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 |
|---|---|---|---|---|---|
A provincial sports team selects two players each from 5 male table tennis players and 3 female table tennis players to participate in a mixed doubles exhibition match. Each side of the match consists of one male and one female player. Calculate the number of different grouping methods possible for this match. | 60 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Combinations"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the function f(x) satisfies f(x+1) = f(3-x), and f(1) = 3, find the value of f(3). Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
In the sequence {a_n}, a_1 = 1, and a_n - a_{n-1} = n (where n is a positive integer and n >= 2). Calculate the value of a_{10}. Express your answer as a single integer. | 55 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence $1$, $\sqrt{3}$, $\sqrt{5}$, $\ldots$, $\sqrt{2n-1}$, $\ldots$, find the term number of $\sqrt{21}$. Express your answer as a whole number. | 11 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given the three numbers $x$, $y = x^x$, and $z = x^{x^x}$ with $0.9 < x < 1.0$, determine the order of increasing magnitude of these numbers. Express your answer by listing the variables in order, separated by commas. | x, z, y | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Given the arithmetic sequence {a_n}, where a_1 = 20, a_n = 54, and S_n = 999, find the value of n. Express your answer as a whole number. | 27 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
During a football game, Matt kicked the ball three times. His longest kick was 43 meters, and the three kicks averaged 37 meters. If the other two kicks were the same length, calculate the distance each of these two kicks travelled. Express your answer as a whole number in meters. | 34 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Solve for \( x \) in the equation \( 4x + 12 = 48 \). Express your answer as a single integer. | 9 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Simple Equations"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{9} = 1$ ($a > 0$) has asymptotes $3x \pm 2y = 0$, find the value of $a$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Given the vectors $\overset{→}{a}=(k,3)$, $\overset{→}{b}=(1,4)$, and $\overset{→}{c}=(2,1)$, and the condition that $\left(2\overset{→}{a}-3\overset{→}{b}
ight)\perp\overset{→}{c}$, determine the value of the real number $k$. Provide your answer as a single number. | 3 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
A savings card has a 6-digit password, with each digit being any number from 0-9. When a person was withdrawing money from an ATM, they forgot the last digit of the password. If they remember that the last digit is an even number, calculate the probability that they will enter the correct digit in no more than 2 attemp... | \frac{2}{5} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given that two distinct points $P$ and $Q$ on the circle $(x-1)^2+(y-2)^2=4$ are symmetric with respect to the line $kx-y-2=0$, find the value of $k$. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Find the value of $\tan 690^{\circ}$. Express your answer as a simplified fraction or decimal, including any necessary radicals. | -\frac{\sqrt{3}}{3} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
Let {a_n} be a geometric sequence with all positive terms, and let S_n denote the sum of its first n terms. If S_4 = 5S_2, find the value of the common ratio q of this sequence. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, the large square has an area of 49 square units, the medium square has an area of 25 square units, and the small square has an area of 9 square units. The region inside the small square is shaded. The region between the large and medium squares is also shaded. Calculate the total area of the shaded regi... | 33 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(1,2)$ and $\overrightarrow{b}=(x,-4)$, if $\overrightarrow{a} \parallel \overrightarrow{b}$, calculate the value of x. Express your answer as a single integer. | -2 | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Geometry -> Other",
"Mathematics -> Linear Algebra -> Vectors"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
How many fractions of the form $\frac{n}{n+1}$ are less than $\frac{7}{9}$, given that $n$ is a positive integer? Express your answer as a whole number. | 3 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Inequalities"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
The operation $\nabla$ is defined by $g \nabla h = g^2 - h^2$. If $g > 0$ and $g \nabla 6 = 45$, find the value of $g$. Express your answer as a single positive number. | 9 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of integer pairs (x, y) that satisfy the equation xy - 3x + 5y = 0. Express your answer as a single integer. | 8 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Evaluate the definite integral $\int_{0}^{1} \left( \sqrt{1-{{x}^{2}}}+\dfrac{1}{2}x \right)dx$. Express your answer as a single mathematical expression. | \dfrac{\pi+1}{4} | big_math | [
"Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
A square piece of paper is folded in half vertically. The folded paper is then cut in half along the dashed line. Three rectangles are formed - a large one and two small ones. Find the ratio of the perimeter of one of the small rectangles to the perimeter of the large rectangle. Express your answer as a fraction or rat... | \frac{3}{4} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Find the maximum value of the function y = 1/4*sin(x - π/6) - cos(x - 2π/3). Express your answer as a single numerical value. | 3/4 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
At 8:00 my watch was four minutes slow. However, it gains time at a constant rate and at 16:00 on the same day it was six minutes fast. At what time did it show the correct time? Express your answer in the format HH:MM (24-hour clock). | 11:12 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
The sequence 1, 4, 9, 16, 25, … can be recognized as a pattern of squares of natural numbers. Find the general term formula for this sequence in terms of n. | $a_n = n^2$ | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Sequences and Series"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $(1+2i)i^{3}+2i^{2}$. Express your answer as a simplified complex number. | -i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Given the definitions:
a = 0.5^0.1,
b = log4(0.1),
c = 0.4^0.1,
and the fact that y = x^0.1 is an increasing function on (0, +∞), determine the ordering of a, b, and c from greatest to least. Express your answer in the form of an ordering (e.g., a > b > c) enclosed in the boxed notation. | a > c > b | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Let omega be a primitive 10th root of unity, defined as omega = cos(pi/5) + i sin(pi/5). Derive the polynomial equation with roots omega, omega^3, omega^7, and omega^9. Express your answer as a polynomial equation, including the variable x and the constant term, and set equal to 0. | x^4 - x^3 + x^2 - x + 1 = 0 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $(5 - i) - (3 - i) - 5i$. Express your answer as a complex number in the form $a + bi$. | 2 - 5i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
A product is sold at 20% profit when discounted by 20% from its original price. What would be the profit, expressed as a percentage, if it were sold at its original price? | 50\% | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Let $i$ be the imaginary unit. If the complex number $2- \frac{a}{2-i}$ (where $a \in \mathbb{R}$) is a pure imaginary number, find the value of $a$. Express your answer as a single real number. | 5 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Given the inequality system about x: {x + 2 > 0, x - a <= 0} has a total of 4 integer solutions, calculate the range of values for a. Express your answer as an interval (e.g., [lower bound, upper bound)). | [2, 3) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $A$ and $B$ are two vertices of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > 0$, $b > 0$), if $P$ is a point on the hyperbola and the symmetric point of $P$ with respect to the $x$-axis is $Q$, and if the slopes of lines $AP$ and $BQ$ are $K_1$ and $K_2$ respectively, and $K_1K_2 = -\frac{4}{9... | \frac{\sqrt{13}}{3} | big_math | [
"Mathematics -> Geometry -> Non-Euclidean Geometry -> Hyperbolic Geometry"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Loki, Moe, Nick, and Ott are good friends. Ott had no money, but the others did. Moe gave Ott one-fifth of his money, Loki gave Ott one-fourth of his money, and Nick gave Ott one-third of his money. Each gave Ott the same amount of money. What fractional part of the group's money does Ott now have? Express your answer ... | \frac{1}{4} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
If $a = -3$ and $b = 2$, calculate the value of the expression $a(b - 3)$. Provide your answer as a single number. | 3 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
In the following equation, each of the letters represents uniquely a different digit in base ten:
\[(YE) \cdot (ME) = TTT\]
Calculate the sum $E+M+T+Y$. Express your answer as a single integer. | 21 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
A beam of light shines from point \( S \), reflects off a reflector at point \( P \), and reaches point \( T \) such that \( P T \) is perpendicular to \( R S \). Given that \(\angle PTS = 90^\circ\) and \(\angle TSR = 26^\circ\), find the measure of angle \( x \). Express your answer in degrees as a whole number. | 32^\circ | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given that F is the focus of the parabola $x^2=4y$, and P is a moving point on the parabola, derive the equation of the trajectory of the midpoint of the line segment PF in the form $x^2=f(y)$. Include all necessary constants and coefficients. | x^2=2y-1 | big_math | [
"Mathematics -> Geometry -> Differential Geometry -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the random variable X follows a normal distribution N(2, σ^2), and P(1 < X < 3) = 0.36, calculate the probability P(X ≥ 3). Express your answer as a decimal value between 0 and 1. | 0.32 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $\frac{x}{(x-1)^2}-\frac{1}{(x-1)^2}$. Express your answer as a simplified algebraic expression. | \frac{1}{x-1} | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence {a_n} where S_n is the sum of the first n terms, and a_3 * a_5 = 12, a_2 = 0. If a_1 > 0, calculate the sum of the first 20 terms, S_20. Express your answer as a single integer. | -340 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $a^{2}+a^{-2}=3$, find the value(s) of $a+a^{-1}$. Express your answer as a mathematical expression, including both possible values if applicable. | \pm\sqrt{5} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
On a bookshelf, there are three types of books: literature, technology, and general knowledge, with a ratio of 5:2:4. If 35 more literature books are added and the number of technology books is tripled, then general knowledge books account for 22%. How many general knowledge books are there? Express your answer as a wh... | 44 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Given the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with left and right vertices $A$ and $B$, and top vertex $C$. If $ riangle ABC$ is an isosceles triangle with base angle $30^{\circ}$, calculate the eccentricity $e$ of the ellipse. Express your answer as a fraction with a radical. | \frac{\sqrt{6}}{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $f(x)$ is an even function and $f(x) = 1 + 2x - x^2$ when $x > 0$, derive the expression for $f(x)$ when $x < 0$. Use the definition of an even function and the given expression to find $f(x)$ when $x < 0$. Express your answer in the format $f(x) = \\boxed{[expression]}$. | 1 - 2x - x^2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Two non-coplanar straight lines a and b form an angle θ. Three straight lines passing through a fixed point O each form a 60° angle with both a and b. Determine the exact value of θ in degrees (°). | 60° | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given a complex number $z$ that satisfies $iz = 3 + 4i$ (where $i$ is the imaginary unit), determine the quadrant in which the point corresponding to the conjugate of $z$ is located in the complex plane. Express your answer as the name of the quadrant (e.g., First quadrant). | First quadrant | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
In a 4x4 grid, each digit from 1 to 4 must appear exactly once in each row, each column, and each of the four smaller 2x2 squares. Calculate the digit that replaces P. Express your answer as a single digit. | 1 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
What is the volume of tetrahedron ABCD with edge lengths AB = 2, AC = 3, AD = 4, BC = sqrt(13), BD = 2*sqrt(5), and CD = 5? Express your answer as a single number. | 4 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Volume"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Simplify and add the square roots: $\sqrt{8}+\sqrt{18}$. Express your answer as a simplified radical expression. | 5\sqrt{2} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
A bag contains several balls of the same size and shape, among which there is 1 ball labeled 0, 1 ball labeled 1, and n balls labeled 2. If the probability of randomly drawing a ball labeled 2 from the bag is 1/2, find the value of n. Express your answer as a whole number. | 2 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Solve for $x$ in the equation $\frac{x^2-1}{x-1}=0$, given that the numerator must be zero and the denominator must not be zero. Express your answer as a single value of $x$ and provide any necessary explanations or justifications. | -1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given a set of numbers $x_1, x_2, …, x_n$ with mean $\overline{x}$ and variance $s^2$, find the mean and variance of another set of numbers $\sqrt{3}x_1+ \sqrt{2}, \sqrt{3}x_2+ \sqrt{2}, …, \sqrt{3}x_n+ \sqrt{2}$. Express your answer in the format $\boxed{\text{Mean} = m, \text{Variance} = v}$. | \text{Mean} = \sqrt{3}\overline{x}+ \sqrt{2}, \text{Variance} = 3s^2 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
A grandfather, who is over 70 years old, and his grandson are celebrating their birthdays today. Several years ago, the grandfather's age was 10 times the age of his grandson. Several years later, he will be 8 times the age of his grandson. Find their current ages. Express your answer as two whole numbers, representing... | 71, 8 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Let the buying price of stock A be $a$. After 4 consecutive limit downs, each representing a 10% drop from the previous day's closing price, the price becomes $0.6561a$. How many limit ups, each representing a 10% increase from the previous day's closing price, are needed at least to avoid a loss? Express your answer a... | 5 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of $\sin 72^\circ \cos 18^\circ + \cos 72^\circ \sin 18^\circ$. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given the hyperbola x^2/a^2 - y^2/b^2 = 1 (a > 0, b > 0) whose right vertex coincides with the focus of the parabola y^2 = 8x and has an eccentricity e = 3/2, derive the equation of this hyperbola. Express your answer in the standard hyperbola equation format, including all necessary variables (x, y, a, b). | \dfrac{x^2}{4} - \dfrac{y^2}{5} = 1 | big_math | [
"Mathematics -> Geometry -> Other",
"Mathematics -> Precalculus -> Functions",
"Mathematics -> Algebra -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
For \( p = 1, 2, \cdots, 10 \), let \( S_{p} \) be the sum of the first 40 terms of an arithmetic sequence with the first term \( p \) and the common difference \( 2p - 1 \). Calculate \( S_{1} + S_{2} + \cdots + S_{10} \). Express your answer as a single integer. | 80200 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Person A, B, and C are given boxes A, B, and C, respectively, containing red, yellow, and blue balls. However, it is known that person A did not get box A; person B did not get box B and did not get a yellow ball; there is no red ball in box A, and box B contains a blue ball. Determine the box number and the color of t... | \text{A, yellow} | big_math | [
"Mathematics -> Discrete Mathematics -> Logic"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
There are 5 students signing up for two extracurricular activity groups, with each student limited to joining one of the groups. Calculate the total number of different sign-up methods. Express your answer as a whole number. | 32 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
In a positive geometric sequence {a_n}, a_1 = 1, and the sum of the first n terms is S_n. If -a_3, a_2, and a_4 form an arithmetic sequence, then find the value of S_7. Express your answer as a whole number. | 127 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
A new complex number is formed by taking the imaginary part of $2i - \sqrt {5}$ as the real part, and the real part of $\sqrt {5}i - 2$ as the imaginary part. Calculate this new complex number. Express your answer in standard complex number format (a + bi). | 2-2i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Let $U=\mathbb{R}$, $A=\{x|x>0\}$, $B=\{x|x>1\}$. Find the intersection of set A and the complement of set B with respect to the universal set U. Express your answer in interval notation or set builder notation. | \{x|0<x\leq 1\} | big_math | [
"Mathematics -> Discrete Mathematics -> Logic"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Carley made treat bags, each containing exactly 1 chocolate, 1 mint, and 1 caramel. The chocolates come in boxes of 50, the mints in boxes of 40, and the caramels in boxes of 25. Carley made complete treat bags with no unused chocolates, mints, or caramels. What is the minimum total number of boxes that Carley could ha... | 17 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $|a-2|$ and $(b+5)^2$ are opposites of each other, find the value of $a+b$. Express your answer as a single integer. | -3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Let the prime number \( p \) satisfy the condition that there exist positive integers \( x \) and \( y \) such that
\[
p-1 = 2x^2,
\]
\[
p^2 - 1 = 2y^2.
\]
Find the number of such prime numbers \( p \). Express your answer as a whole number. | 1 | big_math | [
"Mathematics -> Number Theory -> Prime Numbers"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z= \frac {a+3i}{1-2i}$, where $a\in \mathbb{R}$ and $i$ is the imaginary unit, and $z$ is a pure imaginary number, calculate the value of $|a+2i|$. Express your answer as a single value, including the square root symbol if necessary. | 2\sqrt{10} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Find the exact value of $\tan 45^{\circ}$. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
The function $f(x)$ is an odd function defined on the domain $\mathbb{R}$. When $x > 0$, $f(x) = -x + 2$. Find the expression for $f(x)$ when $x < 0$. Express your answer as a function of $x$. | -x - 2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Find the domain of the function $y= \sqrt{2x+1}+ \sqrt{3-4x}$. Express your answer in interval notation. | [-\frac{1}{2}, \frac{3}{4}] | big_math | [
"Mathematics -> Precalculus -> Functions"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
An old clock's minute hand and hour hand overlap every 66 minutes of standard time. Calculate the time difference between 24 hours of this old clock and 24 hours of standard time. Express your answer in minutes, specifying whether the old clock is fast or slow. | 12 \text{ minutes slow} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
In April 2017, the second-year students of Gezhouba Middle School organized a hiking activity named "The Road Underfoot, The Dream in the Heart". The second-year class (1) was divided into three teams: red, blue, and yellow. If students A and B each randomly choose one of the teams, what is the probability that they ch... | \dfrac{1}{3} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Given a periodic sequence {x_n} that satisfies x_n = |x_{n-1} - x_{n-2}| for n >= 3 and x_1 = 1, x_2 = a >= 0, find the sum of the first 2002 terms of the sequence when the period is minimal. Express your answer as a single integer. | 1335 | big_math | [
"Mathematics -> Discrete Mathematics -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
In right triangle $ABC$, the hypotenuse $BC=4$. Find the value of $AB^{2}+AC^{2}+BC^{2}$. Express your answer as a single integer. | 32 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Three equilateral triangles are cut from the corners of a large equilateral triangle to form an irregular hexagon. The perimeter of the large equilateral triangle is 60 cm. The perimeter of the irregular hexagon is 40 cm. Find the sum of the perimeters of the triangles that were cut from the large triangle. Express you... | 60 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $(x-8)(x-10) = 2^y$, where $x$ and $y$ are integers, find the number of integer solutions. Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given a geometric sequence $\{a_n\}$ with all positive terms, the logarithm of the product of three terms is 6. Specifically, $\lg(a_3 \cdot a_8 \cdot a_{13}) = 6$. Find the product of the first and 15th terms, $a_1 \cdot a_{15}$. Express your answer as a power of 10. | 10^4 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Given a non-zero arithmetic sequence $\{a_n\}$ that satisfies $2a_3 - a_7^2 + 2a_{11} = 0$, and a geometric sequence $\{b_n\}$ with $b_7 = a_7$, calculate the value of $b_5b_9$. Express your answer as a single integer. | 16 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
In a regular tetrahedron ABCD with equal edge lengths, if M is the centroid of the base triangle BCD, and O is a point inside the tetrahedron with equal distances to all faces, find the ratio AO:OM. Express your answer as a ratio of integers. | 3:1 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Let \( S = \left\{\left(s_{1}, s_{2}, \cdots, s_{6}\right) \mid s_{i} \in \{0, 1\}\right\} \). For any \( x, y \in S \) where \( x = \left(x_{1}, x_{2}, \cdots, x_{6}\right) \) and \( y = \left(y_{1}, y_{2}, \cdots, y_{6}\right) \), define:
(1) \( x = y \) if and only if \( \sum_{i=1}^{6}\left(x_{i} - y_{i}\right)^{2} ... | 32 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
A carpenter wishes to cut a wooden 3 x 3 x 3 cube into twenty-seven 1 x 1 x 1 cubes. What is the minimum number of cuts needed if he is allowed to rearrange the pieces after each cut? Express your answer as a single integer. | 6 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given that one element of the solution set of the inequality about $x$, $2x^{2}+ax-a^{2} > 0$, is $2$, find the range of the real number $a$ in interval notation. | (-2,4) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence 1, $\sqrt{3}$, $\sqrt{5}$, $\sqrt{7}$, …, $\sqrt{2n-1}$, …, find the term number of $3\sqrt{5}$ in the sequence. Express your answer as a whole number. | 23 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
A shopping mall is holding a New Year shopping lottery event. First, 160 customers are randomly assigned numbers from 001, 002, 003,..., 160. The method of systematic sampling is used to draw lucky customers. It is known that the smallest two numbers among the drawn lucky customers are 007 and 023. What is the largest ... | 151 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given an acute-angled $\triangle ABC$ with opposite sides $a$, $b$, and $c$ to angles $A$, $B$, and $C$ respectively, and $23\cos ^{2}A+\cos 2A=0$, $a=7$, $c=6$, calculate the length of side $b$ using the cosine rule. Express your answer as a numerical value. | 5 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
If the complex number $z$ satisfies $z(2+i) = \frac{10}{1+i}$, then calculate the conjugate of $z$, $\overline{z}$. Express your answer in the form $a+bi$, where $a$ and $b$ are real numbers. | 1+3i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(2,-1,3)$ and $\overrightarrow{b}=(-4,2,3-x)$, and knowing that $\overrightarrow{a}\perp \overrightarrow{b}$, calculate the value of x. Express your answer as a single value. | -\dfrac{1}{3} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
A two-player game is played on a grid of varying sizes (6x7, 6x8, 7x7, 7x8, and 8x8). The game starts with a piece at the bottom-left corner, and players take turns moving the piece up, right, or diagonally up-right. The game ends when a player cannot make a move. How many of these grid sizes allow the first player to ... | 4 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Johannes told me that it rained seven times during his holiday. When it rained in the morning, it was sunny in the afternoon; when it rained in the afternoon, it was sunny in the morning. There were 5 sunny mornings and 6 sunny afternoons. What is the least number of days that the holiday could have lasted? | 9 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Find the arithmetic square root of the integer 100. Express your answer as a single integer. | 10 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given the ellipse $C: \frac{x^2}{5} + y^2 = 1$, and a point $P$ on $C$ such that $\overrightarrow{PF_{1}}\cdot \overrightarrow{PF_{2}}=0$, find the product $|PF_{1}|\cdot |PF_{2}|$. Express your answer as a numerical value. | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Randomly select a real number $x$ in the interval $[-1,1]$. What is the probability of the event "$2x-1 < 0$"? Express your answer as a fraction in simplest form. | \dfrac {3}{4} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence {a_n} where each term is not equal to 0 and satisfies a_6 - a_7^2 + a_8 = 0, and a geometric sequence {b_n} where b_7 = a_7, calculate the value of b_4 * b_7 * b_10. | 8 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
The front view of a cylinder is a rectangle with an area of 6. Find the lateral area of the cylinder. Express your answer in terms of pi (π). | 6\pi | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.703125 | 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 $a=4$ and $A= \frac{\pi}{3}$, find the maximum area of this triangle. Express your answer as a numerical value, including the square root symbol if necessary. | 4 \sqrt{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Find the binomial coefficient of the third term in the expansion of $(1-2x)^{4}$. Express your answer as a numerical value. | 6 | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
A number $a$ is randomly selected from the set $\{2,3,4,5,6\}$, and a number $b$ is randomly selected from the set $\{1,2,3,5\}$. Find the probability that $b > a$. Express your answer as a simplified fraction. | \frac{1}{5} | big_math | [
"Mathematics -> Applied Mathematics -> Probability"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
In the cube $ABCD-A'B'C'D'$, the midpoint of $AB$ is $M$, and the midpoint of $DD'$ is $N$. Calculate the angle formed by the skew lines $B'M$ and $CN$. Express your answer in degrees (°). | 90 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $\frac{1+3i}{1-i}$ and express the result in the form of a complex number, with the real part first and the imaginary part second, separated by a plus sign if the imaginary part is positive, and a minus sign if it is negative. | -1+2i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
The ancient Chinese calendrical system, "Heavenly Stems and Earthly Branches", also known as "Ganzhi", is used for years' notation. The ten Heavenly Stems are Jia (甲), Yi (乙), Bing (丙), Ding (丁), Wu (戊), Ji (已), Geng (庚), Xin (辛), Ren (壬), and Gui (癸). The twelve Earthly Branches are Zi (子), Chou (丑), Yin (寅), Mao (卯),... | 1950 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given points A(-3,0) and B(3,0) in a Cartesian coordinate system, find the equation of the locus of a moving point M that satisfies the condition |MA|/|MB| = 2. Express your answer as an equation of a circle in standard form. | (x-5)^2 + y^2 = 16 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.