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 quadratic function $y=2(x-3)^2+1$, identify the coordinates of the vertex in the form $(h,k)$. | (3,1) | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Quadratic Functions"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Given the parabola y^2 = 4x, with focus F and a fixed point P(3, 1), find the minimum value of |MP| + |MF| for a moving point M on the parabola. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given the point $M(-2,3)$, find the coordinates of the point symmetric to $M$ with respect to the origin. Express your answer as an ordered pair (x, y). | (2,-3) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Order the following logarithmic expressions from least to greatest:
\[
\log_{\sin 1} \cos 1, \quad \log_{\sin 1} \operatorname{tg} 1, \quad \log_{\cos 1} \sin 1, \quad \log_{\cos 1} \operatorname{tg} 1
\]
Provide your answer in the format: expression 1 < expression 2 < expression 3 < expression 4. | \] | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Logarithmic Functions"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence $\{a_n\}$ with the sum of the first $n$ terms being $S_n$, and it is given that $S_4 = 6$ and $2a_3 - a_2 = 6$, find the value of $a_1$. Express your answer as a single integer. | -3 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Find the maximum value of the function $y=x(3-2x)$ where $0 < x < \frac {3}{2}$. Express your answer as a fraction. | \frac {9}{8} | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the difference between the two roots of the equation $x^2+5x+k=0$ is 3, find the value of $k$. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
The sum of seven consecutive integers is 77. Find the smallest of these integers. Express your answer as a whole number. | 8 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
The function f(x) = 3sin(x + π/6) reaches its maximum value at x = θ. Find the value of tan(θ). | \sqrt{3} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Find the x-axis intercept of the line given by the equation $x + 2y + 1 = 0$. Express your answer as a single number. | -1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Linear Equations"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Among 100 students in a school's reading club, 90 students have read either *Journey to the West* or *Dream of the Red Chamber*, 80 students have read *Dream of the Red Chamber*, and 60 students have read both *Journey to the West* and *Dream of the Red Chamber*. Find the number of students who have read *Journey to th... | 70 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
What is the probability that a random arrangement of the letters in the word 'ARROW' will have both R's next to each other? Express your answer as a simplified fraction. | \frac{2}{5} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Given positive integers a, b, c, where [a, b] = 12 and [b, c] = 15, find the minimum possible value of [a, c]. Express your answer as a single integer. | 20 | big_math | [
"Mathematics -> Number Theory -> Least Common Multiples (LCM)"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
In the Cartesian coordinate system, the endpoints of line segment AB are A(-4, -1) and B(1, 1). After translating line segment AB, we obtain line segment A'B'. If the coordinates of point A' are (-2, 2), then find the coordinates of point B'. Express your answer as a coordinate pair (x, y). | (3, 4) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of $\cos420^\circ+\sin330^\circ$. Express your answer as a single numerical value. | 0 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence {a_n} with sum of the first n terms S_n, and it satisfies a_2 + S_3 = 4 and a_3 + S_5 = 12, find the value of a_4 + S_7. Express your answer as a single integer. | 24 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
If the product of $(x+n)$ and $(x+2)$ does not contain a linear term of $x$, find the value of $n$. Express your answer as a single integer. | -2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Let $f(x)$ be an odd function defined on $\mathbb{R}$, and $f(2)=0$. When $x > 0$, it always holds that $xf'(x)-f(x) < 0$. Find the solution set of the inequality $x^{2}f(x) > 0$. Express your answer as an interval or union of intervals. | (-\infty,-2)\cup(0,2) | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Calculus -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the curve $y=x^4+ax^2+1$ and a tangent line at the point $(-1, a+2)$ with a slope of 8, find the value of $a$. Express your answer as a single numerical value. | -6 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Let $\overrightarrow{a} =(4,3)$, the projection of $\overrightarrow{b}$ on $\overrightarrow{a}$ is $4$, and the projection of $\overrightarrow{b}$ on the $x$-axis is $2$. Find the ordered pair representing vector $\overrightarrow{b}$. Express your answer in the format $(x, y)$. | (2,4) | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
How many positive integers n are there such that 3n^2 + 3n + 7 is a perfect cube? Express your answer as a single integer. | 0 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Given unit vectors $\overrightarrow{a}$ and $\overrightarrow{b}$, if for any real number $x$, the inequality $|{x\overrightarrow{a}+\overrightarrow{b}}|≥\frac{{\sqrt{3}}}{2}$ always holds, find the range of values for the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$. Express your answer as an int... | [\frac{\pi}{3},\frac{2\pi}{3}] | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Anne, Beth, and Chris have 10 candies to divide amongst themselves. Anne gets at least 3 candies, while Beth and Chris each get at least 2. If Chris gets at most 3 candies, determine the set of possible values for the number of candies that Beth could get. Express your answer as a set or range of integers. | \{2, 3, 4, 5\} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence {a_n} with the sum of its first n terms denoted as S_n, if a_1 = 1 and S_5 = 15, calculate the value of the 6th term, a_6. Express your answer as a single integer. | 6 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given a geometric sequence {a_n} where each term is a real number, let S_n denote the sum of the first n terms of this sequence. If S_{10} = 10 and S_{30} = 70, find S_{40}. Express your answer as a single numerical value. | 150 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given the system of inequalities in x $$\begin{cases} x-1≥a^{2} \\ x-4＜2a\end{cases}$$, find the range of the real number a such that the system has a solution. Express your answer as an interval. | (-1, 3) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation of the line $2x-y-3=0$, find the value of $\tan\theta$, where $\theta$ is the slope angle of the line. Express your answer as a single numerical value. | 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Find the distance from the focus to the directrix of the parabola given by the equation y = 4x^2. Express your answer as a fraction or decimal. | 1/8 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
A string has been cut into 4 pieces, all of different lengths. The length of each piece is 2 times the length of the next smaller piece. What fraction of the original string is the longest piece? Express your answer as a simplified fraction. | \frac{8}{15} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
A seven-layered tower has a total of 381 lights, with the number of lights doubling on each successive layer. Calculate the number of lights on the top layer of the tower. Express your answer as a whole number. | 3 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression cos(-23π/4) using the appropriate trigonometric identity. Express your answer as a fraction or radical. | \dfrac{\sqrt{2}}{2} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Given a regular tetrahedron A-BCD with an edge length of 1, and $\overrightarrow{AE} = 2\overrightarrow{EB}$, $\overrightarrow{AF} = 2\overrightarrow{FD}$, calculate the value of $\overrightarrow{EF} \cdot \overrightarrow{DC}$. Express your answer as a single numerical value, including the negative sign if applicable. | -\frac{1}{3} | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given that positive numbers x and y satisfy the equation x+y=xy, find the minimum value of x+2y. Express your answer as an exact algebraic expression, including any necessary radicals or constants. | 3+2\sqrt{2} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
A cone is formed by wrapping a sector with a radius of $4$ and a central angle of $90^{\circ}$. Calculate the radius of the base circle of this cone. | 1 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence $$1, \sqrt{3}, \sqrt{5}, \sqrt{7}, 3, \sqrt{11}, \ldots, \sqrt{2n-1},$$ find the term number of $\sqrt{21}$ in this sequence. Express your answer as a whole number. | 11 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $x > 2$, find the minimum value of the function $y= \frac {x^{2}-4x+8}{x-2}$. Express your answer as a single numerical value. | 4 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
A broadcaster broadcasts a volleyball match, which adopts a best-of-five sets format, meaning the match ends as soon as one side wins three sets. It is known that the two teams are equally matched, and the outcome of each set is independent of the others. If the broadcaster can earn 200,000 yuan for each set broadcaste... | \dfrac{3}{4} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given the quadratic equation $x^{2}-2x+m-1=0$, find the range of the real number $m$ for which the equation has two distinct real roots. Express your answer as an inequality in terms of $m$. | m < 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Given a number x is randomly selected from the interval (0,4), find the probability that 2 < 2^(x-1) < 4. Express your answer as a fraction or decimal. | 1/4 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given the set M={-1, 0, 1} and N={x|x-1<0}, determine the intersection M∩N. Express your answer as a set. | \{-1, 0\} | big_math | [
"Mathematics -> Discrete Mathematics -> Set Theory -> Other"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the following expression:
$$
\frac{bx(a^{2}x^{2} + 2a^{2}y^{2} + b^{2}y^{2})}{bx + ay} + \frac{ay(a^{2}x^{2} + 2b^{2}x^{2} + b^{2}y^{2})}{bx + ay}
$$
Express your answer as a simplified algebraic expression. | (ax + by)^{2} | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number z1 = 2 + 3i and z2 = a + bi (a, b ∈ R, and b ≠ 0), where i is the imaginary unit, if z1z2 is a real number, find the value of a/b. Express your answer as a fraction. | -\frac{2}{3} | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Solve for $x$ in the equation $\frac{x-2}{1-x}=0$. Provide the exact value of $x$. | 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Alberto, Bernardo, and Carlos participated in a race, each running at a constant speed throughout the entire distance. When Alberto crossed the finish line, Bernardo and Carlos were 36 meters and 46 meters behind him, respectively. When Bernardo crossed the finish line, Carlos was 16 meters behind him. Find the length ... | 96 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
In a four-level symbol pyramid, the symbols in the two bottom cells determine the symbol above them. If the symbols are the same, the cell above will be "+", otherwise, it will be "-". How many possible ways are there to fill the bottom row of four cells such that the top of the pyramid results in a "+"? Express your a... | 8 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, square PQRS has side length 40. Points J, K, L, and M are on the sides of PQRS, so that JQ = KR = LS = MP = 10. Line segments JZ, KW, LX, and MY are drawn parallel to the diagonals of the square so that W is on JZ, X is on KW, Y is on LX, and Z is on MY. Find the area of quadrilateral WXYZ. Express your... | 200 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\vec{a}$ and $\vec{b}$ are both unit vectors, and their angle is $60^{\circ}$, find the magnitude of the vector $\vec{a}+3\vec{b}$. Express your answer as a numerical value. | \sqrt{13} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $x^{2}-3x=2$, calculate the value of the algebraic expression $3x^{2}-9x-7$. Express your answer as a single integer. | -1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Quadratic Functions"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
A tangent line is drawn from a point on the circle $x^2+y^2=1$ to intersect the positive x-axis and y-axis at points A and B, respectively. Find the minimum value of the product of $|\overrightarrow{OA}|$ and $|\overrightarrow{OB}|$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the graph of $f(x)$ is symmetric about the point $M(1, -2)$ and there exists an inverse function $f^{-1}(x)$, if $f(2011) = 2008$, calculate the value of $f^{-1}(-2012)$. Express your answer as an integer. | -2009 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the number of arrangements of 6 items taken 2 at a time, denoted as ${A}_{6}^{2}$. Express your answer as a whole number. | 30 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Permutations"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given $\alpha \in (0, \pi)$, if $\sin \alpha + \cos \alpha = \frac{\sqrt{3}}{3}$, calculate the value of $\cos^2 \alpha - \sin^2 \alpha$. Express your answer as a fraction with a square root, if necessary. | -\frac{\sqrt{5}}{3} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the projection of vector $\overrightarrow{b}$ in the direction of vector $\overrightarrow{a}$ is $2$, and $|\overrightarrow{a}| = 1$, find the value of $\overrightarrow{a} \cdot \overrightarrow{b}$. Express your answer as a single numerical value. | 2 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z= \frac{1+2i}{i}$, calculate its conjugate complex number $\overset{.}{z}$. Express your answer in the form $a+bi$, where $a$ and $b$ are real numbers. | 2+i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Given a vector $\overrightarrow {a} = (1, 2, -1)$, calculate the magnitude of the vector $\overrightarrow {a}$. Express your answer as a simplified radical. | \sqrt{6} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets $M=\{(x, y)|x+y=2\}$ and $N=\{(x, y)|x-y=4\}$, find the intersection of sets M and N. Express your answer as an ordered pair (x, y). | (3,-1) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given the ellipse $\dfrac{x^{2}}{4}+y^{2}=1$ with its left and right foci being $F_{1}$ and $F_{2}$ respectively, and a point $M$ on the ellipse such that $\overrightarrow{MF_{1}} \cdot \overrightarrow{MF_{2}}=0$, calculate the distance from point $M$ to the $y$-axis. Express your answer as a single value, including an... | \dfrac{2 \sqrt{6}}{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Four people in a room each write a New Year's card, then collect them together, and each person picks a card written by someone else. Find the number of different ways the four cards can be distributed. Express your answer as a numerical value. | 9 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given a regular triangle ACD with side length 1, let $\overrightarrow {AB}= \overrightarrow {a}$, $\overrightarrow {BC}= \overrightarrow {b}$, $\overrightarrow {AC}= \overrightarrow {c}$. Calculate the magnitude of $\overrightarrow {a}- \overrightarrow {b}+ \overrightarrow {c}$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
The diagram shows a square \(PQRS\). Points \(T, U, V\), and \(W\) lie on the edges of the square, such that \(PT = 1\), \(QU = 2\), \(RV = 3\), and \(SW = 4\). The area of \(TUVW\) is half that of \(PQRS\). Calculate the length of \(PQ\). Express your answer as a single integer. | 6 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
If \( n > 0 \) and \(\sqrt{n^{2} + n^{2} + n^{2} + n^{2}} = 64\), calculate the value of \( n \). Express your answer as a single number. | 32 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
A square floor is tiled with a large number of regular hexagonal tiles, which are either blue or white. Each blue tile is surrounded by 6 white tiles, and each white tile is surrounded by 3 white and 3 blue tiles. Determine the simplified ratio of the number of blue tiles to the number of white tiles, ignoring part til... | 1:2 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Let $P$ be a polynomial with each root real and each coefficient either 1 or -1. What is the maximum possible degree of $P$? Express your answer as a whole number. | 3 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
A wheel of radius 8 rolls along the diameter of a semicircle of radius 25 until it bumps into this semicircle. Find the length of the portion of the diameter that cannot be touched by the wheel. Express your answer as a whole number. | 20 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, point P lies on the y-axis, point Q has coordinates (4,0), and PQ passes through the point R(2,4). Calculate the area of triangle OPQ. Express your answer as a whole number. | 16 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
12 real numbers x and y satisfy the equation 1 + cos^2(2x + 3y - 1) = (x^2 + y^2 + 2(x+1)(1-y))/(x-y+1). Calculate the minimum value of xy. Express your answer as a single fraction in its simplest form. | \frac{1}{25} | big_math | [
"Mathematics -> Calculus -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
What is the largest possible area of an isosceles trapezoid in which the largest side is 13 and the perimeter is 28? Express your answer as a whole number. | 27 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
A moving circle has its center on the parabola $y^2 = -8x$, and it always passes through the point $(-2, 0)$. Find the equation of the common tangent line to the moving circle. Express your answer in the format $x = a$ or $y = b$. | x=2 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
A bag contains eight yellow marbles, seven red marbles, and five black marbles. Without looking in the bag, Igor removes N marbles all at once. If he is to be sure that, no matter which choice of N marbles he removes, there are at least four marbles of one color and at least three marbles of another color left in the b... | 7 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
If the complex number $z$ satisfies $iz=1+2i$, where $i$ is the imaginary unit, then find the coordinates of the point corresponding to the complex number $z$ on the complex plane. Express your answer as an ordered pair of integers (x, y). | (2,-1) | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the measurement result X follows a normal distribution N(1, σ^2) and the probability of X falling in the interval (0, 1) is 0.4, calculate the probability that X falls in the interval (0, +∞). Express your answer as a decimal value between 0 and 1, rounded to 1 decimal place if necessary. | 0.9 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Evaluate the expression x^4 - xy^3 - x^3y - 3x^2y + 3xy^2 + y^4 when x - y = 1. Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Three congruent squares overlap as shown. The area of the three overlapping sections are $2 \mathrm{~cm}^{2}, 5 \mathrm{~cm}^{2}$ and $8 \mathrm{~cm}^{2}$ respectively. The total area of the non-overlapping parts of the square is $117 \mathrm{~cm}^{2}$. Find the side-length of each square. Express your answer in centim... | 7 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Let $i$ be the imaginary unit, and let $ar{z}$ denote the complex conjugate of the complex number $z$. If $z = 1 + i$, calculate the value of $-i \cdot z + i \cdot ar{z}$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Find the minimum value of the function f(x) = x^3 - 3x^2 + 1. | -3 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\tan (\alpha - \frac{\pi }{4}) = \frac{1}{2}$, calculate the value of $\frac{\sin \alpha + \cos \alpha }{\sin \alpha - \cos \alpha }$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors a = (2,1) and b = (sin θ, cos θ), if a is parallel to b, find the value of tan(θ + 45°). Express your answer as a single numerical value. | -3 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Given the set A = {x|1≤x≤2}, find the necessary and sufficient condition on 'a' such that the proposition ∀x∈A, x^2-a≤0 is true. Express your answer as an inequality involving 'a'. | a ≥ 4 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
On February 1, it was 16.2°C outside Jacinta's house at 3:00 p.m. On February 2, it was -3.6°C outside Jacinta's house at 2:00 a.m. If the temperature changed at a constant rate between these times, calculate the rate at which the temperature decreased. Express your answer in °C per hour. | 1.8 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = x^2 + 2x + 4, and the conditions x_1 < x_2 and x_1 + x_2 = 0, compare the values of f(x_1) and f(x_2). Provide your answer as a comparison statement (e.g., <, >, =). | f(x_1) < f(x_2) | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given the binomial (x+a)^6, where a is a natural number (a ∈ N*), find the value of 'a' for which the coefficient of the fourth term in the expansion is the largest. Express your answer as a single natural number. | 1 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
The hyperbola \(x^2 - \frac{y^2}{m} = 1\) has one of its foci at \((-3,0)\). Find the value of \(m\). Express your answer as a single value. | 8 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence $\{a_n\}$ with $a_1 = 1$, $a_2 = 2$, and the recursive relation $2a_{n+1} = a_n + a_{n+2}$, calculate the sum of the first 5 terms of the sequence. Express your answer as a single integer. | 15 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
The temperature of the main peak of Huangshan was -12°C in the morning, rose by 8°C at noon, and then dropped by 10°C at night. What was the temperature of the main peak of Huangshan at night on that day? Express your answer in degrees Celsius (°C) as an integer. | -14 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Sri's car clock, which is not accurate, gains time at a constant rate. One day, he notes that his car clock and his accurate watch both say 12:00 noon. When he is done shopping, his watch says 12:30 and his car clock says 12:35. Later that day, Sri loses his watch. He looks at his car clock, and it says 7:00. What is t... | 6:00 \text{ PM} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given a straight line with a slope of 3 that passes through points A(3, 4) and B(x, 7), find the value of x. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x+2016) = (x^2+1)/(2x) for x>0, find the minimum value of the function f(x). | 1 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Jackie scored 36 points in her last basketball game, raising her average points per game from 20 to 21. How many points must she score in her next game to raise her average points per game to 22? Express your answer as a whole number. | 38 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the result of $2÷3×(-\frac{1}{3})$. Express your answer as a fraction. | -\frac{2}{9} | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the positive real numbers a and b satisfy a + b = 1, find the maximum value of sqrt(a) + sqrt(b). | \sqrt{2} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Given two vectors $\overrightarrow {a} = 2\overrightarrow {e}_{1} - 3\overrightarrow {e}_{2}$ and $\overrightarrow {b} = (1+n)\overrightarrow {e}_{1} + n\overrightarrow {e}_{2}$, if $\overrightarrow {a}$ is parallel to $\overrightarrow {b}$, find the value of $n$. Express your answer as a numerical value. | -\frac{3}{5} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
If $a=2 \int_{-3}^{3} (x+|x|) \, dx$, then in the expansion of $(\sqrt{x} - \frac{1}{\sqrt[3]{x}})^a$, find the total number of terms where the power of $x$ is not an integer. Express your answer as a single integer. | 15 | big_math | [
"Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals",
"Mathematics -> Algebra -> Exponential Functions -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $x$ and $y$ are positive real numbers satisfying $2x + y = 1$, find the maximum value of $xy$. Express your answer as a numerical value. | \frac{1}{8} | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Determine the interval where the function y = -x^2 is monotonically increasing. Express your answer as an interval. | (-\infty, 0] | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given two points $P(1,3)$ and $Q(4,-1)$, calculate the distance between these two points using the distance formula. Express your answer as a single number. | 5 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Let the random variable ξ follow the normal distribution N(3, σ^2). Given P(ξ > 4) = 0.2, find the probability P(3 < ξ ≤ 4). Express your answer as a numerical value. | 0.3 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Roll a die twice. Let X be the maximum of the two numbers rolled. Calculate the expected value E(X). Express your answer as a single numerical value, rounded to two decimal places if necessary. | 4.47 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Find the area of the triangle formed by the tangent line at point (1, -2) on the curve y = ln(x) - 2x and the coordinate axes. Express your answer as a decimal or fraction. | \frac{1}{2} | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Given the set A={1,2,3,4,5} and B={x|x<3}, determine A∩(∁$_{m R}$B). Express your answer as a set of numbers, including all elements of the resulting set. | \{3,4,5\} | big_math | [
"Mathematics -> Set Theory -> Operations on Sets"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Place four identical white balls and five identical black balls into three different boxes, such that each box contains both white and black balls, and each box contains at least two balls. Calculate the total number of different arrangements possible. | 18 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
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