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 a counterexample to the statement, 'If n is a prime number, then exactly one of n-2 and n+2 is prime.' Express your answer as a single prime number. | 37 | big_math | [
"Mathematics -> Number Theory -> Prime Numbers"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
In $\triangle ABC$, $AB=3$, $AC=4$, and $\angle A=60^{\circ}$. Using the cosine theorem, calculate the length of $BC$. Express your answer as the square root of a number. | \sqrt{13} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the inequality $x^2 + ax + 1 \geqslant 0$ holds for all $x \in (0, \frac{1}{2}]$, calculate the minimum value of $a$. Express your answer as a single number. | -\frac{5}{2} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of the algebraic expression $10 \cdot 61 \left( \frac{1}{\log _{2} 100!}+\frac{1}{\log _{3} 100!}+\frac{1}{\log _{4} 100!}+\cdots+\frac{1}{\log _{100} 100!} \right)$. Express your answer as a single integer. | 610 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions",
"Mathematics -> Applied Mathematics -> Other"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Find the coefficient of the term $x^{2}y^{2}$ in the expansion of $(x-2y)^{4}$. Express your answer as a single integer. | 24 | big_math | [
"Mathematics -> Algebra -> Algebra -> Binomial Theorem"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the angle between the plane vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\frac{π}{3}$, and for any real number $\lambda$, $|{\overrightarrow{a}-λ\overrightarrow{b}}|≥|{\overrightarrow{a}-\overrightarrow{b}}|$ always holds, calculate the ratio $\overrightarrow{|a|}:\overrightarrow{|b|}$. Express ... | 2:1 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Find the minimum value of the function $f(x) = x^{2} - 4x + 4$. Express your answer as a single number. | 0 | big_math | [
"Mathematics -> Algebra -> Algebra -> Polynomial Operations"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Find the distance from the intersection point of line $l_1: x + y - 2\sqrt{2} = 0$ and line $l_2: \begin{cases} x= \frac{\sqrt{2}}{2}t \\ y= \frac{\sqrt{2}}{2}t \end{cases}$ to the origin $O$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function y = |x| and its domain A = {-1, 0, 1}, find the intersection of A and the range B of the function. Express your answer as a set, including all elements of the intersection. | {0, 1} | big_math | [
"Mathematics -> Algebra -> Other"
] | 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 as a complex number in the form a+bi. | -2+i | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence $\{a_n\}$ with general term formula $a_n = -2n^2 + \lambda n$ ($n \in \mathbb{N}^*, \lambda \in \mathbb{R}$), find the range of values for $\lambda$ such that the sequence is decreasing. Express your answer as an interval in standard interval notation. | (-\infty,6) | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Determine the smallest positive period of the function f(x) = cos^2 x. Express your answer in terms of pi. | \pi | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation i/z = 1 + i, where i is the imaginary unit, find the value of z - conjugate(z). Express your answer as a simplified complex number. | i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, circles C1 and C2 each have center O. The radius of circle C1 is sqrt(2) and the radius of circle C2 is 2*sqrt(2). Find the area of the shaded region. Express your answer in terms of pi. | 6\pi | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Area"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Determine the maximum number of beads of the same shape and size, among which only one is lighter than the others, that can be definitively identified as the lighter one using a scientific algorithm and a balance scale (without weights) with just two weighings. | 9 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given a 3x3 matrix where each row and each column forms an arithmetic sequence, and the middle element $a_{22} = 5$, find the sum of all nine elements. Express your answer as a single numerical value. | 45 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation sin(α)sin(β) + cos(α)cos(β) = 0, calculate the value of sin(2α) + sin(2β). Express your answer as a single numerical value. | 0 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
A light-year is a unit of distance in astronomy, with 1 light-year being approximately 9500000000000 km. Express this number in scientific notation in the form a × 10^n, including the unit (km). | 9.5 × 10^12 km | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
A moving circle has its center on the parabola $y^2 = 4x$, and it is always tangent to the line $x = -1$. Find the coordinates of the fixed point that the moving circle must pass through. Express your answer as an ordered pair (x, y). | (1, 0) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given that the function f(x) defined on R is monotonically increasing on [-2, +∞) and f(x-2) is an even function, find the range of values of x that satisfy the inequality f(2x) < f(x+2). Express your answer as an interval of x. | (-2, 2) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the angle between non-zero vectors $\overrightarrow {a}$ and $\overrightarrow {b}$ is $\frac {2π}{3}$, and $|\overrightarrow {a}|=|\overrightarrow {a}+\overrightarrow {b}|$, find the minimum value of $\frac {|2 \overrightarrow {a}+t \overrightarrow {b}|}{|2 \overrightarrow {b}|}$. Express your answer as a si... | \frac{\sqrt{3}}{2} | big_math | [
"Mathematics -> Geometry -> Other",
"Mathematics -> Algebra -> Other",
"Mathematics -> Applied Mathematics -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) (x∈ℝ) that satisfies f(x+π) = f(x) + cos(x), and f(x) = -1 when 0 ≤ x < π, calculate the value of f(2017π/3). | -1 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given that plane vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}-2\overrightarrow{b}|=1$ and $\overrightarrow{a}•\overrightarrow{b}=1$, find the magnitude of $|\overrightarrow{a}+2\overrightarrow{b}|$. Express your answer as a numerical value. | 3 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
There are 2016 kangaroos in a zoo, each of which is either grey or pink, with at least one grey and one pink kangaroo. For every kangaroo, calculate the fraction: the number of kangaroos of the other color divided by the number of kangaroos of the same color as this kangaroo (including himself). Find the sum of all the... | 2016 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Determine the range of real numbers a for which the quadratic inequality 2x^2 + (a-1)x + 1/2 > 0 holds true for all x in R. Express your answer as an interval or range of values for a. | (-1, 3) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Quadratic Functions"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = |2x - a| + a, find the value of the real number a such that the solution set of the inequality f(x) ≤ 6 is {x | -2 ≤ x ≤ 3}. Express your answer as a single real number. | 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given the vectors a = (3, 4), b = (4, 3), and c = λa - b on a plane, the angle between c and a is equal to the angle between c and b. Find the value of λ. Express your answer as a single real number. | -1 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the tangent line to the function y = xe^x at the point (1, e). Express your answer in slope-intercept form (y = mx + b). | y = 2ex - e | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given the decomposition reaction of potassium chlorate:
\[\ce{2KClO3(s)} \rightarrow \ce{2KCl(s)} + \ce{3O2(g)}\]
and the molar mass of KClO3 as 122.6 g/mol, calculate the number of moles of oxygen gas produced from the decomposition of 245 g of potassium chlorate. Express your answer in moles, rounded to two decimal... | 3.00 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
The area of the triangle formed by the tangent line to the curve $y=x^{3}$ at the point $(3,27)$ and the two coordinate axes is to be calculated. Find the area of this triangle. Express your answer as a whole number. | 54 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Determine the range of values for a that satisfies the inequality |x-3| + |x+2| > a. Express your answer as an inequality (e.g., a < 5, a > 5, etc.). | a < 5 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
What is the smallest root of the equation $\frac{|x-1|}{x^{2}}=6$? Express your answer as a numerical value. | -\frac{1}{2} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
In the sequence {a_n}, a_1 = 1, a_n - a_{n-1} = n, where n is a positive integer. Calculate the value of a_8. Express your answer as a single integer. | 36 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Given \(A = \{x \in \mathbb{Z} | -2 < x < 4\}\) and \(B = \{x | \frac{2}{x-1} \geqslant 0\}\), find the number of elements in \(A \cap (\mathbb{C}_\mathbb{R}B)\). Express your answer as a single integer. | 3 | big_math | [
"Mathematics -> Discrete Mathematics -> Set Theory -> Other"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the sum of the first 6 terms of a geometric sequence to the sum of the first 3 terms is in the ratio 1 : 2, find the ratio of the sum of the first 9 terms to the sum of the first 3 terms. Express your answer as a ratio. | 3 : 4 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
A rectangular prism has one vertex where the lengths of the three edges are $3$, $4$, and $5$, respectively. Additionally, all eight vertices of the prism are on the same sphere. Find the surface area of this sphere. Express your answer in terms of pi. | 50\pi | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Surface Area"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
If $x, y > 0$ and $x + 2y = 3$, find the minimum value of $\frac{1}{x} + \frac{1}{y}$. Express your answer as a single value, including any necessary radicals or fractions. | 1 + \frac{2\sqrt{2}}{3} | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Given $$\overrightarrow {a}=(\cos x，-2)， \overrightarrow {b}=(\sin x，1)$$ and $$\overrightarrow {a}$$ is parallel to $$\overrightarrow {b}$$, find the value of $\sin 2x$. Express your answer as a single value. | - \frac {4}{5} | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Trigonometry -> Trigonometric Functions -> Other"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex numbers $z_1=\cos \alpha+i\sin \alpha$ and $z_2=\cos \beta+i\sin \beta$, find the real part of the complex number $z_1\cdot z_2$. Express your answer as a trigonometric function of the angles $\alpha$ and $\beta$. | \cos(\alpha+\beta) | big_math | [
"Mathematics -> Algebra -> Complex Numbers -> Other"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
What is the absolute value of -2022? Express your answer as a positive integer. | 2022 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given a sequence $\{a_{n}\}$ that satisfies $2a_{1}+2^{2}a_{2}+2^{3}a_{3}+\ldots +2^{n}a_{n}=n\cdot 2^{n}$, derive the general formula for $\{a_{n}\}$. Express your answer as a mathematical formula in the boxed notation. | a_{n}=\frac{n+1}{2} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Pat gives half of his marbles to his best friend and then a third of those remaining to his sister. If his sister receives 9 marbles, find the number of marbles Pat keeps. Express your answer as a whole number. | 18 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Find the smallest positive angle that is equivalent to -1990 degrees. Express your answer in degrees. | 170^{\circ} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions -> Other"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
There are three volunteers, A, B, and C, who are scheduled to participate in a volunteer activity from Monday to Friday. Each person participates only once, and no more than one person is scheduled per day. It is required that A is scheduled before both B and C. Find the total number of different scheduling methods. Ex... | 20 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Permutations"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given an ellipse $\frac{x^{2}}{6} + \frac{y^{2}}{5} = 1$ and a point $P(2, -1)$ inside it, if a chord passing through point $P$ is bisected by $P$, find the equation of the line containing this chord. Express your answer as a linear equation in the form $Ax + By + C = 0$. | 5x - 3y - 13 = 0 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
In the polar coordinate system, find the axis of symmetry for the curve $\rho=4\sin \left( \theta- \frac{\pi}{3} \right)$. Express your answer as a mathematical expression for the axis of symmetry, using theta (\theta) to represent the axis. | \theta=\frac{5\pi}{6} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given the ellipse $\frac{x^2}{9} + \frac{y^2}{5} = 1$ with foci $F_1$ and $F_2$, and a point $P$ on the ellipse such that the midpoint of line segment $PF_1$ lies on the $y$-axis, find the ratio $\frac{|PF_2|}{|PF_1|}$. Express your answer as a simplified fraction. | \frac{5}{13} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Express 98.83 million in scientific notation. Provide your answer in the form of $a \times 10^b$, where $a$ is the coefficient and $b$ is the exponent. | 9.883 \times 10^7 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence $\{a_n\}$, where $a_1+a_4+a_6+a_8+a_{10}=26$, and $a_1-a_3-a_{11}=-10$, calculate the sum of the first 7 terms, $S_7$. Express your answer as a single integer. | 28 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
The geometric sequence {a_n} is a monotonically increasing sequence, and it is given that a_2a_5 = 6 and a_3 + a_4 = 5. Find the common ratio q. Express your answer as a numerical value. | \frac{3}{2} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
In the complex plane, the complex number $z$ and $\dfrac{2}{1-i}$ correspond to points that are symmetric with respect to the real axis. Find the value of $z$. Express your answer as a complex number in the form $a+bi$. | 1-i | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $x$ and $y$ are positive real numbers satisfying the equation $xy = x + 2y + 6$, find the minimum value of $\frac{1}{x} + \frac{1}{2y}$. Express your answer as a single fraction. | \frac{1}{3} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
The angle between the plane vectors $\overset{→}{a}$ and $\overset{→}{b}$ is $60^{\circ}$, $\overset{→}{a} =(2,0)$, $| \overset{→}{b} |=1$. Calculate the magnitude of the vector $\overset{→}{a} + 2 \overset{→}{b}$. Express your answer as a simplified radical. | 2 \sqrt{3} | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Geometry -> Other",
"Mathematics -> Linear Algebra -> Vectors"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $(1-2i)z=5i$, calculate the value of $|z|$. Express your answer as a single value in simplest form. | \sqrt{5} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
A magic box is defined such that when any pair of real numbers $(a, b)$ is entered into it, it produces a new real number: $a^2 + b + 1$. If the real number pair $(-2, 3)$ is put into it to get a real number $m$, and then the real number pair $(m, 1)$ is put into it, calculate the resulting real number. Express your an... | 66 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, \( PQR \) is a straight line segment and \( QS = QT \). Also, \( \angle PQS = x^\circ \) and \( \angle TQR = 3x^\circ \). If \( \angle QTS = 76^\circ \), find the value of \( x \). Express your answer in degrees. | 38 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
If the function f(x) is differentiable, calculate the limit lim_{Δx→0} [f(1+Δx) - f(1)] / (2Δx) in terms of f''(1). Express your answer as a coefficient of f''(1). | \frac{1}{2}f''(1) | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Given $z=1+i$, simplify the expression $\frac{{z-2}}{z}$. Express your answer as a simplified complex number. | i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Given a solid shape whose front view and side view are both isosceles triangles, and its top view is a circle with a radius of 1cm (including the center). If the area of its lateral surface when unfolded is $2\pi ext{cm}^2$, then calculate the height of this solid. Express your answer in centimeters (cm). | \sqrt{3} | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the result of $100^2 - 2 \times 100 \times 99 + 99^2$. Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Given that i is the imaginary unit, find the exact value of i^2015. Express your answer as a single value, possibly including the imaginary unit i. | -i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given x = √3, y = log₃2, and z = cos 2, order these expressions from least to greatest. Express your answer as an inequality (e.g., x < y < z). | z < y < x | big_math | [
"Mathematics -> Precalculus -> Functions",
"Mathematics -> Calculus -> Other"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
In the geometric sequence {a_n}, a_3 + a_5 = 20 and a_4 = 8. Find the value of a_2 + a_6. Express your answer as a single integer. | 34 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the complex number $z$ satisfies $(2z-i)(2-i)=5$, find the value of $z$. Express your answer as a complex number in the form $a+bi$, where $a$ and $b$ are real numbers. | 1+i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ that satisfy $\overrightarrow{a} \cdot \overrightarrow{b} = 10$ and $|\overrightarrow{a} + \overrightarrow{b}| = 5\sqrt{2}$, and $\overrightarrow{a} = (2, 1)$, find the magnitude of $\overrightarrow{b}$. Express your answer as a single number. | 5 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors a = (1,1,0) and b = (-1,0,1), find the value of k such that k*a + b is perpendicular to a. Express your answer as a fraction or decimal. | \frac{1}{2} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Let f(x) be an odd function defined on the real numbers that is increasing on (0, +∞). Find the range of x such that f(x) > f(x^2 - 2x + 2). Express your answer as an interval. | (1,2) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
The amount 2.5 is split into two nonnegative real numbers uniformly at random, and each number is rounded to its nearest integer. Find the probability that the two integers sum to 3. Express your answer as a fraction in simplest form. | \frac{3}{5} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Figures $I$, $II$, and $III$ are squares. The perimeter of $I$ is $12$ and the perimeter of $II$ is $24$. Find the perimeter of square $III$. Express your answer as a whole number. | 36 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Convert the number 850,000 to scientific notation. Express your answer in the format $a \times 10^{n}$. | $8.5 \times 10^{5}$ | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given that point M is in the second quadrant, and the distance from point M to the x-axis is 1, and to the y-axis is 2, determine the coordinates of point M. Express your answer as an ordered pair (x, y), including negative signs if applicable. | (-2,1) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $x^2 + 5x - 990 = 0$, evaluate the expression $x^3 + 6x^2 - 985x + 1012$. Express your answer as a single integer. | 2002 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given an odd function $f(x)$ defined on $\mathbb{R}$ with its derivative denoted as $f''(x)$, when $x < 0$, $f(x)$ satisfies $2f(x)+xf''(x) < xf(x)$. Calculate the number of zeros of $f(x)$ on $\mathbb{R}$. Express your answer as a single numerical value. | 1 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Other"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
There are four balls that look the same but have different weights, denoted as $a$, $b$, $c$, and $d$. Given the inequalities $a+b=c+d$, $a+d>b+c$, and $a+c<b$, find the exact order of these four balls from heaviest to lightest. Express your answer as a sequence of variables, using '>' to denote 'heavier than' (e.g., d... | d>b>a>c | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given two vectors in the plane $\vec{a}$ and $\vec{b}$ with an angle of $45^\circ$ between them, and $\vec{a}=(1,1)$, $|\vec{b}|=1$, calculate the magnitude of the sum of the two vectors, $|\vec{a}+ \vec{b}|$. Express your answer as a numerical value. | \sqrt{5} | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Five students donated amounts of 8, 10, 10, 4, and 6 Yuan. Calculate the median of this set of data. Express your answer as a single number. | 8 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Let $P$ be a moving point on the line $y=-1$. A tangent line is drawn through point $P$ to the circle $C: (x+3)^{2} + (y-2)^{2} = 4$. Calculate the minimum length of the tangent line. Express your answer as a numerical value. | \sqrt{5} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
To obtain the graph of the function $y=\cos \left( 2x-\frac{\pi }{6} \right)$, determine the horizontal shift required from the graph of the function $y=\cos 2x$. Express your answer in radians and specify the direction (left or right). | \frac{\pi }{12} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
Given the polar equation $\rho = 4\cos\theta$, calculate the area of the shape enclosed by this curve. Express your answer in terms of $\pi$. | 4\pi | big_math | [
"Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Given that cos($$α- \frac {π}{4}$$) = $$\frac { \sqrt {2}}{4}$$, find the value of tan^2($$α+ \frac {π}{4}$$). Express your answer as a fraction. | \frac {1}{7} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
A machine produces 150 items in one minute. How many items would it produce in 10 seconds? Provide your answer as a whole number. | 25 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the number of different arrangements for 7 people to stand in a row where person A cannot stand at the head and person B cannot stand in the middle. Provide your answer as a single integer. | 3720 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Permutations"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
In an opaque bag, there are 9 white chess pieces and some black chess pieces, all of which are identical except for the color. After shaking the bag, a chess piece is randomly drawn from it, its color is noted, and then it is put back into the bag. This process is repeated continuously, and it is found that the frequen... | 21 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Other"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
The price of gasoline rose by 20% during January, fell by 20% during February, rose by 25% during March, and fell by x% during April. If the price at the end of April was the same as it had been at the beginning of January, calculate the value of x. Express your answer as a percentage rounded to the nearest integer. | 17 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Let the vector $\overrightarrow {a} = (-1, 2)$, and vector $\overrightarrow {b} = (m, 1)$. If the vector $\overrightarrow {a} + 2\overrightarrow {b}$ is parallel to $2\overrightarrow {a} - \overrightarrow {b}$, then find the value of $m$. Express your answer as a numerical value. | -\frac{1}{2} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Given the line $l: x-y+4=0$ and the circle $C: \begin{cases} x=1+2\cos\theta \\ y=1+2\sin\theta \end{cases}$, calculate the minimum distance from any point on $C$ to $l$. Express your answer as a numerical value. | 2\sqrt{2}-2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Find the smallest positive integer $n$ that satisfies the inequality $\sqrt{n} - \sqrt{n-1} < 0.01$. Express your answer as a single integer. | 2501 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
A traveler walks a certain distance each day to reach a pass that is 378 miles away. The distance walked each day forms a geometric sequence, where each day he walks half as much as the day before. It takes him 6 days to reach the pass. Calculate the distance walked on the second day. | 96 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Mr. Jones has eight children of different ages. On a family trip his oldest child, who is 9, spots a license plate with a 4-digit number in which each of two digits appears two times. "Look, daddy!" she exclaims. "That number is evenly divisible by the age of each of us kids!" "That's right," replies Mr. Jones, "and th... | 5 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Convert the binary number $1101_2$ to its decimal equivalent. Express your answer as a single integer. | 13 | big_math | [
"Mathematics -> Other -> Other"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Given a point P(2,1) inside the circle (x-1)^{2}+y^{2}=4, find the equation of the shortest chord passing through point P. Express your answer in the standard form of the equation of a line (Ax + By + C = 0). | x+y-3=0 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Given that point P (-a^2 - 1, -a + 3) is symmetric to the origin and a < 0, determine the quadrant in which the symmetric point P1 lies. Express your answer as the name of the quadrant (e.g., first, second, third, fourth). | fourth | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $\frac {ai}{2-i}=1-2i$, where $i$ is the imaginary unit, find the value of the real number $a$ that satisfies this equation. Express your answer as a single real number. | -5 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Sophia did push-ups every day for 7 days. Each day after the first day, she did 5 more push-ups than the day before. In total, she did 175 push-ups. How many push-ups did Sophia do on the last day? Express your answer as a whole number. | 40 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
If $$\int_{0}^{T} x^2 \, dx = 9$$, calculate the value of the constant $T$. Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given $4^x = 9^y = 6$, find the value of $\frac {1}{x} + \frac {1}{y}$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the vertex of angle $\alpha$ is at the origin of coordinates, the initial side is the positive half-axis of the x-axis, and the terminal side falls in the second quadrant. Point A $(x, y)$ is a point on its terminal side, vector $\overrightarrow{m} = (3, 4)$. If $\overrightarrow{m} \perp \overrightarrow{OA}$... | \frac{1}{7} | big_math | [
"Mathematics -> Trigonometry -> Trigonometric Functions"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the tangent line to the curve y = x^2 at x = 1. Express your answer in the form y = mx + b, where m is the slope and b is the y-intercept. | y = 2x - 1 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given the set $A=\{1,2,3,4\}$, calculate the number of non-empty proper subsets of $A$. Express your answer as a whole number. | 14 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
In the complex plane, the point corresponding to the complex number z is (2, -1). Express z in its standard form and calculate the product of z and (2+i). Provide your answer in the standard form a + bi, if applicable. | 5 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
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