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 that the sum of the first n terms of an arithmetic sequence {a_n} is S_n, and a_3 + a_5 + a_7 = 15, find the value of S_9. Express your answer as a single numerical value. | 45 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
An odd function f(x) defined on R satisfies f(x) = 1/(x+1) when x < 0. Determine the value of f(1/2). Provide your answer as a single number. | -2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Given the ellipse x^2/16 + y^2/9 = 1 and the line x/4 + y/3 = 1, find the number of points P on the ellipse such that the distance from P to the line is 6/5. Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Find the coefficient of x in the expansion of the binomial (x^2 + 1/x)^5. Express your answer as a single integer. | 10 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Let $f(x)$ be a differentiable function such that $\lim_{\Delta x\to 0} \frac{f(x_0+3\Delta x)-f(x_0)}{\Delta x}=-3$. Using the definition of a derivative, calculate the value of $f'(x_0)$. Express your answer as a single numerical value. | -1 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Find the coefficient of $x^{4}$ in the expansion of $({2{x^2}-\frac{1}{x}})^5$. Express your answer as a numerical value. | 80 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
In the expansion of $({x-\\frac{1}{x}})({a+y})^6$, the coefficient of the $x^{-1}y^{4}$ term is $-15$. Calculate the value of $a$ that satisfies this condition. Express your answer as a numerical value, possibly with a $\\pm$ symbol. | \\pm 1 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Asha and Sasha studied for different amounts of time each day over a 5-day period. The daily differences in their study times (in minutes) are given as 10, -10, 20, 30, -20. Calculate the average difference in study time per day between Sasha and Asha. Express your answer as a whole number of minutes. | 6 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation of a circle $(x-2)^{2}+y^{2}=3$, find the range of values for the expression $\frac{y}{x}$. Express your answer as an interval. | [-\sqrt{3}, \sqrt{3}] | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Analyze the following propositions about probability and events, and determine the number of false propositions among them.
① Complementary events must be mutually exclusive events.
② If $A$ and $B$ are two events, then $P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)$.
③ If events $A$, $B$, and $C$ are pairwi... | 2 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
A cuboid has three faces meeting at one vertex with areas of $\sqrt{2}$, $\sqrt{3}$, and $\sqrt{6}$, respectively. Calculate the length of the diagonal of this cuboid. Express your answer as a simplified radical. | \sqrt{6} | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the sequence $\{a_n\}$ is a geometric sequence, and if $a_5=2$, calculate the product of the first $9$ terms of the sequence $\{a_n\}$, denoted as $T_9$. Express your answer as a single numerical value. | 512 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z=m-3+(m-1)i$, where $m$ is an integer, the point corresponding to $z$ in the complex plane is located in the second quadrant. Determine the value of $|$$\frac{1}{z}$$|$. Express your answer as a simplified radical. | \frac{\sqrt{2}}{2} | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, $\triangle ABC$ is a right-angled triangle at $C$. Given that $BC = 8$ and $AB = 10$, calculate the area of $\triangle ABC$. Express your answer as a whole number. | 24 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Different patterns can be created by shading exactly three of the nine small triangles shown, no two of which can share a side. Patterns that can be matched by rotations or by reflections are considered the same. How many different patterns can be created? Express your answer as a single integer. | 10 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
If the terminal side of angle α passes through the point P0(-3, -4), calculate the value of tan α. Express your answer as a fraction. | \frac{4}{3} | big_math | [
"Mathematics -> Trigonometry -> Trigonometric Functions -> Tangent"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $i$ is the imaginary unit, and $z(1-i) = 1+i$, calculate the conjugate of the complex number $z$. Express your answer as a simplified complex number. | -i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
If \( 2^n = 8^{20} \), what is the value of \( n \)? Express your answer as a whole number. | 60 | big_math | [
"Mathematics -> Algebra -> Exponential Functions"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given the original parabola equation y = 2x^2 and the translated parabola equation y = 2(x+4)^2 + 1, describe the translation sequence to obtain the translated parabola from the original parabola. Use the format'shift X units [left/right], then shift Y units [up/down]'. | Shift 4 units left, then shift 1 unit up. | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Quadratic Functions"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
The average height of the members of a certain school's honor guard team is 175cm. If being taller than the average height by 2cm is denoted as +2cm, then how would you denote being shorter than the average height by 2cm? Provide your answer in the format of either +x or -x, where x is the deviation from the average in... | -2cm | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Given that \( x = 18 \) is one of the solutions of the equation \( x^{2} + 12x + c = 0 \), find the other solution of this equation. Express your answer as a single integer. | -30 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $\frac{1}{16}a^0+\left (\frac{1}{16a}
ight )^0- \left (64^{-\frac{1}{2}}
ight )- (-32)^{-\frac{4}{5}}$. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the factorization \(x^{2}-x+c=(x+2)(x-3)\), calculate the value of \(c\). Express your answer as a single integer. | -6 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
In a geometric sequence $\{a_n\}$, the first term $a_1$ is 3 and the fourth term $a_4$ is 24. Find the sum of the third, fourth, and fifth terms ($a_3 + a_4 + a_5$). Express your answer as a single integer. | 84 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $8^{\frac{2}{3}} - \log_{10} 100$. Provide your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Precalculus -> Functions",
"Mathematics -> Calculus -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given the plane vectors $\overrightarrow{a}=(1,1)$ and $\overrightarrow{b}=(1,-1)$, calculate the coordinates of the vector $-2 \overrightarrow{a}- \overrightarrow{b}$. Express your answer as an ordered pair (x, y), enclosed in parentheses. | (-3,-1) | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
A TV factory plans to reduce the cost of a certain model of TV by 19% over the course of two years. If the percentage of reduction is the same each year, calculate the annual percentage reduction. Express your answer as a percentage. | 10\% | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function:
$$
\begin{aligned}
y= & |x-1|+|2x-1|+|3x-1|+|4x-1|+|5x-1|
\end{aligned}
$$
Find the value of x that minimizes this function. Express your answer as a single real number. | \frac{1}{3} | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
The parabola $y^{2}=2px$ has its focus at the same point as the right focus of the hyperbola $\dfrac{x^{2}}{3}- \dfrac{y^{2}}{1}=1$. Calculate the value of $p$. Express your answer as a single numerical value. | 2 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Mary's top book shelf holds five books with the following widths, in centimeters: $6$, $\dfrac{1}{2}$, $1$, $2.5$, and $10$. Calculate the average book width. Express your answer in centimeters as a decimal number. | 4 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
After playing 500 games, my success rate at Spider Solitaire is 49%. Assuming that I win every game from now on, how many extra games do I need to play in order for my success rate to increase to 50%? Express your answer as a whole number. | 10 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
If line $l_1: 2x-y-1=0$ is parallel to line $l_2: 2x+(a+1)y+2=0$, then what is the value of $a$ that makes the slopes of these two lines equal? Express your answer as a single integer. | -2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given the parametric equation of line $l$ as $$\begin{cases} x=1+3t, \\ y=2+4t, \end{cases}$$ (where $t$ is the parameter), find the distance between point $(1, 0)$ and line $l$. Express your answer as a fraction. | \frac{6}{5} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = e^(x-1) - ax (a > 1) with a minimum value f(x_0) on the interval [0, a] and x_0 < 2, find the range of the real number a. Express your answer as an interval. | (1, e) | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = sin(ωx + π/4) with ω > 0, which has a minimum positive period of π, find the value of f(π/8). Express your answer as a single numerical value. | 1 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence $\{a_n\}$, the sum of the first $n$ terms is $S_n$. If $a_1=1$ and the common difference $d=2$, and $S_{k+2}=28+S_k$, find the value of $k$. Express your answer as a whole number. | 6 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Define the function f(x) on R that satisfies the equation f(x) + (x + 1/2) * f(1 - x) = 1. Calculate the value of 2016 / (f^2(0) * f^3(1)). Express your answer as a single integer. | -63 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function y = 1/2*x^2 - ln(x), find the interval where the function is monotonically decreasing. Express your answer in standard interval notation. | (0, 1) | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Find the asymptotic line equation of the hyperbola $3x^2 - y^2 = 1$. Express your answer in the form $y = \pm kx$. | y = \pm \sqrt{3}x | big_math | [
"Mathematics -> Geometry -> Non-Euclidean Geometry -> Hyperbolic Geometry"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
According to the ancient Chinese mathematical book "Nine Chapters on the Mathematical Art," an income of $7$ yuan is denoted as $+7$. How would an expenditure of $5$ yuan be denoted using the concept of negative numbers? Express your answer as a single integer with a negative sign if necessary. | -5 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Points \(M\) and \(N\) are the midpoints of two sides of the rectangle. Calculate the fraction of the rectangle's area that is shaded. Express your answer as a simplified fraction. | \frac{1}{4} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
The scientific notation for $2.3\mu m$ in meters is to be determined, given that $1\mu m = 0.000001m$. Express your answer in scientific notation (e.g., $2.3 \times 10^{-6}m$). | 2.3 \times 10^{-6}m | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Point M is on the circle (x-5)^{2}+(y-3)^{2}=9. Calculate the shortest distance from point M to the line 3x+4y-2=0. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
In the spring of 2008, a middle school in Liuzhou organized a fundraising activity among the students. On the first day, only 10 people donated, with an average donation of 10 yuan per person. After active promotion, starting from the second day, the number of donors doubled each day, and the average donation per perso... | 8000 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Determine the number of real solutions to the equation f(x) = f(-x), given that the function f(x) satisfies the equation f(x) + 2f(1/x) = 3x for all non-zero real numbers x. | 0 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets $M=\{x|x^{2}-\frac{x}{2}>0\}$ and $N=\{x|\log_{10} x\leqslant 0\}$, determine the intersection of sets M and N, $M\cap N$. Express your answer as an interval, using interval notation. | (\frac{1}{2},1] | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Express the number 274,000,000 in scientific notation. Your answer should be in the form a × 10^b, where a is a number between 1 and 10 (not including 10) and b is an integer. | 2.74 \times 10^{8} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Find the monotonic decreasing interval of the function $f(x) = \lg(-x^2 + x + 6)$. Express your answer as an interval in standard notation. | (\frac{1}{2}, 3) | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Find the slope angle of the tangent to the curve y = (1/3)x^3 - 2 at the point (1, -5/3). Express your answer in radians, using pi for the mathematical constant. | \frac{\pi}{4} | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given sets $A=\{x|x>0\}$ and $B=\{-2,-1,1,2\}$, calculate the intersection of the complement of $A$ in the real numbers and $B$. Express your answer as a set of numbers, including all elements of the intersection. | \{-2,-1\} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
If circle C is tangent to both lines x-y=0 and x-y-4=0, and the center of the circle lies on the line x+y=0, then find the equation of circle C in standard form. | (x - 1)^2 + (y + 1)^2 = 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Let the set A consist of positive integers that include only the prime factors 2, 3, or 5. If the sum of the reciprocals of all elements in set A can be expressed as m/n, where m, n are coprime positive integers, find the value of m + n. | 19 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Let $a=\log_{3}2$, $b=\ln3$, $c=\log_{2}3$. Determine the correct order of the values from greatest to least. Express your answer as an ordered list of variables (e.g., c > b > a). | c > b > a | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Find the area of the figure enclosed by the curve $y=x^2$ and the curve $y^2=x$. Express your answer as a single number. | \frac{1}{3} | big_math | [
"Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Given the linear regression equation y = -0.7x + a, and the information that the mean of x is 1 and the mean of y is 0.3, predict the value of y when x = 2. Express your answer as a single number. | -0.4 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Sally picks four consecutive positive integers. She divides each integer by four, and then adds the remainders together. What is the sum of the remainders? Express your answer as a single integer. | 6 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a>0$, find the range of values for the real number $a$ such that the minimum value of the function $f(x) = e^{x-a} - \ln(x+a) - 1$ $(x>0)$ is 0. Express your answer as a set or interval of real numbers. | \{\frac{1}{2}\} | big_math | [
"Mathematics -> Calculus -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of \\(\\tan 15^{\\circ} + \\cot 15^{\\circ}\\). Express your answer as a single number. | 4 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Given the following ratios:
- The weight of 1 pig is equal to the weight of 6 chickens,
- The weight of 3 chickens is equal to the weight of 4 ducks,
- The weight of 2 ducks is equal to the weight of 6 fish.
Calculate how many fish are equal in weight to 2 pigs. Provide your answer as a whole number. | 48 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Let $P_1 P_2 P_3 P_4 P_5 P_6$ be a regular hexagon with edge-centered distance (distance from the center to the midpoint of a side) of 2. If $Q_i$ is the midpoint of the side $P_i P_{i+1}$ for $i=1,2,3,4$, calculate the area of quadrilateral $Q_1 Q_2 Q_3 Q_4$. Express your answer as a product of a number and a square r... | 3\sqrt{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Polygons"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
A number, when divided by 10, leaves a remainder of 9; when divided by 9, leaves a remainder of 8; when divided by 8, leaves a remainder of 7; and so on until, when divided by 2, leaves a remainder of 1. Calculate this number. Express your answer as a single integer. | 2519 | big_math | [
"Mathematics -> Number Theory -> Congruences"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\angle A = 25^{\circ}$, find the measure of its complement angle. Express your answer in degrees as a whole number. | 65^{\circ} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
There are four statements on a card, and only these four:
There is exactly one false statement on this card.
There are exactly two false statements on this card.
There are exactly three false statements on this card.
There are exactly four false statements on this card.
(Assume each statement on the card is either true... | 3 | big_math | [
"Mathematics -> Discrete Mathematics -> Logic"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
In the right-angled, isosceles triangle FGH, FH = sqrt(8). Arc FH is part of the circumference of a circle with center G and radius GH. Calculate the area of the shaded region. Express your answer in terms of pi. | \pi - 2 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
In a triangle with consecutive integer side lengths where the largest angle is twice the smallest angle, calculate the cosine of the smallest angle. Express your answer as a fraction. | \frac{3}{4} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given a geometric sequence {a_n} with a common ratio of 2, where all terms are positive numbers, and a_3 * a_11 = 16, calculate the value of log_2 a_10. Express your answer as a single numerical value. | 5 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Judy had 35 hits during the softball season, consisting of 1 home run, 1 triple, and 5 doubles, with the rest being singles. Calculate the percentage of her hits that were singles. Express your answer as a percentage, including the percentage sign (%). | 80% | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the hyperbola that passes through the point (2,-2) and has the same asymptotes as x^2/2 - y^2 = 1. Express your answer in the standard form of a hyperbola equation. | \frac{y^2}{2} - \frac{x^2}{4} = 1 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Two circles \( C_1 \) and \( C_2 \) are given by \( C_1: x^2 + y^2 = a^2 \) and \( C_2: (x-b)^2 + (y-c)^2 = a^2 \) respectively. If they are tangent to each other, calculate the value of \( \frac{b^2+c^2}{a^2} \). Express your answer as a single number. | 4 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
For the quadratic function y = mx^2 - 4x + 1, find the value of m given that the minimum value of the function is -3. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Algebra -> Algebra -> Quadratic Functions"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Given a right triangle with side lengths of 3 cm and 4 cm, find the length of the median to the hypotenuse. Express your answer in centimeters (cm) as a decimal number. | 2.5 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Given a plane that intersects a sphere at a distance of 1 from the center of the sphere, the area of the resulting circular section is $2\pi$. Find the volume of the sphere. Express your answer as a multiple of $\pi$. | 4\sqrt{3}\pi | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given an ellipse x^2/a^2 + y^2/b^2 = 1 (a>b>0) with left and right foci F1 and F2 respectively. Point A is an endpoint of the minor axis of the ellipse, and cos(∠F1AF2) = 5/6. Calculate the eccentricity of the ellipse. Express your answer as a fraction or decimal. | \frac{\sqrt{3}}{6} | big_math | [
"Mathematics -> Geometry -> Differential Geometry -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
In the geometric sequence $\left\{ a_n \right\}$, $a_2=9$, $a_5=243$. Find the sum of the first 4 terms of $\left\{ a_n \right\}$. Express your answer as a single numerical value. | 120 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
The first three terms of an arithmetic sequence {a_n} are a-1, a+1, and a+3. Using the common difference calculated from these terms, find the general formula for the sequence. Express your answer in the format a_n =..., including the variables 'a' and 'n'. | a_n = a + 2n - 3 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Given sets $A=\{-1,0,1,2,3\}$ and $B=\{-3,-1,1,3,5\}$, find the intersection $A \cap B$. Express your answer as a set, including all elements that are present in both sets A and B. | \{-1, 1, 3\} | big_math | [
"Mathematics -> Discrete Mathematics -> Logic"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Nine balls, numbered 1 through 9, are placed in a bag. Person A draws a ball from the bag, which has the number a. After placing it back, person B draws another ball from the bag, which has the number b. Find the probability that the inequality a - 2b + 10 > 0 holds. Express your answer as a simplified fraction. | \frac{61}{81} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the line that passes through the point (2, -1) and is parallel to the line 3x + y - 2 = 0. Express your answer in the standard form Ax + By + C = 0. | 3x + y - 5 = 0 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Veronica put on five rings: one on her little finger, one on her middle finger, and three on her ring finger. The three rings on her ring finger must be taken off in a fixed order. In how many different orders can she take off all five rings? Express your answer as a single integer. | 20 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $3^{a}=4$, calculate $\log_{2}3$ in terms of $a$. Express your answer as a single expression. | \frac{2}{a} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Let $a$ be a real number in $\mathbb{R}$ and $(a-i) \cdot i$ (where $i$ is the imaginary unit) be a positive real number. Find the value of $a$. Express your answer as a single real number. | 0 | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
The graph of the function y = ax + b passes through the first, second, and third quadrants, and intersects the x-axis at the point (-2, 0). Find the solution set for the inequality ax > b, expressing your answer as an inequality in terms of x. | x > 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression: $\sqrt {(π-4)^{2}}+π$. Express your answer as a single numerical value. | 4 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
During the "National Day" holiday, a class is planning to organize an outing. The class monitor conducted a public opinion survey on several outing spots provided by the students. What statistical measure would be most relevant for identifying the most preferred outing spot from the data collected? | \text{Mode} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given a function \( y = f(x) \) that satisfies \( f(3 + x) = f(3 - x) \) for all real numbers \( x \), and the equation \( f(x) = 0 \) has exactly 6 distinct real roots, find the sum of these 6 roots. Express your answer as a single number. | 18 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the sequence $\{a_n\}$ is an arithmetic sequence and satisfies $\overrightarrow{BA}=a_{3} \overrightarrow{OB}+a_{2015} \overrightarrow{OC}$, if $\overrightarrow{AB}=λ \overrightarrow{AC}(λ∈R)$, and point $O$ is a point outside line $BC$, calculate the value of $a_{1}+a_{2017}$. Express your answer as a singl... | 0 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function y = 2/x, calculate the increment of the function when x changes from 2 to 1.5. Express your answer as a simplified fraction. | 1/3 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Student A must participate in competitions of three different subjects, along with two other students selected from students B, C, and D. Calculate the total number of different participation schemes possible, considering the arrangements of the three students in the three subjects. Express your answer as a single nume... | 18 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Permutations"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
$\triangle ABC$ and $\triangle DEF$ have a similarity ratio of $1:4$. Find the ratio of the perimeters of $\triangle ABC$ and $\triangle DEF$. Express your answer as a ratio. | 1:4 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Similarity"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
In rectangle \( ABCD \), point \( F \) is the midpoint of side \( CD \), and the length of side \( BC \) is three times the length of \( BE \). Calculate the ratio of the area of the rectangle to the area of the shaded portion. Express your answer as a single numerical value. | 3 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Find the range of the function \( f(x) = \arctan(x) + \frac{1}{2} \arcsin(x) \) over its domain \( [-1, 1] \). Express your answer as an interval. | [-\frac{\pi}{2}, \frac{\pi}{2}] | big_math | [
"Mathematics -> Calculus -> Other",
"Mathematics -> Precalculus -> Functions"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
The Gaussian Guardians intramural basketball team had the following points scored by each player: Daniel (7 points), Curtis (8 points), Sid (2 points), Emily (11 points), Kalyn (6 points), Hyojeong (12 points), Ty (1 point), and Winston (7 points). Find the total number of points scored by the team. Express your answer... | 54 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Find the smallest three-digit number N that is the sum of three different two-digit palindromic numbers, where N itself is not a palindromic number. Calculate the sum of all digits of N. Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Given the system of equations:
1. x + y + z = 25
2. x + y = 19
3. y + z = 18
Find the value of y. Express your answer as a single integer. | 12 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given the equations $\triangle + \triangle = ※$, $○ = □ + □$, and $\triangle = ○ + ○ + ○ + ○$, calculate the value of $※ ÷ □$. Express your answer as a single integer. | 16 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Given a sequence $\{a_n\}$ satisfying $a_1=1$ and
$$
a_{n+1}= \begin{cases} 2a_{n}, & \text{if } n \text{ is an odd positive integer} \\
a_{n}+1, & \text{if } n \text{ is an even positive integer} \end{cases},
$$
determine the term number in the sequence that has a value of 254. Express your answer as a single integer. | 14 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
A TV station is broadcasting 5 different advertisements, among which there are 3 different commercial advertisements and 2 different Olympic promotional advertisements. It is required that the last advertisement must be an Olympic promotional advertisement, and the two Olympic promotional advertisements cannot be broad... | 36 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Find the imaginary part of the complex number $z=(-1+i)^2$. Express your answer as a single number. | -2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given the set $M = \{x \in \mathbb{N}^* \mid x(x-3) < 0\}$, find the total number of subsets of set $M$. Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the perimeter of a triangle is 15 cm, and each of two sides is equal to twice the length of the third side, find the length of the shortest side. Express your answer in centimeters as a whole number. | 3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
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