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 |
|---|---|---|---|---|---|
In triangle $\triangle ABC$, if ${a^2}+{b^2}={c^2}+\sqrt{3}ab$, then find the measure of angle $C$ in degrees. Express your answer as a whole number. | 30 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Let \( a \) and \( b \) be positive constants not equal to 1. Find the number of real values of \( x \) that satisfy the equation \( (\log _{a} x)(\log _{b} x) = \log _{a} b \). Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Suppose hops, skips and jumps are specific units of length. If $b$ hops equals $c$ skips, $d$ jumps equals $e$ hops, and $f$ jumps equals $g$ meters, then express the number of skips equivalent to one meter in terms of the variables $b$, $c$, $d$, $e$, $f$, and $g$. Provide your answer as a fraction or ratio. | \frac{cef}{bdg} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
In the Gauss 2004 Olympics, there are six competitors and eight events. The top three competitors in each event receive gold, silver, and bronze medals respectively. Each competitor scores 5 points for each gold medal, 3 points for each silver medal, and 1 point for each bronze medal. If one of the competitors had a to... | 4 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Determine the number of real solutions for the system of equations:
\[\left\{\begin{array}{l}x + y = 2, \\ xy - z^{2} = 1.\end{array}\right.\]
Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
A box contains 6 white balls and 5 black balls. If 3 balls are drawn at once and it is known that their colors are the same, what is the probability that these balls are white? Express your answer as a simplified fraction. | \frac{2}{3} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Arrange the letters a, a, b, b, c, c into three rows and two columns, with the requirement that each row has different letters and each column also has different letters. Calculate the total number of different arrangements. | 12 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
In a circle with a radius of 2cm, find the central angle in radians of a sector with an area of 4cm^2. Express your answer as a whole number. | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
How many lattice points (x, y) satisfy the inequality (|x|-1)^2 + (|y|-1)^2 < 2 in a rectangular coordinate system? Express your answer as a single integer. | 16 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
From a fish pond, 1200 fish were caught, marked, and then released back into the pond. After a suitable period of time, 1000 fish were caught again, among which 100 were marked. Estimate the total number of fish in the pond. Express your answer as a whole number. | 12000 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Given that \( n \) is a two-digit natural number, when \( n \) is divided by 9, the remainder is 1, and when \( n \) is divided by 10, the remainder is 3. Calculate the remainder when \( n \) is divided by 11. Express your answer as a single integer. | 7 | big_math | [
"Mathematics -> Number Theory -> Congruences"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the mean (average) of the numbers 6, 8, 9, 11, and 16. Express your answer as a single number. | 10 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the solution set of the inequality x^2 + x - c < 0 is (-2, 1), find the value of c. Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Quadratic Functions"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
The random variable $X$ follows a normal distribution $N\left( 3,{{\sigma }^{2}} \right)$, and $P\left( X\leqslant 4 \right)=0.84$. Calculate the probability $P(2 < X < 4)$. Express your answer as a decimal value between 0 and 1, rounded to 2 decimal places. | 0.68 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given a rectangle with length a and width b, its area is 6, and its perimeter is 12. Calculate the value of a^2 + b^2. | 24 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Given the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ (where $a > 0$, $b > 0$), the left and right vertices are A($-a, 0$) and B($a, 0$) respectively. The equations of the asymptotes are $l_1: y = \frac{b}{a}x$ and $l_2: y = -\frac{b}{a}x$. Point P lies on $l_1$ and has coordinates $(m, \frac{b}{a}m)$. If $PA$ is... | 2 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given $a > 0$ and $b > 0$ with the condition that $\frac{1}{a} + \frac{2}{b} = 1$, calculate the minimum value of $a + 2b$. Express your answer as a single number. | 9 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence $\{a_n\}$, the sum of the first four terms is $20$, the sum of the last four terms is $60$, and the sum of the first $n$ terms is $100$. Find the value of $n$. Express your answer as a whole number. | 10 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a>1$, $b>1$, and $\lg(a+b) = \lg a + \lg b$, calculate the value of $\lg(a-1) + \lg(b-1)$. Express your answer as a single numerical value. | 0 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Find the maximum value of the function f(x) = (-2x^2 + x - 3)/x, given that x > 0. Express your answer as a single numerical value. | 1 - 2\sqrt{6} | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\cos (\alpha+\beta)=-1$ and $\tan \alpha=2$, find the value of $\tan \beta$. Express your answer as a single value. | -2 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given a sequence $\{a_n\}$ that satisfies $a_{n+1}-a_n=2$ and $a_1=-5$, find the value of $\left| a_1 \right| + \left| a_2 \right| + \cdots + \left| a_6 \right|$. Express your answer as a single integer. | 18 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the line symmetric to the line $x+y+1=0$ with respect to the point $(1,2)$. Express your answer in the standard form of a line equation (Ax + By + C = 0). | x+y-7=0 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Let $f$ be the function defined by $f(x)=ax^2-\sqrt{2}$ for some positive $a$. If $f(f(\sqrt{2}))=-\sqrt{2}$, calculate the value of $a$. Express your answer as a single value. | \frac{\sqrt{2}}{2} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
When $b=1$, the equation $a(3x-2)+b(2x-3)=8x-7$ has infinitely many solutions. Substitute $b=1$ into the equation and solve for $a$. Express your answer as a single numerical value. | 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the random variable ξ follows a normal distribution N(1, σ^2), and given that P(ξ > 2) = 0.16, find the probability P(0 < ξ < 1). Express your answer as a decimal value between 0 and 1, rounded to 2 decimal places. | 0.34 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Given two arithmetic sequences {a_n} and {b_n} with the sum of their first n terms being A_n and B_n respectively, and A_n/B_n = (7n+45)/(n+3), find the value of a_7/b_7. Express your answer as a simplified fraction. | \frac{17}{2} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given $\cos (\pi-\alpha)= \frac {4}{5}$, and $\alpha$ is an angle in the third quadrant, calculate the value of $\tan 2\alpha$. Express your answer as a single fraction or decimal, including the negative sign if applicable. | \frac{24}{7} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence {a_n}, its sum of the first n terms is S_n. It is known that a_2 = 2, S_5 = 15, and b_n = 1 / (a_{n+1}^2 - 1). Find the sum of the first 10 terms of the sequence {b_n}. Express your answer as a single fraction in simplest form. | = 175/264 | big_math | [
"Mathematics -> Algebra -> Sequences and Series",
"Mathematics -> Algebra -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
The volume of a cube is given as 8 cm³. What is the length of a side of the cube? Express your answer as a whole number. | 2 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Volume"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
A pyramid has a square base with an area of 1440 cm². Each of the pyramid's triangular faces is identical and has an area of 840 cm². Calculate the height of the pyramid in centimeters. | 40 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation x + y + z = 1, find the minimum value of the expression x^2 + y^2 + z^2. Express your answer as a fraction. | \frac{1}{3} | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Eight distinct points are placed on a circle. Each pair of points is joined with a line segment. How many line segments would there be in total? Express your answer as a whole number. | 28 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $(2-x)^5 = a_0 + a_1x + a_2x^2 + \ldots + a_5x^5$, calculate the value of $\frac {a_{0}+a_{2}+a_{4}}{a_{1}+a_{3}+a_{5}}$. Express your answer as a simplified fraction. | -\frac{122}{121} | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Determine the solution set for the inequality $3x^2 - x + 2 < 0$. Express your answer using proper set notation. | \emptyset | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the line that passes through the center of the circle $(x-2)^2+y^2=1$ and is parallel to the line $2x-y+1=0$. Express your answer in the standard form of a line equation (Ax + By + C = 0). | 2x-y-4=0 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the complex conjugate of a complex number z is \overline{z}, and it satisfies the equation \overline{z}(2+3i) = (2-i)^2, where i is the imaginary unit, find the imaginary part of the complex number z. Express your answer as a fraction. | \frac{17}{13} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
A rectangular solid has one vertex where the three edge lengths are 3, 4, and 5, respectively, and all 8 vertices of it lie on the same spherical surface. Calculate the surface area of this sphere. Express your answer in terms of π. | 50\pi | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Surface Area"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Given positive numbers x and y that satisfy 2x + y - 1 = 0, find the minimum value of the expression (x + 2y) / xy. Express your answer as a single number. | 9 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, a semi-circle has diameter \(XY\). Rectangle \(PQRS\) is inscribed in the semi-circle with \(PQ = 12\) and \(QR = 28\). Square \(STUV\) has \(T\) on \(RS\), \(U\) on the semi-circle, and \(V\) on \(XY\). Calculate the area of square \(STUV\). Provide your answer as a whole number. | 16 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Solve the equation (x-1)(x-2) = 0 for its roots. Express your answer in the format x1 =, x2 =. | x1 = 1, x2 = 2 | big_math | [
"Mathematics -> Algebra -> Algebra -> Equations and Inequalities"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the sum of the first five terms of an arithmetic sequence {a$_n$} is 15, and a$_6$ = 6, find the value of a$_{2019}$. Express your answer as a single numerical value. | 2019 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $i$ is the imaginary unit, the complex number $\frac {a+2i}{1+i}$ is a real number. Find the value of the real number $a$. Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $\square \times 3a = -3a^{2}b$, solve for the algebraic expression that fits in the "$\square$". Express your answer as a simplified algebraic expression. | -ab | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Using 100 boxes to pack cups, with each box containing a different number of cups and none of the boxes being empty, calculate the minimum number of cups required. Express your answer as a single integer. | 5050 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
During a holiday, an office at a school needs to arrange a duty roster from the 1st to the 6th, with six designated people participating. Each person is required to be on duty for one day, but person A and person B cannot be adjacent, and person C and person D also cannot be adjacent. Calculate the number of different ... | 336 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
If four distinct positive real numbers $a$, $b$, $c$, $d$ satisfy $(a^{2012} - c^{2012})(a^{2012} - d^{2012}) = 2012$ and $(b^{2012} - c^{2012})(b^{2012} - d^{2012}) = 2012$, then calculate the value of $(ab)^{2012} - (cd)^{2012}$. Express your answer as an integer. | -2012 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given that the sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$ is $S_n$, and $\frac {S_{5}}{5} - \frac {S_{2}}{2} = 3$, find the common difference $d$ of the sequence. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of solutions (x, y) to the equation 3x + y = 100, where x and y are positive integers. Express your answer as a single integer. | 33 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Determine the number of real roots for the equation $x + \sqrt{x-2} = 4$. Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Evaluate the expression \( 4^1 + 3^2 - 2^3 + 1^4 \). Provide your answer as a single number. | 6 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
The complex number $z$ satisfies the equation $\bar{z} \cdot (1+2i)=4+3i$. Determine the value of $z$ in standard form $a+bi$, including both real and imaginary parts. | 2+i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Given a function $y = f(x)$ defined on $\mathbb{R}$ that is an even function satisfying $f(2 + x) = f(2 - x)$ and has at least one zero in the interval $(0, 4)$ with $f(0) = 0$, determine the minimum number of zeros the function $y = f(x)$ has in the interval $(-8, 10]$. Provide your answer as a single integer. | 9 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given a sequence $\{a_n\}$ that satisfies the recurrence relation $a_n = a_{n-1} + a_{n-2}$ for $n > 2$, $n \in \mathbb{N}^*$, and given $a_{2015} = 1$ and $a_{2017} = -1$, let $S_n$ be the sum of the first $n$ terms of $\{a_n\}$. Calculate the value of $S_{2020} - S_{2016}$. Express your answer as a single integer. | -15 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
A pitcher is filled to three fourths of its capacity with pineapple juice. The juice is then poured into 5 cups in equal amounts. What percentage of the total capacity of the pitcher does each cup receive? Express your answer as a percentage, including the percentage sign (%). | 15% | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
If the complex number z satisfies the equation zi = 1 - i, calculate the value of z. Express your answer as a complex number in the form a + bi. | -1 - i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Determine the range of values for the real number $m$ such that the equation $x^{3}-3x+m=0$ has at least one root within the interval $[0,2]$. Express your answer as an interval or a combination of intervals that describe the possible values of $m$. | [-2,2] | big_math | [
"Mathematics -> Calculus -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given that $12^x = 18^y = 6^{xy}$ and x, y are non-zero real numbers, find the sum of x and y. Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Algebra -> Exponential Functions"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Person A and Person B practice shooting, with the probabilities of hitting the target being $\dfrac{1}{2}$ and $\dfrac{1}{3}$, respectively. If A and B each shoot once, what is the probability that the target is hit? Express your answer as a simplified fraction. | \dfrac{2}{3} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Given the quadratic equation x^2 - 2x - 4 = 0, calculate the value of (x1 + x2) / (x1 * x2), where x1 and x2 are the roots of the equation. Express your answer as a single fraction or integer. | -\frac{1}{2} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $a=(2,1)$, $b=(1,m)$, $c=(2,4)$, and the relationship $(2a-5b) \perp c$, calculate the value of $m$. Express your answer as a fraction or decimal. | \dfrac{3}{10} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Last summer, 30% of the birds living on Town Lake were geese, 25% were swans, 10% were herons, and 35% were ducks. What percent of the birds that were not swans were geese? Express your answer as a percentage. | 40 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
The ancient Chinese mathematical text "Zhang Qiu Jian Suan Jing" records: "There is a horse whose pace slows down, covering half the distance each subsequent day. After seven days, the horse has travelled a total of 700 li." How far did the horse travel on the 7th day? Express your answer as a fraction in li. | \dfrac{700}{127} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Among the rectangles inscribed in a circle with a diameter of 4, find the largest possible area. Express your answer as a single number. | 8 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Point \( R \) is the midpoint of the line segment \( P Q \) and \( S \) is the midpoint of the line segment \( Q R \). If \( P \) has coordinates \( (2, 1) \) and \( S \) has coordinates \( (14, 7) \), then find the coordinates of \( Q \). Express your answer as a coordinate pair \( (x, y) \) with whole numbers. | (18,9) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Ali chooses one of the stones from a group of 2005 stones, marks this stone in a way that Betül cannot see the mark, and shuffles the stones. At each move, Betül divides stones into three non-empty groups. Ali removes the group with more stones from the two groups that do not contain the marked stone (if these two grou... | 11 | big_math | [
"Mathematics -> Discrete Mathematics -> Algorithms"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
A certain hospital has a cure rate of $\frac{1}{5}$ for a specific disease. If the first four patients were not cured, then what is the cure rate for the fifth patient? Express your answer as a simplified fraction. | \frac{1}{5} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence {a_n}, where S_n is the sum of its first n terms, and a_2 = 3a_4 - 6, find the value of S_9. Express your answer as a single numerical value. | 27 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Sequences and Series"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
When two dice are rolled simultaneously, what is the probability that the sum of the numbers facing up is 9? Express your answer as a simplified fraction in the form $\boxed{\frac{a}{b}}$. | \frac{1}{9} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the line l with a slope of α is perpendicular to the line x+2y-λ=0, find the value of tan(π + α). | 2 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
The polynomial $x^{2}+x-6$ can be factored into $\left(x+a\right)\left(x+b\right)$, where $a$ and $b$ are both integers. Calculate the value of $\left(a+b\right)^{2023}$. Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations -> Other"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
In an experiment conducted by an experimenter, there are 5 procedures to be implemented in sequence, where Procedure A can only appear in the first or the last step, and Procedures C and D must be adjacent. Calculate the total number of possible sequences of the experiment. | 24 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the result of $(-\frac{1}{2})×(-2)$. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
A person plans to process 408 parts in 15 days. In the first three days, he processes 24 parts each day. Calculate the minimum number of parts he needs to process each day for the remaining days to complete the task ahead of schedule. Express your answer as an integer. | 29 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the inequality x - (m^2 - 2m + 4)y + 6 > 0, which represents one of the two planar regions bounded by the line x - (m^2 - 2m + 4)y + 6 = 0, and the point (1, 1) is within this region, find the range of the real number m that satisfies this condition. Express your answer in interval notation. | (-1, 3) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Let D be a rectangular area with sides of lengths 2 and 4. Let E be the area within D that is below the curve of the function y = x^2. If 30 points are randomly thrown into area D, calculate the approximate number of points that fall within area E. Express your answer as a whole number. | 10 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
The sums of three whole numbers taken in pairs are 12, 17, and 19. Find the middle number. Express your answer as a whole number. | 7 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Given $z=i(2+i)$, find the value of $\bar{z}$ (the complex conjugate of $z$). Express your answer in the form $a + bi$ and include it in the boxed notation. | -1-2i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Given vector $\overrightarrow{a}=(4,0,-2)$ and the relation $\overrightarrow{a}-\overrightarrow{b}=(0,1,-2)$, calculate the vector $\overrightarrow{b}$. Express your answer as a vector in the form $(x, y, z)$. | (4, -1, 0) | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Given the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ (where $a>0$, $b>0$), the left and right foci are denoted as $F_1$ and $F_2$, respectively. Point $P(3,4)$ lies on an asymptote of the hyperbola. If $|\overrightarrow{PF_1} + \overrightarrow{PF_2}| = |\overrightarrow{F_1F_2}|$, find the equation of the hyperbo... | \frac{x^2}{9} - \frac{y^2}{16} = 1 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
A rhombus has a side length of 5. One of its diagonals is not longer than 6, and the other is not shorter than 6. Find the maximum value of the sum of the lengths of the two diagonals. Express your answer as a single number. | 14 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Each of p, q, r, s, t, u, v is to be replaced with 1, 2, or 3 so that p, q, and r are all different, q, s, and t are all different, and r, u, and v are all different. What is the maximum possible value of the sum s + t + u + v? Express your answer as a single numerical value. | 9 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Given sets A = {1, 2, 3,..., 10} and B = {1, 2, 3, 4}, where C is a subset of A such that C ∩ B ≠ ∅, calculate the number of subsets C that satisfy this condition. | 960 | big_math | [
"Mathematics -> Applied Mathematics -> Combinatorics"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Given an isosceles triangle with two sides measuring $3cm$ and $7cm$, calculate the perimeter of the triangle. Express your answer in centimeters (cm) as a whole number. | 17 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Polygons"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the number of positive integers less than 1000 that are divisible by neither 5 nor 7. Provide your answer as a single integer. | 686 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems",
"Mathematics -> Statistics -> Probability -> Counting Methods -> Other"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $$\overrightarrow {a}$$=(2,3) and $$\overrightarrow {b}$$=(-1,2), find the value of μ such that $$μ \overrightarrow {a}+ \overrightarrow {b}$$ is perpendicular to $$\overrightarrow {a}- \overrightarrow {b}$$. Provide your answer as a single value. | \frac{1}{9} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Given the circles defined by the equations $x^{2}+y^{2}-4x=0$ and $(x-a)^{2}+(y+3)^{2}=9$, find the range of values for $a$ such that the circles have exactly two common tangents. Express your answer as an interval. | (-2,6) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
A survey of 200 students was conducted to determine their favorite cafeteria food. The results showed that 25% of students chose cookies, 30% chose pizza, and 35% chose pasta. Calculate the number of students who chose sandwiches as their favorite food. Express your answer as a whole number. | 20 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the proposition 'There exists $x_0 \in \mathbb{R}$, such that $x_0^2 - 3mx_0 + 9 < 0$' is false, find the range of the real number $m$. Express your answer as an interval in the form $[a, b]$. | [-2, 2] | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
From 3 o'clock to 6 o'clock, the hour hand on a clock rotates a certain number of degrees. Calculate the degree of rotation of the hour hand during this time interval. Express your answer in degrees (°) as a whole number. | 90 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the statement "There exists an $x_0\in\mathbb{R}$ such that $x_0^2 + ax_0 - 4a < 0$" is a false proposition, find the range of values for the real number $a$ that satisfy the inequality $x^2 + ax - 4a \geq 0$ for all $x\in\mathbb{R}$. Express your answer as an interval. | [-16, 0] | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given $\\tan (\\alpha+ \\frac {\\pi}{4})=3$, use the tangent addition formula to find the value of $\\tan \\alpha$. Express your answer as a single value. | \\frac{1}{2} | big_math | [
"Mathematics -> Trigonometry -> Trigonometric Functions -> Other"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given the circle M: $x^2 + y^2 - 2x + ay = 0$ (where $a > 0$) has chords of equal length when intersected with the x-axis and y-axis, find the length of the chord intercepted by the circle M on the line $x + y = 0$. Express your answer as a numerical value. | 2\sqrt{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
In the Cartesian coordinate system, triangle ABC has vertices A(m, 1, 9), B(10, -1, 6), and C(2, 4, 3) and is an isosceles triangle. If m is an integer, find the value of m. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
A right triangle has one leg with a length of 11, and the lengths of the other two sides are also natural numbers. Calculate the perimeter of the triangle. Express your answer as a single integer. | 132 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
There are 120,000 red squirrels living in Scotland. This represents 75% of their total UK population. Calculate the difference in the number of red squirrels living in Scotland and the rest of the UK. Express your answer as a whole number. | 80000 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
A \(2 \times 2 \times 2\) cube has three faces that are partially shaded. Each shaded face is half-shaded, as shown in the diagram. Calculate the fraction of the total surface area of the cube that is shaded. Express your answer as a simplified fraction. | \frac{1}{4} | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Surface Area"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Given the ellipse $C$: $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$, $(a > b > 0)$, whose left and right vertices are $A_1$ and $A_2$ respectively, and the circle with diameter $A_1A_2$ is tangent to the line $bx-ay+2ab=0$, calculate the eccentricity of the ellipse $C$. Express your answer as a fraction or decimal. | \dfrac{\sqrt{6}}{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Given the inequality $kx^{2}-6kx+k+8\geqslant 0$ holds for any $x \in \mathbb{R}$, find the range of $k$. Express your answer as an interval. | [0, 1] | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence {a_n}, the sum of the first n terms is denoted as S_n. If S_{10} = 10 and S_{20} = 40, then find the value of S_{30}. Express your answer as a single numerical value. | 90 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
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