instruction_seed stringlengths 40 1.39k | answer stringlengths 1 95 | source stringclasses 1
value | domain listlengths 1 3 | llama8b_solve_rate float64 0 0.98 | _source stringclasses 1
value |
|---|---|---|---|---|---|
Given the universal set $U = \mathbb{R}$, $A = [0, 2]$, and $B = \{y | y = 2^x, x > 0\}$, find the intersection of $A$ and the complement of $B$ with respect to $U$, denoted as $A \cap C_U B$. Express your answer as an interval or a combination of intervals. | [0, 1] | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Find the coordinates of the intersection point of the line $x+2y-4=0$ and the line $2x-y+2=0$. Express your answer as an ordered pair (x, y). | (0,2) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
The diameter of the "flying blade" is 0.0000009 mm. Express this number in scientific notation as 9 x 10^n mm. Find the value of the exponent n. Provide your answer as an integer. | -7 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
A senior high school has 1600 students across three grades: 1st, 2nd, and 3rd. There are 400 students in the 3rd grade. If a stratified sampling method is used to draw a sample of 80 students from the entire high school, how many students should be drawn from the 3rd grade? Express your answer as a whole number. | 20 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given the equations \(2^{x}=15\) and \(15^{y}=32\), find the value of \(xy\). Express your answer as a whole number. | 5 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\alpha \in (0, 2\pi)$, $\sin{\alpha} > 0$, and $\cos{\alpha} < 0$, determine the range of values for angle $\alpha$. Express your answer as an interval. | (\frac{\pi}{2}, \pi) | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Let the universal set $U=\mathbb{Z}$, the set $M=\{1, 2\}$, and $P=\{x \mid |x| \leq 2, x \in \mathbb{Z}\}$. Calculate the intersection of $P$ and the complement of $M$ in $U$, denoted as $P \cap C_{U}M$. Express your answer as a set of integers using standard set notation. | \{-2, -1, 0\} | big_math | [
"Mathematics -> Other -> Other"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Alicia earns 20 dollars per hour, of which 1.45% is deducted to pay local taxes. Calculate the amount of Alicia's wages that are used to pay local taxes in cents per hour. Express your answer as a whole number. | 29 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
There are 7 light-emitting diodes (LEDs) in a row, each of which can emit red or green light when lit. If exactly 3 LEDs are lit at a time, and adjacent ones cannot be lit simultaneously, calculate the total number of different messages that this row of LEDs can represent, considering the different positions and colors... | 80 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the universal set $U=\{1,2,3,4,5\}$, and $M=\{1,2\}$, find the complement of $M$ with respect to $U$, denoted as $\complement_U M$. Express your answer as a set of elements using standard set notation. | \{3,4,5\} | big_math | [
"Mathematics -> Discrete Mathematics -> Set Theory -> Other"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Solve the equation x^2 - 1 = 0 and represent the solution set. Express your answer as a set of values, including all solutions to the equation. | \{-1,1\} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Let S<sub>n</sub> be the sum of the first n terms of the arithmetic sequence {a<sub>n</sub>}, and 4 + a<sub>5</sub> = a<sub>6</sub> + a<sub>4</sub>. Calculate S<sub>9</sub>. Express your answer as a single number. | 36 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number z that satisfies the equation z * (2 + i) = 1, find the real part of z. Express your answer as a numerical value. | \frac{2}{5} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\triangle ABC$ and $\triangle {A'}{B'}{C'}$ are similar figures with a similarity ratio of $1:3$, calculate the ratio of their areas. Express your answer as a ratio in the format a:b. | 1:9 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Given a complex number $z=\frac{1+3i}{1-i}$, calculate the conjugate of $z$, denoted as $\overline{z}$. Express your answer as a complex number in the form $a + bi$. | -1 - 2i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
A gold rod is 5 feet long, with one end thicker than the other. Cutting 1 foot from the thicker end weighs 4 jin, and cutting 1 foot from the thinner end weighs 2 jin. If the weights of each foot form an arithmetic sequence, find the sum of the weights of the second and fourth feet. Express your answer in jin. | 6 \text{ jin} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
A number is a squared number when, upon being added to the number obtained by reversing its digits, the result is a perfect square. For example, 164 and 461 are squared numbers because 164 + 461 = 625 = 25^2. How many squared numbers are there between 10 and 100? Express your answer as a single integer. | 8 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation sinθ + cos(θ + π/6) = 1, find the value of sin(θ + π/3). Express your answer as a single numerical value. | 1 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Given a geometric sequence, where S_n is the sum of its first n terms, S_3 = 10, and S_6 = 20, calculate the value of S_9. Express your answer as a single numerical value. | 30 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
A circle has an inscribed triangle whose sides are $5\sqrt{3}$, $10\sqrt{3}$, and $15$. Find the measure of the angle subtended at the center of the circle by the shortest side. Express your answer in degrees as a whole number. | 60 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\vec{a}=(\sin \theta, -2)$ and $\vec{b}=(1, \cos \theta)$, and $\vec{a} \perp \vec{b}$, find the value of $\sin 2\theta + \cos^2\theta$. Express your answer as a single numerical value. | 1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
There are 8758 people in the northern village, 7236 people in the western village, and 8356 people in the southern village. A total of 378 people are needed from these three villages. How many people should be collected from the western village? Express your answer as a whole number. | 112 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function g(x+2) = 2x + 3, find the value of g(3). Express your answer as a single integer. | 5 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given the monomials $-3a^{m}b^{2}$ and $\frac{1}{2}{a^2}{b^n}$, their sum will also be a monomial if the exponents of $a$ and $b$ in both terms are the same. Find the value of $mn$ that satisfies this condition. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the asymptotes of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ are perpendicular to each other, find the eccentricity of the hyperbola. Express your answer as a single value. | \sqrt{2} | big_math | [
"Mathematics -> Geometry -> Non-Euclidean Geometry -> Other"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Given the piecewise function $f(x) = \\begin{cases} 2x-1 & (x < \\frac{1}{2}) \\\\ f(x-1)+1 & (x \\geq \\frac{1}{2}) \\end{cases}$, calculate the value of $f\\left(\\frac{1}{4}\\right) + f\\left(\\frac{7}{6}\\right)$. Express your answer as a single fraction. | -\\frac{1}{6} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
If the coefficient of $x^3$ in the expansion of $(1+2x)^n$ is equal to four times the coefficient of $x^2$, calculate the value of $n$. Express your answer as a single integer. | 8 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Binomial Theorem"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Given \(a\in \mathbb{R}\), the line \(l_1\): \(x+2y=a+2\) and the line \(l_2\): \(2x-y=2a-1\) intersect the circle \(E\): \((x-a)^2+(y-1)^2=4\) at points \(A\), \(C\) and \(B\), \(D\) respectively. Calculate the area of the quadrilateral \(ABCD\). Express your answer as a single number. | 8 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
In an isosceles right-angled triangle with a leg length of $2$, a point is chosen such that the distance from this point to the right-angled vertex of the triangle is not greater than $1$. Calculate the probability that this condition is satisfied. Express your answer as a fraction involving $\pi$. | \dfrac {\pi}{8} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the angle between vectors $\overrightarrow {a}$ and $\overrightarrow {b}$ is $\frac {2π}{3}$, $\overrightarrow {a}\cdot \overrightarrow {b}$ = -3, $|\overrightarrow {b}|$ = 2, find the magnitude of vector $\overrightarrow {a}$. Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given the line equation ax + by + c = 0, where ab < 0 and bc < 0, determine the quadrant that the line does not pass through. Express your answer as the name of the quadrant (e.g., First, Second, Third, Fourth). | Fourth | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given $(a-\text{i})\text{i}=-b+2\text{i}$ where $a,b \in \mathbb{R}$, find the sum $a+b$. Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
If \( 3 \times 3 \times 5 \times 5 \times 7 \times 9 = 3 \times 3 \times 7 \times n \times n \), what is the value of \( n \)? Express your answer as a single integer. | 15 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
If $1.2 \times 10^{6}$ is doubled, what is the result? Express your answer in scientific notation. | 2.4 \times 10^{6} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given the parabola equation y^2 = 16x and point A(1, 2), with P being a point on the parabola and F the focus of the parabola, calculate the minimum value of |PF| + |PA|. Express your answer as a single numerical value. | 5 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given a real number $a \lt 0$, and a function $f(x)=\left\{{\begin{array}{l}{2x+a, x＜1}\\{-x-2a, x≥1}\end{array}}\right.$. If $f\left(1-a\right)=f\left(1+a\right)$, then find the value of $a$. Express your answer as a single real number. | -\frac{3}{4} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given two lines $ax-y-2=0$ and $(2-a)x-y+1=0$ are parallel, find the value of $a$. | 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Given $a$, $b > 0$ and $a+b=5$, find the maximum value of $\sqrt{a+1}+\sqrt{b+3}$. Express your answer as a single value. | 3\sqrt{2} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
If $\overrightarrow{a} = (2,3)$ and $\overrightarrow{b} = (4, -1+y)$, and $\overrightarrow{a}$ is parallel to $\overrightarrow{b}$, find the value of $y$. Express your answer as a single integer. | 7 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Find the decreasing interval of the function y = x^2(x - 3). Express your answer in standard interval notation (e.g., (a, b)). | (0, 2) | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Find the maximum value of the function $$ \frac{1}{\left|x+1\right|+\left|x+2\right|+\left|x-3\right|}. $$ Express your answer as a fraction. | \frac{1}{5} | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the complex number $z$ satisfies $z(1+i)=4-2i$, calculate the magnitude of $z$, denoted as $|z|$. Express your answer as a numerical value, including the square root symbol if necessary. | \sqrt{10} | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression: \(6 \cdot 37 \quad(\sqrt{3}+\sqrt{2})^{2 \log _{(\sqrt{3}-\sqrt{2})}^{\sqrt{5}}}\). Express your answer as a single number. | 1110 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of integer solutions for the inequality \( |x| < 3 \pi \). Express your answer as a whole number. | 19 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x)= \dfrac {x^{2}+a}{x}$, and $f(1)=4$, find the value of $f(-1)$. Express your answer as a single integer. | -4 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number z = (3m-2) + (m-1)i and the condition 2/3 < m < 1, identify the quadrant of the complex plane where z is located. Express your answer as the quadrant number (I, II, III, or IV). | IV | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a$ and $b$ are positive real numbers satisfying $a+b=4$, find the maximum value of $\log_2 a + \log_2 b$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Applied Mathematics -> Other",
"Mathematics -> Algebra -> Other"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Among the functions $y= \frac {1}{x^{2}}$, $y=-x^{2}$, $y=x^{2}+x$, how many are power functions according to the definition that only functions of the form $y=x^{\alpha}$ are considered power functions? Express your answer as a whole number. | 1 | big_math | [
"Mathematics -> Algebra -> Functions"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
In an art gallery, a 2-meter high painting, BT, is mounted on a wall with its bottom edge 1 meter above the floor. A spotlight is mounted at S, 3 meters out from the wall and 4 meters above the floor. Calculate the size of angle TSB in degrees, rounded to 1 decimal place. | 26.6 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
A figure is composed of three isosceles right-angled triangles, each with legs of length 4 units. Calculate the total area of the figure. Express your answer in square units as a whole number. | 24 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Let $S$ be the set of points on the rays forming the sides of a $120^{\circ}$ angle, and let $P$ be a fixed point inside the angle on the angle bisector. Consider all distinct equilateral triangles $PQR$ with $Q$ and $R$ in $S$. Count the number of distinct equilateral triangles that can be formed. Express your answer ... | 3 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\sin \alpha = -\dfrac{4}{5}$ and $\alpha$ is an angle in the third quadrant, calculate the value of $\sin (\alpha + \dfrac{\pi}{4})$. Express your answer as a fraction with a radical if necessary, simplified to its simplest form. | -\dfrac{7\sqrt{2}}{10} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Given the quadratic equation $\left(m-2\right)x^{2}+3mx+1=0$, what is the condition on m that makes this equation a quadratic equation in x? Express your answer as an inequality or equality condition on m. | m \neq 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the tangent line to the curve $y=x^3-3x^2+1$ at point $P(0,1)$. Express your answer as an equation of a line. | y=1 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
A certain school organized a practical activity for handcrafting. The slant height of a cone made by a student is $30$ cm, and the radius of the base circle is $10$ cm. Calculate the central angle of the unfolded diagram of the lateral surface of this cone. Express your answer in degrees. | 120 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Construct a one-variable quadratic equation with roots 3 and 4. Express your answer in the form ax^2 + bx + c = 0. | x^2 - 7x + 12 = 0 | big_math | [
"Mathematics -> Algebra -> Algebra -> Quadratic Functions"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Determine the digit A in the hundreds place of the total amount $\overline{1A2}$, given that 11 guests were paid the same amount and the total amount must be divisible by 11. Express your answer as a single digit. | 3 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
There are 1000 numbers arranged in a row, such that any three consecutive numbers satisfy the condition that the middle number is equal to the sum of the numbers before and after it. If the first and second numbers are both 1, calculate the sum of these 1000 numbers. Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
A rectangular solid with a volume of 4 has one face with an area of 1, and all 8 vertices of the solid lie on the surface of sphere O. Calculate the minimum surface area of sphere O. Express your answer in terms of pi (π). | 18\pi | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Surface Area"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
At the Gaussian Store, puzzles cost $10 each or $50 for a box of 6 puzzles. If a customer would like exactly 25 puzzles, what is the minimum possible cost in dollars? | 210 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
A certain brand of computer is reduced by 40%, resulting in a selling price of a yuan per unit. Calculate the original price of each unit of this brand of computer in terms of a. Express your answer as a fraction or decimal multiple of a. | \frac{5}{3}a | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
What is the largest decimal number that can be represented by a four-digit binary number? Express your answer as a single numerical value. | 15 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given real numbers $a > 0$ and $b > 0$, if $2a + b = 1$, calculate the minimum value of $\frac {1}{a} + \frac {2}{b}$. Express your answer as a single numerical value. | 8 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Let $S_n$ be the sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$, where $a_1=1$ and the common difference $d=2$. If $S_{k+2}-S_k=24$, then find the value of $k$. Express your answer as a whole number. | 5 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Let $i$ be the imaginary unit. Calculate the distance from the point corresponding to the complex number $z = \\frac{i}{i + 1}$ on the complex plane to the origin. Express your answer as a simplified fraction or radical. | \\frac{\\sqrt{2}}{2} | big_math | [
"Mathematics -> Algebra -> Complex Numbers -> Other"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Given the circle equation $(x+1)^2+y^2=2$, find the center and radius of the circle. Express your answer in the format (center, radius), where the center is in the format (x, y) and the radius is a numerical value. | ((-1, 0), \sqrt{2}) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of the expression ${\sin ^{2}}{70^{\circ }} + \tan {225^{\circ }} + {\sin ^{2}}{20^{\circ }}$. Provide your answer as a single number. | 2 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Determine the internal angle that guarantees two isosceles triangles are similar. Express your answer as a single angle value in degrees, including the degree symbol (°). | 60° | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
The length of a circular arc is equal to the side length of the inscribed square of its circle. Find the radian measure of its central angle. Express your answer in radians. | \sqrt{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Find the coefficient of x^3 in the expansion of (1-2x)(1+x)^5. Express your answer as a single numerical value. | -10 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations -> Other"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $Z = \frac{a-1+2ai}{1-i}$ corresponds to a point in the second quadrant, find the range of the real number $a$. Express your answer as an inequality. | a > \frac{1}{3} | big_math | [
"Mathematics -> Algebra -> Complex Numbers -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given two convex polygons P1 and P2 with n1 and n2 sides, respectively, where n1 ≤ n2, and they do not share any line segment, find the maximum number of intersections between these two polygons. Express your answer as a single number or a formula. | n1n2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
In a geometric sequence, $a_{2}=2$ and $a_{6}=8$. Find the value of $a_{10}$. Express your answer as a single number. | 32 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation:
\[
\frac{\cos \alpha \cos \frac{\beta}{2}}{\cos \left(\alpha - \frac{\beta}{2}\right)} + \frac{\cos \beta \cos \frac{\alpha}{2}}{\cos \left(\beta - \frac{\alpha}{2}\right)} = 1
\]
Derive the relationship between cos α and cos β, and calculate the value of cos α + cos β. Express your answer as a sing... | 1 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
A school plans to select 3 students from 5 candidates to serve as volunteers for the athletics, swimming, and ball games. It is known that Student A cannot serve as a volunteer for the swimming competition. Calculate the total number of different arrangements. Express your answer as a whole number. | 48 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Combinations"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = x / (3m + 6054), find the sum of f(1) + f(2) +... + f(m + 2017). Express your answer as a single fraction. | \frac{m+2017}{6} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence $\{a_n\}$, if $a_9= \frac{1}{2}a_{12}+6$, calculate the sum of the first 11 terms of the sequence $S_{11}$. Express your answer as a single numerical value. | 132 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
The sum of three numbers is 98. The ratio of the first to the second is 2/3, and the ratio of the second to the third is 5/8. Calculate the value of the second number. Express your answer as a whole number. | 30 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Find the value of a for which the equation ||x-2|-1| = a has exactly three integer solutions. | 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
A high school with 24 classes in the second grade is conducting a survey using a systematic sampling method. 4 classes are selected for the survey, and the sum of the selected class numbers is 52. Using the systematic sampling method, determine the smallest class number drawn. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence ${a_n}$, where $a_2=4$ and $a_4=2$, find the sum of the first 5 terms, $S_5$. Express your answer as a single numerical value. | 15 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
A line $l$ passes through the point $P(-1,2)$ and has a slope of $45^{\circ}$. Find the equation of line $l$ in standard form (Ax + By + C = 0). | x - y + 3 = 0 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
A polygon has each of its interior angles equal to 140°. Calculate the number of its sides. Express your answer as a whole number. | 9 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Polygons"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
A number is randomly selected from the interval [-π, π]. What is the probability that the value of the function y = cos(x) falls within the range [-√3/2, √3/2]? Express your answer as a simplified fraction or decimal within the boxed format. | \frac{2}{3} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Convert the binary number $101101_{(2)}$ to an octal number. Express your answer in octal notation (e.g., $XX_{(8)}$). | 55_{(8)} | big_math | [
"Mathematics -> Other -> Other"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Let $U=\mathbb{R}$, $A=\{x|x>0\}$, $B=\{x|x>1\}$. Find the intersection of $A$ and the complement of $B$ in $U$, denoted as $A\cap \complement_{U}B$. Express your answer in set notation. | \{x|0<x\leq 1\} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $x - y = 3xy$, calculate the value of $\frac{1}{x} - \frac{1}{y}$. Express your answer as a single number. | -3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
A company schedules 6 employees for duty during the New Year's holiday (January 1 to January 3), with 2 people scheduled each day and each person working one day. Calculate the probability that employee A does not work on January 1. Express your answer as a simplified fraction. | \frac{2}{3} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
The number $2.5252525\ldots$ can be written as a fraction. When reduced to lowest terms, find the sum of the numerator and denominator of this fraction. Express your answer as a whole number. | 349 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Given that 4 empty mineral water bottles can be exchanged for 1 bottle of mineral water, and there are 16 empty mineral water bottles, if no money is paid, what is the maximum number of bottles of mineral water that can be drunk? Express your answer as a whole number of bottles. | 5 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Cut a piece of iron wire that is $$\frac {4}{5}$$ meters long into 3 equal parts. What fraction of the total length does each part represent? Express your answer as a simplified fraction. | \frac{1}{3} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Point $P(-2,-3)$ is moved $1$ unit to the left and then $3$ units up. What are the coordinates of the resulting point? Express your answer as an ordered pair $(x, y)$. | (-3,0) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
The line $y=kx$ is perpendicular to the line $y=2x+1$. Find the value of $k$. Express your answer as a single number. | -\frac{1}{2} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given the circle O equation: $x^2 + y^2 = 4$ and the circle C equation: $x^2 + y^2 + 4x - 4y + 4 = 0$, find the equation of line $l$ with respect to which the two circles are symmetric. Express your answer as a linear equation in the form $Ax + By + C = 0$. | x - y + 2 = 0 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
In a news event competition held at school, 40 students who knew that the "Zhurong rover" successfully reached Mars had a frequency of 0.8. Calculate the total number of students participating in the competition. Express your answer as a whole number. | 50 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
In the range of 0°-360°, find the angle that has the same terminal side as -510°. Express your answer in degrees (°). | 210° | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions -> Other"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
A service lift can carry a maximum of either 12 identical large packages or 20 identical small packages. If 9 large packages are already in the lift, what is the greatest number of small packages that can also be carried? Express your answer as a whole number. | 5 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Given that sin(π - α) = -√5/3 and α ∈ (π, 3π/2), find the value of sin(π/2 + α/2). Express your answer as a fraction with a radical. | -\frac{\sqrt{6}}{6} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence $a_n = 1 + \frac{1}{2} + \frac{1}{3} +... + \frac{1}{3n-1}$, calculate the value of $a_{n+1} - a_n$. Express your answer as a sum of fractions. | \frac{1}{3n} + \frac{1}{3n+1} + \frac{1}{3n+2} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Select 4 students to participate in a relay race from a group of 8 students, which includes 6 boys and 2 girls, using stratified sampling by gender. If girls are not to be the last runner, find the number of different arrangements. Express your answer as a single numerical value. | 720 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0 | 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.