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What is the standard error of the difference between the means of two independent samples, given that the variances of the samples are \(\sigma_1^2\) and \(\sigma_2^2\) and the sample sizes are \(n_1\) and \(n_2\)? Provide the formula and explain why this formula involves summing the variances of the two samples before... | automath |
A fair coin is tossed five times. What is the probability that at least one of the tosses shows a head, given that the first toss is a tail? | nvidia_math |
Tony lifts weights as a form of exercise. He can lift 90 pounds with one arm in the exercise known as "the curl." In an exercise known as "the military press," he can lift over his head twice the weight that he can curl. His favorite exercise is known as "the squat" and he can squat 5 times the weight that he can li... | tiger_lab_math |
Jack has $43 in his piggy bank. He also gets an allowance of $10 a week. If Jack puts half of his allowance into his piggy bank every week, after 8 weeks how much will Jack have in his piggy bank? | tiger_lab_math |
Given the polynomial expression:
3(x^2 + 5x - 1) - 2(x^3 - 2x^2 - x + 3)
After fully simplifying this expression, find the product of the coefficients of all the terms. | nvidia_math |
The expression $(5+2\sqrt6)^{x-1}+(5-2\sqrt6)^{x-1}$ can be simplified to the form $2(5^n)$ where $n$ is a function of $x.$ Find the value of $n$ when $x=2$. | nvidia_math |
The Ponde family's Powerjet pumps X gallons of water per hour. At this rate, how many gallons of water will it pump in 45 minutes?
If we know the answer to the above question is 315, what is the value of unknown variable X? | tiger_math |
A group of friends want to share some sweets equally among themselves. If they have 4 fewer sweets than twice their number of friends, what is the smallest possible number of friends they can have so that each friend gets at least one sweet? | nvidia_math |
Consider the quadratic equation
$(cx+d)(dx+c) = 50x^2 + \triangle \cdot x + 50$,
where $c$, $d$, and $\triangle$ are distinct integers.
What is the minimum possible value of $\triangle$, the coefficient of $x$? | nvidia_math |
What will come in place of the x in the following Number series? 11, 14, 19, 22, 27, 30, 35, x
Answer Choices: (A) 29 (B) 31 (C) 23 (D) 35 (E) 38 | tiger_lab_math |
How many integers between 1 and 300 are multiples of either 3, 5, or 7, but not of both 4 and 7? | tiger_math |
A pet store holds cats and dogs. If the difference between the number of cats and the number of dogs is 11. What could be the ratio of Cats to dogs in the pet store?
Answer Choices: (A) 1:3 (B) 1:4 (C) 1:5 (D) 44:55 (E) 4:6 | tiger_lab_math |
On Rudolph's car trip across town, he traveled 2 more than 5 miles and encountered 3 less than 17 stop signs. How many stop signs per mile did Rudolph encounter on his trip across town? | tiger_lab_math |
The two congruent sides of an isosceles triangle are each 8 cm long. If the perimeter of the triangle is 25 cm, what is the length of the base in centimeters? | tiger_math |
A certain organism begins as three cells. Each cell splits and becomes two cells at the end of two days. At the end of another two days, every cell of the organism splits and becomes two cells. This process lasts for a total of 8 days, and no cells die during this time. How many cells are there at the end of the $8^\te... | tiger_math |
A solid spherical bowling ball with a diameter of 30 cm is drilled with four cylindrical holes. The first three holes have depths of 8 cm and diameters of 2 cm, 2 cm, and 3 cm, respectively. The fourth hole has a depth of 10 cm and a diameter of 2.5 cm. Find the volume of the fitted bowling ball in cubic centimeters, r... | tiger_math |
Steve invests in a circus production. The cost includes an overhead of 81,000 dollars, plus production costs of 7,000 dollars per performance. A sold-out performance brings in a certain amount of money. Steve needs 9 sold-out performances to break even. How much money does a sold-out performance bring in? | tiger_math |
What is the smallest positive integer that is both a multiple of $9$ and a multiple of $6$? | tiger_math |
In a recent exam, the average score of a class of 20 students was 85 points. However, the teacher noticed that one student had a score that was much higher than the others. If the median score was 15 points lower than the average score, and the highest score was exactly twice the median score, what was the highest scor... | nvidia_math |
Is it true that for any \( \alpha \in (0,1) \), the interpolation inequality
\[
\|\nabla u\|_p \leq C \|\nabla u\|_2^{b} \|u\|_{C^{0,\alpha}}^c
\]
holds for some constants \( C, b, c > 0 \) and for all functions \( u \) in \( W^{1,2} \cap C^{0,\alpha} \) when \( 2 \leq p \leq n \)? If false, can you provide an expli... | automath |
If a van moving at the speed of x km/hr travels a distance of y km, which of the following cannot be equal to the distance traveled?
Answer Choices: (A) x-y km (B) y-x km (C) (x-y) /2 km (D) y/x km (E) x/2 km | tiger_lab_math |
What is the smallest five digit number that is divisible by 16, 36, 45, and 54?
Answer Choices: (A) 10200 (B) 10400 (C) 10600 (D) 10800 (E) 11200 | tiger_lab_math |
Patty decides that to convince her brother and sister to do her chores in exchange for cookies. Patty agrees to give them 3 cookies for every chore they do. Each kid normally has 4 chores to do per week. Patty has $15 to buy cookies. Each pack of cookies contains 24 cookies and costs $3. How many weeks can Patty go wit... | tiger_lab_math |
If $x+\frac{1}{x}=5$, then what is the value of $x^{2}+\frac{1}{x^{2}}$? | tiger_math |
In a series of five football matches between Germany and Argentina , the probability of Germany winning a match is 1/3 and the probability that the match ends in a draw is 1/4. If a win gets the team 2 points, a draw 1 point and a loss 0 Points, What is the probability that Germany will end up in 5 draws in the series?... | tiger_lab_math |
Suppose there is a fantasy island where magical stones double in number each day. Alex starts with a single magical stone. After keeping it for a week, he gives twice the number of stones he currently has to Bella. Bella, in turn, gives half of these stones back to Alex the next day. If after all transactions, Alex has... | tiger_math |
A road has four intersections. At the first intersection, there are 40 road signs showing different types of information mounted on the street poles. At the second intersection, 1/4 times more road signs than the first intersection are mounted on the street poles. The number of road signs at the third intersection is t... | nvidia_math |
If $x$ is an odd number, then find the largest integer that always divides the expression\[(10x+2)(10x+6)(5x+5)\]
| tiger_lab_math |
Evaluate the limit
\[
\lim_{(x,y) \to (0,0)} \frac{x^2 y^2}{x^3 + y^3}.
\]
Does this limit exist, and if so, what is its value? If not, provide a reasoning or examples showing why the limit does not exist. | automath |
A circular path with a width of 8 feet is built around a square garden with a side length of 24 feet. The path makes four complete turns around the garden, overlapping itself in some areas. What is the area of the path in square feet? | nvidia_math |
What is the greatest possible positive integer value of $x$ if $\displaystyle\frac{x^4}{x^2} < 10$? | tiger_math |
In a regular \( n \)-sided polygon inscribed in a unit circle, let \( A_1, A_2, \ldots, A_n \) be the vertices of the polygon. If we denote the lengths of the diagonals (including the sides) from vertex \( A_1 \) to all other vertices \( A_k \) for \( k = 2, 3, \ldots, n \) as \( A_1A_2, A_1A_3, \ldots, A_1A_n \), what... | automath |
The area of the square formed on the diagonal of a rectangle as its side is 108 1/3 % more than the area of the rectangle. If the perimeter of the rectangle is 28 units, find the difference between the sides of the rectangle?
Answer Choices: (A) 6m (B) 9m (C) 5m (D) 2m (E) 3m | tiger_lab_math |
Tim has some cans of soda. Jeff comes by, and takes x cans of soda from Tim. Tim then goes and buys another half the amount of soda cans he had left. If Tim has 24 cans of soda in the end, Tim had 22 cans of soda at first. What is the value of unknown variable x? | tiger_math |
The number 504 is the product of three consecutive positive integers and can also be expressed as the sum of the cubes of three consecutive positive integers. Find the sum of those six integers. | nvidia_math |
Julian has 6 containers with 30 markers in each. He observed that 3/5 of the markers in 3 containers were still sealed. In 2 other containers, only 1/4 of the markers were used, while the last container was partially used. How many unused markers does Julian have? | nvidia_math |
A fair six-sided die is rolled twice. What is the probability that the sum of the two rolls is not a multiple of 3? Express your answer as a common fraction. | nvidia_math |
The amortization schedule for a 15-year mortgage with monthly payments of $800 shows that the total amount paid over the life of the loan is $144,000. If the interest rate is $x$ percent, compounded monthly, what is the amount borrowed? (Give your answer to the nearest dollar.) | nvidia_math |
Some friends each bought the same number of baseball cards, which come in packs of 20. They have 108 packs of baseball cards in all. How many baseball cards did each friend buy? | tiger_math |
Siddharth wants to borrow Rs.9000 at rate of interest 6% p.a. at S.I and lend the same amount at C.I at same rate of interest for two years. What would be his income in the above transaction?
Answer Choices: (A) s.21.60 (B) s.32.40 (C) s.21.18 (D) s.21.98 (E) s.21.29 | tiger_lab_math |
Given that $(x + y)^2 = 1$ and $xy = -4$, what is the value of $x^2 + y^2$? | tiger_math |
Solve for $m$, $n$, and $p$ in the equation $x(2x-5)=-2$, where the solutions are expressed in the form $\frac{m+\sqrt{n}}{p}$ and $\frac{m-\sqrt{n}}{p}$. Find $m+n+p$. | nvidia_math |
The price of buying a wooden toy at the new Craftee And Best store is $20, and the cost of buying a hat is $10. If Kendra went to the shop with a $100 bill and bought two wooden toys and three hats, calculate the change she received. | tiger_math |
What is the value of $f(f(2)) + f(f(4))$ for the function defined as \[f(x) = \left\{
\begin{array}{cl}
x+1 &\text{ if }x>0, \\
x-1 &\text{ if }x \le 0.
\end{array}
\right.\] | tiger_math |
Among all quadrilaterals $ABCD$ with right angles at $B$ and $C$, find the maximum value of $\sin A + \sin B \sin C + \sin D$.
Please note that I've created a new problem that still deals with trigonometric functions and optimization, but now involves a quadrilateral instead of a triangle, and includes right angles at... | nvidia_math |
the area of a square garden is q square feet and the perimeter is p feet . if q = p + 21 , what is the perimeter of the garden in feet ? Please write a program to solve it | tiger_lab_math |
Given an ordered field \( R \) that satisfies the properties of a field and an ordered ring, what is the necessary condition for the ordering \( \preceq \) on \( R \) to be considered a total ordering? Provide an example of an ordered field that meets this criterion. | automath |
What are the necessary and sufficient conditions on a basepoint-free line bundle \( L \) on a projective variety \( V \) to ensure that the morphism \( \pi: V \to \mathbb{P}^n \) defined by \( L \) induces a birational morphism from \( V \) to its image, and specifically, can you provide an example where these conditio... | automath |
The owner of a furniture shop charges his customer 30% more than the cost price. A customer paid Rs. 8450 for a computer table. What was the cost price of the computer table? | tiger_math |
Given a function \( f(x) = x^2 \), calculate the first derivative \( \frac{dy}{dx} \) where \( y = f(x) \). Additionally, evaluate \( \frac{d(f(x))}{dx} \) and discuss whether it is acceptable to express the derivative as \( df/dx \) in this context. What conclusions can you draw about the notation used for differentia... | automath |
What is the units digit of the sum of the cubes of the first 2015 even, positive integers? | nvidia_math |
Ashley earns $2500 per month and wants to save one-third of her monthly earnings to buy a house with a down payment of $12000. How many months of saving will it take her to have enough for the down payment? | nvidia_math |
During a school play, Jonah staffed the snack bar. He served 0.25 pitcher of lemonade during the first intermission, 0.4166666666666667 pitcher during the second, and 0.25 pitcher during the third. How many pitchers of lemonade did Jonah pour in all? | tiger_math |
Given a smooth algebraic surface \( X \) and a foliation \( F \) on \( X \), let \( K_F \) be the canonical bundle of the foliation. If the Kodaira dimension \( \kappa(K_F) \) is defined as the Kodaira dimension of \( K_F \), under what conditions does the Kodaira dimension \( \kappa(K_F) \) remain invariant for a fami... | automath |
The number 720 is divided into two parts such that the ratio of the two parts is 5:7. Find the number of possible ways to divide 720 into two parts with this ratio, given that each part must be a whole number greater than 0. | tiger_math |
Ava watched 18 movies, Ethan watched 15 movies, and Lily watched 20 movies. Ava and Ethan watched 2 of the same movies, and all three had watched the same 2 movies as well. In total, how many different movies have Ava, Ethan, and Lily watched? | nvidia_math |
What is the average of the absolute values of the differences of the cubes of the first five even positive integers and the squares of the first five odd positive integers? | nvidia_math |
Let $\theta$ be the angle between the planes $2x + y - 2z + 3 = 0$ and $6x + 3y + 2z - 5 = 0.$ Find $\cos \theta.$ | tiger_math |
Given the vectors \( \mathbf{v_1} = [2, -1, 1] \) and \( \mathbf{v_2} = [1, 1, 1] \), describe the parallelogram formed by these two vectors at the origin. What are the coordinates of the four vertices of this parallelogram? | automath |
Alyssa has 36 books. Nancy has 7 times more books than Alyssa. How many books does Nancy have ? Let's write a Python program to solve it. | tiger_lab_math |
A right pyramid has a square base with perimeter 24 inches. Its apex is 9 inches from each of the other vertices. What is the height of the pyramid from its peak to the center of its square base, in inches? | nvidia_math |
A cistern can be filled by a tap in 5 hours while it can be emptied by another tap in 10hours. If both the taps are opened simultaneously then after how much time will the cistern get filled?
Answer Choices: (A) 20hr (B) 10hr (C) 5hr (D) 4hr (E) 15hr | tiger_lab_math |
Given the input graph represented by the adjacency matrix:
\[
\begin{bmatrix}
0 & 4 & 0 & 0 & 0 \\
4 & 0 & 7 & 0 & 0 \\
0 & 8 & 0 & 9 & 0 \\
0 & 0 & 7 & 0 & 6 \\
0 & 2 & 0 & 9 & 0
\end{bmatrix}
\]
using Dijkstra’s algorithm starting from vertex 0, what is the shortest distance from the source vertex 0 to vertex 3? | automath |
A savings account offers an annual interest rate of 4%, compounded quarterly. If $2000 is deposited into the account, what is the maximum number of years the money can be left in the account without exceeding a total balance of $3000? | nvidia_math |
My father has 45 chickens, and there are 8 more chickens than ducks. How many chickens and ducks does my father have in total? | tiger_math |
Santino has 2 papaya trees and 3 mango trees. If each papaya tree produces 10 papayas and each mango tree produces 20 mangos, how many fruits does Santino have in total? | tiger_lab_math |
Ava and three colleagues are assigned to package 240 boxes of merchandise. Her colleagues each package 37 boxes. How many boxes does Ava need to package? | nvidia_math |
Let's say we have a composite function f(x) = (3x^2 + 5x - 2)^3, where g(x) = 3x^2 + 5x - 2 and h(x) = x^3. Use the chain rule to find f'(x) at x = 2. | tiger_lab_math |
Let $A$ be the matrix
$$A=\left\lgroup\begin{array}{cc}
1&2\\
2&1\\
\end{array}\right\rgroup$$and define $A_n = A^{n}$ for $n\geq 1.$
Compute
$$\sum_{n=1}^{\infty}\left|\frac{\det(A_{2n+1})}{\det(A_{2n})}\right|$$ | nvidia_math |
What is the greatest integer less than 100 for which the greatest common factor of that integer and 18 is 3? | nvidia_math |
A cone has a height of 15 cm and a radius of 4 cm. A smaller cone is cut off at the top of the original cone, such that the height of the remaining frustum is 2/3 of the original height. The lateral surface area of the remaining frustum is what fractional part of the lateral surface area of the original cone? | nvidia_math |
In a circle of radius 10 cm, an equilateral triangle is inscribed such that the length of each side is a function of the angle formed by the radii to the vertices. Find the value of the angle such that the area of the triangle is half the area of the circle. | nvidia_math |
A number is doubled and 5 is added. If the resultant is trebled, it becomes 105. What is that number?
Answer Choices: (A) 9 (B) 15 (C) 18 (D) None of these (E) Cannot be determined | tiger_lab_math |
Find the sum of the series \( S = \frac{1}{1 \times 5} + \frac{1}{5 \times 9} + \frac{1}{9 \times 13} + \ldots + \frac{1}{221 \times 225} \). | automath |
A square is inscribed in the ellipse
\[\frac{x^2}{10} + \frac{y^2}{25} = 1,\]so that its sides are parallel to the coordinate axes. Find the area of the square. | tiger_math |
A bicycle store is running a promotion. Every time a customer buys a bicycle, they will receive 2 bike clamps free with it. If the store sells 19 bikes in the morning and 27 bikes in the afternoon, how many bike clamps has the store given to customers? | tiger_lab_math |
Find the value of $25^{5/2}$ in simplified radical form. | tiger_math |
Juniper, the Irish Setter, has 4 bones. Her master gives her enough bones to double her number of bones. Unfortunately, the neighbor's dog steals away two of Juniper's bones. How many bones does Juniper have remaining? | tiger_lab_math |
If p=17 and q=11 are the two prime factors used in the RSA encryption algorithm, and the public key is (e=7, n=187), what is the corresponding private key (d)? | tiger_lab_math |
Liam is a piano student learning four new pieces this semester. His teacher has him spend 45 minutes on learning the first piece. The second piece he learns in two-thirds the time of the first piece. The third piece takes him as long as the first piece, and the fourth piece is the most challenging, taking him twice as ... | nvidia_math |
In a kilometer race, A beats B by 50 meters or 10 seconds. What time does A take to complete the race?
Answer Choices: (A) 900 (B) 190 (C) 277 (D) 262 (E) 223 | tiger_lab_math |
Consider the polynomial \( f(n) = n^2 - 5 \). What is the least positive prime \( p \) such that \( f(n) \equiv 0 \pmod{p} \) has no solutions for \( n \) in the integers? | automath |
If a machine operates at a power of 500 watts for 10 seconds, how many joules of energy does it use? | automath |
Erin counts six trolls hiding by the path in the forest, 6 less than four times that number of trolls hiding under the bridge, and half as many trolls hiding in the plains as under the bridge. How many trolls did she count in total? | nvidia_math |
A variable \( X \) represents the speeds of cars measured by a radar unit, which follows a normal distribution with a mean \( \mu = 30 \) and a standard deviation \( \sigma = 4 \). What is the probability that a randomly selected car is traveling at a speed greater than 38 mph? | automath |
Find the $r$ that satisfies $\log_{16} (r+16) = \frac{5}{4}$. | tiger_lab_math |
If John pays $1000 per episode for the first half of the season and the episodes in the second half are 120% more expensive, and there are a total of 22 episodes, what is the total cost of the entire season? | tiger_math |
The equation $$x^2 + 9y^2 - 12x - 18y = k$$represents a non-degenerate ellipse if and only if $k < b$. Find the value of $b$. | nvidia_math |
Solve the equation
$5x^2-kx+1=0$
determine $k$ such that the difference of the roots be equal to unity.
Write another problem inspired by this one:
The roots $x_1$ and $x_2$ of the equation
$$x^2-3ax+a^2=0$$
are such that
$x_1^2+x_2^2=1.75$.
Determine $a$.
$$x^2-3ax+a^2=0$$
are such that
$x_1^2+x_2^2=1.75$.
Determine... | nvidia_math |
Given the moment-generating function (MGF) of a random variable \( X \) defined as \( M_X(t) = E(e^{tX}) \), determine the moment-generating function for a Poisson random variable with parameter \( \lambda \). What is the expression for \( M_X(t) \)? | automath |
Betty’s herb garden has 2 more than twice as many oregano plants as basil. If there are 5 basil plants, what is the total number of plants in the garden? | tiger_math |
Suppose you take a random sample of 10 items from a lot known to have an unknown proportion of defective items, modeled by a beta prior distribution $\theta \sim \mathsf{Beta}(1,19)$. If you observe 1 defective item in your sample, what is the posterior distribution of the defect rate $\theta$, and what is the posterio... | automath |
Consider a discrete transfer function defined by the zeros \( Z = \{1, 2\} \) and poles \( P = \{0, 0.5, 0.75\} \) with a gain \( K = 2 \). Calculate the value of the transfer function \( H(z) \) at \( z = 3 \). What is the result? | automath |
Let $a$, $b$, and $c$ be real numbers such that
$$a^2 + b^2 + c^2 = ab + bc + ca.$$
Find the sum of all possible values of $a^3 + b^3 + c^3 - 3abc$. | nvidia_math |
There are x times as many swordfish as pufferfish in an aquarium exhibit. If there are 90 fish total, There are 15 pufferfish. What is the value of unknown variable x? What is the value of unknown variable x? | tiger_math |
My friend has a collection of 15 rare books, including 4 novels by Jane Austen. If she arranges the books randomly on a shelf, what is the probability that all four Austen novels will be placed together? | nvidia_math |
Given three noncoplanar vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\) in \(\mathbb{R}^3\), which satisfy the Jacobi identity for the cross product:
\[
\vec{a} \times (\vec{b} \times \vec{c}) + \vec{b} \times (\vec{c} \times \vec{a}) + \vec{c} \times (\vec{a} \times \vec{b}) = \vec{0},
\]
what is the geometric inter... | automath |
$DEF$ is a triangle where $D = (0,0), E = (24, 20)$, and both the coordinates of $F$ are integers. What is the minimum area $\triangle DEF$ can have? | nvidia_math |
Given that $a$ is an odd multiple of $1183$, find the greatest common divisor of $2a^2+29a+65$ and $a+13$. | tiger_lab_math |
On dividing a number by 68, we get 269 as quotient and 0 as remainder. On dividing the same number by 67, what will be the remainder? | tiger_math |
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