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 |
|---|---|---|---|---|---|
The sums of three whole numbers taken in pairs are 12, 17, and 19. Find the middle number. Express your answer as a whole number. | 7 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Given $\sin\alpha = \sin\theta + \cos\theta$, and $\sin^2\beta = 2\sin\theta \cdot \cos\theta$, find the relationship between $\sin^2\alpha$ and $\sin^2\beta$. Express your answer as an equation. | \sin^2\alpha = 1 + \sin^2\beta | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Find the value of a such that in the expansion of (a/x - x^(2/3))^7, the coefficient of the x^3 term is 14. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function y = ax^2 + bx + c (a!= 0), its graph passes through the points (-1, 3) and (1, 1). If 0 < c < 1, calculate the range of values for a. Express your answer as an interval (lower bound, upper bound). | (1,2) | big_math | [
"Mathematics -> Algebra -> Quadratic Functions -> Other"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the line passing through point P(-1, 2) and perpendicular to the line $x+2y-1=0$. Express your answer in the standard form of a line equation (Ax + By + C = 0). | 2x-y+4=0 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
If $(2x+ \sqrt{3})^{4}={a}_{0}+{a}_{1}x+{a}_{2}{x}^{2}+{a}_{3}{x}^{3}+{a}_{4}{x}^{4}$, calculate the value of $({a}_{0}+{a}_{2}+{a}_{4})^{2}-({a}_{1}+{a}_{3})^{2}$. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = x^4cos(x) + mx^2 + x (m ∈ ℝ), if the derivative f'(x) has a maximum value of 10 on the interval [-2, 2], then find the minimum value of the derivative f'(x) on the interval [-2, 2]. Express your answer as a single integer. | -8 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Two poles of heights 20 and 80 units are 100 units apart. Lines are drawn from the top of each pole to the base of the other. Find the height of the intersection point of these lines. Express your answer in units as a whole number. | 16 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $x$ and $y$ are real numbers, and they satisfy $(2x-1) + i = -y - (3-y)i$, find the values of $x$ and $y$. Express your answer as a pair of values in the format $(x, y)$. | (-\frac{3}{2}, 4) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Round 3.8963 to the nearest 0.01. Express your answer as a decimal number rounded to two decimal places. | 3.90 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
A grocer makes a display of cans in which the top row has one can and each lower row has two more cans than the row above it. If the display contains 100 cans, how many rows does it contain? Express your answer as a whole number. | 10 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the locus of the center of a moving circle that passes through point A(-2, 0) and is tangent to the line x = 2. Express your answer as an equation. | y^2 = -8x | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the number of subsets of the set {y ∈ ℤ | 0 < y ≤ 4}. Express your answer as a single integer. | 16 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given the hyperbola C: x^2/a^2 - y^2/b^2 = 1 (a > 0, b > 0) with foci F1 and F2, a point P on the hyperbola C satisfies PF1 ⊥ PF2, |PF1| = 3, and |PF2| = 4. Calculate the eccentricity of the hyperbola C using the necessary formulas and calculations. Express your answer as a single number. | 5 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
If $y=(\log_23)(\log_34)\cdots(\log_n[n+1])\cdots(\log_{31}32)$, calculate the value of y. Express your answer as a single number. | 5 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Logarithmic Functions"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
In the Cartesian coordinate system, find the coordinates of the point symmetric to point P(-2, 3) with respect to the x-axis. Express your answer as an ordered pair (x, y), including negative signs if applicable. | (-2, -3) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Given the set {a,b,c,d,e}, find the probability that a randomly chosen subset of this set is a subset of {a,b,c}. Express your answer as a simplified fraction. | \dfrac{1}{4} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of the expression \(6 \times 2017 - 2017 \times 4\). Express your answer as a single integer. | 4034 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Given the linear equation mx - y = 3 and the solution set x = 1, y = 3, calculate the value of m. Express your answer as a single integer. | 6 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
The diagram shows three triangles which are formed by the five line segments \(A C D F, B C G, G D E, A B\), and \(E F\) so that \(A C = B C = C D = G D = D F = E F\). Also, \(\angle C A B = \angle E F D\). Find the size, in degrees, of \(\angle C A B\). Express your answer as a whole number. | 60 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $f'(1)=1$, evaluate the limit $\lim\limits_{\Delta x\to0}\frac{f(1+3\Delta x)-f(1)}{\Delta x}$. Express your answer as a single numerical value. | 3 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Given sets A={x|-1 < x ≤ 5} and B={-1,2,3,6}, find A∩B. Express your answer as a set of elements using proper set notation. | {2,3} | big_math | [
"Mathematics -> Sets -> Set Operations"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $3m+2n=0$ and the condition $mn\neq 0$, find the value of $\frac{m}{n}-\frac{n}{m}$. Express your answer as a simplified fraction. | \frac{5}{6} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Express the wavelength of the infrared light, 0.000 000 94m, in scientific notation. Include the unit (m) in your answer. | 9.4\times10^{-7}m | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $a^{2}-2b-1=0$, calculate the value of the polynomial $-2a^{2}+4b+2025$. Express your answer as a single integer. | 2023 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Count the number of decreasing whole numbers between 100 and 500. A whole number is called decreasing if each digit of the number is less than the digit to its left. Express your answer as a whole number. | 10 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given a geometric sequence {a_n} that satisfies a_3a_{11}-4a_7=0, find the value of the product a_1 * a_2 *... * a_{12} * a_{13}. Express your answer as a power of 2 in the format 2^x, where x is an integer. | 2^{26} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Among the real numbers $2$, $0$, $\sqrt{5}$, $\frac{π}{3}$, $\sqrt[3]{27}$, $0.1010010001\ldots$ (with one more $0$ between every two $1$'s), how many are irrational numbers? Express your answer as a whole number. | 3 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
What is the value of \(2-2\{2-2[2-2(4-2)]\}\)? Express your answer as an integer. | -10 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Anne, Bronwyn, and Carl ran a race at constant speeds. When Anne finished, Bronwyn had 15 m left to run and Carl had 35 m left to run. When Bronwyn finished, Carl had 22 m left to run. Calculate the length of the race in meters (m). | 165 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Find the minimum value of the function $f(x)=x+\frac{1}{x-2}$ where $x > 2$. Express your answer as a single number. | 4 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the result of the expression \(2014 imes\left(\frac{1}{19}-\frac{1}{53}
ight)\). Express your answer as a single integer. | 68 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Fractions"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Let the sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$ be $S_n$, with $S_{m-1}=-2$, $S_m=0$, and $S_{m+1}=3$. Find the value of $m$. Express your answer as a whole number. | 5 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the line l: ax + by + 1 = 0 always bisects the circumference of the circle M: x^2 + y^2 + 4x + 2y + 1 = 0, find the minimum value of (a-2)^2 + (b-2)^2. Express your answer as a single numerical value. | 5 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
For the curve y = sin(x) + cos(x), find the slope angle of the tangent line at x = π/4. Express your answer in radians. | 0 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
From 5 volunteers, select 4 to take on the roles of host, planner, actor, and music composer, where Volunteer A cannot be the music composer. Calculate the total number of different selection and arrangement methods. Express your answer as a single numerical value. | 96 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets $P=\{1, 3, 5, 7\}$ and $Q=\{x|2x-1>11\}$, find the intersection $P\cap Q$. Express your answer as a set of integers using standard set notation. | \{7\} | big_math | [
"Mathematics -> Set Theory -> Other"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the number of distinct ways to paint the six faces of a cube using seven different colors, such that no two faces have the same color. Consider the symmetry of the cube in your calculation. Express your answer as a single integer. | 210 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Pilots of two fighter jets, A and B, shoot at the same enemy plane. Suppose the probabilities of hitting the target are 0.4 and 0.5, respectively. Calculate the probability that exactly one of them hits the target. Express your answer as a decimal value between 0 and 1, rounded to 1 decimal place. | 0.5 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given the lengths of two line segments as $a$ units and $b$ units, determine the correct inequality relationship between their arithmetic mean and geometric mean. Express your answer as an inequality. | \frac{a+b}{2} \geq \sqrt{ab} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
In the geometric sequence {a_n}, the equation aa_5a_7 = -3 sqrt(3) holds. Find the value of a_2a_8. Express your answer as a single numerical value. | 3 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of dissimilar terms in the expansion of (a + b + c)^10, where each term is of the form a^i b^j c^k and i + j + k = 10 with i, j, k >= 0. Express your answer as a numerical value. | 66 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Given a tetrahedron S-ABC, find the probability that a randomly selected point P within it satisfies the condition that the volume of the tetrahedron P-ABC is less than half the volume of the tetrahedron S-ABC. Express your answer as a simplified fraction. | \frac{7}{8} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x+1) = 3x + 2, find the expression for f(x). Express your answer in the format f(x) =... | f(x) = 3x - 1 | big_math | [
"Mathematics -> Algebra -> Functions -> Other"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
On a twenty-question test, each correct answer is worth 5 points, each unanswered question is worth 1 point, and each incorrect answer is worth 0 points. Determine the score that is NOT possible to achieve in this test. Express your answer as a single numerical value. | 97 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Find the last digit of $2008^{2008}$. Express your answer as a single digit. | 6 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Given a geometric sequence {a_n} with real terms, and the sum of the first n terms is S_n. If S_{10} = 10 and S_{30} = 70, calculate the value of S_{40}. Express your answer as a single numerical value. | 150 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
A fair cube die with faces numbered 1 through 6 is thrown three consecutive times. Calculate the probability that the numbers shown on the top faces form an arithmetic sequence. Express your answer as a simplified fraction. | \frac{1}{12} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given sets $A=\{0,1,2,3\}$ and $B=\{x|x < 3, x\in \mathbb{N}\}$, find the intersection $A \cap B$. Express your answer using standard set notation. | \{0,1,2\} | big_math | [
"Mathematics -> Discrete Mathematics -> Logic"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given $M \cup \{1,2\} = \{1,2,3\}$, count the number of sets $M$ that satisfy the condition. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Three planes are mutually perpendicular, and their three intersecting lines meet at point O. A point P in space has distances 3, 4, and 5 from the three planes, respectively. Calculate the length of OP, expressing your answer as a simplified radical. | 5\sqrt{2} | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Find the maximum value of the function f(x) = sin(x) - cos(x). Express your answer as a single numerical value. | \sqrt{2} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Convert -150 degrees to radians, given that 1 degree equals pi/180 radians. Express your answer in radians. | - \dfrac {5\pi}{6} | big_math | [
"Mathematics -> Precalculus -> Functions -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given a hyperbola E with vertices A and B, and a point M on E forming an isosceles triangle ABM with a 120° vertex angle, calculate the eccentricity of E. Express your answer as a numerical value, including the square root symbol if necessary. | \sqrt{2} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence defined by a_{1} = 2 and a_{n+1} = a_{n} + 2n, calculate the value of a_{100}. Provide your answer as a single number. | 9902 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $2m-3n=-2$, find the value of the algebraic expression $4m-6n+1$. Express your answer as a single integer. | -3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $\frac{{\cos({\frac{π}{4}-α})}}{{\cos({\frac{π}{4}+α})}}=\frac{1}{2}$, calculate the value of $\tan \alpha$. Express your answer as a single fraction or decimal. | -\frac{1}{3} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $a+b+5i=9+ai$ where $a, b \in \mathbb{R}$, solve for the value of $b$. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Algebra -> Complex Numbers -> Other"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
A high school sophomore participates in a social practice activity, using 3 different buses, with a total of 5 supervising teachers. The requirement is that each bus must have at least one supervising teacher. Calculate the total number of different allocation schemes. Express your answer as a single integer. | 150 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given 0 < α < π/2, if sin(2α - π/6) = -1/3, then find the value of sin(α + π/6). Express your answer as a fraction or decimal. | \frac{\sqrt{3}}{3} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given complex numbers ${z_1}=1+2i,{z_2}=\frac{{3+i}}{{1+i}},{z_3}=-1-2i$ correspond to three vertices of a square in the complex plane, calculate the complex number corresponding to the fourth vertex of this square. Express your answer in the form $a+bi$, where $a$ and $b$ are real numbers. | -2+i | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Find the remainder when $91^{91}$ is divided by $100$. Express your answer as a single integer. | 91 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Thomas has constant speeds for both running and walking. When a down-escalator is moving, Thomas can run down it in 15 seconds or walk down it in 30 seconds. One day, when the escalator was broken (and stationary), it took Thomas 20 seconds to run down it. Calculate the time, in seconds, it would take Thomas to walk do... | 60 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given sets $A=\{-1,1,2,4\}$ and $B=\{x||x-1|\leqslant 1\}$, find the intersection of sets A and B, $A\cap B$. Express your answer as a set of values. | \{1, 2\} | big_math | [
"Mathematics -> Discrete Mathematics -> Logic"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Given cos(x) + sin(x) = sqrt(2)/3, calculate sin(2x)/cos(x - pi/4). Express your answer as a single fraction or number. | -\frac{7}{3} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\vec{a} =(-3,4)$ and $\vec{b} =(1, m)$, if $\vec{a}\perp(\vec{a} -\vec{b})$, calculate the value of $m$. Express your answer as a single numerical value. | 7 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
The diameter of larger viruses ranges from 300 to 450nm. Given the conversion factor 1nm = 10^{-9}m, express 450nm in scientific notation. Ensure that the coefficient is between 1 and 10, and include units in your answer. | 4.5 \times 10^{-7}m | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
A regular triangle is initially painted black. Each transformation consists of dividing the black triangle into four smaller congruent triangles and painting the middle smaller triangle white. After applying this transformation five times, what fraction of the original triangle remains black? Express your answer as a f... | \frac{243}{1024} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
A train of length $a$ meters travels over a bridge of length $b$ meters at a speed of $v$ meters per second. Express the time it takes for the train to cross the bridge using an algebraic expression involving the variables $a$, $b$, and $v$. Provide your answer in the form $\boxed{\text{algebraic expression}}$. | \frac{a+b}{v} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
In the space Cartesian coordinate system, find the symmetric point of point P(1, 2, -3) with respect to the xOy coordinate plane. Express your answer as a coordinate (x, y, z). | (1, 2, 3) | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Find the domain of the function $y=\sqrt{{\log_{\frac{1}{2}}(4x-3)}}$. Express your answer as an interval. | \left( \frac{3}{4},1 \right] | big_math | [
"Mathematics -> Precalculus -> Functions"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression: $\frac{1000^2}{252^2-248^2}$. Provide your answer as a single number. | 500 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
A school has a total of 24 classes across three grades. To understand the psychological state of the students, each class is numbered from 1 to 24. Now, using systematic sampling, 4 classes are selected for the survey. If the smallest number drawn is 3, calculate the largest number drawn. Express your answer as a whole... | 21 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
A real number $p$ is randomly chosen from the interval $[0,5]$. Calculate the probability that the equation $x^2 + px + 1 = 0$ has real roots. Express your answer as a decimal value between 0 and 1, rounded to one decimal place. | 0.6 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
How many squares are exactly four greater than a prime number? Express your answer as a whole number. | 1 | big_math | [
"Mathematics -> Number Theory -> Prime Numbers"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
If $1.2 \times 10^{6}$ is doubled, what is the result? Express your answer in scientific notation. | 2.4 \times 10^{6} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Given the expressions $a=\log_{2}0.3$, $b=2^{0.3}$, and $c=0.3^{0.2}$, order the values of a, b, and c from greatest to least. Express your answer as a sequence of the variables (e.g., b > c > a). | b > c > a | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Logarithmic Functions"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the sum of \(\frac{1}{2} + \frac{1}{2} + \frac{1}{3} + \frac{1}{3} + \frac{1}{3}\). Provide your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Fractions"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
One hundred students each send messages to 50 different students. What is the least number of pairs of students who send messages to each other? Express your answer as a whole number. | 50 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Let the random variables ξ ~ B(2, p) and η ~ B(3, p). Given that P(ξ ≥ 1) = 5/9, calculate the value of P(η ≥ 2). Express your answer as a simplified fraction. | \frac{7}{27} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Given $x > 0$, $y > 0$, let $a = \frac{x+y}{1+x+y}$ and $b = \frac{x}{1+x} + \frac{y}{1+y}$. Determine the inequality relationship between $a$ and $b$. Express your answer using standard inequality notation (e.g., $a < b$, $a > b$, etc.). | a < b | big_math | [
"Mathematics -> Applied Mathematics -> Other",
"Mathematics -> Algebra -> Other",
"Mathematics -> Other"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Three people independently attempt to crack a code, with their respective probabilities of doing so being $\frac{1}{5}$, $\frac{1}{3}$, and $\frac{1}{4}$. Calculate the probability that the code is cracked. Express your answer as a simplified fraction. | \frac{3}{5} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given the two circles $x^2+y^2=m$ and $x^2+y^2+6x-8y-11=0$, find the range of values for the real number $m$ such that the two circles have at least one common point. Express your answer as an interval (e.g., [a, b]). | [1, 121] | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given complex number ${z}_{1}=-1+3i,{z}_{2}=a+b{i}^{3}(a,b∈R)$ and $z_{1}=z_{2}$, where $i$ is the imaginary unit, find the value of $b^{a}$. | -\frac{1}{3} | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Given a complex number $z$ that satisfies the equation $zi + z = 2$, find the value of $z$. Express your answer as a complex number in the form $a + bi$, and include the boxed notation around your answer. | 1 - i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Given the functional equation f(a+b) = f(a) * f(b) and the initial condition f(1) = 1, calculate the sum of the series f(2)/f(1) + f(3)/f(2) +... + f(2005)/f(2004). Express your answer as a single integer. | 2004 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Given the set M defined by the polynomial equation x^4 + 4x^2 - 12x + 8 = 0, where x ∈ R^+, find the number of elements in set M (|M|). Express your answer as a non-negative integer. | 0 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Azmi has four blocks, each in the shape of a rectangular prism and each with dimensions $2 \times 3 \times 6$. She carefully stacks these four blocks on a flat table to form a tower that is four blocks high. Calculate the number of possible heights for this tower, considering all combinations of block dimensions. | 14 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given that $x^2 + 3x + 8 = 7$, calculate the value of the algebraic expression $3x^2 + 9x - 2$. Express your answer as a single numerical value. | -5 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Algebraic Expressions"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
A shooter's probability of hitting the 10, 9, and 8 rings in a single shot are respectively 0.2, 0.3, and 0.1. Find the probability that the shooter scores no more than 8 in a single shot. Express your answer as a decimal value between 0 and 1. | 0.5 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a > 0$, $b > 0$, and $\ln (a+b)=0$, find the minimum value of $\frac {1}{a}+ \frac {1}{b}$. Express your answer as a single numerical value. | 4 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Roll a die twice in succession. Given that the first roll results in an odd number, calculate the probability that the second roll also results in an odd number. Express your answer as a simplified fraction. | \frac{1}{2} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Given a periodic sequence $\left\{x_{n}\right\}$ that satisfies $x_{n} = |x_{n-1} - x_{n-2}|$ for $n \geq 3$, if $x_{1} = 1$ and $x_{2} = a \geq 0$, what is the sum of the first 2002 terms when the period of the sequence is minimized? Express your answer as a single integer. | 1335 | big_math | [
"Mathematics -> Discrete Mathematics -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets $M = \{x | x^2 - 2x < 0\}$ and $N = \{x | x > 1\}$, determine the set $M \cap \lnot_R N$. Express your answer in interval notation. | (0, 1] | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = 2x^2 - 4(1-a)x + 1, find the range of values for the real number a such that the function is increasing on the interval [3,+\infty). Express your answer as an interval using standard interval notation. | [-2,+\infty) | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
How many multiplicatively perfect integers exist between 2 and 30? A multiplicatively perfect integer is defined as a positive integer that is equal to the product of its proper divisors. Provide your answer as a whole number. | 9 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Using systematic sampling to select a sample of size 20 from 160 students, the 160 students are numbered from 1 to 160, and are evenly divided into 20 groups (1-8, 9-16, ⋯). If the number to be drawn from the 15th group is 116, calculate the number determined by drawing lots from the first group. Express your answer as... | 4 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Find the greatest common divisor (GCD) of 360 and 504 using the Euclidean algorithm. Express your answer as a single integer. | 72 | big_math | [
"Mathematics -> Number Theory -> Greatest Common Divisors (GCD)"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
A triangle has a base of 15 inches. Two lines are drawn parallel to the base, terminating in the other two sides, and dividing the triangle into three equal areas. Find the length of the parallel closer to the base. Express your answer in inches, including the square root symbol if necessary. | 5\sqrt{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the condition $2x^2 - 6x + y^2 = 0$, find the maximum value of $x^2 + y^2 + 2x$. Express your answer as a single numerical value. | 15 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
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