instruction_seed stringlengths 40 1.39k | answer stringlengths 1 95 | source stringclasses 1
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In the complex plane, consider a regular 20-gon inscribed in the unit circle with vertices corresponding to complex numbers \( z_{1}, z_{2}, \cdots, z_{20} \). Determine the number of distinct points corresponding to the complex numbers \( z_{1}^{1995}, z_{2}^{1995}, \cdots, z_{20}^{1995} \). Express your answer as a w... | 4 | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
The point P(2,-5) is symmetric with respect to the x-axis at point Q. Find the coordinates of point Q. Express your answer as a coordinate (x, y). | (2,5) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
What is the largest prime factor of \(106^{2} - 15^{2}\)? Express your answer as a single integer. | 13 | big_math | [
"Mathematics -> Number Theory -> Prime Numbers"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
In $\triangle ABC$, $\angle B=90^{\circ}$, $\vec{AB}=(1,2)$, $\vec{AC}=(3,\lambda)$. Calculate the value of $\lambda$. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of $(-3)^0$. Express your answer as a single numerical value. | 1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given the polar equation of a circle $\rho=\sqrt{2}(\cos \theta+\sin \theta)$, find the coordinates of the center of this circle in polar coordinates. | \left(1, \frac{\pi}{4}\right) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Two students went to an organization for a job interview together. Before the interview, the person in charge of the organization told them, "We are going to hire 3 people from the interviewees, without considering the level of the candidates, the probability of both of you being hired at the same time is $\frac{1}{15}... | 10 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
A foreign investor plans to invest in 3 different projects in 4 candidate cities, with no more than 2 projects in the same city. Calculate the total number of different investment schemes the foreign investor can choose from. Express your answer as a single integer. | 60 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given $\cos\left(\alpha - \frac{\pi}{4}
ight) = \frac{\sqrt{2}}{4}$, calculate the value of $\sin 2\alpha$. Provide your answer as a single value. | -\frac{3}{4} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the plane vectors $\overrightarrow {a}$ and $\overrightarrow {b}$ satisfy $\overrightarrow{a}=(\sqrt{3},1)$, $|\overrightarrow {b}|=4$, and $(\overrightarrow {a}-2\overrightarrow {b})\bot \overrightarrow {a}$, calculate the magnitude of the vector $\overrightarrow{a}-\overrightarrow{b}$. Express your answer ... | 4 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given the mapping $f: (x, y) \rightarrow (x+2y, 2x-y)$, find the ordered pair $(x, y)$ that maps to $(3, 1)$ under $f$. Express your answer as an ordered pair $(x, y)$. | (1, 1) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $x$, $y$, $a \in \mathbb{R}^+$, and $\sqrt{x} + \sqrt{y} \leq a \sqrt{x+y}$ always holds, find the minimum value of $a$. Express your answer as a single value. | \sqrt{2} | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Define a new operation "$*$" within the range of positive integers, where $k*n$ represents the sum of $n$ consecutive positive integers starting from $k$. Given the equation $3*n=150$, calculate the value of $n$ that satisfies this equation. Express your answer as a single positive integer. | 15 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
An RNA molecule consists of 100 bases, and each base can be one of four different types: A, C, G, U. These bases can appear in any order. Calculate the total number of different RNA molecules. Express your answer as an exponential expression, clearly indicating the base and exponent. | 4^{100} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Other"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Four different products are arranged in a row for an exhibition, with the requirement that products A and B must have at least one other product between them. Calculate the number of different arrangement methods that satisfy this condition. | 12 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = 1/3*x^3 + x, find the solution set of the inequality f(2-x^2) + f(2x+1) > 0. Express your answer in interval notation, enclosing the interval in parentheses. | (-1, 3) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a$, $b$, $c \geqslant 0$, and $a^{2}+b^{2}+c^{2}=3$, find the maximum value of $ab+bc+ca$. Express your answer as a single number in the form $\boxed{[answer]}$. | 3 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Find the condition on b for the function f(x) = x^3 - 3bx + 3b to have a local minimum. Express your answer as an inequality or equality condition on b, using standard mathematical notation. | b > 0 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Find the minimum value of the function $y=\log_{2}\left(x+\frac{1}{x-1}+5\right)$ where $x>1$. Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
The origin and the point (1,1) are on opposite sides of the line x+y-a=0. Find the range of values for a. Express your answer in interval notation. | (0,2) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression: \(\frac{1^{2}-3^{2}+5^{2}-7^{2}+\cdots+97^{2}-99^{2}}{1-3+5-7+\cdots+97-99}\). Provide your answer as a single numerical value. | 100 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function y = 1/2 * (x^2 - 100x + 196 + |x^2 - 100x + 196|), calculate the sum of the function values when the variable x takes on the 100 natural numbers 1, 2, 3,..., 100. Express your answer as a single integer. | 390 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions",
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
If sin(α/2) = √3/3, calculate the value of cos(α) using the double angle formula cos(α) = 1 - 2sin^2(α/2). Express your answer as a fraction. | 1/3 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given the set $M=\{\left(x,y\right)\left|\right.x+y\leqslant 1,x\in \mathbb{N},y\in \mathbb{N}\}$, find all elements in set M. Express your answer as a set of ordered pairs. | \{\left(0,0\right),\left(0,1\right),\left(1,0\right)\} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given that July 3, 2030 is a Wednesday, determine the day of the week for July 14, 2030. Provide your answer as the full day of the week. | Sunday | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the line that passes through the point (1,0) and is parallel to the line x-2y-2=0. Express your answer in the standard form of a line equation (Ax + By + C = 0), including all necessary coefficients and constants. | x - 2y - 1 = 0 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
What is the smallest number of rectangles, each measuring 2 cm by 3 cm, that can fit together without overlap to form a rectangle whose sides are in the ratio 5:4? Express your answer as a single integer. | 30 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Dana creates a "rectangular spiral" by drawing line segments of lengths, in cm, of \(1,1,2,2,3,3,4,4, \ldots\). Dana's pen runs out of ink after the total of all the lengths he has drawn is \(3000 \, \text{cm}\). Find the length of the longest line segment that Dana draws. Express your answer as a single integer (lengt... | 54 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Determine the number of different values of y that occur in the ordered pairs (x, y) that satisfy the equation (x - y)^2 + x^2 = 25, where x and y are integers and x >= 0. Express your answer as a whole number. | 5 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Find the minimum value of n such that the expansion of (x^2 + 1/(2x^3))^n contains a constant term. Express your answer as a single integer. | 5 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given the following expressions: $①\sqrt{32}$; $②6$; $③\sqrt{-12}$; $④\sqrt{-m}\left(m\leqslant 0\right)$; $⑤\sqrt{a^{2}+1}$; $⑥\sqrt[3]{5}$, how many of these expressions represent quadratic radicals? Provide your answer as a whole number. | 3 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
A stone is dropped into a well and the report of the stone striking the bottom is heard 7.7 seconds after it is dropped. Assume that the stone falls 16t^2 feet in t seconds and that the velocity of sound is 1120 feet per second. Calculate the depth of the well in feet, rounded to the nearest whole number. | 784 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Given that f(x) is an odd function with a domain of R, and it satisfies the property f(x+4) = f(x) for all x in R, find the value of f(43) if f(1) = 3. Express your answer as a single number. | -3 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence {a_n}, given that a_7 = 19 and a_2 + a_8 = 26, calculate the sum of the first 5 terms (S_5) of the sequence. Express your answer as a single integer. | 35 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Find the solution set of the inequality $\dfrac{x}{2x-1} > 1$. Express your answer in interval notation. | \left( \dfrac{1}{2},1 \right) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
If $f(x)$ is differentiable at $x=x_{0}$, find the value of $\lim_{{\Delta x\to 0}}\frac{{f({{x_0}+\Delta x})-f({{x_0}})}}{{2\Delta x}}$. Express your answer in terms of $f'(x_0)$. | \frac{1}{2}f'(x_0) | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given a complex number $z=1+2i$ (where $i$ is the imaginary unit), find the imaginary part of its conjugate $\overline{z}$. Express your answer as a single integer. | -2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Consider sphere O which is the circumscribed sphere around a cube with edge length 4. If a plane passes through the midpoint of an edge of the cube and intersects sphere O, calculate the area of the smallest such section. Express your answer in terms of pi (π). | 4\pi | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x)= \begin{cases} (3-a)x-3, & x\leqslant 7 \\ a^{x-6}, & x > 7 \end{cases}$, find the range of values for the real number $a$ that ensures the function is monotonically increasing. Express your answer as an interval. | [\frac{9}{4},3) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
If $\left (a+\frac{1}{a} \right )^2=3$, then find the value of $a^3+\frac{1}{a^3}$. Express your answer as a single number. | 0 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the ratio of the speeds of the hour hand and the minute hand on a clock face. Express your answer as a simplified ratio, where the first number represents the speed of the hour hand and the second number represents the speed of the minute hand. | 1:12 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
In triangle $\triangle ABC$, $AB=2$, $AC=\sqrt{2}$, and $B=30^{\circ}$. Calculate the value of angle $A$ in degrees, listing all possible values if multiple solutions exist. | 105^{\circ} \text{ or } 15^{\circ} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $y=\sqrt{ax^2+4x+1}$ with a range of $[0,+\infty)$, determine the range of values for $a$ that satisfies this condition. Express your answer using interval notation. | [0,4] | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
A regular n-gon is inscribed in a circle with radius R, and the area of this n-gon is 3R^2. Find the value of n. Express your answer as a whole number. | 12 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given the equations p = 2, p * q = 20, p * q * r = 202, and p * q * r * s = 2020, calculate the sum p + q + r + s. Express your answer as a single number, rounded to one decimal place if necessary. | 32.1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
A triangle has one vertex at the pole, and the polar coordinates of the other two vertices are $P_1(5, 109^{\circ})$ and $P_2(4, 49^{\circ})$, respectively. Calculate the area of triangle $P_1OP_2$. Express your answer in simplest radical form. | 5 \sqrt{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
The opposite of a number is defined as the number that, when added to the original number, results in zero. What is the opposite of 2023? Express your answer as an integer. | -2023 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given the equations $3^{x-2}+ \frac {x}{3}=2$ and $\log_{3}(x-1)+x=6$, find the sum of their roots $a$ and $b$. Express your answer as a single integer. | 7 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
The English word "hello" has 5 letters: h, e, l, l, o. Calculate the total number of possible arrangements of these letters, taking into account the identical letters "l". Then, subtract the correct arrangement to find the number of possible incorrect arrangements. Express your answer as a whole number. | 59 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
The common ratio of a geometric sequence $\{a_n\}$ is 2, and all terms of the sequence are positive. Given that $a_3a_{11}=16$, find the value of $a_5$. Express your answer as a single numerical value. | 1 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Given that \\(\\sin \\alpha = \\frac{3}{5}\\) where \\(\\frac{\\pi}{2} < \\alpha < \\pi\\), and \\(\\tan (\\pi - \\beta) = \\frac{1}{2}\\), calculate the value of \\(\\tan (\\alpha - 2\\beta)\\). Express your answer as a simplified fraction. | \\frac{7}{24} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
A recipe calls for \( 4 \frac{1}{2} \) cups of flour. If you only make half of the recipe, calculate the exact amount of flour you need in cups, expressing your answer as a mixed number. | 2 \frac{1}{4} | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Fractions"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of $\frac{2^{8}}{8^{2}}$. Express your answer as a single number. | 4 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Four families visit a tourist spot that has four different routes available for exploration. Calculate the number of scenarios in which exactly one route is not visited by any of the four families. | 144 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Find the center of the circle given by the equation $x^2+y^2+2y=1$. Express your answer as a coordinate pair (x, y). | (0, -1) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Given the set $A = \{1, 2, 3\}$, find the number of proper subsets of $A$. Express your answer as a whole number. | 7 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
To understand the performance of a certain type of tire, 8 samples were randomly selected for testing. The maximum mileage for each sample (in units of 1,000 km) were: 96, 112, 97, 108, 99, 104, 86, and 98. Determine the median value of the given mileage values. Express your answer as a single number, including one dec... | 98.5 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a} = (1-t, 2t-1, 0)$ and $\overrightarrow{b} = (2, t, t)$ (where $t \in \mathbb{R}$), calculate the minimum value of $|\overrightarrow{b} - \overrightarrow{a}|$. Express your answer as a numerical value, including the square root symbol if necessary. | \sqrt{2} | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Geometry -> Other",
"Mathematics -> Applied Mathematics -> Other"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Two wheels are spun, and two numbers are selected by the pointers. Calculate the probability that the sum of the two selected numbers is even. Express your answer as a fraction in simplest form. | \frac{1}{2} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Determine the coefficient of the term containing $x^2$ in the binomial expansion of ${\left( \frac{x^2}{2} - \frac{1}{\sqrt{x}} \right)}^6$. Express your answer as a fraction. | \frac{15}{4} | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Determine the smallest positive period of the function $y=\sin (-\frac{x}{2}+\frac{\pi }{4})$. Express your answer in terms of π. | 4\pi | big_math | [
"Mathematics -> Precalculus -> Functions -> Trigonometric Functions"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Talulah plants 60 tulip bulbs. When they flower, she notes that half are yellow; one third of those which are not yellow are red; and one quarter of those which are neither yellow nor red are pink. The remainder are white. What fraction of the tulips are white? Express your answer as a simplified fraction. | \frac{1}{4} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Points P1(x1,y1) and P2(x2,y2) move on two parallel lines l1: x-y-5=0 and l2: x-y-15=0 respectively. Find the minimum distance from the midpoint P of P1P2 to the origin O. Express your answer as a numerical value, including the square root symbol if necessary. | 5\sqrt{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equations x^2 y z^3 = 7^3 and x y^2 = 7^9, calculate the product x y z. Express your answer as a power of 7. | 7^4 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Find the coefficient of x^2 in the expansion of (1+x)(1-2x)^5. Express your answer as a single integer. | 30 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Evita wants to write the numbers 1 to 8 in the boxes of the grid shown, so that the sums of the numbers in the boxes in each row are equal and the sums of the numbers in the boxes in each column are equal. She has already written numbers 3, 4, and 8, as shown. What number should she write in the shaded box? Express you... | 5 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Find the derivative of the function y = (x-2)^2 at x = 1. Express your answer as a single number. | -2 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Let $\overrightarrow{a}$ and $\overrightarrow{b}$ be non-zero vectors, where $|\overrightarrow{b}| = 2|\overrightarrow{a}|$. Four pairs of vectors $\overrightarrow{x_{1}}$, $\overrightarrow{x_{2}}$, $\overrightarrow{x_{3}}$, $\overrightarrow{x_{4}}$ and $\overrightarrow{y_{1}}$, $\overrightarrow{y_{2}}$, $\overrightarr... | \frac{\pi}{3} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $x^{2}-4|x|+5-m=0$, find the range of the real number $m$ such that the equation has four distinct real solutions. Express your answer as an interval (lower bound, upper bound). | (1,5) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given a point M on the parabola $y^2=12x$ whose distance to the focus is 8, find the x-coordinate of point M. Express your answer as a single numerical value. | 5 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of sets A that satisfy the condition {a, b} ⊆ A ⊆ {a, b, c, d, e}. Express your answer as a whole number. | 8 | big_math | [
"Mathematics -> Applied Mathematics -> Combinatorics"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
A woman weaves fabric in such a way that each day she weaves twice as much as the previous day. If she weaves a total of 5 feet of fabric in 5 days, what is the minimum number of days she needs to weave at least 25 feet of fabric? Express your answer as a whole number. | 8 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of \( (x + 1)^{3} \) when \( x = -2 \). Provide your answer as a single integer. | -1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Polynomial Operations"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
The ratio of the number of young, middle-aged, and elderly employees in a certain unit is 7:5:3. If a stratified sampling method is used to draw a sample and there are 14 young employees in the sample, calculate the total sample size. Provide your answer as a whole number. | 30 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Other"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
The volume of the rectangular prism shown is 60 cm³. The base of the prism is a rectangle with a length of 4 cm and a width of 5 cm. Find the height of the prism. Express your answer in centimeters as a whole number. | 3 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Volume"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
A question can be solved by A, B, and C with probabilities of $\frac{1}{2}$, $\frac{1}{3}$, and $\frac{1}{4}$ respectively. If A, B, and C independently solve the question, find the probability that exactly one person solves it. Express your answer as a fraction or decimal. | \frac{11}{24} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z= \frac {a}{1-2i}+bi$ ($a, b \in \mathbb{R}$) is an "ideal complex number", derive the relationship between a and b in the form of an equation, based on the definition that the real part and the imaginary part of a complex number are opposites of each other. | 3a+5b=0 | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
There are 5 math books, 4 physics books, and 5 chemistry books on a bookshelf. How many different ways are there to pick one book? Express your answer as a whole number. | 14 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = 3sin(2x), if the graph of function f(x) is shifted to the right along the x-axis by π/8 units, find the analytical expression of the new function y = g(x). Express your answer as a function of x, using the format g(x) =.... | g(x) = 3sin(2x - π/4) | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Two circles $(x^2+y^2+2ax+a^2-4=0)$ and $(x^2+y^2-4by-1+4b^2=0)$ have exactly three common tangents. If $a \in R$ and $ab \neq 0$, find the minimum value of $(\frac{1}{a^2 }+ \frac{1}{b^2 })$. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
In triangle $\triangle ABC$, if $\overrightarrow {BD}=2 \overrightarrow {DC}$, and $\overrightarrow {AD}=x \overrightarrow {AB}+y \overrightarrow {AC}$, find the value of $\frac {y}{x}$. Express your answer as a single numerical value. | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other",
"Mathematics -> Algebra -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given the polar equation $\rho = 4\cos \theta$, convert it to Cartesian coordinates and find the diameter of the resulting circle. Express your answer as a single numerical value. | 4 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Given $a+b+c=0$ and $a^2+b^2+c^2=4$, calculate the value of $a^4+b^4+c^4$. Express your answer as a single integer. | 8 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram shown, \(PQR\) is a straight line segment. Given that \(\angle SQP = 75^\circ\) and \(\angle QRS = 30^\circ\), find the measure of \(\angle QSR\). Express your answer in degrees. | 45^\circ | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Given the binomial expansion $(a - x)^5 = a_0 + a_1 x + a_2 x^2 + \dots + a_5 x^5$, if $a_2 = 80$, calculate the sum of coefficients $a_0 + a_1 + a_2 + \dots + a_5$. Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
A computer user has a budget of 500 yuan to purchase single sheet software priced at 60 yuan each and boxed disks priced at 70 yuan each. The user must buy at least 3 pieces of software and at least 2 boxes of disks. How many different purchasing methods are possible within these constraints? Express your answer as a s... | 7 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given the experimental range is $[628, 774]$ and the good point is 718, use the 0.618 method to find the addition point value to be tested at this time. Express your answer as a single numerical value. | 684 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(1,-2,1)$ and $\overrightarrow{b}=(1,0,2)$, calculate the difference $\overrightarrow{a}-\overrightarrow{b}$. Express your answer as a vector in the form $(x, y, z)$. | (0, -2, -1) | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
The sum of the binomial coefficients in the expansion of $(1+x)^n$ is 128. Find the value of $n$. Express your answer as a whole number. | 7 | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z$ satisfies $z=\frac{1-i}{1+i}$ ($i$ is the imaginary unit), calculate the modulus of $z$, denoted as $|z|$. | 1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence with sums S_{5}=10 and S_{10}=50, calculate the sum of the first 15 terms, S_{15}. Express your answer as a single integer. | 120 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
A student attempted a problem by calculating the sum of two polynomials A and B, but mistakenly treated it as the difference A - B. As a result, they obtained 9x^2 - 2x + 7. If B = x^2 + 3x - 2, then calculate the correct sum A + B. Express your answer as a polynomial in standard form, including all terms and coefficie... | 11x^2 + 4x + 3 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence $\{a_n\}$, $a_2 + a_3 = 9$ and $a_4 + a_5 = 21$. Find the common difference of the sequence. Express your answer as a single integer. | 3 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given that \\( \\{a_n\\} \\) is an arithmetic sequence, if \\( a_3 + a_5 + a_{12} - a_2 = 12 \\), then what is the value of \\( a_7 + a_{11} \\)? Express your answer as a single number. | 12 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
The sum of the first 3 terms of an arithmetic sequence is 12, and the fourth term is 0. Find the sixth term of the sequence. Express your answer as a single integer. | -4 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Find the conjugate of the complex number $\left( \frac{5}{2-i} \right)^{2}$. Express your answer as a complex number in the form a + bi. | 3-4i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Given $\sin\alpha = \sin\theta + \cos\theta$, and $\sin^2\beta = 2\sin\theta \cdot \cos\theta$, find the relationship between $\sin^2\alpha$ and $\sin^2\beta$. Express your answer as an equation. | \sin^2\alpha = 1 + \sin^2\beta | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the two numbers \(a\) and \(b\) both lie between -5 and 10 inclusive, calculate the largest possible value of \(a - b\). Express your answer as a single integer. | 15 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
At a certain intersection, the traffic signal lights are red for $30$ seconds, green for $25$ seconds, and yellow for $5$ seconds every minute. What is the probability of seeing the yellow light when you look up at the traffic light? Express your answer as a simplified fraction. | \frac{1}{12} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Find the symmetric point of (1, 1, 1) with respect to the z-axis. Express your answer as a coordinate (x, y, z). | (-1, -1, 1) | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
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