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What is the maximum degree of the polynomial interpolation function that can be formed using 12 distinct equidistant points sampled from the Runge function \( R(x) = \frac{1}{1 + 25x^2} \) over the interval \([-1, 1]\), and how does the approximation error behave at the endpoints of the interval as the number of points... | automath |
Given the outer product \( T_{mn} \) of the matrices \( T_m \) and \( T_n \) defined as:
\[
T_m = \begin{pmatrix} 1 & 2 \\ 4 & 5 \end{pmatrix}, \quad T_n = \begin{pmatrix} 2 & 3 \\ 0 & 1 \end{pmatrix},
\]
compute the resulting matrix from the outer product \( T_{mn} = T_m \otimes T_n \). What are the entries of the r... | automath |
Toby is dining out with his friend. They each order a cheeseburger for $3.65. He gets a milkshake for $2 and his friend gets a coke for $1. They split a large fries that cost $4. His friend also gets three cookies that cost $.5 each. The tax is $.2. They agree to split the bill. If Toby arrived with $15, how much chang... | tiger_lab_math |
During a park cleanup event, Alex and her friends collected leaves. Alex collected three times as many leaves as Charlie did. Charlie got tired early and only collected a quarter of what Emily did. If Emily collected 48 leaves, how many did Alex collect? | nvidia_math |
Together, 15 type A machines and 7 type B machines can complete a certain job in some hours. Together, 8 type B machines and 15 type C machines can complete the same job in 11 hours. It would take one type A machine, one type B machine, and one type C machine working together 44 hours to complete the job. How many hour... | tiger_math |
Compute the limit \( L = \lim_{x \to 0} \left( \frac{1}{x} - \frac{1}{e^x - 1} \right) \). What is the value of \( L \)? | automath |
Passage: World Expos are also massive in scale, sometimes 300 or 400 hectares in size (Montreals Expo 67 was 410 hectares, Osakas Expo 70 was 330 hectares, Sevilles Seville Expo 92 was 215 hectares and Shanghais Expo 2010, 528 hectares). Pavilions participating at a World Expo can also be large, sometimes 5,000 to 10,0... | tiger_lab_math |
What is the formula to compute the Root Mean Square Error (RMSE) from the residuals of a TBATS forecasting model in R, and how can it be confirmed that the variance component of the model output corresponds to the RMSE? | automath |
If i equals the sum of the even integers from 2 to 200, inclusive, and k equals the sum of the even integers from 20 to 200, inclusive, what is the value of i - k?
Answer Choices: (A) 10 (B) 99 (C) 9 (D) 88 (E) 90 | tiger_lab_math |
Mrs. Hilt bought 4 pizzas. The first 2 pizzas had 8 slices each, the third pizza had 10 slices, and the fourth pizza had 12 slices. How many total slices of pizza did she have? | tiger_math |
What is the minimum value of the functional
\[
I(\mu) = \frac{1}{2}\iint (V(x) + V(y) + W(x,y)) \, d\mu(x) \, d\mu(y)
\]
for a probability measure \(\mu\) on \(\mathbb{R}^d\) under the assumptions that \(W\) is a Riesz kernel and the external potential \(V\) is continuous and satisfies the limit condition \(V(x) \to... | automath |
Little John had $10.10. He spent $3.25 on sweets and gave some money to each of his two friends. After that, he had $2.45 left. How much money did Little John give to each friend? | tiger_math |
Travis goes through 2 boxes of cereal a week. If each box costs $3.00 and he eats 2 boxes every week for an entire year, 52 weeks, how much does he spend on cereal? | nvidia_math |
What is the value of \( \sin\left(\frac{3\pi}{2}\right) \) and explain how you determined it using the unit circle? | automath |
A bag contains 15 identical red balls and 10 identical blue balls. If three balls are drawn at random, what is the probability that at least two of them are red? Answer Choices: (A) 0.5 (B) 0.6 (C) 0.7 (D) 0.8 | tiger_lab_math |
Given f(x) = 3x - 5, for the value of x = 3, 2 * [f(x)] - 10 = f(some function of x). What is the function of x in the equation? | tiger_math |
The area of a triangle will be when a = 1m, b = 2m, c = 3m, a, b, c being lengths of respective sides.
Answer Choices: (A) 0 sq m (B) 2 sq m (C) 9 sq m (D) 20 sq m (E) 5 sq m | tiger_lab_math |
Fifteen more than a quarter of a number is 27. What is the number? | tiger_lab_math |
Osborn is testing a new way to get dressed in the morning on school days so he goes faster and can get up later. He tracks his time for the week and on Monday is takes him x minutes. On Tuesday it takes him 4 minutes. On Wednesday it takes him 3 minutes. On Thursday it takes him 4 minutes. Before Friday arrives he look... | tiger_math |
In a game involving a loaded 6-sided die, the probabilities of rolling each face are given by the vector \(\boldsymbol{\mu} = [0.2857, 0.1429, 0.1429, 0.1429, 0.1429, 0.1429]\). If the die is rolled 35 times, what is the probability of rolling exactly 5 ones, 5 twos, 5 threes, 5 fours, and 15 fives? | automath |
Let \( N = 10^{2000} + 2000 \) and let \( S \) be the set of consecutive integers from \( 1 \) to \( N \). In how many different ways can three consecutive integers be removed from \( S \) such that the average of the remaining integers is an integer? | automath |
Increasing the radius of a cylinder by $6$ units increased the volume by $y$ cubic units. Increasing the height of the cylinder by $6$ units also increases the volume by $y$ cubic units. If the original height is $2$, then the original radius is:
$\text{(A) } 2 \qquad \text{(B) } 4 \qquad \text{(C) } 6 \qquad \text{(D)... | tiger_math |
Express the set of students at your school who are either not sophomores or not taking discrete mathematics in terms of sets A and B, where A is the set of sophomores and B is the set of students in discrete mathematics. | automath |
Jamie collects glass balls. He had x red balls and two times more blue balls. Later on, he lost 6 of the red balls, so he decided to buy some yellow balls to fill up his collection. How many yellow balls did he buy if, after all, he had 74 balls in total?
If we know the answer to the above question is 32, what is the v... | tiger_math |
In a board game played on a $25 \times 25$ grid, two players alternate moves with the goal of positioning a pawn in the bottom left corner at (0,0). The first player places the pawn anywhere in the top row, and the second player can then move the pawn in one of three directions: south, west, or south-west, for any numb... | automath |
A cuboidal water tank has a base measuring 7.5 meters by 3 meters. If water flows into the tank at a rate of 10 liters per second for 50 minutes, what will be the height of the water in the tank, expressed in meters, rounded to one decimal place? | automath |
Determine the greatest integer less than or equal to the square root of 25 multiplied by 2. | tiger_math |
On dividing a number by 3, the quotient is 16 and the remainder is 4. What is the dividend? | tiger_math |
In a fuel station the service costs $1.75 per car, every liter of fuel costs 0.65$. Assuming that a company owns 12 cars and that every fuel tank contains 50 liters and they are all empty, how much money total will it cost to fuel all cars?
Answer Choices: (A) 320$ (B) 380$ (C) 411$ (D) 450$ (E) 480$ | tiger_lab_math |
Given a finite set of nodes \(\mathbb{X}\) with a distance function \(d(x, y) \in \{0, 1\}\) for each pair \((x, y) \in \mathbb{X}\), positive integers \(S\) and \(H\), what is the minimum possible value of the summation
\[
\sum_{s=1}^S \sum_{x,y \in \mathbb{X}_s} d(x,y)
\]
when partitioning \(\mathbb{X}\) into disj... | automath |
The sum of two numbers is 20, and their difference is some value. The difference of their squares is 200. What is the difference between the two numbers? | tiger_math |
Let $p$ and $q$ be the two distinct solutions to the equation $$(x-3)(x+3) = 21x - 63.$$If $p > q$, what is the value of $p - q$? | nvidia_math |
a pupil ' s marks were wrongly entered as 83 instead of 63 . due to the average marks for the class got increased by half . the number of pupils in the class is ? Let's write a program. | tiger_lab_math |
Ice-ColdIce-cream factory produces only tricolor ice-cream products, where each ice-cream has three stripes of different colors. The factory uses the colors pink, purple, orange, silver, blue and white. How many different ice-cream products have at least one stripe out of the following colors: pink, purple or orange (a... | tiger_lab_math |
The parabola given by the equation \[\frac{y^2}{16}-\frac{x^2}{25} = 1\]has asymptotes $y = \pm nx,$ where $n$ is positive. Find $n.$ | nvidia_math |
Zookeeper Johnson is caring for 10 monkeys and 8 kangaroos. How many total legs are there among the animals in her care? | nvidia_math |
Let's consider a problem where we need to form different positive integers using a given set of digits, but with a twist.
Problem:
Given the three digits 2, 4, and 7, how many different positive integers can be formed using these digits if each digit must be used exactly once, and the integer must be a multiple of 3?... | nvidia_math |
Find the $y$-coordinate of the point on the unit circle that has an angle of $\frac{5\pi}{2}$. | tiger_math |
What is the value of \( n! \) for \( n = 7 \) computed using a for loop in MATLAB, and can you provide the MATLAB code to perform this calculation? | automath |
A committee of five people is formed from a group of ten candidates, and it is required that at least one of the members of the committee knows all the others. If each candidate knows exactly four other candidates, find the smallest possible number of candidates who know all the members of the committee. | nvidia_math |
"statement 1": Jack only collects the ones with 2 spots ,
and Jill only collects the ones with 7 spots, "statement 2" :Jack only collects the ones with less than 5 spots , and Jill only collects the ones with 7 spots, "options: " Entailment or contradiction or neutral? Please write a program to solve it | tiger_lab_math |
John decides to take up illustration. He draws and colors 10 pictures. It takes him 2 hours to draw each picture and 30% less time to color each picture. How long does he spend on all the pictures? | tiger_lab_math |
Given the following dataset for a linear regression problem:
```
x = [6.1101, 5.5277, 8.5186, 7.0032, 5.8598, ...] # (representing 'x' values)
y = [17.592, 9.1302, 13.662, 11.854, 6.8233, ...] # (representing 'y' values)
```
If we define the cost function for linear regression using the matrix \( X \) (with a colum... | automath |
Consider a 7x7 grid of lattice points. How many non-congruent equilateral triangles can be drawn, such that their vertices are lattice points on this grid? | nvidia_math |
Jude bought three chairs for his house, all at the same price. He also bought a table that costs $50 and two sets of plates at $20 for each set. After giving the cashier $130, Jude got a $4 change. How much did each of the chairs cost? | tiger_math |
Solve for $x$: $x+3x = 500-(4x+5x)$. | tiger_math |
Let \( f(x) = x^2 + (2x - 4) \sqrt{x + 3} \). Find the value of \( f^{\prime}(1) \). | automath |
A population of microbes doubles every 20 minutes. If the initial population is 1 microbe, what will the population be after 4 hours? | automath |
The number x of cars sold each week varies with the price y in dollars according to the equation x = 800,000 – 50y. What would be the total weekly revenue Q, in dollars, from the sale of cars priced at $ 15,000?
Answer Choices: (A) 50,000 (B) 750,000 (C) 850,000 (D) 7,500,000 (E) Q=750,000,000 | tiger_lab_math |
Sarah, an avid birder, discovers a new park where four unique bird species are commonly seen: the Red Robin, Blue Jay, Golden Finch, and Emerald Sparrow. She wants to observe each species, but due to her schedule, she can only watch one species per hour for four consecutive hours on a Saturday morning. How many differe... | tiger_math |
What is the value of $\dbinom{50}{2}$? | tiger_math |
A trained covered x km at 40 kmph and another 2x km at 20 kmph. Find the average speed of the train in covering the entire 3x km.
Answer Choices: (A) 18 kmph (B) 17 kmph (C) 24 kmph (D) 19 kmph (E) 12 kmph | tiger_lab_math |
Let A = {1, 2} and B = {3, 4}. Calculate (AxB) ∩ (BxA). | tiger_lab_math |
How many integers from 1 to 10 are factors of the six-digit number 352,836? | tiger_math |
Let \( y_1(x) = x \) and \( y_2(x) = x^2 \) on the interval \( I = [-1, 1] \). Calculate the Wronskian \( W(y_1, y_2) \) and determine if the functions \( y_1 \) and \( y_2 \) are linearly independent on \( I \). | automath |
Simplify $3(2x^2 - 5x + 1) + 2(x^2 + 3x - 2) - (x^2 - 2x - 3)$. | nvidia_math |
If the inverse of the matrix \[\begin{pmatrix} a & b \\ c & d \end{pmatrix}\] is the zero matrix, and we know that a = 5 and c = -3, find the value of b + d. | tiger_math |
Problem:
A regular polygon with an odd number of sides is inscribed in a circle. The length of the largest diagonal is $d$ and the radius of the circle is $r$. What is the largest possible ratio of $d$ to $r$?
(Note: I've tried to create a problem that requires a different approach to solve, while still being related ... | nvidia_math |
Emily is 30 years younger than her mother. The sum of their ages is 80 years. How many years old is Emily? | nvidia_math |
a big container is 30 % full with water . if 18 liters of water is added , the container becomes 3 / 4 full . what is the capacity of the big container ? Please respond by writing a program in Python. | tiger_lab_math |
Let \( f(z) = \frac{1}{z^2 + 1} \). Evaluate the integral
\[
\int_C f(z) \, dz
\]
where \( C \) is the contour defined by the unit circle \( |z| = 1 \). Use Cauchy's nth derivative formula as necessary. What is the value of the integral? | automath |
Mark has the option of getting a $300 lens with a 20% discount or a $220 lens. How much money does he save by buying the cheaper lens? | tiger_lab_math |
How many non-isomorphic groups of order 16 are there, and provide a list of all possible subgroups for each group? | tiger_lab_math |
A Cuban cigar would cost 3 dollar less than 1.5 times a French cigar, had the French cigar cost 0.7 dollar less than it does now. An Arabian cigar costs 50 cents more than 1.5 times the Cuban cigar. The three cigars together cost 74.7 dollars. What is the price of the French cigar?
Answer Choices: (A) 19.7$. (B) 23$. (... | tiger_lab_math |
A cube is composed of 27 smaller congruent cubes, where each smaller cube has a side length of $\frac{1}{3}$ of the large cube. If the large cube is painted with a total surface area of 100 square units, and each smaller cube is then separated and its surface area measured, what is the total surface area of all 27 smal... | nvidia_math |
A bag contains 50 tickets numbered 1,2,3,4......50 of which five are drawn at random and arranged in ascending order of magnitude.Find the probability that third drawn ticket is equal to 30.
Answer Choices: (A) 551/15134 (B) 551/15139 (C) 551/15132 (D) 551/15196 (E) 551/15766 | tiger_lab_math |
The difference between the number of boys and girls in a tree planting event is 400. If there are 600 boys at the event, and the number of girls is more than the number of boys, what's 60% of the total number of boys and girls at the event? | tiger_lab_math |
A sphere has a volume of $P\text{ cm}^3$ and a surface area of $Q\text{ cm}^2$. If $\dfrac{P}{Q}=15$, what is the radius of the sphere, in cm? | nvidia_math |
Find the Fourier series of the periodic function f(x) = 3x, where -π < x < π. | tiger_lab_math |
For a certain quadratic equation $y = ax^2 + bx + c$, the graph is a parabola with a vertex at (-4,0) and it passes through the point (1,-75). What is the value of $a$? | tiger_math |
License plates from different states follow different alpha-numeric formats, which dictate which characters of a plate must be letters and which must be numbers. Florida has license plates with an alpha-numeric format like the one pictured. North Dakota, on the other hand, has a different format, also pictured. Assumin... | tiger_math |
Evaluate the determinant of the matrix:
\[\begin{bmatrix} y - X & y & y \\ y & y - 2 & y \\ y & y & y - 2 \end{bmatrix}\]
If the determinant is equal to -6y + 2, find the value of X. | tiger_math |
In the city park, there are various attractions for kids and their parents. The entrance ticket to the park is $5, but for each attraction, you have to pay separately - such ticket costs $2 for kids and $4 for parents. A family with a certain number of children, their parents, and her grandmother paid $55 for visiting ... | tiger_math |
(64 - 12)^2 + 4 x 64 x 12 = ?
Answer Choices: (A) 5246 (B) 4406 (C) 5776 (D) 5126 (E) 5128 Please respond by writing a program in Python. | tiger_lab_math |
What is the particular solution of the differential equation
\[
x'' = \left( \begin{array}{ccc} -50&10 \\ 30&-30 \\ \end{array} \right)\ x + \left( \begin{array}{ccc} 5 \\ 0 \\ \end{array} \right)\ \sin10t
\]
in the form
\[
x_p(t) = \begin{pmatrix} a \\ b \end{pmatrix} \cos(10t) + \begin{pmatrix} c \\ d \end{pmatr... | automath |
In the diagram, $RSP$ is a straight line and $\angle QSP = 75^\circ$. What is the measure of $\angle PQR$, in degrees?
[asy]
draw((.48,-.05)--(.48,.05)); draw((.52,-.05)--(.52,.05)); draw((1.48,-.05)--(1.48,.05)); draw((1.52,-.05)--(1.52,.05));
draw((1.04,.51)--(1.14,.49)); draw((1.03,.47)--(1.13,.45));
draw((0,0)--... | tiger_math |
Two standard fair four-sided dice are rolled. What is the probability that the product of the numbers on the top faces is 12? Express your answer as a common fraction. | nvidia_math |
What is the result of rounding down 0.999 and rounding up 2.001, and then adding the two results together? | tiger_math |
What is the expected number of trials needed until an event occurs with a probability of \( p \)? Please express your answer in terms of \( p \). | automath |
An equilateral triangle and a square have the same perimeter of 12 inches. What is the ratio of the side length of the triangle to the side length of the square? Express your answer as a common fraction. | nvidia_math |
What is the boy's speed in kilometers per hour if he runs 1.5 km in 45 minutes? | tiger_math |
a rectangular plot measuring 40 meters by 50 meters is to be enclosed by wire fencing . if the poles of the fence are kept 5 meters apart . how many poles will be needed ? Let's write a program. | tiger_lab_math |
The Shopkeeper increased the price of a product by 20% so that customer finds it difficult to purchase the required amount. But somehow the customer managed to purchase only 70% of the required amount. What is the net difference in the expenditure on that product?
Answer Choices: (A) A) 12.5% (B) B) 13% (C) C) 13.15% (... | tiger_lab_math |
A machine produces 25 items per hour. If the machine operates for 10 hours, how many items will it produce in total? | tiger_math |
The number $n$ is an odd number between 10 and 20. If you divide $n$ by 7, the remainder is 2. What is the value of $n$? | tiger_math |
A small company employs 3 men and 5 women. If a team of 3 employees is to be randomly selected to organize the company retreat, what is the probability that the team will have exactly 2 men?
Answer Choices: (A) 5/7 (B) 1/7 (C) 2/7 (D) 3/7 (E) 1/2 | tiger_lab_math |
In a hexagon $ABCDEF$, three interior angles $A$, $B$, and $C$ are $60^{\circ}$, $85^{\circ}$, and $95^{\circ}$ respectively. The remaining three angles are equal. Determine the measurement of each of the equal angles. | tiger_math |
In $\triangle ABC$, we have $AB = BC = 25$ and $AC = 24$. Point $D$ is the foot of the perpendicular from $A$ to $\overline{BC}$. Find $BD$. | nvidia_math |
Ashley has five pairs of shoes, eight dresses, and four scarves. How many different combinations can she create if a combination consists of a pair of shoes, a dress, and a scarf? | nvidia_math |
If the sum of products of numbers from 1 to 2023 is expressed as:
$$1 \cdot 2023 + 2 \cdot 2022 + 3 \cdot 2021 + \dots + 2022 \cdot 2 + Y \cdot 1$$
and it is equal to $2023 \cdot 1012 \cdot z$, what is the value of the unknown variable Y? | tiger_math |
An airplane flew 600 miles against a headwind, taking 3 hours, and then returned the same distance with the wind in 2 hours. Can you determine the speed of the plane in still air and the speed of the wind? | automath |
A pastry chef is making brownies but is running out of butter. The recipe he is following calls for 2 ounces of butter for every 1 cup of baking mix; however, it allows 2 ounces of coconut oil to be substituted for the 2 ounces of butter if necessary. The chef would like to use as much butter as possible before switc... | tiger_lab_math |
Determine the largest prime factor of $n$, where $n$ is the smallest integer such that $n!$ is divisible by 1000. | nvidia_math |
What is the greatest common divisor of $1729$ and $1768$? | nvidia_math |
Determine the units digit of the product of the first three positive composite numbers. | tiger_math |
two pipes a and b can separately fill a cistern in 60 minutes and 120 minutes respectively . there is a third pipe in the bottom of the cistern to empty it . if all the three pipes are simultaneously opened , then the cistern is full in 60 minutes . in how much time , the third pipe alone can empty the cistern ? | tiger_math |
Mia and Ryan are building towers using blocks. Mia uses 6 blocks to build each of her towers, and Ryan uses 8 blocks for his. If they have 96 blocks to start with and Mia builds 7 towers, how many towers can Ryan build? | nvidia_math |
Consider the following production functions defined for non-negative integers \( K \) and \( L \):
1. \( Q_1(K,L) = A \min(BK, CL) \)
2. \( Q_2(K,L) = A \max(BK, CL) \)
3. \( Q_3(K,L) = BK + CL \)
Let \( A = 1 \), \( B = 2 \), and \( C = 3 \).
Calculate \( Q_1(5, 10) \), \( Q_2(5, 10) \), and \( Q_3(5, 10) \). What... | automath |
What is the length of the longest diagonal of a rectangle with side lengths of 17 inches and 20 inches? | nvidia_math |
The function $f(x)$ satisfies
\[f(x+y) = f(x)f(y)\]for all real numbers $x$ and $y.$ If $f(2) = 3$ and $f(4) = 5$, find $f(6)$. | nvidia_math |
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