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 |
|---|---|---|---|---|---|
Tarek wants to shade two further squares on the given diagram so that the resulting pattern has a single axis of symmetry. Calculate the total number of different ways he can complete his pattern to achieve this symmetry. Express your answer as a whole number. | 6 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
A taxi charges a fare of 10 yuan for a distance within 4 kilometers (including 4 kilometers), and an additional 1.5 yuan for each kilometer beyond 4 kilometers. If a passenger paid 16 yuan for a taxi ride, how many kilometers did the passenger travel? Express your answer in kilometers, rounded to the nearest whole numb... | 8 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
The temperature of a high mountain decreases by 0.7 degrees Celsius for every 100 meters increase in elevation from the base. If the temperature at the summit is 14.1 degrees Celsius and the temperature at the base is 26 degrees Celsius, find the relative height of the mountain. Express your answer in meters as a whole... | 1700 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
The Sunshine Company sells an electronic product with a cost price of 21 yuan. If the product is sold at 90% of its marked price and still yields a 20% profit, then what is the marked price of this electronic product? Express your answer in yuan as a whole number. | 28 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Lakshmi is travelling at 50 km/h. How many kilometres does she travel in 30 minutes? Express your answer in kilometres. | 25 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
In a line of 5 students consisting of 2 boys and 3 girls, if the two boys must stand next to each other, calculate the number of different ways they can be arranged. Express your answer as a whole number. | 48 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the value of the expression $x^{2}+3x+5$ is $7$, calculate the value of the expression $3x^{2}+9x-2$. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
The ages of the 16 male soccer team members of a school are as follows: 17, 17, 18, 18, 16, 18, 17, 15, 18, 18, 17, 16, 18, 17, 18, 14. Calculate the mode of these team members' ages. Express your answer as a whole number, including the unit 'years old'. | 18 \text{ years old} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Given that \( a^{100} \) is a 120-digit number and \( a^{b} \) is a 10-digit number, where \( b \in \mathbb{N} \), find the value of \( b \). Express your answer as a single integer. | 8 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Given a sequence {a_n} where a_1 = 2, and for all positive integers n, it always holds that a_n + a_{n+1} = 2n. Calculate the value of a_{10}. Express your answer as a single integer. | 8 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
In an increasing geometric sequence $\{a_n\}$, the sum of the first 3 terms is 13 and the product of the first 3 terms is 27. Calculate the sum of the first 5 terms. Express your answer as a single numerical value. | 121 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Determine the interval on which the function $f(x)=x^{2}-4\ln (x+1)$ is monotonically decreasing. Express your answer as an interval in the format (a, b) or [a, b] or a combination of these. | (-1,1) | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = 2sin(ωx + π/3) (ω > 0), the graph is shifted to the left by π/6 units. The shifted graph passes through the point (π/2, 1). Determine the minimum value of ω. Express your answer as a single numerical value. | 3/4 | big_math | [
"Mathematics -> Precalculus -> Functions",
"Mathematics -> Calculus -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
In the geometric sequence {a_n}, the sum of the first n terms is given by S_n = 5^(n+1) + a. Find the value of a. Express your answer as a single integer. | -5 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the complex number expression: $\frac{4-2i}{(1+i)^{2}}$. Express your answer as a complex number in the form a+bi, using the boxed notation. | -1-2i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Solve the inequality $3 \leq |5-2x| < 9$ and express the solution set in interval notation. | (-2,1] \cup [4,7) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(2,1)$ and $\overrightarrow{b}=(0,-1)$, calculate the resulting vector $2\overrightarrow{b}+3\overrightarrow{a}$. Express your answer as an ordered pair (x, y). | (6,1) | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
In a geometric sequence {a_n}, if a_1 = 3 and a_5 = 75, then find the value of a_3. | 15 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Let the function \( y = f(x) \) satisfy the following condition for all real numbers \( x \):
\[ f(3 + x) = f(3 - x) \]
Suppose the equation \( f(x) = 0 \) has exactly 6 different real roots. Find the sum of these 6 real roots. Express your answer as a single numerical value. | 18 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Given that ω > 0, the graph of the function f(x) = cos(ωx) is translated to the right by π/2 units to obtain the graph of the function g(x) = sin(ωx - π/4). Find the minimum value of ω. | ω = 3/2 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given points A(1,2,1), B(4,11,4), and D(1,2,1), if point P satisfies \u03c4AP=2\u03c4PB, calculate the magnitude of \u03c4PD. Express your answer as a numerical value, including the square root symbol if necessary. | 2\sqrt{11} | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
The geometric sequence $\{a_n\}$ is an increasing sequence, and the product of its first $n$ terms is $T_n$ ($n\in\mathbb{N}^*$). Given that $T_{13} = 4T_{9}$, find the product of $a_{8}$ and $a_{15}$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence {a_n} with a_1 = 13, and the sum of the first 3 terms equal to the sum of the first 11 terms (S_3 = S_{11}), find the value of n when the sum of the first n terms (S_n) is at its maximum. Express your answer as a whole number. | 7 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence $\{a_n\}$ satisfies $a_1=1$ and $a_{n-1}=2a_n$ $(n\geqslant 2,n\in \mathbb{N}^{*})$, find the sum of the first 6 terms of the sequence $\{a_n\}$. Express your answer as a fraction. | \frac{63}{32} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $\frac{5}{x+1}=A-\frac{2x-3}{x+1}$, solve for the value of $A$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
The concentration of a drug in the air is given by the function y(t) = { 10t for 0 <= t <= 0.1, (1/16)^(t-1/10) for t > 0.1 }. Find the minimum time (in hours) after the drug is released when the concentration drops below 0.25 milligrams per cubic meter. Express your answer as a single decimal value. | 0.6 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
From 9 cards labeled with $1, 2, \ldots, 9$, two cards are drawn randomly without replacement, one at a time. Calculate the probability that the numbers on the two drawn cards have different parity (one is odd and the other is even). Express your answer as a simplified fraction. | \dfrac{5}{9} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Other"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
A picture, together with its frame, forms a square with a side length of 80 cm. The frame has a width of 4 cm. Calculate the percentage of the area of the square that is covered by the frame. Express your answer as a percentage, rounded to the nearest whole number. | 19\% | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Given that both sequences {a_n} and {b_n} are arithmetic sequences, if (a_1 + b_1 = 7, a_3 + b_3 = 21), find (a_5 + b_5). Express your answer as a single integer. | 35 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $2x+3y+4z=1$, find the minimum value of $x^2+y^2+z^2$. Express your answer as a fraction. | \frac{1}{29} | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Given the set M = {-1, 0, 1} and N as the set of natural numbers, find the intersection M ∩ N. Express your answer as a set, with elements separated by commas. | {0, 1} | big_math | [
"Mathematics -> Set Theory -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $a^{2}+b^{2}=c^{2}+ \sqrt {3}ab$, use the cosine rule to find the value of angle $C$. Express your answer in radians as a fraction of $\pi$. | \dfrac {\pi}{6} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Find the distance from the focus of the parabola $y^2 = 5x$ to its directrix. Express your answer as a fraction or decimal. | \frac{5}{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Suppose the number of passengers traveling from place A to B daily, X, is a random variable that follows the normal distribution N(800, 50^2). Calculate the probability p0 that the number of passengers from A to B does not exceed 900 on any given day. Express your answer as a decimal value between 0 and 1, rounded to 4... | 0.9772 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(2,1)$ and $\overrightarrow{b}=(-2,4)$, find the magnitude of the vector $\overrightarrow{a}-\overrightarrow{b}$. Express your answer as a single number. | 5 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the perimeter of a sector of a circle is $10$, and the central angle is $3 \text{ rad}$, determine the area of the sector. Express your answer as a single number. | 6 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
A certain service department has $n$ service targets. Each target's need for service is independent, and the probability of each service target requiring service within a day is $p$. Calculate the average number of targets that the department needs to serve per day. Express your answer as a formula in terms of $n$ and ... | np | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Other"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence {a_n}, the sum of the first n terms is S_n. If \overrightarrow{OB} = a_4 \overrightarrow{OA} + a_{2013} \overrightarrow{OC}, and points A, B, C are collinear (with O being a point not on this line), calculate the value of S_{2016}. Express your answer as a single integer. | 1008 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
In a four-level symbol pyramid, the symbols in the two bottom cells determine the cell above them: if they are the same, the cell above is "+", otherwise, it is "-". 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 answer as a single i... | 8 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
In the geometric sequence $\{a_n\}$, if $a_{2010} = 8a_{2007}$, then find the value of the common ratio $q$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
During the period of creating a "National Civilized City", a municipality requires various units to dispatch personnel to street intersections to persuade and educate citizens to cross the road in an orderly manner. The education bureau assigned five workers, A, B, among others, to three different intersections, with a... | 36 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Combinations"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
What is the condition on $x$ for the fraction $\frac{x}{x-1}$ to be meaningful? Express your answer as an inequality or a condition on $x$. | x \neq 1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Three CDs are bought at an average cost of $15 each. If a fourth CD is purchased, the average cost becomes $16. Find the cost of the fourth CD. Express your answer in dollars as a whole number. | 19 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Find the area enclosed by the curve $y=x^2+2$ and the line $5x-y+2=0$. Express your answer as a fraction or decimal. | $\frac{125}{6}$ | big_math | [
"Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
The heights of two trees are in the ratio 3:4. The top of one tree is 16 feet higher than the top of the other tree. Find the height of the taller tree. Express your answer in feet as a whole number. | 64 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence {a_n} where a_1 + a_5 = 6, calculate the sum of the first 5 terms, S_5. Express your answer as a single number. | 15 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Given points A(1, 0), B(-1, $\sqrt{3}$), and O as the origin, point C is in the second quadrant, and $\angle AOC=135°$. Let $\overrightarrow{OC} = -\overrightarrow{OA} + \lambda \overrightarrow{OB}$ ($\lambda \in \mathbb{R}$). Calculate the real number $\lambda$. | \frac{\sqrt{3}+1}{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(-1,2)$ and $\overrightarrow{b}=(2,m)$, if $\overrightarrow{a}\parallel\overrightarrow{b}$, then find the value of $m$. Express your answer as a single integer. | -4 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $\overrightarrow {BC} + \overrightarrow {AB} - \overrightarrow {AC}$. Express your answer as a simplified vector expression. | \overrightarrow {0} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
If the radius of a sphere is $r$, find the maximum lateral surface area of a cylinder inscribed in the sphere. Express your answer in terms of $r$. | 2\pi r^2 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Let point O be located inside a regular tetrahedron ABCD. G is the centroid of the tetrahedron. The line OG intersects the faces of the tetrahedron at points A1, B1, C1, D1. Find the value of (A1O/A1G) + (B1O/B1G) + (C1O/C1G) + (D1O/D1G). Express your answer as a single numerical value. | 4 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $-2 < a < -1$ and $-1 < b < 0$, find the range of $M = a + b$. Express your answer as an inequality representing the range of M. | -3 < M < -1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Given that point P(1+m,3) is in the second quadrant, determine the condition for m that satisfies this geometric requirement. Express your answer as an inequality. | m < -1 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
In the sequence {a_n}, a_1 = 3, and a_{n+1} = 3a_n / (a_n + 3). Find the value of a_4. Express your answer as a fraction. | \frac{3}{4} | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Given sets A={x | -2 < x < 1} and B={x | -1 ≤ x ≤ 2}, find the intersection A∩B. Express your answer in interval notation, using brackets or parentheses as necessary to denote the type of interval. | [-1,1) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
If the line \( y = x - 3 \) is tangent to the curve \( y = e^{x+a} \), find the value of the real number \( a \). Express your answer as a single integer. | -4 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(1,3)$ and $\overrightarrow{b}=(-2,m)$, if $\overrightarrow{a}$ is parallel to $\overrightarrow{a}+2\overrightarrow{b}$, then find the value of $m$. Express your answer as a single integer. | -6 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
The base three representation of $x$ is
\[12112211122211112222\]
Find the first digit of the base nine representation of $x$. Express your answer as a single digit. | 5 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
In square ABCD, the coordinates of A and C are (1,-2) and (4,1), respectively. AB is parallel to the x-axis. Find the coordinates of point B. Express your answer as a coordinate pair (x, y). | (4,-2) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z$ satisfies $(3-z)i=1-3i$, calculate the value of $z$. Express your answer as a complex number in the form $a+bi$. | 6+i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Two individuals, A and B, participate in an exam that consists of a pool of 10 questions. A can answer 6 of these questions correctly, while B can answer 8 correctly. For each test, 3 questions are randomly selected from the pool. The passing criterion is to answer at least 2 questions correctly. Calculate the probabil... | \frac{44}{45} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given that |a-2| and |m+n+3| are opposite numbers, find the value of a+m+n. Express your answer as a single integer. | -1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Two squares have integer side lengths in cm. The difference between their areas is 19 cm^2. Find the sum of their perimeters. Express your answer in cm as a whole number. | 76 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given point $A(-2,1)$ is symmetric to point $B$ with respect to the origin, find the coordinates of point $B$. Express your answer as an ordered pair $(x, y)$. | (2,-1) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
The average and median of seven numbers \(60, 100, x, 40, 50, 200, 90\) are both \(x\). Determine the value of \(x\). Express your answer as a single number. | 90 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given a sequence ${a_n}$ where all terms are positive numbers, $a_1=2$, and $a_{n+1} - a_n = \frac{4}{a_{n+1}+a_n}$, if the sum of the first $n$ terms of the sequence ${\frac{1}{a_{n+1} + a_n}}$ is 5, calculate the value of $n$. Express your answer as a whole number. | 120 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given that \( x, y, z \) are real numbers and \( x^2 + y^2 + z^2 = x + y + z \), calculate the sum of the maximum and minimum values of \( x + y + z \). Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the two foci of an ellipse are F1(0,-4) and F2(0,4), and the shortest distance from a point on the ellipse to F1 is 2, find the equation of this ellipse. Express your answer in standard form. | \frac{x^2}{20} + \frac{y^2}{36} = 1 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Two people, A and B, are working together to type a document. Initially, A types 100 characters per minute, and B types 200 characters per minute. When they have completed half of the document, A's typing speed triples, while B takes a 5-minute break and then continues typing at his original speed. By the time the docu... | 18000 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Translate the equation y = -2x + 3 down by 2 units. What is the resulting equation of the line? Express your answer in slope-intercept form (y = mx + b). | y = -2x + 1 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Other"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $-1 < a < b < 1$ and $2 < c < 3$, find the range of $(a-b)c$. Express your answer as an interval, using parentheses to denote the interval. | (-6, 0) | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Within the range of 0° to 360°, find the angle that has the same terminal side as 750°. Express your answer in degrees (°). | 30° | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions -> Other"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Given the center coordinates (-3, 4) and radius $\sqrt{3}$, write the standard equation for the circle in the form $(x-a)^{2}+(y-b)^{2}=r^{2}$. | $(x+3)^{2}+(y-4)^{2}=3$ | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the equations x/a + y/b + z/c = 1 and a/x + b/y + c/z = 0, find the value of the algebraic expression x^2/a^2 + y^2/b^2 + z^2/c^2. Express your answer as a single numerical value. | 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the terminal side of angle α passes through point P(-3,4), calculate the value of cos α. Express your answer as a simplified fraction. | -\frac{3}{5} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
In an opaque box, there are 2 white ping-pong balls of the same texture and size, and 3 yellow ping-pong balls. What is the probability of randomly drawing a yellow ping-pong ball? Express your answer as a fraction. | \frac{3}{5} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the coordinates of point $A$ are $A(5,4)$, if point $A$ is moved $3$ units down and then $4$ units to the left, what are the coordinates of the new point $A'$? Express your answer as a coordinate pair $(x, y)$. | (1, 1) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, $\triangle QRS$ is an isosceles right-angled triangle with $QR=SR$ and $\angle QRS=90^{\circ}$. Line segment $PT$ intersects $SQ$ at $U$ and $SR$ at $V$. If $\angle PUQ=\angle RVT=y^{\circ}$, calculate the value of $y$ in degrees, rounding to one decimal place if necessary. | 67.5 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
A fish farmer initially catches 50 fish from a pond, marks each of them, and then releases them back into the pond. After that, the farmer catches 200 fish from the pond and finds that only two of them were marked earlier. Assuming that the fish are uniformly distributed in the pond, estimate the total number of fish i... | 5000 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence {a_n}, given a_3 = 16 and a_9 = 80, calculate the value of a_6. Express your answer as a single integer. | 48 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Two numbers a and b with 0 ≤ a ≤ 1 and 0 ≤ b ≤ 1 are chosen at random. The number c is defined by c = 2a + 2b. The numbers a, b, and c are each rounded to the nearest integer to give A, B, and C, respectively. Find the probability that 2A + 2B = C. Express your answer as a fraction in lowest terms. | \frac{7}{16} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given two random variables X and Y, where X follows a binomial distribution with parameters n=8 and p=1/2, and Y follows a normal distribution with mean μ and variance σ^2. If μ = E(X) and P(Y < 0) = 0.2, calculate the probability P(4 ≤ Y ≤ 8). Express your answer as a decimal value between 0 and 1, rounded to 1 decima... | 0.3 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Find the interval of decrease for the function $y = \ln(-x^2 - 2x + 8)$. Express your answer as an interval, e.g., (a, b). | (-1, 2) | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Solve for \(x\) in the equation \(2\left(2^{2x}
ight) = 4^x + 64\). Express your answer as an integer. | 3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of $\sqrt{0.09}$. Provide your answer as a single number. | 0.3 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
In an equilateral triangle $ABC$ with side length $1$, $\\overrightarrow{BD} = \\frac{1}{3} \\overrightarrow{BA}$, and $E$ is the midpoint of $CA$. Calculate the value of $\\overrightarrow{CD} \\cdot \\overrightarrow{BE}$. Express your answer as a single numerical value. | -\\frac{1}{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given the parabola $y^{2}=2px\left(p \gt 0\right)$ with focus $F$, if point $P\left(1,m\right)$ lies on the parabola and $|PF|=3$, find the value of $p$. Express your answer as a single numerical value. | 4 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Given the set $A=\{x\in\mathbb{Z}|\log_{2}k < x < 2\}$, if set $A$ contains at least $3$ elements, find the range of the real number $k$. Express your answer as an interval in standard notation. | \left(0, \frac{1}{2}\right) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
A program is run with the following loop statement:
i = 1
Do
i = i + 2
S = 2i + 3
i = i - 1
Loop While i < 8
What is the final value of S after the loop exits? | 21 | big_math | [
"Mathematics -> Discrete Mathematics -> Algorithms"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
A computer that cost 8100 yuan in the year 2000 decreases in price by $\frac{1}{3}$ every year. Calculate the price of the computer in 2004. Express your answer as a whole number in yuan. | 1600 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\frac{\pi}{3}$, $|\overrightarrow{a}|=4$, and $|\overrightarrow{b}|=1$, calculate $|\overrightarrow{a}-4\overrightarrow{b}|$. Express your answer as a single number. | 4 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $\frac{2x+m}{x-1}=1$, find the range of values for $m$ such that the solution for $x$ is negative. Express your answer as an inequality in terms of $m$. | m > -1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Given the hyperbola $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$ $(a > 0, b > 0)$ with an eccentricity of $\frac{5}{4}$, its foci are $F_1$ and $F_2$. $P$ is a point on the hyperbola such that $\overrightarrow{PF_1} \cdot \overrightarrow{PF_2} = 0$. If the area of $\triangle PF_1F_2$ is equal to $9$, calculate the value of ... | 7 | big_math | [
"Mathematics -> Geometry -> Differential Geometry -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence {a_n}, where a_1 = 2 and a_1 + a_2 + a_3 = 18, find the value of a_4 + a_5 + a_6. Express your answer as a single integer. | 54 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Let $S_n$ be the sum of the first $n$ terms of the sequence $\{a_n\}$, where $a_n = a_{n-1} + 2$ for $n \geq 2$. Given that $S_3 = 9$, find the value of $a_1$. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Given proposition P: ∃x₀ ∈ ℝ, x₀² + 2x₀ + 2 ≤ 0, what is the negation of proposition P? Express your answer as a logical statement using proper notation for universal and existential propositions. | ∀x ∈ ℝ, x² + 2x + 2 > 0 | big_math | [
"Mathematics -> Discrete Mathematics -> Logic"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Find the value of $\sin \frac{\pi}{6}$. Express your answer as a fraction or decimal. | \frac{1}{2} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions -> Other"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
A community has about 3000 people. A stratified random sampling survey of the residents includes 12 young people, 34 middle-aged people, and 14 elderly people. Estimate the number of elderly people in the community based on the proportions from this sample. Express your answer as a whole number. | 700 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
A basketball player made 5 baskets during a game. Each basket was worth either 2 or 3 points. How many different numbers could represent the total points scored by the player? Express your answer as a whole number. | 6 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Find the minimum value of the function y = x^2 + 2/x on the interval (0, +∞). Express your answer as a single numerical value. | 3 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
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