instruction_seed stringlengths 40 1.39k | answer stringlengths 1 95 | source stringclasses 1
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Given an ellipse $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ (with $a > b > 0$), a perpendicular line is drawn from the left focus $F_1$ to the x-axis, which intersects the ellipse at points P and Q. If triangle $\triangle{PQF_2}$ (with $F_2$ being the right focus) is an equilateral triangle, calculate the eccentricity o... | \frac{\sqrt{3}}{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
In a parking lot, there are 24 vehicles, among which one car has 4 wheels, and one motorcycle has 3 wheels. The parking lot only contains cars and motorcycles, and together, these vehicles have a total of 86 wheels. Calculate the number of motorcycles in the parking lot. | 10 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
On a 3x3 grid with colored squares, how many different routes are there from square S to square T that pass through every other square exactly once? Each step is to an adjacent square of a different color. Provide your answer as a single integer. | 2 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Find the minimum value of the function $f(x)=|3-x|+|x-7|$. Express your answer as a single number. | 4 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Let the set $U=\{x|x < 5, x\in\mathbb{N}^{*}\}$, and $M=\{x|x^{2}-5x+6=0\}$. Calculate the complement of $M$ in $U$, denoted as $\complement_{U}M$, and express your answer as a set of integers using standard set notation. | \{1,4\} | big_math | [
"Mathematics -> Discrete Mathematics -> Logic"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation of the ellipse $\dfrac {x^{2}}{a^{2}}+ \dfrac {y^{2}}{b^{2}}=1$ with an eccentricity of $\dfrac { \sqrt {3}}{2}$, find the value of $\dfrac {a}{b}$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
What is the result of the calculation ${(3\sqrt{2})}^{2}$? Express your answer as a numerical value. | 18 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Find the greatest common divisor (GCD) of 136 and 1275. Provide your answer as a single integer in the form \boxed{ }. | 17 | big_math | [
"Mathematics -> Number Theory -> Greatest Common Divisors (GCD)"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
There is a sequence, the first term of which is 1. For all n >= 2, the product of the first n terms of this sequence is n^2. Find the sum of the third and fifth terms of this sequence. Express your answer as a simplified fraction. | \frac{61}{16} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
The ages of three contestants in the Pascal Contest are 14 years, 9 months; 15 years, 1 month; and 14 years, 8 months. Calculate the average (mean) age of the contestants, expressing your answer in years and months. | 14 years, 10 months | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Find the domain of the function $f(x)=\log_{2}(1-2x) + \frac{1}{x+1}$. Express your answer in interval notation. | (-\infty,-1)\cup(-1, \frac{1}{2}) | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Let vector $\overrightarrow {m} = (2x-1, 3)$, and vector $\overrightarrow {n} = (1, -1)$. If $\overrightarrow {m} \perp \overrightarrow {n}$, find the value of the real number $x$. Express your answer as a single real number. | 2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
In the geometric sequence {a_n}, the value of a_5 is given as 2. Find the product of a_1 and a_9. Express your answer as a single number. | 4 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Given $a>0$, $b>0$, and $\log_{4}\left(\frac{1}{a}+\frac{1}{b}
ight)=\log_{2}\sqrt{\frac{1}{ab}}$, find the minimum value of $\frac{1}{a}+\frac{1}{b}$. Express your answer as a single numerical value. | 4 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\cos \alpha = \frac{5}{13}$ and $\alpha \in (\frac{3\pi}{2}, 2\pi)$, calculate the value of $\cos (\alpha - \frac{\pi}{4})$. Express your answer as a fraction with a radical, and include the boxed notation around your final answer. | -\frac{7\sqrt{2}}{26} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
How many squares are exactly four greater than a prime number? Express your answer as a whole number. | 1 | big_math | [
"Mathematics -> Number Theory -> Prime Numbers"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
The first four terms of an arithmetic sequence are $a, x, b, 2x$. Find the ratio of $a$ to $b$. Express your answer as a simplified fraction. | \frac{1}{3} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Find the total number of non-negative integer solutions to the equation $x_1+2x_2+x_3+x_4+\ldots+x_{10}=3$. Express your answer as a single integer. | 174 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
The product of three positive integers is 42, and the sum of two of these integers is 9. Find the third integer. Express your answer as a single positive integer. | 3 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
A 9-ounce shaving lotion contains 50% alcohol. How many ounces of water need to be added to reduce the alcohol concentration to 30%? Express your answer as a whole number of ounces. | 6 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\alpha \in (0, \frac{\pi}{2})$, and $\tan 2\alpha = \frac{\cos \alpha}{2-\sin \alpha}$, calculate the value of $\tan \alpha$. Express your answer as a fraction, including any necessary square roots. | \frac{\sqrt{15}}{15} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
The temperature in the morning was $-2^{\circ}\mathrm{C}$, rose by $13^{\circ}\mathrm{C}$ at noon, and then dropped by $8^{\circ}\mathrm{C}$ at midnight. What was the temperature at midnight? Express your answer in degrees Celsius ($^{\circ}\mathrm{C}$) as an integer. | 3 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
What is the square root of $\sqrt{9}$, considering both the positive and negative roots? Express your answer using the ± symbol and in simplest radical form. | \pm\sqrt{3} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a$ and $b$ are non-zero real numbers, analyze the relationship between the statements "$a > b$" and "$\frac{1}{a} < \frac{1}{b}$". Classify the relationship, stating whether "$a > b$" is sufficient, necessary, both, or neither for concluding "$\frac{1}{a} < \frac{1}{b}$". Provide your answer in a clear and ... | Neither | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the geometric sequence $\{a_n\}$ is increasing, and $S_n$ is the sum of the first $n$ terms of $\{a_n\}$. If $a_1$ and $a_3$ are the two roots of the equation $x^2 - 5x + 4 = 0$, calculate the value of $S_5$. | 31 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Quadratic Functions"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation (m-2)x^2 - 3x = 0, find the condition or range of values for m that ensures the equation remains a quadratic equation in terms of x. Express your answer as an inequality or condition for m, using mathematical notation. | m \neq 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of integer solutions for the inequality \( |x| < 3 \pi \). Express your answer as a single integer. | 19 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
A high school has 400 male students and 300 female students in the first grade. To investigate the preferences of first-grade students for liberal arts or sciences in the second grade, a random sample of 35 students will be selected using stratified sampling. Calculate the number of male students to be sampled. Express... | 20 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x) = \begin{cases} \sin x, & x \geq 0 \\ -x^{2}-1, & x < 0 \end{cases}$, find the range of $k$ such that $f(x) \leq kx$. Express your answer as an interval, including the endpoints. | [1, 2] | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Let the domain of the function y=f(x) be R and satisfy y=f(x+2) is an odd function. Find the value of f(2). | 0 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Given \(x, y, z \in \mathbb{R}^{+}\), and \(\frac{1}{x}+\frac{2}{y}+\frac{3}{z}=1\), find the minimum value of \(x+\frac{y}{2}+\frac{z}{3}\). Express your answer as a single number. | 9 | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets $A=\{1,2,3\}$ and $B=\{4,5\}$, let $M=\{x|x=a+b, a\in A, b\in B\}$. Calculate the number of proper subsets of set $M$. Express your answer as a single numerical value. | 15 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = cos(pi/2 + x) + sin^2(pi/2 + x), with x in [-pi, 0], find the maximum value of f(x). Express your answer as a single numerical value. | \frac{5}{4} | big_math | [
"Mathematics -> Precalculus -> Functions",
"Mathematics -> Calculus -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Cassandra sets her watch to the correct time at noon. At the actual time of 1:00 PM, she notices that her watch reads 12:57 and 36 seconds. Assuming that her watch loses time at a constant rate, what will be the actual time when her watch first reads 10:00 PM? Express your answer in the format HH:MM PM, without seconds... | 10:25 PM | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
The parametric equations of a line are given as $$\begin{cases} x=-t\cos20° \\ y=3+t\sin20° \end{cases}$$ (where $t$ is a parameter). Calculate the inclination angle of the line. Express your answer in degrees (°). | 160° | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
Find the minimum value of the function $f(x)=\sum_{n=1}^{19}{|x-n|}$. Express your answer as a single integer. | 90 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given a sequence $\{a_n\}$ where $a_1=1$ and $a_{n+1}=2a_n+1$, find the value of $a_4$. Express your answer as a single integer. | 15 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the center of circle $C$ is $(2,-3)$ and it passes through the point $(0,0)$, find the equation of the circle in standard form. | (x-2)^2 + (y+3)^2 = 13 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
If $|x| \leqslant \frac{\pi}{4}$, find the minimum value of the function $f(x)=\cos^2x+\sin x$. Express your answer as a single number in the form $\boxed{[answer]}$. | \frac{1-\sqrt{2}}{2} | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given a tetrahedron with face areas denoted by S_i (where i=1,2,3,4), and the perpendicular distances from an arbitrary point inside the tetrahedron to the i-th face as H_i (where i=1,2,3,4), it holds that S_i = ik (i=1,2,3,4). Calculate the sum 1H_1 + 2H_2 + 3H_3 + 4H_4 in terms of the volume V of the tetrahedron and ... | \frac{3V}{k} | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the sum of two numbers and their quotient are both 6, calculate the product of these two numbers minus their difference (greater minus smaller). Express your answer as a simplified fraction. | \frac{6}{49} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets $A=\{x|0<x\leq3, x\in \mathbb{N}\}$ and $B=\{x|y= \sqrt{x^{2}-9}\}$, calculate the set $A\cap(\complement_{\mathbb{R}}B)$. Express your answer as a set of integers, including all elements of the resulting set. | \{1,2\} | big_math | [
"Mathematics -> Set Theory -> Set Operations"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Given the line $l_1: x-2y-2=0$ and the line $l_2: x+y=0$, find the equation of the line $l_3$ that is symmetric to $l_1$ with respect to $l_2$. Express your answer as an equation of a line in standard form. | 2x - y - 2 = 0 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given that 0 < x < 1/2, find the maximum value of the function y = x(1-2x). Express your answer as a single numerical value. | \frac{1}{8} | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the complex number z and (z+2)^2 + 8i are both purely imaginary numbers, find the value of z. Express your answer as a complex number in the form bi, where b is a real number. | z = 2i | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Let the function $f(x)= \begin{cases} \left( \frac {1}{2}\right)^{x}, & x\leqslant 0 \\ f(x-1), & x > 0\end{cases}$. Calculate the value of $f(1+\log _{2}3)$. Express your answer as a single number. | \frac{4}{3} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the inequality |x+2| - |x+3| > m, find the range of real numbers m for which the inequality has solutions. Express your answer as an inequality involving m. | m < 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Xiao Shen is $a$ years old this year, and Xiao Wang is $\left(a-8\right)$ years old. After $\left(n+3\right)$ years, what will be the age difference between Xiao Shen and Xiao Wang? Express your answer in years as a whole number. | 8 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Find the difference between the two smallest integers greater than 1 that, when divided by any integer k where 2 <= k <= 11, give a remainder of 1. Express your answer as a single integer. | 27720 | big_math | [
"Mathematics -> Number Theory -> Congruences"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given that point P(m,2) lies on the y-axis, what is the value of m? Express your answer as a single number. | 0 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
A square has sides of length 10, and a circle centered at one of its vertices has radius 10. Calculate the area of the union of the regions enclosed by the square and the circle. Express your answer as the sum of a numerical value and a pi term. | 100 + 75\pi | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Solve the following equation and express your answer as a single value:
$$\sin(45^\circ)\sin(105^\circ) + \sin(45^\circ)\sin(15^\circ)$$ | \frac{\sqrt{3}}{2} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Jing purchased eight identical items for a total cost of $26. Calculate the cost per item in dollars, rounding to 2 decimal places if necessary. | 3.25 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
A rectangular colored paper has a length of 20 cm and a width of 15 cm. First, cut out the largest square possible, then from the remaining paper, cut out another largest square. Calculate the length of the remaining rectangular piece of paper. Express your answer in centimeters (cm). | 10 \text{ cm} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation a^2 + 3b = 33, where a and b are positive integers, calculate the value of ab. Express your answer as a single integer. | 24 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
A class meeting is preparing to select 4 students to speak from 7 students, including A and B, with the requirement that at least one of A and B must participate. When A and B both participate, their speeches cannot be adjacent. Calculate the total number of different speaking orders. | 600 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $y=f(x)$ is an increasing function on $\mathbb{R}$, and $f(2m) < f(9-m)$, find the range of the real number $m$. Express your answer in interval notation. | (-\infty, 3) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
In △QRS, point T is on QS with ∠QRT = ∠SRT. Suppose that QT = m and TS = n for some integers m and n with n > m and for which n + m is a multiple of n - m. Suppose also that the perimeter of △QRS is p and that the number of possible integer values for p is m^2 + 2m - 1. Calculate the value of n - m. Express your answer... | 4 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the result of $\left(-3\right)^{2}\div 4\times \frac{1}{4}$. Express your answer as a fraction in simplest form. | \frac{9}{16} | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given two lines $l_{1}$: $ax+(a+2)y+1=0$ and $l_{2}$: $x+ay-2=0$, find the real number(s) $a$ for which $l_{1}$ is parallel to $l_{2}$. Express your answer as a numerical value or a set of numerical values. | -1, 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a} = (1, n)$ and $\overrightarrow{b} = (-1, n-2)$, if $\overrightarrow{a}$ and $\overrightarrow{b}$ are collinear, find the value of $n$. Express your answer as a single numerical value. | 1 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given the polar equation of a line $\rho \sin (\theta +\frac{\pi }{4})=1$, calculate the distance from the pole to this line. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Given three points $P(5,2)$, $F_1(-6,0)$, and $F_2(6,0)$, find the length of the minor axis of the ellipse that has $F_1$ and $F_2$ as foci and passes through point $P$. Express your answer as a single number. | 6 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Solve the equation $(3x-1)(2x+4)=1$ for $x$. Express your answer as a fraction or with a square root, separating the two roots by 'or' if necessary. | \frac{-5 - \sqrt{55}}{6} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given two real numbers x and y are randomly selected from the interval [-1,0], calculate the probability that y < 3x. Express your answer as a simplified fraction. | \frac{1}{6} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
In the geometric sequence {a_n}, it is known that a_4 = 5. Find the product of a_3 and a_5. Express your answer as a single integer. | 25 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Fill the 25 integers from -10 to 14 into a 5x5 grid such that the sum of the numbers in each row, each column, and each diagonal is the same. Calculate this common sum. Express your answer as a single integer. | 10 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Solve the equation $2x+1=x-1$ for $x$. Express your answer as a single value. | -2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.765625 | 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"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = 4 + 2a^(x-1), find the coordinates of the fixed point P through which the graph of the function always passes. Express your answer as an ordered pair (x, y). | (1, 6) | big_math | [
"Mathematics -> Algebra -> Functions"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of \(10^{2} + 10 + 1\). Express your answer as a single integer. | 111 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
A person contracted the flu, and after two rounds of infection, a total of 196 people were infected. Calculate the average number of people infected by one person in each round of infection. Express your answer as a whole number. | 13 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
A store is promoting a certain product by offering a discount. The product is priced at $30$ dollars. The promotion is as follows: if you buy no more than $5$ items, you pay the original price; if you buy more than $5$ items at once, the excess items are discounted by $20\%$. If you have $270$ dollars, calculate the ma... | 10 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the greatest common divisor (GCD) of 1554 and 2405 using the division and remainder method. Express your answer as a single integer. | 37 | big_math | [
"Mathematics -> Number Theory -> Greatest Common Divisors (GCD)"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Express the number 17,600 in scientific notation. Provide your answer in the form $a \times 10^{b}$, where $a$ is the coefficient and $b$ is the power of 10. | 1.76 \times 10^{4} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
A bicycle travels at a constant speed of 15 km/h. A bus starts 195 km behind the bicycle and catches up to the bicycle in 3 hours. Calculate the average speed of the bus in km/h. Express your answer as a whole number. | 80 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given the one-variable quadratic equation $x^{2}-3x-4=0$ with two real roots $x_{1}$ and $x_{2}$, find the value of $x_{1}x_{2}-x_{1}-x_{2}$. Express your answer as a single integer. | -7 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Quadratic Functions"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence $\{a_{n}\}$ where $a_{3}+a_{4}+a_{5}=12$, find the sum of the first seven terms, $a_{1}+a_{2}+\ldots+a_{7}$. Express your answer as a single integer. | 28 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex equation $\frac{2-mi}{1+2i}=A+Bi$, where $m, A, B \in \mathbb{R}$ and $A+B=0$, find the value of $m$. Express your answer as a single real number. | -\frac{2}{3} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
The distance from the right focus of the hyperbola x^2/3 - y^2 = 1 to the line x - sqrt(3)y = 0 is to be determined. Calculate this distance and provide your answer as a single number. | 1 | big_math | [
"Mathematics -> Geometry -> Non-Euclidean Geometry -> Hyperbolic Geometry"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
What is the greatest multiple of 4 that is less than 100? Express your answer as a whole number. | 96 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Find the imaginary part of the complex number $\frac{5i}{2-i}$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
In a WeChat group's red packet snatching event, the total amount of the red packet is 9 yuan, and it is randomly divided into 1.49 yuan, 1.31 yuan, 2.19 yuan, 3.40 yuan, and 0.61 yuan, making 5 parts for A, B, and other 3 people to snatch, with each person only allowed to snatch once. Calculate the probability that the... | \frac{2}{5} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Find the coefficient of x^3 in the expansion of (2-x)(1-x)^4. Express your answer as an integer. | -14 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations -> Other"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Given three line segments of lengths 4, 6, and a, apply the triangle inequality theorem to find the valid range of values for a that allows the formation of a triangle. Express your answer as an inequality or a range. | 2 < a < 10 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Solve the inequality $4-x^2<0$ and express the solution set in interval notation. | (-\infty,-2) \cup (2,\infty) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Determine the number of zeros at the end of the product \(25 \times 25 \times 25 \times 25 \times 25 \times 25 \times 25 \times 8 \times 8 \times 8\). Express your answer as a whole number. | 9 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Given the parabola $y^2 = 12x$, find the equation of its directrix. Express your answer in the form $x = c$, where $c$ is a constant. | x = -3 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Let $z=\frac{5}{{2-i}}$ (where $i$ is the imaginary unit), then calculate the value of $z\cdot \overline{z}$. Express your answer as a single number. | 5 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
A magazine contains six photos: three photos of famous people and three photos of them as babies, but the baby photos are not labeled. Readers need to match the baby photos to the correct famous person. Assuming each photo match is equally likely, what is the probability that a reader randomly selects the correct baby ... | \frac{1}{6} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Combinations"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
A piece of rope is randomly cut into two parts. Calculate the probability that the longer segment is at least x times the shorter segment. Express your answer as a fraction or ratio involving x. | \frac{2}{x+1} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given that angle α = 2kπ - π/5 (k ∈ ℤ), if angle θ has the same terminal side as angle α, calculate the value of y = sin θ / |sin θ| + cos θ / |cos θ| + tan θ / |tan θ|. Express your answer as a single integer. | -1 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Given the functional equation \( f(a+b) = f(a) \cdot f(b) \) and \( f(1) = 1 \), calculate the sum of the sequence:
\[
\frac{f(2)}{f(1)} + \frac{f(3)}{f(2)} + \frac{f(4)}{f(3)} + \cdots + \frac{f(1988)}{f(1987)}
\]
Express your answer as a single integer. | 1987 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Given a geometric sequence {a_n} with the first term a_1 < 0, derive the necessary and sufficient condition for the common ratio q that makes the sequence increasing. Express your answer as an inequality. | 0 < q < 1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
A computer executes the following program:
1. Initial values: x = 3, S = 0.
2. x = x + 2.
3. S = S + x.
4. If S >= 10000, go to step 5; otherwise, go back to step 2.
5. Print the final value of x.
What is the final value of x printed by the program? | 201 | big_math | [
"Mathematics -> Discrete Mathematics -> Algorithms"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given the operation \( k
abla m = k(k-m) \), calculate the value of \( (5
abla 1) + (4
abla 1) \). Provide your answer as a single number. | 32 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Five line segments are drawn inside a rectangle, creating a decagon (10-sided polygon). Find the sum of the six marked angles. Express your answer in degrees as a whole number. | 1080^\circ | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Polygons"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given the two equations $x^2+ax+b=0$ and $x^2+bx+a=0$ have only one common root, calculate the value of $a+b$. Express your answer as a single integer. | -1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Suppose the function $y = f(x)$ has an inverse $y = f^{-1}(x)$, and the graph of the function $y = x - f(x)$ passes through the point $(1,2)$. Calculate the point through which the graph of the function $y = f^{-1}(x) - x$ must pass. Express your answer as an ordered pair $(x,y)$. | (-1,2) | big_math | [
"Mathematics -> Algebra -> Functions -> Other"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Given that the domain of the function $y=f(x)$ is $[-1,3]$, determine the domain of the function $y=f(3x-2)$. Express your answer in interval notation. | [\frac{1}{3},\frac{5}{3}] | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
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