} - \frac{y^2}{14} = 1 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Xiao Shen is $a$ years old this year, and Xiao Wang is $\left(a-8\right)$ years old. After $\left(n+3\right)$ years, what will be the age difference between Xiao Shen and Xiao Wang? Express your answer in years as a whole number. | 8 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
In an anti-terrorism drill, the probability of a shooter hitting the target is 0.8. Each shot is independent of the others. If the shooter fires 99 times, what is the most likely number of times he hits the target? Express your answer as a whole number or a range of whole numbers if there are multiple most likely outco... | 79 \text{ or } 80 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Other"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Faruq subtracted 5 from a number and then divided by 4. Next, he subtracted 4 from the original number and then divided by 5. He got the same final answer both times. Find the original number. Express your answer as a single integer. | 9 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, $ABCD$ is a rectangle with dimensions 8 x 9. If the area of the circle is equal to the area of the shaded region, calculate the radius of the circle. Express your answer as a simplified mathematical expression. | \frac{6}{\sqrt{\pi}} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of $\sin\frac{2023\pi}{3}$. Express your answer as a fraction or a radical. | \frac{\sqrt{3}}{2} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Given proposition P: ∃x₀ ∈ R, x₀² + 2x₀ + 3 ≤ 0, what is the negation of proposition P (∉P)? Express your answer as a logical statement in the format of a universal proposition. | ∀x ∈ R, x² + 2x + 3 > 0 | big_math | [
"Mathematics -> Discrete Mathematics -> Logic"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the right focus of the hyperbola $\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1\left(a \gt 0,b \gt 0\right)$ coincides with the focus of the parabola $y^{2}=2px\left(p \gt 0\right)$, the directrix of the parabola intersects the hyperbola at points $A$ and $B$, and the asymptotes of the hyperbola intersec... | \sqrt{2} | big_math | [
"Mathematics -> Geometry -> Differential Geometry -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given that \(S_{n}\) is the sum of the first \(n\) terms of an arithmetic sequence \(\{ a_{n}\}\), and \(2(a_{1}+a_{3}+a_{5})+3(a_{8}+a_{10})=36\), find \(S_{11}\). Express your answer as a single integer. | 33 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
In the Cartesian coordinate system, given points $A(-1,0)$ and $B(1,2)$, the line segment $AB$ is translated to line segment $CD$. If the coordinates of point $C$ corresponding to point $A$ are $(1,-2)$, then find the coordinates of point $D$ corresponding to point $B$. Express your answer as a coordinate pair $(x, y)$... | (3,0) | big_math | [
"Mathematics -> Geometry -> Coordinate Geometry -> Other"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of the expression $-3^{2}+\left(-2\right)^{3}$. Provide your answer as a single integer. | -17 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Starting from a vertex of an n-sided polygon, connect this vertex with each of the other vertices. If this divides the polygon into 7 triangles, find the value of n. Express your answer as a whole number. | 9 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangulations"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
In triangle $\triangle ABC$, let $a$, $b$, and $c$ be the lengths of the sides opposite angles $A$, $B$, and $C$ respectively. Given the inequality $\frac{b}{{b+c}}+\frac{c}{{a+b}}≥1$, determine the range of angle $A$ in radians. | (0, \frac{\pi}{3}] | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the linear function y=kx+3, where k is a constant and k!=0, and the graph passes through the point (-2,0), solve the equation k(x-5)+3=0 for x. Express your answer as a single value of x. | x = 3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the two roots of the quadratic equation $x^{2}-10x+24=0$ are the lengths of the diagonals of a rhombus, find the area of this rhombus. Express your answer as a whole number. | 12 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Quadratic Functions"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
The number of different integer solutions (x, y) for |x|+|y|=n forms an arithmetic sequence with a first term of 4 and a common difference of 4. Calculate the 10th term of this sequence, which corresponds to the number of integer solutions for |x|+|y|=10. Express your answer as a single integer. | 40 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
In the given triangle ABC, AB = 4 and AC = 8. If M is the midpoint of BC and AM = 3, what is the length of BC? Express your answer as a numerical value with a square root if necessary. | 2 \sqrt{31} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
In $\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively. If $a=2$, $b=2\sqrt{3}$, and $\angle A=30^{\circ}$, calculate the possible values of $\angle B$ in degrees. | 60^{\circ} \text{ or } 120^{\circ} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Given 10 points on the positive x-axis and 5 points on the positive y-axis, calculate the maximum number of intersection points of the line segments formed by connecting these points within the first quadrant. Express your answer as a whole number. | 450 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors a = (1, -1), b = (1, 2), c = (x, 1), and vector c satisfies 2a ⊥ (b + c), find the value of x. Express your answer as a single numerical value. | 2 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the sum of $a$ and its absolute value is 0, find the value of $a - |2a|$. Express your answer as a single algebraic expression in terms of $a$. | 3a | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
February 1, 2012, was a Wednesday. What day of the week was June 1, 2012? Express your answer as the name of the day of the week (e.g., Monday, Tuesday, etc.). | Friday | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, triangle FHI and point G on FH are such that GH = FI. The points M and N are the midpoints of FG and HI respectively. If ∠NMH = α°, find an expression for ∠IFH in terms of α. Express your answer as an algebraic expression, including the degree symbol (°). | 2α° | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given vector $\overrightarrow{a}=(2\tan \alpha,\tan \beta)$, vector $\overrightarrow{b}=(4,-3)$, and $\overrightarrow{a}+ \overrightarrow{b}= \overrightarrow{0}$, calculate the value of $\tan (\alpha+\beta)$. Express your answer as a simplified fraction. | \frac{1}{7} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given that one root of the equation $x^2 - kx - 6 = 0$ is $-2$, find the value of the real number $k$. Express your answer as a single real number. | 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
In a three-dimensional Cartesian coordinate system, find the coordinates of the point symmetrical to point P(3, -2, 1) with respect to the y-axis. Express your answer in the format (x, y, z). | (-3, -2, -1) | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Find the solution set of the inequality $${x^{2}}-3x+2 < 0$$. Express your answer in interval notation. | \left( 1,2 \right) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Molly has a set of cards numbered 1 to 12. She places eight of them at the vertices of an octagon so that the sum of every pair of numbers at opposite ends of an edge of the octagon is a multiple of 3. Identify the numbers that Molly did not place on the vertices of the octagon. Express your answer as a list of numbers... | 3, 6, 9, 12 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow {a}=(3,4)$ and $\overrightarrow {b}=(2,-1)$, if the vector $\overrightarrow {a}+x \overrightarrow {b}$ is perpendicular to $\overrightarrow {b}$, calculate the value of $x$. Express your answer as a single numerical value, including the negative sign if applicable. | - \frac {2}{5} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Given sets $A = \{x|\log_{2}x < 2\}$ and $B = \{x|x^{2} < 9\}$, find the intersection $A \cap B$. Express your answer in interval notation. | (0,3) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
For lines $l_1 : kx - y - 3 = 0$ and $l_2 : x + (2k + 3)y - 2 = 0$ to be perpendicular to each other, find the value of $k$. Express your answer as a single number. | -3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Determine the maximum value of the function y = 2 - 9x - 4/x, given that x > 0. Express your answer as a single number. | -10 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
In triangle ABC, ∠C = 3∠A, a = 27, and c = 48. Find the length of side b. Express your answer as a whole number. | 35 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = (x^2 + ax - 1)e^(x-1), where x = -2 is an extremum point, find the minimum value of f(x). | -1 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the sum \(C_{2}^{2} + C_{3}^{2} + C_{4}^{2} + \ldots + C_{10}^{2}\). Express your answer as a single integer. | 165 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Combinations"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Find the maximum value of the function f(x) = sin(x) + x on the interval [0, 2π]. Provide your answer as a single value. | 2π | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
At a certain primary school in my city, there are two 3rd-grade classes, Class A and Class B. Class A has 30 boys and 20 girls, while Class B has 25 boys and 25 girls. Now, each class needs to randomly select 20% of the students from both boys and girls for a survey. Calculate the total number of boys selected from bot... | 11 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x)=\left\{\begin{array}{l}{(x-m)}^{2}-2, x＜0\\ 2{x}^{3}-3{x}^{2}, x≥0\end{array}\right.$, find the range of real number $m$ such that the minimum value of $f(x)$ is $-1$. Express your answer as an inequality or condition for $m$. | m \geq 1 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Kevin needs 12 hours to shovel all of the snow on his driveway. Individually, Dave needs 8 hours to shovel all of Kevin's snow, John needs 6 hours to shovel all of Kevin's snow, and Allison needs 4 hours to shovel all of Kevin's snow. If Kevin, Dave, John, and Allison all work together, how many minutes do they need to... | 96 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
From four students A, B, C, and D, two are to be selected to participate in a legal knowledge quiz. Calculate the probability of student A being selected. Express your answer as a simplified fraction. | \frac{1}{2} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Combinations"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
In the interval (1, 3), find the probability that a randomly chosen number x satisfies the inequality log<sub>2</sub>(2x - 1) > 1. Express your answer as a simplified fraction. | \frac{3}{4} | big_math | [
"Mathematics -> Applied Mathematics -> Probability"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Given that December 21, 2013, is a Saturday, determine the day of the week for January 31, 2014. Express your answer as a day of the week (e.g., Monday, Tuesday, etc.). | Friday | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\cos \alpha = -\frac{1}{2}$ and $\alpha$ is an obtuse angle, calculate the value of $\tan \alpha$. Express your answer as a single value, including a negative sign if applicable. | -\sqrt{3} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
The sum of the squares of three consecutive natural numbers exceeds eight times their sum by 2. Find the sum of the squares of these three natural numbers. Express your answer as a single integer. | 194 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
In a semicircle with radius $r$, an inscribed trapezoid is constructed such that its base is the diameter and the other three sides are chords of the circle. Find the length of the top base of the trapezoid that maximizes its area. Express your answer in terms of $r$. | r | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Two dice (each die has sides numbered from 1 to 6) are rolled sequentially and land on a horizontal surface, recording the numbers facing up as x and y. Let event A be “x+y is an even number”, and event B be “one of x or y is an even number, and x!= y”. Calculate the conditional probability P(B|A). Express your answer ... | 1/3 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Find the minimal value of integer n that guarantees: Among n sets, there exist at least three sets such that any of them does not include any other; or there exist at least three sets such that any two of them include the other. Express your answer as a single integer. | 5 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Giselle had run 50 metres, which is 25% of the total race distance. Find the total distance of the race in metres. Express your answer as a whole number. | 200 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
A television station in a certain city wants to survey the viewership rating of the TV drama "People's Sense of Justice" by using stratified sampling based on the population of three districts. The population ratio of the three districts is 2:3:5. If the district with the most people samples 60 people, what is the tota... | 120 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of $\sin 810^{\circ}+\cos (-60^{\circ})$. Express your answer as a fraction. | \dfrac{3}{2} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
In a cube, determine the number of lines containing the edges that are perpendicular to the line containing the edge AA₁. Express your answer as a whole number. | 8 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence $\{a_n\}$, if $a_2+a_4+a_9+a_{11}=32$, then find the value of $a_6+a_7$. Express your answer as a single integer. | 16 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
If the sum of the first $3n$ positive integers is $150$ more than the sum of the first $n$ positive integers, calculate the sum of the first $4n$ positive integers. Express your answer as a single integer. | 300 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Given a function $y=f(x)$ ($x \in \mathbb{R}$) that satisfies $f(x+2)=f(x)$, and when $x \in (-1,1]$, $f(x)=|x|$, find the number of real roots of $\log_{3}|x|-f(x)=0$. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
In the Cartesian coordinate system, point P $(m-3, m+1)$ is in the second quadrant. Find the range of values for $m$ that satisfies the conditions for the second quadrant. Express your answer as an inequality in the format of a<b<m<c, where a, b, c are the boundary values. | -1<m<3 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Given the linear functions $y = (m^2 - 4)x + (1 - m)$ and $y = (m + 2)x + (m^2 - 3)$, find the value of $m$ such that their y-intercepts are symmetric about the x-axis. Express your answer as a single numerical value. | -1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Find the minimum positive period of the function $y=\tan \frac{\pi }{4}x$. Express your answer as a single number. | 4 | big_math | [
"Mathematics -> Precalculus -> Functions -> Periodic Functions"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Find the reciprocal of the real number -2. Express your answer as a fraction. | -\frac{1}{2} | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets $A=\{x|\ln x\geqslant 0\}$ and $B=\{x|x^{2} < 9\}$, find the intersection $A\cap B$. Express your answer as an interval, including any relevant brackets or parentheses. | [1,3) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Xiaoming and Xiaohua solve the same quadratic equation, but Xiaoming mistakenly takes the coefficient of the linear term and gets the roots as 2 and -3, while Xiaohua mistakenly takes the constant term and gets the roots as -2 and 5. Using this information, formulate the original quadratic equation in standard form. | x^2 - 3x - 6 = 0 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
The diameter of a certain virus is approximately 0.000000105 meters. Express this diameter in scientific notation in the format a × 10^b, where a is the coefficient and b is the exponent. | 1.05 × 10^(-7) | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given $a, b \in \mathbb{R}^{+}$ and the equation $\dfrac{1}{a} + \dfrac{2}{b} = 1$, find the minimum value of $a + b$. Express your answer as a single mathematical expression. | 3 + 2\sqrt{2} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Find the minimum value of the function $y=x^{2}-4x+1$ on the interval $\left[0,3\right]$. Express your answer as a single number within the boxed notation. | -3 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Euler's formula $e^{ix} = \cos x + i\sin x$ (where $i$ is the imaginary unit) was discovered by the famous Swiss mathematician Euler. It extends the domain of the exponential function to complex numbers and establishes the relationship between trigonometric functions and the exponential function. Euler's formula is hig... | Third quadrant | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Ten different numbers (not necessarily integers) are written down. Any number that is equal to the product of the other nine numbers is then underlined. What is the maximum number of numbers that can be underlined? Express your answer as a whole number. | 2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
From a fish pond, 1200 fish were caught, marked, and then released back into the pond. After a suitable period of time, 1000 fish were caught again, among which 100 were marked. Estimate the total number of fish in the pond. Express your answer as a whole number. | 12000 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Given that three times Dick's age plus Tom's age equals twice Harry's age, and twice the cube of Harry's age equals three times the cube of Dick's age plus the cube of Tom's age, with the additional constraint that their ages are coprime, calculate the sum of the squares of their ages. Express your answer as a single i... | 42 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Let \( a \) be a positive integer such that \( a < 100 \). Given that \( a^{3} + 23 \) is divisible by 24, determine how many such values of \( a \) exist. Express your answer as a whole number. | 5 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given $a=2^{0.1}$, $b=\left(0.5\right)^{0.8}$, $c=\left(0.5\right)^{0.5}$, determine the ordering of $a$, $b$, and $c$ from largest to smallest. Express your answer using the symbols > and < (e.g., a > c > b). | a > c > b | big_math | [
"Mathematics -> Algebra -> Exponential Functions"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
A line $l$ passing through points $M(m, 3)$ and $N(1, m)$ is perpendicular to a line with a slope of $-1$. Find the value of $m$. Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other",
"Mathematics -> Algebra -> Other"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given a set of data pairs \((3,y_{1})\), \((5,y_{2})\), \((7,y_{3})\), \((12,y_{4})\), \((13,y_{5})\) corresponding to variables \(x\) and \(y\), the linear regression equation obtained is \(\hat{y} = \frac{1}{2}x + 20\). Calculate the value of \(\sum\limits_{i=1}^{5}y_{i}\). Express your answer as a single integer. | 120 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Given functions f(x) and g(x) satisfy: f(2+x) = f(2-x), g(x+1) = g(x-1), and f(x) is a decreasing function on the interval [2, +∞), let h(x) = f(x) * |g(x)|. Compare h(-2) and h(4) using ≤ or ≥ notation. | h(-2) ≤ h(4) | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Applied Mathematics -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
If $X \sim B(10, 0.8)$, calculate the value of $D(2X + 1)$. Express your answer as a single number. | 6.4 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
The first terms of both an arithmetic sequence {a_{n}} and a geometric sequence {b_{n}} are 1, with a common difference and common ratio of 2. Calculate the value of b_{a_{1}}b_{a_{3}}b_{a_{5}}. Express your answer as a whole number. | 4096 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
If the mutually prime integers A, B, and C satisfy the equation A log_{200} 5 + B log_{200} 2 = C, find the value of A + B + C. Express your answer as a single integer. | 6 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given that sin(2x) = 1/7, find the numerical value of sin^4(x) + cos^4(x). Express your answer as a numerical value. | \frac{97}{98} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Given the set $U=\{1,2,3,4,5\}$, $A=\{1,3\}$, and $B=\{1,2,4\}$, find the set $(\complement _{U}B)\cup A$. Express your answer as a set of elements using standard set notation. | \{1,3,5\} | big_math | [
"Mathematics -> Discrete Mathematics -> Set Theory -> Other"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $(-a^{2})^{3}\\div (-a)^{2}$. Provide your answer as a simplified algebraic expression. | -a^{4} | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given sets M and N defined as follows:
- M = {x | x ≤ 1 or x ≥ 3}
- N = {x | log2(x) ≤ 1}
Find the intersection of sets M and N, expressing your answer as an interval in standard notation. | (0,1] | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the random variable X follows a normal distribution with mean 2 and variance σ^2, and P(0 < X < 4) = 0.3, calculate the probability that X is less than 0. Express your answer as a decimal value between 0 and 1, rounded to 2 decimal places. | 0.35 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Find the valid range of the independent variable x for the function y = 2 / sqrt(x-1). Express your answer as an inequality using mathematical notation. | x > 1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Consider the following code which calculates a result based on a loop:
```plaintext
S = 0
A = 1
while S <= 36:
A += 1
S += A
```
What is the final value of S after the loop execution? | 45 | big_math | [
"Mathematics -> Discrete Mathematics -> Algorithms"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
In the sequence $\{a_n\}$, $a_1=3$, $a_2=7$, and for $n\geq1$, $a_{n+2}$ is equal to the units digit of $a_n \cdot a_{n+1}$. Calculate the value of $a_{2010}$. Express your answer as a single integer. | 9 | big_math | [
"Mathematics -> Discrete Mathematics -> Sequences and Series"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
The positive integers are arranged in rows and columns as shown below.
| Row 1 | 1 |
| Row 2 | 2 | 3 |
| Row 3 | 4 | 5 | 6 |
| Row 4 | 7 | 8 | 9 | 10 |
| Row 5 | 11 | 12 | 13 | 14 | 15 |
| Row 6 | 16 | 17 | 18 | 19 | 20 | 21 |
|... |
More rows continue to list the positive integers in order, with each new row contain... | 16 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Find the integer \( n \) that satisfies the inequality:
\[
\frac{3}{10} < \frac{n}{20} < \frac{2}{5}
\]
Express your answer as a single integer. | 7 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = x(x+1)(x+2)(x-3), calculate the slope of the tangent line to f(x) at x = 0. Express your answer as a single numerical value. | -6 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
A rhombus has its diagonals in the ratio of 1:2. If the shorter diagonal is 4 cm long, calculate the length of each side of the rhombus. Express your answer in centimeters (cm), simplifying any radical expressions if necessary. | 2\sqrt{5} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Given the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is $60^\circ$, $|\overrightarrow{a}|=2$, and $|\overrightarrow{b}|=5$, calculate the value of $|2\overrightarrow{a}-\overrightarrow{b}|$. Express your answer as a numerical value, including the square root symbol if necessary. | \sqrt{21} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets A = {(x,y)|y = e^x} and B = {(x,y)|y = log_{1/2}x}, find the number of intersection points between the two curves. Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the volume of the circumscribed sphere of the prism $ABCD-A_1B_1C_1D_1$ is $\frac{4}{3}\pi$, and the base $ABCD$ is a square with side length 1, calculate the lateral area of the prism $ABCD-A_1B_1C_1D_1$. | 4\sqrt{2} | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Volume"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
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