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 the geometric sequence $\{a_n\}$, the sum of the first $n$ terms is $S_n$. Given that $S_3 = 7$ and $S_6 = 63$, find the common ratio $q$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
From the numbers $1, 2, \cdots, 20$, 3 different numbers are randomly selected. Calculate the probability that these 3 numbers form an arithmetic sequence. Express your answer as a simplified fraction. | \frac{3}{38} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the equation $\sin \omega x + 1 = 0$ has exactly one solution in the interval $(0, \frac{\pi}{2})$ for $x$, find the maximum value of the positive number $\omega$. Express your answer as a single integer. | 7 | big_math | [
"Mathematics -> Trigonometry -> Trigonometric Functions -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given the circle $C$: $(x-1)^{2}+(y-2)^{2}=4$, find the maximum distance between point $P(3,3)$ and a point $M$ on the circle $C$. Express your answer as a simple algebraic expression, including the square root symbol if necessary. | \sqrt{5}+2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
The line $kx-y+1-3k=0$ passes through a fixed point. Find the coordinates of this fixed point. Express your answer as an ordered pair (x, y). | (3,1) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Given an angle that is $30^{\circ}$ less than its complement, find the size of this angle in degrees. Express your answer as a whole number. | 30^{\circ} | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Given the circle equations $C_1: x^2+y^2-6x+4y+12=0$ and $C_2: x^2+y^2-14x-2y+14=0$, find the number of lines that are tangent to both circles. Express your answer as a whole number. | 3 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Given \( S = (x-1)^{4} + 4(x-1)^{3} + 6(x-1)^{2} + 4(x-1) + 1 \), simplify the expression S. Express your answer as a simplified algebraic expression. | x^4 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Given a hyperbola with its left and right foci being F1 and F2 respectively, and the length of chord AB on the left branch passing through F1 is 5. If 2a = 8, calculate the perimeter of triangle ABF2. Express your answer as a whole number. | 26 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given non-zero vectors a and b satisfying |b| = 2|a| and a ⊥ (√3a + b), find the angle between a and b. Express your answer in radians, and if applicable, as a fraction of π. | \frac{5\pi}{6} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Given point A(3,-1,0) and vector $\overrightarrow{AB}=(2,5,-3)$, calculate the coordinates of point B. Express your answer in the format (x, y, z). | (5,4,-3) | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
The Ministry of Finance adjusted the securities transaction stamp duty rate from 3‰ to 1‰. If a person purchases stocks worth 100,000 yuan after the adjustment, how much less will they pay in securities transaction stamp duty compared to before the adjustment? Express your answer in yuan as a whole number. | 200 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Nine balls numbered 1, 2, …, 9 are placed into a bag. Person A draws a ball from the bag, and its number is $a$. After putting it back, person B draws another ball from the bag, and its number is $b$. Calculate the probability that the event $a-2b+10>0$ occurs. Express your answer as a fraction in simplest form. | \frac{61}{81} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
From a group of 6 students, 4 are to be selected to participate in competitions for four subjects: mathematics, physics, chemistry, and biology, with the restriction that two specific students, A and B, cannot participate in the biology competition. Calculate the total number of different selection plans possible under... | 240 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(\cos \theta,1)$ and $\overrightarrow{b}=(1,-1)$, find the minimum value of $|\overrightarrow{a}-\overrightarrow{b}|$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Let $p$ and $q$ be real numbers, and $\overrightarrow{a}$, $\overrightarrow{b}$ be two non-collinear vectors. Given $\overrightarrow{AB} = 2\overrightarrow{a} + p\overrightarrow{b}$, $\overrightarrow{BC} = \overrightarrow{a} + \overrightarrow{b}$, and $\overrightarrow{CD} = (q-1)\overrightarrow{a} - 2\overrightarrow{b}... | -2 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Other"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
A haunted house has six windows. Calculate the total number of ways Georgie the Ghost can enter the house by one window and leave by a different window. Express your answer as a numerical value. | 30 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Permutations"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Rowena is able to mow 2/5 of a lawn in 18 minutes. If she began the job at 10:00 a.m., and mowed at this same constant rate, what time did she finish mowing the entire lawn? Express your answer in 12-hour time format (e.g., 10:45 a.m.). | 10:45 a.m. | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Given the ellipse C: x^2/16 + y^2/4 = 1 with left and right foci F1 and F2, respectively. Find the number of points P on the ellipse C that satisfy angle F1 P F2 = pi/2. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Given the universal set $U=\{1,2,3,4\}$ and $\complement_{U}A=\{2\}$, find the number of proper subsets of set $A$. Express your answer as a whole number. | 7 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Gail is a server at a restaurant. On Saturday, Gail gets up at 6:30 a.m., starts work at \( x \) a.m. and finishes at \( x \) p.m. Calculate the total number of hours Gail works on Saturday. Express your answer as a simple algebraic expression or a numerical value if possible. | 12 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Solve the equation \sqrt{4 x^{2}+2 x+1}-\sqrt{4 x^{2}+14 x+5}=6 x+2 and determine the number of real roots. Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Given the system of equations in terms of x and y:
\[
\left\{
\begin{array}{l}
x + 2y = 5m \\
x - 2y = 9m
\end{array}
\right.
\]
and the solution satisfies the equation 3x + 2y = 19, calculate the value of m. Express your answer as a single numerical value. | 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Systems of Equations"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
How many positive two-digit whole numbers are divisible by 7? Express your answer as a whole number. | 13 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Given a sequence $\{a_n\}$ satisfies $2na_n = (n-1)a_{n-1} + (n+1)a_{n+1}$ (for $n \geq 2$ and $n \in \mathbb{N}^*$), with initial terms $a_1=1$ and $a_2=3$, calculate the maximum value of $\frac{a_n}{n}$. Express your answer as a single number. | \frac{3}{2} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Given the parabola equation $y=ax^2$ where $a<0$, find the equation of its directrix. Express your answer as an equation involving $y$ and any necessary variables or constants. | y = -\frac{1}{4a} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Consider the following code which calculates a result based on a loop:
```plaintext
S = 0
A = 1
while S <= 36:
A += 1
S += A
print(S)
```
What is the final value of S after the loop execution? | 45 | big_math | [
"Mathematics -> Discrete Mathematics -> Algorithms"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Given a geometric sequence of positive terms \(\{a_{n}\}\) with the sum of the first \(n\) terms denoted by \(S_{n}\), and the following equation:
\[ 2^{10} S_{30} + S_{10} = \left(2^{10} + 1\right) S_{20} \]
Determine the common ratio \(q\) of the sequence. Express your answer as a fraction. | \frac{1}{2} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Let x1, x2, x3,..., x9 be distinct positive integers such that x1 < x2 <... < x9 and their sum is 220. When the value of x1 + x2 +... + x5 is maximized, find the minimum value of x9 - x1. Express your answer as a single integer. | 9 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, two circles are centered at O. Point S is on the larger circle. Point Q is the intersection of OS and the smaller circle. Line segment PR is a chord of the larger circle and tangent to the smaller circle at Q. Note that OS is the perpendicular bisector of PR. Given PR = 12 and QS = 4, calculate the radi... | 6.5 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
A square has points marked on its sides, with exactly 3 points on each side. Calculate the minimum number of unique points marked on the square. Express your answer as a whole number. | 8 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given a sequence \\(\\{a_n\\}\\) that satisfies \\(a_{p+q}=a_{p}+a_{q}\\) for any \\(p, q \\in \\mathbb{N}^*\\) and \\(a_{2}=-6\\), calculate the value of \\(a_{10}\\). Express your answer as an integer. | -30 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Given a complex number $z_1 = 1 + i$ and its corresponding point $z_2$ in the complex plane that is symmetric about the imaginary axis, determine the value of $z_1z_2$. Express your answer as a single number in the form $\boxed{[answer]}$. | -2 | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $f(x)$ is an odd function defined on $\mathbb{R}$, and $f(x) = 2^x$ when $x < 0$, determine the range of $f(x)$. Express your answer as an interval in interval notation. | (-1, 1) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
In a singing competition, a set of sample data is obtained from the scores given by 10 judges. After removing the highest score and the lowest score, which statistical measure of the data set will definitely remain unchanged? Provide your answer as the name of the statistical measure. | Median | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Given an acute triangle $\triangle ABC$ with angles $A$, $B$, $C$ opposite sides $a$, $b$, $c$ respectively, and $b=3$, $c=1$, the area of $\triangle ABC$ is $\sqrt{2}$. Find the value of $a$. Express your answer as a numerical value. | 2\sqrt{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
A shop owner sells a product priced at 80% above the cost price, resulting in a selling price of 360 yuan. However, the product must be sold at a price not less than 130% of the cost price. Calculate the maximum discount that can be given to buy this item. Express your answer in yuan as a whole number. | 100 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Suppose the function f(x) is an even function that is differentiable and has a period of 5 on R. Find the slope of the tangent line to the curve y = f(x) at x = 5. | 0 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
In the "Fire Safety" knowledge quiz selection, there are a total of 20 questions. For each question, answering correctly scores 10 points, while answering incorrectly or not answering deducts 5 points. To pass the selection, one must score no less than 80 points. What is the minimum number of questions that must be ans... | 12 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Solve the inequality x + 1 > 3. Express your answer as an inequality in the form x > a or x < a. | x > 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Find the units digit of the product \(5^3 imes 7^{52}\). | 5 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
If the complex number $z$ satisfies $\left(3-4i\right)z=1$, calculate the modulus of $z$, $|z|$. | \frac{1}{5} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Given $a_1 > a_2 > a_3 > 0$, find the range of values for $x$ that makes $(1-a_ix)^2 < 1$ (for $i=1,2,3$) true. Express your answer as an interval in the form $(\text{lower bound}, \text{upper bound})$. | (0, \frac{2}{a_1}) | big_math | [
"Mathematics -> Algebra -> Inequalities"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
At Euclid Middle School, the mathematics teachers are Mrs. Germain, Mr. Newton, and Mrs. Young. There are 11 students in Mrs. Germain's class, 8 students in Mr. Newton's class, and 9 students in Mrs. Young's class taking the AMC 8 this year. Calculate the total number of mathematics students at Euclid Middle School tak... | 28 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Jing purchased eight identical items for a total cost of $26. Calculate the cost per item in dollars, rounded to two decimal places. | 3.25 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\\alpha$ is an angle in the second quadrant and $\\sin \\alpha = \\frac{5}{13}$, find the value of $\\cos \\alpha$. Express your answer as a fraction. | -\\frac{12}{13} | big_math | [
"Mathematics -> Trigonometry -> Trigonometric Functions"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
There are 200 identical steel pipes to be stacked into a triangular pyramid shape. To have as few pipes remaining as possible, calculate the number of pipes that will be left over. Provide your answer as a whole number. | 10 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
In the positive sequence {a_n}, if a_1 = 1, and for all n in the set of positive integers, it satisfies na_{n+1} - (n+1)a_n = 0, calculate the value of a_{2014}. Express your answer as a single integer. | 2014 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Consider an arithmetic sequence {a_n}. Let S_n denote the sum of its first n terms. If a_{m-1}+a_{m+1}-a_m^2=0 (m≥2, m∈ℕ*), and S_{2m-1}=58, then find the value of m. Express your answer as a whole number. | 15 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors a and b that satisfy |a|=2, |b|=1, and a⋅(a-b)=3, find the angle between a and b. Express your answer in radians as a fraction of π. | \frac{\pi}{3} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
At an observation station C, the distances to two lighthouses A and B are 300 meters and 500 meters, respectively. Lighthouse A is observed at 30° north by east from station C, and lighthouse B is due west of station C. Find the distance between the two lighthouses A and B. Express your answer in meters as a whole numb... | 700 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Find the coefficient of x in the expansion of (sqrt(x) - 2/x)^8. Express your answer as a single integer. | 112 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
What number should be subtracted from 21 to give 8? Express your answer as a whole number. | 13 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the angle between vectors a and b is 60°, |a| = 1, |b| = 2, find |3a + b|. Express your answer as a square root of a numerical value. | \sqrt{19} | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Geometry -> Other",
"Mathematics -> Vector Calculus -> Other"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Find the number that, when added to -2023, results in 0. Express your answer as a single integer. | 2023 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Find the conjugate of the complex number $\frac {5}{2-i}$. Express your answer in the standard form $a+bi$, where $a$ and $b$ are real numbers. | 2-i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. Given that $A=60^{\circ}$, $a=4\sqrt{3}$, and $b=4\sqrt{2}$, calculate the measure of angle $B$ in degrees. Express your answer in the form $\boxed{B^{\circ}}$. | 45^{\circ} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the solution set for the inequality $ax^2 + bx + 2 < 0$ is $\{x|x < -\frac{1}{2}, \text{ or } x > \frac{1}{3}\}$, calculate the value of $\frac{a-b}{a}$. Express your answer as a fraction. | \frac{5}{6} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Quadratic Functions"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z$ that satisfies the equation $2z + \overline{z} = 3 - 2i$, where $i$ is the imaginary unit, find the value of $z$. Express your answer as a complex number in the form $a + bi$, where $a$ and $b$ are real numbers. | 1 - 2i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
If the radius of circle $C$ is $1$, the center of the circle is in the first quadrant, and it is tangent to the line $4x-3y=0$ and the $x$-axis, then find the standard equation of the circle. Express your answer in the format $(x-h)^2 + (y-k)^2 = r^2$, where $(h,k)$ is the center of the circle and $r$ is the radius. | (x-2)^2 + (y-1)^2 = 1 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given a sequence $\{a_n\}$ satisfies $a_{n+1}=\frac{1}{1-a_n} (n\in\mathbb{N}^*)$, and $a_8=2$, calculate the value of $a_1$. Express your answer as a single value. | \frac{1}{2} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
A certain unit has 350 young employees, 250 middle-aged employees, and 150 elderly employees. In order to understand the health of the employees in the unit, stratified sampling is used to extract a sample. If there are 7 young employees in the sample, calculate the total sample size. | 15 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Xiao Ming is playing a stone-throwing game. For the first round, he walks 1 meter to place 2 stones, for the second round, he walks 2 meters to place 4 stones, and so forth. On the nth round, he walks n meters to place $2^n$ stones. When Xiao Ming has walked a total of 36 meters, calculate the total number of stones he... | 510 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Find the values of $k$ for which the quadratic equation $2x^2 - kx + x + 8 = 0$ has real and equal roots. Express your answer as a list of values. | 9, -7 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
The geometric sequence {a<sub>n</sub>} satisfies a<sub>1</sub>=3, a<sub>1</sub>+a<sub>3</sub>+a<sub>5</sub>=21. Find the product a<sub>2</sub>a<sub>6</sub>. Express your answer as a single integer. | 72 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Express the data "6.13 trillion" in scientific notation. Provide your answer in the format $a \times 10^{b}$, where $a$ is a decimal number and $b$ is an integer. | 6.13 \times 10^{12} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Given the monomial $-3\pi xy^{2}z^{3}$, what are its coefficient and degree? Express your answer with the coefficient as a product of its numerical value and pi (if applicable), and the degree as a whole number. | -3\pi, 6 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
The sum of the first n terms of a geometric sequence is 24, and the sum of the first 3n terms is 42. Find the sum of the first 2n terms. Express your answer as a single numerical value. | 36 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
A high school with three grades has a total of 2800 students. A stratified sampling method is required to select a sample of size 40. It is known that there are 910 students in the first grade; the number of students sampled from the second grade accounts for 3/10 of the total sample size. Calculate the number of stude... | 15 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
In an arithmetic sequence {a_n}, it is given that a_3 + a_4 + a_5 = 12. Find the sum of the first seven terms of this sequence, a_1 + a_2 +... + a_7. Express your answer as a single integer. | 28 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = 2cos^2(πx - π/6) - 1, stretch the horizontal coordinates of all points on the graph to twice their original length, keep the vertical coordinates unchanged, and then shift the resulting graph to the right by φ (φ > 0) units. Determine the minimum value of φ such that the final graph correspond... | \frac{1}{6} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Let $z=i^{2023}-1$ (where $i$ is the imaginary unit). Calculate the value of $z^{2}-2\overline{z}$. Provide your answer as a single value. | 2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets $M=\{x|x\leqslant 0\}$ and $N=\{-2,0,1\}$, find the intersection $M\cap N$. Express your answer as a set using proper set notation. | \{-2,0\} | big_math | [
"Mathematics -> Discrete Mathematics -> Other"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
In a race involving three horses X, Y, and Z where there are no ties, the odds of X winning are 3 to 1, and the odds of Y winning are 2 to 3. Calculate the odds of Z winning. Express your answer as a simplified ratio (e.g., a to b). | 3/17 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given $a\in \mathbb{R}$, $i$ is the imaginary unit. If $\frac{{a-i}}{{3+i}}$ is a real number, find the value of $a$. Express your answer as a single real number. | -3 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Given the set $P=\{x|x^{2}\leqslant 4\}$ and $M=\{m\}$, and it's given that $P \cap M = M$, find the range of values for $m$. Express your answer in interval notation, including all necessary endpoints and symbols. | [-2,2] | big_math | [
"Mathematics -> Algebra -> Inequalities"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Given the ellipse $\dfrac{x^2}{16} + \dfrac{y^2}{7} = 1$, with left and right foci $F_1$ and $F_2$, a line passing through $F_1$ intersects the ellipse at points $A$ and $B$. Calculate the perimeter of $ riangle ABF_2$. | 16 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
A systematic sampling method is used to draw a sample of size 8 from a group of 72 students. Calculate the interval between segments. Express your answer as a whole number. | 9 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given two non-collinear plane vectors $\overrightarrow a$ and $\overrightarrow b$, let $\overrightarrow m=\overrightarrow a-2\overrightarrow b$ and $\overrightarrow n=3\overrightarrow a+k\overrightarrow b$ ($k\in\mathbb{R}$). If $\overrightarrow m$ is parallel to $\overrightarrow n$, find the value of $k$. Express your... | -6 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
There are some identical square pieces of paper. If a part of them is paired up to form rectangles with a length twice their width, the total perimeter of all the newly formed rectangles is equal to the total perimeter of the remaining squares. Additionally, the total perimeter of all shapes after pairing is 40 centime... | 280 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Find the absolute value of -8. Express your answer as a whole number. | 8 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
The radius of the first circle is 1 inch, that of the second 1/2 inch, that of the third 1/4 inch, and so on indefinitely. Calculate the sum of the areas of all the circles. Express your answer in terms of pi. | \frac{4\pi}{3} | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
If the number of sides of a polygon is doubled, the sum of its interior angles is 2160 degrees. What was the original number of sides of the polygon? Express your answer as a whole number. | 7 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Distribute 5 students into 3 groups: Group A, Group B, and Group C. If Group A must have at least two people, and Groups B and C must each have at least one person, how many different distribution schemes are possible? Express your answer as a single integer. | 80 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Define the operation $$\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc$$. Then, find the complex number $z$ that satisfies $$\begin{vmatrix} 1 & -1 \\ z & zi \end{vmatrix} = 4 + 2i$$. Express your answer in the form $a+bi$, where $a$ and $b$ are real numbers. | 3-i | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Other"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
In $ riangle ABC$, it is known that $a^{2}=b^{2}+c^{2}+bc$. Find the measure of angle A in radians. | \dfrac {2\pi}{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation 3 × 3 × 5 × 5 × 7 × 9 = 3 × 3 × 7 × n × n, solve for n. Express your answer as a single numerical value. | 15 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence {a_n}, where a_1 = -1 and the common difference d = 2, and a_n = 15, find the value of n. Express your answer as a whole number. | 9 | big_math | [
"Mathematics -> Algebra -> Algebra -> Sequences and Series"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Find the smallest value of $n$ for which the inequality $1+ \frac{1}{\sqrt{2}}+ \frac{1}{\sqrt{3}}+…+ \frac{1}{\sqrt{n}} > \sqrt{n}$ holds true. Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Let a and b satisfy the conditions:
\[
\begin{cases}
a^3 - 6a^2 + 15a = 9 \\
b^3 - 3b^2 + 6b = -1
\end{cases}
\]
Calculate the value of (a - b)^2014. | 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
From 9 students, select 4 to participate in a debate competition, ensuring that at least two out of the three students A, B, and C are selected. Calculate the total number of different selection methods. Express your answer as a single integer. | 51 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
In the Cartesian coordinate system, point A has coordinates (-2, 3). Find the coordinates of point B, which is symmetric to point A with respect to the x-axis. Express your answer as a coordinate pair (x, y). | (-2, -3) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Commercial vinegar is a 5.00% by mass aqueous solution of acetic acid, CH3CO2H (M = 60.0 g/mol). Given the density of vinegar is 1.00 g/mL, calculate the molarity of acetic acid in vinegar. Express your answer in molarity (M), rounded to three decimal places. | 0.833 \text{ M} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
A local government convened a meeting with the leaders of 4 companies. Company A had 2 representatives attend the meeting, while the other 3 companies each had 1 representative attend. During the meeting, 3 individuals gave speeches (the order of the speeches is not considered). What is the probability that these 3 ind... | 0.6 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Combinations"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
A sample with a capacity of 100 has its data grouped and the parameters of each group are as follows: $(0,10]$, 12; $(10,20]$, 13; $(20,30]$, 15; $(30,40]$, 24; $(40,50]$, 16; $(50,60]$, 13; $(60,70]$, 7. Calculate the frequency of this sample data falling within $(10,40]$. Express your answer as a decimal, rounded to ... | 0.52 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Given that the even function $f(x)$ is increasing on the interval $[0, +\infty)$, determine the relationship between $f(-1)$ and $f(a^2-2a+3)$. Express your answer as an inequality using standard notation. | f(-1) < f(a^2-2a+3) | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Precalculus -> Functions"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x) = 2x^2 + 4x - 5$, where $x \in [t, t+2]$, the maximum value of the function $f(x)$ forms a function $y = g(t)$. Calculate the minimum value of the function $y = g(t)$. Express your answer as a single integer. | -7 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $F\_1$ and $F\_2$ are the left and right foci of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a > 0, b > 0)$, respectively. A circle with the coordinate origin $O$ as the center and $OF\_1$ as the radius intersects the hyperbola at point $P$ in the first quadrant. When the area of $\triangle PF\_1F\... | \sqrt{2} | big_math | [
"Mathematics -> Geometry -> Differential Geometry -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow {m} = (x-1, 1)$ and $\overrightarrow {n} = (1, y)$, if $\overrightarrow {m}$ is perpendicular to $\overrightarrow {n}$, find the minimum value of $2^x + 2^y$. Express your answer as a single value. | 2\sqrt{2} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Find the area of the closed region bounded by the equations xy=1, y=x, and x=3. Express your answer as a mathematical expression involving the natural logarithm and numerical values. | 4-\ln3 | big_math | [
"Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
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