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Given the function \( f(x, y, z) = \exp(-(x - ky)^2) \exp(-z^2) \), where \( k \) is a constant representing the slope, and considering that the function outlines a tube whose axis is tilted in the \( xz \)-plane, determine the points at which this function attains a value of \( c \) (where \( c \) is a positive consta... | automath |
Two circles are described by the equations:
\[(x-2)^2 + (y-3)^2 = 4\]
and
\[(x+1)^2 + (y-4)^2 = 9.\]
If the circles are externally tangent to each other and to a given line $y = mx + b$, then find all possible values of $m$. | nvidia_math |
Consider the abelian group \( G = \mathbb{Z}_2 \times \mathbb{Z}_3 \times \mathbb{Z}_5 \). Let \( g_1 = (1, 1, 0) \) and \( g_2 = (0, 1, 1) \). Determine the order of the subgroup generated by \( g_1 \) and \( g_2 \), and verify if \( G \) is isomorphic to the direct product \( \langle g_1 \rangle \times \langle g_2 \r... | automath |
Consider the algebraic variety defined by the equation x^3 - y^2 = 0. Find the singular points on this algebraic variety. | tiger_lab_math |
Find the value of $\log_{5}{625}+\log_{5}{\frac{1}{25}}$. | tiger_math |
In triangle MJK, if the measure of angle MKL is determined to be 120 degrees, and the measure of angle MJK is 60 degrees, how many degrees greater is the measure of angle MKL than the measure of angle MJK? | automath |
Given that a disk has \( B \) blocks and \( F \) of those blocks are free, and that each disk address requires \( d \) bits, derive the inequality condition under which using a free list to manage free disk space requires less storage than using a bit map. Additionally, calculate this condition explicitly in terms of \... | automath |
May can knit x scarves using one yarn. She bought 2 red yarns, 6 blue yarns, and 4 yellow yarns. She will be able to make 36 scarves in total. What is the value of unknown variable x? | tiger_math |
Given a system consisting of a 300 g ball and a 500 g ball connected by a 60 cm long massless rigid rod, rotating about the center of the rod with an angular velocity of 5 rad/s, what is the rotational kinetic energy of the system? | automath |
For every integer \( n \geq 1 \), construct a graph on \( n \) vertices such that each vertex has degree at least \( \frac{1}{2}n - 1 \) and the graph is not Hamiltonian. Additionally, provide a clear justification for why this graph does not contain a Hamiltonian cycle. | automath |
The sum of two numbers is 50 and their difference is 6. What is their product? | tiger_math |
Susan is making jewelry with a repeating pattern that has 3 green beads, 5 purple beads, and twice as many red beads as green beads. If the pattern repeats x times per bracelet and 5 times per necklace, how many beads does she need to make 1 bracelets and 10 necklaces?
If we know the answer to the above question is 742... | tiger_math |
For some positive integers $a$ and $b$, the product \[\log_a(a+2) \cdot \log_{a+2} (a+4) \dotsm \log_{b-4} (b-2) \cdot\log_{b-2} b\]contains exactly $406$ terms, and its value is $2.$ Compute $a+b.$ | tiger_math |
Find the matrix that corresponds to reflecting over the vector $\begin{pmatrix} 3 \\ 2 \end{pmatrix}.$ | nvidia_math |
The basketball team sold 50 cupcakes for $2 each and 40 cookies for $0.5 each. Then the money from their sales was used to buy two basketballs that cost $40 each. The rest of the money was used to buy 20 bottles of energy drinks. How much does a bottle of energy drink cost? | tiger_lab_math |
the average of runs of a cricket player of 20 innings was 32 . how many runs must he make in his next innings so as to increase his average of runs by 8 ? Please respond by writing a program in Python. | tiger_lab_math |
Irene shares half of a small apple with her dog every day. A small apple weighs a certain amount. She can currently buy apples for $2.00 a pound. She will spend $7 so that she and her dog have enough apples to last for 2 weeks. What is the weight of a small apple in pounds? | tiger_math |
a certain mixture of nuts consists of 5 parts almonds to 2 parts walnuts , by weight . what is the number of pounds of almonds in 150 pounds of the mixture ? Let's write a Python program. | tiger_lab_math |
Evaluate the quadruple integral
\[
I = \int_0^1 \int_0^1 \int_0^1 \int_0^1 \frac{1}{1 - xyzw} \, dw \, dz \, dy \, dx.
\]
What is the value of \( I \) in terms of \(\pi\)? | automath |
An integer-valued function $f$ is called tenuous if $f(x) + f(y) > x^2 + y^2$ for all positive integers $x$ and $y.$ Let $g$ be a tenuous function such that $g(1) + g(2) + \dots + g(20)$ is as small as possible. Compute the minimum possible value for $g(15).$ | tiger_math |
In a zoo, a hare is x inches tall, while a camel is 24 times taller than the hare. How tall is the camel in feet?
If we know the answer to the above question is 28, what is the value of unknown variable x? | tiger_math |
Excluding stoppages, the average speed of a bus is 80 km/hr, and including stoppages, the average speed of the bus is 40 km/hr. For how many minutes does the bus stop per hour? | tiger_math |
The graph of \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\]has its foci at $(0,\pm 4),$ while the graph of \[\frac{x^2}{a^2}-\frac{y^2}{b^2} = 1\]has its foci at $(\pm 6,0).$ Compute the value of $|ab|.$ | nvidia_math |
What is the sum of all values of $x$ for which $\sqrt{(x+3)^{2}} = 7$?
Determine the sum of all values of $x$ for which $|x+3| = 5$. | nvidia_math |
if x is a positive integer with fewer than 3 digits , what is the probability r that x * ( x + 1 ) is a multiple of either 4 or 5 ? Let's write a Python program to solve it. | tiger_lab_math |
Taylor and his friends decide to bring their pets together for a walk. 3 of his friends come first, each having twice as many pets as Taylor has. Another two of his friends have 2 pets each. If Taylor has 4 pets, what's the total number of pets they have together? | tiger_math |
A cylindrical container is used to store oil. The container has a height of 8 feet and a radius of 3 feet. If the oil costs $\$$12 per cubic foot, what would be the cost of filling a spherical tank with a diameter of 12 feet to its full capacity? | nvidia_math |
Evaluate $\sqrt{12 +\!\sqrt{12 + \!\sqrt{12 + \!\sqrt{12 + \cdots}}}}$. | nvidia_math |
A bag has 5 red marbles, 6 white marbles, and Y blue marbles. Three marbles are drawn from the bag (without replacement). If the probability that they are all the same color is 1/27, what is the value of Y? | tiger_math |
Calculate the value of the integral
\[
I = \int_{1}^{2} \frac{1}{\sqrt{x - 1}} \, dx.
\]
What is the result? | automath |
Let \( f: \mathbb{R} \setminus \{3\} \to \mathbb{R} \setminus \{2\} \) be defined by the function \( f(x) = \frac{2x + 3}{x - 3} \). Determine whether \( f \) is invertible, and if so, find the inverse function \( f^{-1}(y) \). | automath |
At the beginning of 1992, Maria's stock portfolio was worth $600,000. During 1992, the portfolio's value increased by 95%. During the next year, the portfolio increased its worth by only 5%. What was Maria's portfolio worth, in dollars, by the end of 1993?
Answer Choices: (A) 1,180,000 (B) 1,200,000 (C) 1,200,300 (D) 1... | tiger_lab_math |
There are 20 students in a class. Only one-fourth of the students stayed in the classroom while the rest went to the playground. Of those who went to the playground, one-third are boys. How many girls are there on the playground from this class? | tiger_lab_math |
Find the $75$th digit after the decimal point when $\frac{7}{18}$ is expressed as a decimal. | nvidia_math |
What is the degree of the cyclotomic polynomial \(\Phi_p(x)\) for a prime \(p\), and how many distinct roots does it have in the complex plane? | automath |
list r contains 5 numbers that have an average value of 60 . if the median of the numbers in the list is equal to the mean and the largest number is equal to 20 more than two times the smallest number , what is the smallest possible value in the list ? Let's write a Python program. | tiger_lab_math |
Six men and six women have to sit around a circular table so that no 2 women are together. In how many ways can that be done?
Answer Choices: (A) 5!*6! (B) 6!*6! (C) 5!*7! (D) 6!*7! (E) 7!*7! | tiger_lab_math |
"statement 1": Steve Purchased a house for $ 15000 and he like to resell it with 20 % profit, "statement 2" :Steve Purchased a house for $ less than 75000 and he like to resell it with 20 % profit, "options: " Entailment or contradiction or neutral? Let's write a program. | tiger_lab_math |
On the three shelves of a bookcase there are books whose thicknesses are $5$, $8$ and $12$ cm respectively. If all the books were placed on one shelf, what would be the total thickness of all the books in centimeters? | nvidia_math |
a train length 100 m going with speed 10 m / sec another train 150 m long going with speed 15 m / sec . the distance between two train is 50 m . then how much time second train will take to cross the first train ? Please write a program to solve it | tiger_lab_math |
How many zeros are at the end of the integer resulting from the product \( 20^{50} \times 50^{20} \)? | automath |
Let $f(x)$ be a polynomial of degree 2 such that $f(2)=1$, $f(3)=-2$ and $f(4)=7$. Let $g(x)=x^2-mx-n$ be a quadratic function such that the product $f(x)\cdot g(x)$ has a triple root at $x=2$. Determine $m+n$. | nvidia_math |
Consider a playoff scenario among the top 4 teams in a sports tournament. The teams are seeded #1 through #4. The playoff consists of two rounds of games, with the losers of each game being eliminated and receiving a corresponding rank. The winners advance to the next round, with the winner of the final game receiving ... | nvidia_math |
Lily and Ava are shopping at a toy store that sells stuffed animals at $6 each, puzzles at $8 each, and stickers at $2 each. Lily buys a stuffed animal and two stickers, while Ava buys a puzzle and a stuffed animal. How much do Lily and Ava spend in total? | nvidia_math |
What are the semantic differences between the symbols $\angle ABC$ and $\measuredangle ABC$ in the context of angles, and can you provide a reference or a mathematical source that illustrates the common usage of one of these symbols over the other? Additionally, how do these notations differ in usage between English an... | automath |
How many integers between 324,700 and 448,600 have tens digit 1 and units digit 3?
Answer Choices: (A) 10,300 (B) 8,030 (C) 1,253 (D) 1,252 (E) 1,239 | tiger_lab_math |
A club is going to get additional members so that they will have 5 more than twice their current number of their members. If the club has 10 members now, how many additional members do they need? | tiger_lab_math |
A cyclist covers 25 miles in 2 hours. What is her average speed in miles per hour? | nvidia_math |
Sally has a gold credit card with a certain spending limit, and a platinum card with twice the spending limit of the gold card. Currently, she has a balance on her gold card that is 1/3 of the spending limit on that card, and she has a balance on her platinum card that is 1/9 of the spending limit on that card. If Sall... | tiger_lab_math |
Given the potential models for tasks completed \( t \) as a function of hours worked \( h \) and experience \( x \), where \( t \) could follow a linear, quadratic, or logarithmic relationship, which model would you expect to yield the highest predictive accuracy based on the Bayesian information criterion (BIC) if app... | automath |
The state of Virginia had 3.79 inches of rain in March, 4.5 inches of rain in April, 3.95 inches of rain in May, 3.09 inches of rain in June and 4.67 inches in July. What is the average rainfall amount, in inches, in Virginia? | tiger_lab_math |
If $0.75= \frac{y}{186+x}$, find the smallest value of $x$, if $x$ and $y$ are positive integers. | tiger_math |
Madeline wants to drink x ounces of water in a day. Her water bottle can hold 12 ounces of water. She refills her water bottle 7 times. She needs to drink 16 more water. What is the value of unknown variable x? What is the value of unknown variable x? | tiger_math |
If $x = X$, The value of $2x + 3$ is 9. What is the value of unknown variable X? | tiger_math |
If the population of Mojave was 4000 ten years ago and has since tripled, with a predicted 40% increase in the next five years, what will be the population of Mojave in five years? | tiger_math |
Anna wants to grow some lettuce in the garden and would like to grow enough to have at least 12 large salads. She estimates that, despite her best efforts, half of the lettuce will be lost to insects and rabbits. Each lettuce plant is estimated to provide 3 large salads. How many lettuce plants should Anna grow in the ... | nvidia_math |
Is the function \( f(x,y) = \cos(\sqrt{xy}) \) uniformly continuous on the domain \( D = \{(x,y) \in \mathbb{R}^2 : x \geq 0, y \geq 0\} \)? If it is not uniformly continuous, provide a specific example of points \( (x_1, y_1) \) and \( (x_2, y_2) \) in \( D \) such that \( \lVert (x_1, y_1) - (x_2, y_2) \rVert < \delt... | automath |
Find the modular inverse of $4$, modulo $21$.
Express your answer as an integer from $0$ to $20$, inclusive. Please write a program to solve it | tiger_lab_math |
A cylinder is inscribed in a cone, and the cone has a volume of 60 cubic centimeters. A second cone is then inscribed within the cylinder. What is the volume in cubic centimeters of the inner cone? | nvidia_math |
In a regression model where the error term is specified to be autoregressive, describe the conditions under which the autoregressive coefficients, denoted as $\phi_1, \phi_2, \ldots, \phi_p$, ensure that this model is stationary. Specifically, what criteria must these coefficients satisfy for the process to maintain st... | automath |
Maria has 19 cookies. She decided to give her friend 5 of them, and half of the rest to her family. From the rest, Maria decided to eat 2 cookies. How many cookies will she have left? Please write a program to solve it | tiger_lab_math |
Alexis is heading to the airport for a flight. A private car service to the airport costs $10 more than a shuttle bus ride. A shuttle bus ride costs $5 more than a public transportation ticket. The private car service costs $40. If Alexis takes public transportation to the airport and buys a snack for 15% of the origin... | nvidia_math |
If (x + 1) times (2x + 3) is an odd integer, then x must be:
Answer Choices: (A) an odd integer (B) an even integer (C) a prime number (D) a composite number (E) a negative integer | tiger_lab_math |
Given the equation
$$A = 0.0193\left(\frac{x+0.05}{0.05}\right)^2 + 0.0876\left(\frac{x+0.05}{0.05}\right) + \left(\frac{Y}{0.2}\left(0.75 + 0.028\left(\frac{x+0.05}{0.05}\right)\right)\right) + 1.356,$$
can you simplify this expression and solve for \( Y \) in terms of \( A \) and \( x \)? What is the resulting equ... | automath |
Determine x such that $\log_x 3 + \log_x 5 = 1$. | nvidia_math |
evaluate 49 / . 07 ? Let's program in Python in the response. | tiger_lab_math |
a car traveling at a certain constant speed takes 2 seconds longer to travel 1 kilometer than it would take to travel 1 kilometer at 225 kilometers per hour . at what speed , in kilometers per hour , is the car traveling ? Let's write a Python program. | tiger_lab_math |
Consider the function \( f: \mathbb{R} \to \mathbb{R} \) defined by \( f(x) = 3x + 2 \). Is the function bijective? If so, find its inverse function. | automath |
Let \( R \) be a commutative ring with identity, and let \( S \) be a multiplicative subset of \( R \). If \( S^{-1}I \subseteq S^{-1}J \) for ideals \( I \) and \( J \) in \( R \), under what conditions can we conclude that \( I \subseteq J \)? Provide an example or counterexample that illustrates your reasoning. | automath |
Three married couples are to be seated around a circular table, each person seated next to their spouse. In addition, for each couple, the wife always sits to the left of her husband. How many ways can the six people be seated around the table? | nvidia_math |
A certain bakery sells a total of 480 muffins and cakes on a certain day. The number of muffins sold was 3/5 of the total number of muffins and cakes sold, while the number of cakes sold was 2/5 of the total number sold. If the profit on each muffin is $0.50 and the profit on each cake is $0.75, what is the total profi... | nvidia_math |
Consider a European call option with a strike price of $50$, a time to expiration of $5$ years, and a stock price following geometric Brownian motion with an initial price of $40$, a risk-free interest rate of $5\%$, and a stochastic volatility given by the Heston model with parameters: $\sigma_0=0.2$, $\theta=0.2$, $\... | tiger_lab_math |
An ellipse has a major axis of length 20 cm and a minor axis of length 12 cm. A sector of the ellipse is shaded, and the probability of a randomly selected point within the ellipse falling within the shaded sector is $\frac{1}{3}$. What is the area, in sq cm, of the shaded sector? Express your answer in terms of $\pi$. | nvidia_math |
Given a differentiable function \( f : \mathbb{R} \to \mathbb{R} \) that has a local maximum at \( \xi = 3 \) and a local minimum at \( \xi = 5 \), what can be said about the values of the derivative \( f'(\xi) \) at \( \xi = 3 \) and \( \xi = 5 \)? Provide a detailed justification for your answer. | automath |
In Fifi's closet, she hangs all of her clothes on colored plastic hangers. She has clothes hanging on x pink hangers, 4 green hangers, one less blue hanger than there are green hangers, and one less yellow hanger than there are blue hangers. What is the total number of colored hangers in Fifi's closet?
If we know the... | tiger_math |
a vase has 2 dozens of assorted flowers of which 8 are daisies. what is the ratio of the daisies to the whole arrangement?
Answer Choices: (A) 1/2 (B) 1/3 (C) 1/12 (D) 8/12 (E) 1/24 | tiger_lab_math |
Determine the number of intersections between the graphs of the polar equations $r = 3 + \cos(2\theta)$ and $r = 2 - \sin(\theta)$. | nvidia_math |
If the average (arithmetic mean) of a and b is 110, and the average of b and c is 150, what is the value of a − c?
Answer Choices: (A) −220 (B) −100 (C) 100 (D) −80 (E) It cannot be determined from the information given | tiger_lab_math |
Consider a matrix \( X \in \mathbb{R}^{N \times d} \) representing a dataset with \( N \) observations and \( d \) features. Given a subset size \( k \) and using a projection determinantal point process (DPP) that favors diversity among selected columns, derive the upper bound on the expected approximation error of th... | automath |
Find the Fourier series of the piecewise function:
$$
f(x) = \begin{cases}
-1 & -\pi < x < 0 \\
1 & 0 < x < \pi \\
\end{cases}
$$
and determine the value of the series at $x = \frac{\pi}{4}$. | tiger_lab_math |
Kyle knows that $4 = 2^{5r+1}$. What is the value of $r$? Express your answer as a common fraction. | nvidia_math |
Let $h(x) = 2x^2 - 5x + 1$ and $k(x) = x^3 - 2x^2 - 3x + 1$. Find $(h \circ k)(x) - (k \circ h)(x)$ when $x = 2$. | nvidia_math |
Evaluate the integral \( \int \sqrt{1 - x^2} \, dx \) and express the result in terms of \( x \). | automath |
Let \( f(x) = 3\log_2(x - 1) + 4 \). Determine the domain, range, and the equation of the vertical asymptote for this function. | automath |
Annika hikes at a constant rate of 10 minutes per kilometer. She has hiked 2.75 kilometers east from the start of a hiking trail when she realizes that she has to be back at the start of the trail in 45 minutes. If Annika continues east, then turns around and retraces her path to reach the start of the trail in exactly... | tiger_lab_math |
Tom decides to renovate a house. There are 3 bedrooms and each bedroom takes x hours to renovate. The kitchen takes 50% longer than each bedroom. The living room took twice as much time as everything else combined. Everything took 54. What is the value of unknown variable x? | tiger_math |
The numbers 2, 4, 6, and 8 are a set of four consecutive even numbers. Suppose the sum of five consecutive even numbers is 320. What is the smallest of the five numbers? | nvidia_math |
Joe worked at a bar. The bar had a glass counter on one side and a wooden counter on the other. When Joe slid a glass along the glass counter, he noticed it slid 8.4 cm and 1.3 cm on the wood counter. This is because the _____ is smoother.? "Option 1": glass counter, "Option 2" :wooden counter Please respond by writin... | tiger_lab_math |
Consider the function $g(x)$ defined as:
\[g(x) = \left\{
\begin{array}{cl}
x^3 &\text{ if }x \text{ is an integer}, \\
2x+1 &\text{ if }x \text{ is not an integer}.
\end{array}
\right.\]
Find $g(g(g(1.5)))$. | nvidia_math |
Given a Rayleigh random variable \( Z \) with the probability density function \( p(z) = \frac{z}{\sigma^2} e^{\frac{-z^2}{2\sigma^2}} \) for \( z \ge 0 \), what is the value of \( E[Z^2] \) in terms of \( \sigma^2 \)? | automath |
in how many ways seven letters between A & B be arranged from alphabets so that repetition is not allowed?
Answer Choices: (A) 24P7 * 2! (B) 24P7 * 3! (C) 24P7 * 4! (D) 24P7 * 5! (E) 24P7 * 6! | tiger_lab_math |
Find the constant term in the expansion of $\left(\sqrt{x} - \frac{1}{\sqrt{x}}\right)^{2024}$. | nvidia_math |
33 campers went rowing in the morning, 34 campers went rowing in the afternoon, and some campers went rowing in the evening. 24 more campers went rowing in the afternoon than in the evening. How many campers went rowing in the evening? | tiger_math |
What is the probability that a triangle formed by randomly selecting three vertices from a regular octagon has at least one of its sides coinciding with a side of the octagon? | automath |
Consider the integral:
$$ I = \int_{-1}^{1} C_{n_1-l_1}^{l_1+1}(x) C_{n_2-l_2}^{l_2+1}(x) C_{n_3-l_3}^{l_3+1}(x) (1-x^2)^{(l_1+l_2+l_3+1)/2} dx $$
where \( C_n^\lambda(x) \) are Gegenbauer polynomials. For non-negative integers \( n_1, n_2, n_3, l_1, l_2, l_3 \), find a condition on \( n_1, n_2, n_3, l_1, l_2, l_3 \)... | automath |
Given a circle \( C \) of radius \( 3a \) and an annulus \( A \) with an outer radius of \( 5a \) and an inner radius of \( 4a \), calculate the ratio of the area of circle \( C \) to the area of the annulus \( A \). What does this ratio reveal about their comparative sizes? | automath |
A trebuchet consists of a uniform rod of length 3 m, with a particle of mass 60 kg attached at one end and a particle of mass 0.12 kg attached at the other end. The rod rotates around a frictionless axle that is located 14 cm from the larger mass. If the system is released from a horizontal position, what is the maximu... | automath |
If a < b < c , which of the following must be true?
(I) a < b^2
(II) b − a > c
(III) a^2 < b^2 < c^2
Answer Choices: (A) II only (B) I only (C) None (D) III only (E) II and III | tiger_lab_math |
What are the dimensions of a rectangle with a perimeter of 36 cm that maximizes the volume swept out when it is revolved around one of its sides? | automath |
If x is an integer, then x(x - 1)(x - l) must be evenly divisible by three when l is any of the following values EXCEPT
Answer Choices: (A) -4 (B) -2 (C) -1 (D) 2 (E) 5 | tiger_lab_math |
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