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 |
|---|---|---|---|---|---|
Given the function y=2x^{2}-6x+3, where x∈[-1,1], find the minimum value of y. Express your answer as a single number. | -1 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Given the inequality $|2a-b|+|a+b| \geq |a|(|x-1|+|x+1|)$ holds for any non-zero real numbers $a$ and $b$, find the range of the real number $x$ that satisfies this inequality for all $a$ and $b$. Express your answer as an interval or describe the range. | [-\frac{3}{2}, \frac{3}{2}] | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
A group of students went boating, but there weren't enough boats in the park. If each boat holds 4 people, there will be 10 people left without a boat. If each boat holds 5 people, there will be 1 person left without a boat. Calculate the total number of people in the group. Provide your answer as a whole number. | 46 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Form a four-digit number using the digits 1, 2, and 3, where each digit must be used at least once, and the same digit cannot be adjacent to itself. How many such four-digit numbers can be formed? Express your answer as a single integer. | 18 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
The line $3x+4y=b$ is tangent to the circle $x^{2}+y^{2}-2x-2y+1=0$. Find the possible values of $b$. Express your answer as two numerical values separated by 'or'. | 2 or 12 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Let the sum of the first $n$ terms of a geometric sequence $\\{a_n\\}$ be $S_n$. If $S_n = 2^{n+1} + \\lambda$, find the value of $\\lambda$. Express your answer as a single integer. | -2 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
The sum of three consecutive odd numbers is 128 more than the smallest one. Find the first odd number in the sequence. Express your answer as a single integer. | 61 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Five student volunteers are going to two communities to participate in the normalization of epidemic prevention and control publicity activities. Each student will only go to one community, and each community must have at least one student. Calculate the number of different arrangements possible under these conditions. | 30 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
In the same plane rectangular coordinate system, curve $C$ is transformed into curve $2x'^{2}+8y'^{2}=1$ after an expansion transformation $\begin{cases} x'=5x, \\ y'=3y \end{cases}$. Determine the equation of curve $C$. Express your answer as an equation. | 50x^{2}+72y^{2}=1 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the plane vector $\overrightarrow{a}=(1,3)$, $|\overrightarrow{b}|=2$, and $|\overrightarrow{a}-\overrightarrow{b}|=\sqrt{10}$, calculate the value of $({2\overrightarrow{a}+\overrightarrow{b})\cdot(\overrightarrow{a}-\overrightarrow{b})}$. Express your answer as a single number. | 14 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given a geometric sequence $\{a_{n}\}$ such that $\frac{{a}_{5}-{a}_{1}}{{a}_{3}-{a}_{1}}=3$, calculate the value of $\frac{{a}_{10}-{a}_{2}}{{a}_{6}+{a}_{2}}$. Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Sequences and Series"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Seven cookies with a radius of 1 inch each are cut from a circular dough. The adjacent two cookies are externally tangent, and all the cookies are internally tangent to the circular dough. The remaining dough is re-formed into a circular cookie of the same thickness. Calculate the radius of the new circular cookie made... | \sqrt{2} | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given the set {a, b, c, d, e}, how many sets M exist such that M is a proper subset of {a, b, c, d, e}? Express your answer as a whole number. | 31 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
What is the additive inverse (opposite) of the number 8? Express your answer as a single number. | -8 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Let the random variable X follow the normal distribution N(0,1). If P(X>1) = p, find the probability P(X>-1). Express your answer as a probability value between 0 and 1. | 1-p | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = sqrt(x), find its derivative f'(x) using the limit definition of a derivative. Express your answer as a mathematical expression in the form of f'(x) = [answer]. | \frac{1}{2 \sqrt{x}} | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Given the line equation $(3k-1)x+(k+2)y-k=0$, find the coordinates of the fixed point that all lines pass through when $k$ varies. Express your answer as a coordinate (x, y) with fractions in simplest form. | \left( \frac{2}{7}, \frac{1}{7} \right) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Let \(\vec{a}, \vec{b}, \vec{c}\) be three unit vectors in the same plane, and \(\vec{a} \perp \vec{b}\). Calculate the maximum value of \((\vec{c} - \vec{a}) \cdot (\vec{c} - \vec{b})\). Express your answer as a single number. | 1 + \sqrt{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation of a circle (x+2)^2 + (y-5)^2 = 9, find the maximum value of (x-1)^2 + (y-1)^2. Express your answer as a single integer. | 64 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
A man travels m feet due north at 2 minutes per mile and returns due south to his starting point at 2 miles per minute. Calculate the average rate in miles per hour for the entire trip. Express your answer as a whole number, rounded if necessary. | 48 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
An agricultural planting company invested 330,000 yuan to purchase a large agricultural machinery and equipment, expecting an annual return rate of 10%. If the company withdraws funds of 82,500 yuan at the end of each year, how many years will it take for the company to recover all the principal and interest? Use the f... | 6 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Find the smallest positive period for the function f(x) = sin(x)cos(x). Express your answer in terms of pi. | \pi | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Suppose $a$ and $b$ are single-digit positive integers chosen independently and at random. Calculate the probability that the point $(a,b)$ lies above the parabola $y=ax^2-bx$. Express your answer as a simplified fraction. | \frac{19}{81} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
The product of 2, 3, 5, and y is equal to their sum. Find the value of y. Express your answer as a fraction. | \frac{10}{29} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $1+i^3$. Express your answer as a simplified complex number in the form a+bi. | 1 - i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
A woman's barracks is located 300 meters from a straight road. A man's barracks is located on this road, 500 meters from the woman's barracks. It is desired to build a restaurant on this road, equidistant from both barracks. Calculate the distance along the road where the restaurant should be built. Express your answer... | 200 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
A die is rolled six times. Calculate the probability of scoring at least 5 points at least five times. Express your answer as a fraction in simplest form. | \frac{13}{729} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence $\{a_n\}$, the common difference $d=2$, and $a_{2007}=2007$. Find the value of $a_{2009}$. Express your answer as a single integer. | 2011 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
A fire broke out in a building. After the fire truck arrived at the scene, it was found that it could only get as close as 5 meters to the base of the building. If the maximum extension of the fire truck's ladder is 13 meters, calculate the maximum height the ladder can reach on the building. Express your answer in met... | 12 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Six students each have a point labeled on them: (2, -3), (0, -1), (-2, 0), (2, 3), (-2, -3), (3, -2). How many of these points are located in the fourth quadrant? Provide your answer as a whole number. | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
If the value of $20$ quarters and $10$ dimes equals the value of $10$ quarters and $n$ dimes, calculate the value of $n$. Express your answer as a whole number. | 35 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Find the maximum number of consecutive integers whose sum is 45. Express your answer as a whole number. | 90 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
In a basketball game at a certain middle school, Alex scored $\frac{1}{4}$ of the total points, Ben scored $\frac{2}{7}$ of the total points, Charlie scored 15 points, and the remaining seven players each scored no more than 2 points. Calculate the total score of these seven players. Express your answer as a single int... | 11 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Find a counterexample to the proposition "if $|a-1| > 1$, then $a > 2$". Specifically, provide a numerical value of $a$ that satisfies the condition $|a-1| > 1$ but does not satisfy $a > 2$. | -1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Alisha wrote an integer in each square of a 4x4 grid, such that integers in squares with a common edge differed by 1. She wrote a 3 in the top-left corner and a 9 somewhere in the grid. How many distinct integers did she write in the grid? Express your answer as a single integer. | 7 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the solution set of the inequality $x^2 - ax - b < 0$ is $(2, 3)$, find the solution set of the inequality $bx^2 - ax - 1 > 0$. Express your answer as an interval, with the endpoints specified as fractions or decimals. | \left(- \frac {1}{2}, - \frac {1}{3}\right) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given a sequence {a_n} satisfying a_1 = 1, a_2 = 1, a_{n+1} = |a_n - a_{n-1}| for n >= 2, calculate the sum of the first 2011 terms of this sequence, S_{2011}. Express your answer as a single integer. | 1341 | big_math | [
"Mathematics -> Discrete Mathematics -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given a regular tetrahedron with a volume of 54, if a tetrahedron is formed by the centroids of its four faces, what is the volume of the new tetrahedron? Express your answer as a whole number. | 2 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the product of two positive integers p and q is 75, calculate the sum of all possible values of p. Express your answer as a single numerical value. | 124 | big_math | [
"Mathematics -> Number Theory -> Factorization"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Given $tan(θ + \frac{π}{4}) = 3$, calculate the value of $cos 2θ$. Express your answer as a fraction or decimal value. | \frac{3}{5} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Given a function $f\left(x\right)=\left\{\begin{array}{l}{{x}^{2}-1,}&{x≥1}\\{x-2,}&{x＜1}\end{array}\right.$, find the value of $a$ such that $f\left(f\left(a\right)\right)=3$. Express your answer as a single value, including any necessary mathematical notation. | \sqrt{3} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Given that point $P$ moves on the curve $y = -\sin x$ where $x \in [0, \pi]$, and point $Q$ moves on the line $x - 2y - 6 = 0$, find the abscissa of point $P$ when the distance $|PQ|$ is minimized. Express your answer in radians. | \frac{2\pi}{3} | big_math | [
"Mathematics -> Calculus -> Applications of Derivatives"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
A railway freight station is organizing and dispatching 8 coal freight trains into two groups, each containing 4 trains. The conditions are that trains A and B cannot be in the same group, train A departs first, and train B departs last. If the group containing train A departs first, find the total number of different ... | 720 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
ABCD is a rectangle. P is any point on AB. PS is perpendicular to BD, PR is perpendicular to AC, AF is perpendicular to BD, and PQ is perpendicular to AF. Find the length equal to PR + PS. | AF | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Factor the polynomial $a^2 + 2b - b^2 - 1$. Express your answer as a factored polynomial. | (a+b-1)(a-b+1) | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Three equilateral triangles with sides of length 1 are shown shaded in a larger equilateral triangle. The total shaded area is half the area of the larger triangle. Find the side-length of the larger equilateral triangle. Express your answer as a single value, including the square root symbol if necessary. | \sqrt{6} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $i$ is the imaginary unit, calculate the value of the complex number $i^{2015}$. Express your answer as a complex number, including the imaginary unit $i$ if necessary. | -i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given the events E1: "miss", E2: "hit the target", E3: "the number of rings hit is greater than 4", and E4: "the number of rings hit is at least 5" related to an archer shooting at a target, determine the number of pairs of events that are mutually exclusive but not contradictory. Provide your answer as a whole number. | 2 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence \(a_1, a_2, \ldots\) consists of integers that satisfy the equation:
\[
a_1 \cdot a_2 + a_2 \cdot a_3 + \cdots + a_{n-1} \cdot a_n = \frac{(n-1) \cdot n \cdot (n+1)}{3} \quad (n=2, 3, \ldots)
\]
Find the number of such sequences. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
In triangle \( \triangle ABC \), \( \angle A = 80^\circ \). Points \( D, E, F \) are on sides \( BC, AB \), and \( AC \) respectively, with \( BD = BE \) and \( CD = CF \). Find the measure of \( \angle EDF \) in degrees. | 50^\circ | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Three valves A, B, and C allow water to flow into a tank at constant rates specific to each valve. When all three valves are opened, it takes 1 hour to fill the tank. When only valves A and C are opened, it takes 1.5 hours to fill the tank. If only valves B and C are opened, it takes 2 hours to fill the tank. Find the ... | 1.2 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $(a-2i)i = b-i$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit, calculate the complex number $a+bi$ in the standard form. | -1 + 2i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Observe the pattern in the unit digits of powers of 3: $3^{1}=3$,$3^{2}=9$,$3^{3}=27$,$3^{4}=81$,$3^{5}=243$,$\ldots $, Find the unit digit of the sum $3^{1}+3^{2}+3^{3}+\ldots +3^{2023}$. Express your answer as a single digit. | 9 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $f(x) = x^7 + ax^5 + bx - 5$, and $f(-3) = 5$, calculate the exact value of $f(3)$. Express your answer as a single integer in the form $\boxed{[answer]}$. | -15 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence ${a_n}$ satisfying $a_{n+1} = \frac{4a_n + 3}{4}$ (n ∈ ℕ*), and $a_1 = 1$, find the value of $a_{21}$. Express your answer as a whole number. | 16 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Find the sum of the numbers 1.000, 0.101, 0.011, and 0.001. Express your answer as a decimal number with all significant digits. | 1.113 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Decimals"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
What is the maximum number of consecutive positive integers in which all numbers have prime factorizations with only odd powers of their prime factors? | 7 | big_math | [
"Mathematics -> Number Theory -> Factorization"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
In triangle $ABC$, it is known that $(a+b+c)(b+c-a)=bc$. Calculate the degree measure of angle $A$. Express your answer in degrees. | 120 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Find the difference between the sum of the first eighty positive even integers and the sum of the first eighty positive odd integers. Express your answer as a single numerical value. | 80 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
The line with equation $\frac{x}{3} + \frac{y}{4} = 1$ forms a triangle with the two coordinate axes. Find the perimeter of this triangle. Express your answer as a whole number. | 12 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
In a dunk performance involving four representative teams from Xichang City, Mianning County, Butuo County, and Zhaojue County, where each team sends one player, calculate the probability that the player from the Xichang representative team does not perform the dunk in the first or last position. Express your answer as... | \frac{1}{2} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
The area of the kombucha in the jar doubles every day. If it takes 19 days to fill the entire jar, how many days does it take to fill half of the jar? Express your answer as a whole number. | 18 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
The total weight of a box, 20 plates and 30 cups is $4.8 \mathrm{~kg}$. The total weight of the box, 40 plates and 50 cups is $8.4 \mathrm{~kg}$. Calculate the total weight of the box, 10 plates and 20 cups. Express your answer in kilograms (kg). | 3 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
A high school senior needs to choose 3 out of 6 colleges to apply to, but two of the colleges have their exams scheduled at the same time and cannot be applied to simultaneously. Calculate the number of different ways this student can choose the colleges to apply to. | 16 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
In the Cartesian coordinate system, point M is located in the fourth quadrant. The distance from point M to the x-axis is 12, and the distance to the y-axis is 4. Determine the coordinates of point M. Express your answer as an ordered pair (x, y), including the negative sign for the y-coordinate. | (4, -12) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
A triathlete competes in a triathlon in which the swimming, biking, and running segments are all of the same length. The triathlete swims at a rate of 3 kilometers per hour, bikes at a rate of 20 kilometers per hour, and runs at a rate of 10 kilometers per hour. Calculate the triathlete's average speed, in kilometers p... | 6 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Given the universal set U = {0,1,2,3}, set A = {0,1,2}, and set B = {0,2,3}, find the intersection of A and the complement of B with respect to U. Express your answer as a set using standard set notation. | {1} | big_math | [
"Mathematics -> Discrete Mathematics -> Logic -> Other"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
In a geometric sequence $\{a_n\}$ where each term is positive, if $a_5a_6=81$, calculate the sum $\log_{3}a_1+\log_{3}a_2+\ldots+\log_{3}a_{10}$. Express your answer as a single number. | 20 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence $a_{n}=\frac{1}{\sqrt{n}+\sqrt{n+1}}$ and $S_{n}=8$, calculate the value of $n$. Express your answer as a whole number. | 80 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Given a sequence $\{a_n\}$ where $a_1=1$ and $a_{n+1}=a_n+3$, find the value of $n$ when $a_n=2008$. Express your answer as a whole number. | 670 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\sin\left(\alpha + \frac{\pi}{2}\right) = \frac{1}{3}$, and $\alpha\in \left(0, \frac{\pi}{2}\right)$, find the value of $\tan\alpha$. Express your answer as a simplified radical. | 2\sqrt{2} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given that {a_n} is an arithmetic sequence and a_1 + a_5 + a_9 = 8π, calculate the value of cos(a_3 + a_7). Express your answer as a single fraction or decimal, including the negative sign if applicable. | -\frac {1}{2} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Find the imaginary part of the complex number (1-i)(3+i). Express your answer as a single integer. | -2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the two foci of an ellipse and the endpoints of its minor axis precisely form the four vertices of a square, calculate the eccentricity of the ellipse. Express your answer as a simplified fraction or radical. | \frac{\sqrt{2}}{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
The northern village has 8,758 people, the western village has 7,236 people, and the southern village has 8,356 people. A total of 378 people are to be conscripted from the three villages according to their population size. How many people should be conscripted from the western village? Express your answer as a whole n... | 112 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets $M=\{-1,0,1,2\}$ and $N=\{x | x < 0 \text{ or } x > 1\}$, find the number of elements in $M \cap N$. Express your answer as a whole number. | 2 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the angle between vectors $\overrightarrow {a}$ and $\overrightarrow {b}$ is 120°, and $|\overrightarrow {a}| = |\overrightarrow {b}| = 4$, find the magnitude of the sum of vectors $\overrightarrow {a}$ and $\overrightarrow {b}$. Express your answer as a single number. | 4 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
The graph of the function y=f(x) passes through points A(1,3) and B(3,1). Calculate the average rate of change between these two points. Express your answer as a single number. | -1 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
To obtain the graph of the function y = sin(2x - π/3), what horizontal shift needs to be applied to the graph of the function y = sin(2x)? Express your answer as a single value with units (e.g., π/6 units to the right). | π/6 units to the right | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
For the quadratic equation $ax^{2}-x+2=0$ to have two distinct real roots, find the range of real number $a$. Express your answer as an inequality, including any necessary conditions or restrictions on $a$. | a < \frac{1}{8} \text{ and } a \neq 0 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $iz = 4 + 3i$, where $i$ is the imaginary unit, find the complex number $z$. Express your answer in the form $a + bi$, where $a$ and $b$ are integers. | 3 - 4i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Solve the inequality $x^2-x-6<0$ and express the solution set in interval notation using set brackets, e.g., $\{x|a<x<b\}$. | \{x|-2<x<3\} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Consider a sequence of complex numbers \( z_{1} = 0 \), and for \( n \geq 1 \), \( z_{n+1} = z_{n}^{2} + i \). Calculate the modulus of \( z_{111} \). Express your answer as a single value. | \sqrt{2} | big_math | [
"Mathematics -> Complex Analysis -> Other"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
The sum of two positive numbers is $5$ times their difference. Find the ratio of the larger number to the smaller number. Express your answer as a simplified ratio. | \frac{3}{2} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Given two vectors $a=(\cos\theta,1,\sin\theta)$ and $b=(\sin\theta,1,\cos\theta)$, determine the angle between the vectors $a+b$ and $a-b$. Express your answer in degrees, including the degree symbol (\u00B0). | 90\u00B0 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
The sides of the convex quadrilateral \(ABCD\) are \(AB = 3\), \(BC = 4\), \(CD = 12\), and \(DA = 13\). The angle \(\angle CBA\) is a right angle. Find the area of the quadrilateral. Express your answer as a whole number. | 36 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
A monkey in a zoo becomes lucky if it eats three different fruits. Given 20 oranges, 30 bananas, 40 peaches, and 50 tangerines, what is the largest number of monkeys that can be made lucky? Provide your answer as a whole number. | 40 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
In a class of 60 students, a systematic sampling method is used to draw a sample of size 5. It is known that the students with seat numbers 3, 15, 39, and 51 are all in the sample. Using the given seat numbers, determine the seat number of the remaining student in the sample. | 27 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
If the sum of interior angles of a polygon is $540^{\circ}$, find the number of sides of the polygon. Express your answer as a whole number. | 5 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Polygons"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given the piecewise function f(x) = { x^2 - 1, x <= 0; f(x-2), x > 0 }, calculate the value of f[f(1)]. Express your answer as a single integer. | -1 | big_math | [
"Mathematics -> Functions -> Other"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Square \(PQRS\) is divided into four identical squares. One of these squares is shaded. Calculate the percentage of square \(PQRS\) that is shaded. Express your answer as a percentage, including the percentage sign (%). | 25\% | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Given $(1-2x)^7 = a + a_1x + a_2x^2 + \ldots + a_7x^7$, calculate the value of $a - a_1 + a_2 - a_3 + \ldots - a_7$. Express your answer as a single integer. | 2187 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given that the random variable ξ follows a normal distribution N(3, σ^2), if P(ξ > m) = a, then find P(ξ > 6 - m) in terms of a. | 1 - a | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Estimate the value of $\sqrt{17}$ to be between two consecutive integers. Express your answer in the form "between $a$ and $b$", where $a$ and $b$ are integers. | between 4 and 5 | big_math | [
"Mathematics -> Precalculus -> Other"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Given sets A = [-2, 4) and B = {x | x^2 - ax - 4 ≤ 0}, and knowing that B ⊆ A, calculate the range of the real number a. Express your answer as an interval. | [0,3) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the imaginary unit $i$ and a complex number $z$ that satisfies $\frac {2}{1+i}= \bar{z}+i$, determine the value of $z$. Express your answer as a complex number in the form $a+bi$. | 1+2i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $\sqrt{\frac{x}{1-\frac{x-1}{x}}}$ given that $x < 0$. Express your answer as a simplified algebraic expression. | -x | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Given a sequence $\{a_n\}$ with the general term $a_n=(-1)^n(3n-2)$, find the sum $a_1+a_2+…+a_{10}$. Express your answer as a single integer. | 15 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Given a complex number $z$ satisfying $$\frac {1+i}{1-i} \cdot z = 3+4i,$$ calculate the conjugate of $z$ in the form $a+bi$, where $a$ and $b$ are real numbers. | 4+3i | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Given a regular tetrahedron with side length a, find the minimum number of points n that must be placed on its surface such that there are at least two points whose distance is no greater than a/2. Express your answer as a single integer. | 7 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
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