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Let \( f(n) = 3n^2 + 2n + 1 \) and \( g(n) = n^2 \). Determine whether \( f(n) \) is \( O(g(n)) \), \( \Omega(g(n)) \), or \( \Theta(g(n)) \) by providing an appropriate constant and \( n_0 \) for each case. | automath |
Find the number of positive integers that can be expressed as the sum of two perfect squares. | nvidia_math |
Find the value of $\frac{4 \cdot 6! + 20\cdot 5!}{7!}$ | tiger_math |
Alice wants 1000 folded paper cranes. She folds x% by herself, and a friend folds a fifth of the remaining paper cranes for her. Alice still needs to fold 400 paper cranes. What is the value of unknown variable x? | tiger_math |
Two congruent rectangular prisms each have dimensions 6 cm by 8 cm by 10 cm. The width of one prism and the length of the other are both decreased by the same nonzero number of centimeters, while their heights remain unchanged. The resulting surface areas are equal. How many centimeters is the decrease? Express your an... | nvidia_math |
Stacy has 2 more than triple as many berries as Steve. Steve has one half as many berries as Skylar. If Skylar has 20 berries, how many berries does Stacy have? | tiger_math |
The median of the set {3, 8, x, 12, 17} is equal to the median of the set {4, 9, y}. What is the value of y? | nvidia_math |
A rectangle and a regular hexagon have equal perimeters. The length of the rectangle is 10 cm, and its width is 8 cm. What is the area of the regular hexagon in square centimeters? | nvidia_math |
Consider the piecewise function:
\[g(x) = \left\{
\begin{array}{cl}
x^3 &\text{ if }x>2, \\
3x &\text{ if }x \le 2.
\end{array}
\right.\]
Find $g(g(g(1)))$. | nvidia_math |
What is the approximate number of miles in one degree of latitude and one degree of longitude at the equator? | automath |
An engineering student has to secure 36% marks to pass. He gets 130 marks and fails by 14 marks. The maximum No. of marks obtained by him is?
Answer Choices: (A) 300 (B) 400 (C) 500 (D) 600 (E) 700 | tiger_lab_math |
What is the relationship between the expressions $\sqrt{-(m^2-i\epsilon)X^2}$ and $\sqrt{im(-X^2 + i\epsilon)}$ as derived in the context of the Feynman propagator, and can you provide a detailed explanation of the error in equating these two forms? | automath |
As $n$ ranges over the positive integers, what is the maximum possible value for the greatest common divisor of $11n+3$ and $6n+1$? | nvidia_math |
Linda is building a new hotel with two wings. The first wing has 9 floors and each floor has 6 halls each with 32 rooms. The second wing has 7 floors each with 9 halls with 40 rooms each. How many rooms are in the hotel total? | tiger_math |
Determine the sum of all single-digit replacements for $z$ such that the number ${24{,}z38}$ is divisible by 6. | nvidia_math |
Maria baked a large batch of 36 cookies. She had two with breakfast and one with dinner every day for 4 days. Her brother took 2 cookies per day for 4 days to eat at school. They gave 6 cookies to their neighbors as a welcome gift. How many cookies were left? | nvidia_math |
If a chlorine ion has a mass of \( 5.86 \times 10^{-26} \) kg and its position is measured with an uncertainty of \( 1.00 \, \mu m \), what is the minimum uncertainty in its velocity according to the Heisenberg uncertainty principle? Additionally, if the chlorine ion achieves this velocity, what is its kinetic energy i... | automath |
Find the analytic continuation of the function f(z) = sqrt(z-1) from z=1 to z=4. | tiger_lab_math |
In a non-inertial frame of reference where a planet is in circular orbit around a stationary star, if the position of the planet at time \( t \) is given by the vector \( r(\cos(\omega t),\sin(\omega t),0) \), and the fictitious acceleration due to the non-inertial frame is \( rw^2(\cos(\omega t),\sin(\omega t),0) \), ... | automath |
A convex pentagon $ABCDE$ is given such that
$AB+BC=AC$, $BC+CD=BD$, $CD+DE=CE$, and $DE+EA=DA$.
If the length of edge $AE$ is $5$, what is the length of edge $BD$? | nvidia_math |
Given a length-doubling pseudo-random generator \( G: \{0,1\}^n \to \{0,1\}^{2n} \) that produces outputs indistinguishable from the uniform distribution over \( \{0,1\}^{2n} \), what is the probability that a randomly chosen bit string in \( \{0,1\}^{2n} \) is a valid output of \( G \)? | automath |
The function $g$ is defined on real numbers as follows:
\[g(x) = \left\{
\begin{array}{cl}
x^2 & \text{if $x < 2$}, \\
g(x - 2) + 1 & \text{if $x \ge 2$}.
\end{array}
\right.\]Find the minimum value of the function. | nvidia_math |
What is the value of the integral \( \int \sec(2t) \tan(2t) \, dt \)? | automath |
John joins a country club with 3 other members of his family. The fee to join is $4000 per person. There is also a monthly cost of $1000 per person. John pays half the cost. How much does John pay for the first year? | tiger_lab_math |
Sam is hired for a 25-day period. On days that he works, he earns $\$$50. For each day that he does not work, $\$$25 is subtracted from his earnings. At the end of the 25-day period, he received $\$$750. How many days did he not work? | tiger_math |
Given a mass \( m = 3 \, \text{kg} \) attached to a horizontal spring with spring constant \( k = 200 \, \text{N/m} \), if the system is displaced \( x = 0.2 \, \text{m} \) from the equilibrium position and released, what will be the maximum speed of the mass during its oscillation? | automath |
Cadence has worked for her new company x months longer than she worked for her old company. If she worked at her old company for 3 years earning $5000 per month, and she's now earning 20% more in the new company, calculate the total amount of money she's earned in both companies in the period she's worked for them?
If ... | tiger_math |
Find all solutions to the equation $\sqrt{3-4t} = 7$. | nvidia_math |
A profit of Rs. 1000 is divided between X and Y in the ratio of 1/2 : 1/3. What is the difference between their profit shares? | tiger_math |
The area of a square equals the square of a length of the side of the square. The perimeter of a square equals the sum of the lengths of all four sides. The sum of the areas of two squares is 65, while the difference in their areas is 33. Find the sum of their perimeters. | nvidia_math |
in a certain company , a third of the workers do not have a retirement plan . 40 % of the workers who do not have a retirement plan are women , and 40 % of the workers who do have a retirement plan are men . if 120 of the workers of that company are men , how many of the workers are women ? Let's program in Python in t... | tiger_lab_math |
Brad wanted to set up a lemonade stand to earn some money. Every gallon of lemonade would yield 16 glasses. He figured it would cost him $3.50 to make every gallon of lemonade. He made 2 gallons to sell and decided he would sell each glass for $1.00 He drank 5 glasses while sitting in his stand. He sold all but 6 ... | tiger_lab_math |
Given the equation \( x = x \log(y) + \log(y) \), can you verify whether the solution \( y = 10^{\frac{x}{x+1}} \) is correct? Additionally, analyze if \( y = 10^{\frac{x}{2x}} \) could be a valid alternative solution and justify your reasoning. | automath |
Jason is making pasta. He fills the pot with 41 degree water. Each minute the temperature of the water increases by 3 degrees. Once the water reaches 212 degrees and is boiling, Jason needs to cook his pasta for 12 minutes. Then it will take him 1/3 that long to mix the pasta with the sauce and make a salad. How many m... | tiger_math |
Suppose a population of plants follows a logistic growth model governed by the equation
$$\frac{dy(t)}{dt} = ry(t)\left ( 1 -\frac{y(t)}{K} \right )$$
where \(r\) is the intrinsic growth rate, \(K\) is the carrying capacity, and \(y(t)\) is the population size at time \(t\). Given that the initial population size \(y... | automath |
Let $\mathbf{a}$ and $\mathbf{b}$ be vectors such that $\|\mathbf{a}\| = 3,$ $\|\mathbf{b}\| = 4,$ and $\|\mathbf{a} \times \mathbf{b}\| = 9.$ Find $|\mathbf{a} \cdot \mathbf{b}|.$ | tiger_math |
In a triangular array of 15 dots, 7 are red and 8 are blue. Let T be the number of lines formed by joining any two dots of different colors. What is the average value of T if all possible colorings of the dots are considered? | nvidia_math |
In a Geometry exam, Madeline got 2 mistakes which are half as many mistakes as Leo. Brent scored 25 and has 1 more mistake than Leo. What is Madeline's score? | nvidia_math |
Let $f(x) = -2x^2 + 12x - 5$. If we express $f(x)$ in the form $a(x - r)(x - s)$, then what is $a(r+s)$? | nvidia_math |
Alex was 11 years old 5 years ago. How old will he be 8 years from now? | nvidia_math |
What is the simplified value of the sum: $-1^{2004} + (-1)^{2005} + 1^{2006} -1^{2007}$? | tiger_lab_math |
Compute the range of $y=|x+7|-|x-2|$. | nvidia_math |
Find the values of $x$ for which the expression $\frac{3x}{2x^2 - 6x + 4}$ is undefined, and determine the sum of these values. | tiger_math |
Mark constructed a deck that was 30 feet by 40 feet. It cost $3 per square foot. He then paid an extra $x per square foot for sealant. How much did he pay?
If we know the answer to the above question is 4800, what is the value of unknown variable x? | tiger_math |
Given two sets of points \( a \) and \( b \) in \( \mathbb{R}^d \), let their respective centroid points be \( \mu_a \) and \( \mu_b \). If the distance squared between the centroids is defined as \( d^2(\mu_a, \mu_b) = \sum_{i} (a_i - \mu_a)^2 + \sum_{j} (b_j - \mu_b)^2 - 2 \langle \mu_a, \mu_b \rangle \), and if \( |... | automath |
Lily's grandmother gave her $60 to buy supplies for a bake sale. Lily bought 5 bags of flour and 3 jars of jam. Each bag of flour cost $3 and each jar of jam cost $4. How much money does Lily have left? | nvidia_math |
Determine the ratio of the volumes of the two rectangular prisms $A$ and $B$ if the ratios of the lengths of their corresponding sides are $1:\sqrt2$, $2:1$, and $\sqrt2:3$. Express your answer as a common fraction. | nvidia_math |
What is the sum of the first 50 letters in the pattern ABCABCABC...? | tiger_math |
"statement 1": In Winchard ' s company 60 % of the employees earn less than $ 50,000 a year , 60 % of the employees earn more than $ 40,000 a year , 11 % of the employees earn $ 43,000 a year and 5 % of the employees earn $ 49,000 a year, "statement 2" :In Winchard ' s company 30 % of the employees earn less than $ 50... | tiger_lab_math |
Lily completed 25 art projects in 3 months. Emma completed three times as many art projects as Lily in the same period. How many art projects did they both complete in total? | nvidia_math |
Determine if the following vectors are parallel:
$$\begin{pmatrix} 1 \\ -3 \\ 2 \end{pmatrix} \text{ and } \begin{pmatrix} -2 \\ 6 \\ -4 \end{pmatrix}$$ | tiger_math |
What is the ratio of the volume of a sphere with radius 6 units to the area of its circumscribed cube? Express your answer as a common fraction in simplest form. | nvidia_math |
The speed of a boat in upstream is 100 kmph and the speed of the boat downstream is 112 kmph. Find the speed of the boat in still water and the speed of the stream?
Answer Choices: (A) 70, 10 kmph (B) 106, 6 kmph (C) 50, 60 kmph (D) 45, 55 kmph (E) 45, 65 kmph | tiger_lab_math |
Given the matrix \( A = \begin{pmatrix} 3 & 4 \\ 2 & 5 \end{pmatrix} \), compute the inverse of \( A \) in \( \mathbb{Z}/12\mathbb{Z} \) using the method of inverting the matrix over \( \mathbb{Q} \) and then adjusting the results modulo 12. Verify that the result corresponds to the correct inverse in \( \mathbb{Z}/12\... | automath |
The Guests at a football banquet consumed a total of 325 pounds of food. if no individual guest consumed more than 2 pounds of food, What is the minimum number of guests that could have attended the banquet?
Answer Choices: (A) 160 (B) 161 (C) 162 (D) 163 (E) 164 | tiger_lab_math |
If Princess Daphne purchased three necklaces and a set of earrings for a total of $240,000, and all three necklaces were priced equally, while the earrings were three times the price of a single necklace, what was the cost of a single necklace? | tiger_math |
Colin ran his first mile in 6 minutes. He ran his next two miles in 5 minutes each and finished his 4th mile in 4 minutes. What was the average time it took him to run a mile? | tiger_math |
After getting two successive discounts, a shirt with a list price of Rs. 300 is available at Rs. 210. if the second discount is 12.5%, find the first discount.
Answer Choices: (A) 15% (B) 20% (C) 25% (D) 12.5% (E) None of these | tiger_lab_math |
Integer m has 4 different prime factors and n has 3 different prime factors. If m and n have a certain greatest common factor, and mn has 5 different prime factors, what is the greatest common factor of m and n? | tiger_math |
What is the percentage increase in your earnings from last year to this year, if you earned $72,400 last year and $75,500 this year? | automath |
What are the coordinates of the points of inflection for the curve \( y = e^{-x^2} \)? | automath |
Find the sum of the squares of the values of x that satisfy the equation |5x| - 10 = 40. | tiger_math |
A man purchased a watch for Rs.400 and sold it at a gain of 20% of the selling price. The selling price of the watch is:
Answer Choices: (A) 232 (B) 480 (C) 178 (D) 277 (E) 1981 | tiger_lab_math |
The average monthly salary of 8 workers and one supervisor in a factory was $430. The supervisor's salary was a certain amount per month. When the supervisor retired, a new person was appointed and then the average salary of 9 people was $420 per month. The salary of the new supervisor is $780. What was the salary of t... | tiger_math |
A certain type of bacteria grows in a way that every day, its population is multiplied by a constant factor. On the first day, the bacteria's population is 100, and by the seventh day, it has reached 180,000. However, due to a limited food supply, it is discovered that the bacteria population cannot exceed a certain th... | nvidia_math |
A farmer wants to maximize his profits by planting two types of crops: wheat and corn. He has 200 acres of land and 400 units of fertilizer. He expects to earn a profit of $100 per acre of wheat and $200 per acre of corn. Wheat requires 2 units of fertilizer per acre, and corn requires 4 units of fertilizer per acre. W... | tiger_lab_math |
In a given population dynamics scenario, if a Galton-Watson process with mean offspring $\lambda$ exhibits a steady-state behavior described by the convergence of $Z_n(\lambda)/\lambda^n$ to a limit $W(\lambda)$, what conditions must be met for the convergence of the functional $\lambda \mapsto Z_n(\lambda)/\lambda^n$ ... | automath |
Emily plans to travel to Japan. She requires 8 different vaccinations and a medical consultation. Each vaccination costs $30, and the consultation costs $200, but her insurance will cover 75% of these medical expenses. The trip itself costs $1500. How much will she have to pay? | nvidia_math |
Problem:
A customer needs to rent a car for a certain number of days. Company A charges a fixed rate of $\$40$ per day, while Company B charges a base rate of $\$20$ plus an additional $\$0.25$ per mile driven. If the customer plans to drive an average of $x$ miles per day, for what minimum number of days will Company... | nvidia_math |
Ava works as a librarian at a nearby school. One morning, she organizes 120 books on a shelf. By recess, some books had been checked out. She then adds 25 more books to the shelf. By the end of the school day, 20 more books had been checked out. If the shelf had 75 books remaining by the end of the school day, how many... | nvidia_math |
Emily has containers that she uses to store paint. They store 5 liters more than three times as much as a small bucket. If she has 3 containers and a bucket stores 15 liters, how much paint can she store? | nvidia_math |
TWO WORKERS A AND B ARE ENGAGED TO DO A WORK. A WORKING ALONE TAKES 8 HOURS MORE TO COMPLETE THE JOB THAN IF BOTH WORKED TOGETHER. IF B WORKED ALONE, HE WOULD NEED 4 1/2 HOURS MORE TO COMPLETE THE JOB THAN THEY BOTH WORKING TOGETHER. WHAT TIME WOULD THEY TAKE TO DO THE WORK TOGETHER?
Answer Choices: (A) 4 HOURS (B) 5 H... | tiger_lab_math |
There are 12 more green apples than red apples in a bowl. There are 16 red apples. How many apples are there in the bowl? Please respond by writing a program in Python. | tiger_lab_math |
A company produces widgets using labor and raw materials. The cost of labor is $50 per hour and the cost of raw materials is $10 per widget. The company can produce up to 100 widgets per day. The demand function for the widgets is given by P=250-2Q, where P is the price per widget and Q is the quantity sold per day. Th... | tiger_lab_math |
Rajesh walked 10 kilometers less than 4 times the distance that Hiro walked. Together they walked 25 kilometers. How many kilometers did Rajesh walk? | tiger_lab_math |
Sure! Here's an example question for a math student to solve:
A shipping company needs to determine the most efficient way to transport goods from a warehouse to 5 different stores using a network of roads. The distances between the warehouse and the stores are as follows:
- Store 1: 10 km
- Store 2: 8 km
- Store 3: ... | tiger_lab_math |
What approximate value should come in place of the question mark (?) in the following equation ?
1⁄7 of 1596 + 3015 = ? – 2150
Answer Choices: (A) 7200 (B) 48000 (C) 5380 (D) 58000 (E) 6300 | tiger_lab_math |
What is the correct expression for a function \( f: \mathbb{R} \to \mathbb{R} \) to denote that \( f \) is a constant function, and how would you negate that expression? | automath |
Aisha likes to mix and match her scarves and hats. She has three times as many scarves as hats and thinks of each unique pairing of scarf and hat as a distinct look. She owns four hats. How many different looks can Aisha create? | nvidia_math |
Convert $1024_{16}$ to base 2. | nvidia_math |
Let \( A \) be a positive definite \( n \times n \) matrix. Suppose \( B = A^{-1} \) and we define the matrix \( C \) such that \( C_{ij} = (B_{ij})^2 \). If \( C \) is invertible, what is the relationship between the inverse of \( C \) and the original matrix \( A \)? Specifically, can you express the inverse of \( C ... | automath |
Let \( S = \{ a, b, c \} \) be a finite set, and define a right operation \( \rightarrow \) on \( S \) such that for any elements \( x, y \in S \), \( x \rightarrow y = y \). Find the Cayley table for the algebraic structure \( \struct{S, \rightarrow} \) and verify the characteristic that each column contains exactly o... | 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) 1 (B) 3 (C) 4 (D) 2 (E) 5 | tiger_lab_math |
Given a multiple linear regression model where the dependent variable \( Y \) is estimated as a linear function of two independent variables \( X_1 \) and \( X_2 \), if the estimated coefficients for \( X_1 \) and \( X_2 \) are 3 and -2, respectively, what will be the predicted value of \( Y \) when \( X_1 = 4 \) and \... | automath |
A man spends 1/3rd of his salary on food. he spends 1/4th on rent and 1/5th on cloths. If he is left with 1760, then who much salary does he earn
Answer Choices: (A) 8223 (B) 8123 (C) 8323 (D) 8423 (E) 8523 | tiger_lab_math |
What is the value of the discriminant for the quadratic equation \( r^2 + 5r + 2 = 0 \), and what can you conclude about the nature of its roots based on that value? | automath |
What is the magnetic flux through the circular cross-sectional area of a solenoid that is 2.9 cm in diameter and 15 cm in length, has 300 turns, and carries a current of 5 A? | automath |
Given the second-degree equation in three dimensions
$$3x^2 - 4y^2 + 6z^2 + 8yz - 2zx + 4xy + 14x - 10y - 4z + 5 = 0,$$
determine the geometric nature of the surface it represents and write the equation in standard form, identifying the type of quadric surface, including the necessary coefficients and parameters in... | automath |
Given the identity function defined as `let id = fun x -> x`, consider the expression `let a = id 0 in id true`. What type does the overall expression infer to in a language with type schemes, and how does the generalization of the type for `id` facilitate this inference? | automath |
If p is a natural number and p! ends with g trailing zeros, then the number of zeros that (5p)! ends with will be
Answer Choices: (A) a) (p+g) trailing zeros (B) b) (5p+y) trailing zeros (C) c) (5p+5y) trailing zeros (D) d) (p+5y) trailing zeros (E) e) none of them above | tiger_lab_math |
The length of the rectangular field is double its width. Inside the field, there is a square-shaped pond 8 m long. The area of the pond is related to the area of the field. The length of the field is 16 m. What is the ratio of the area of the pond to the area of the field? | tiger_math |
Emily and Emma live 1.5 miles apart. It takes Emily 6 minutes to walk directly to Emma's house. At the same rate, how many minutes would it take Emily to walk the 2.75 miles from her own house to David's house? | nvidia_math |
Given the joint density function \(\psi_{(X,Y)}(x,y) = \frac{8xy}{5}\) for \((x,y) \in T\) where \(T = \{(x,y) \mid 1 < x+y < 2, x > 0, y > 0\}\), find the probability distribution function \(F_Z(z)\) of the random variable \(Z = X + Y\) and determine \(E[Z]\). What is the value of \(E[Z]\)? | automath |
Daria the botanist is studying the growth patterns of a rare fern that doubles its height each week. Initially, Daria observes that the fern is 4 centimeters tall. She wants to predict its height after it has grown continuously for 5 weeks. Additionally, every week she measures its height and rounds it up to the neares... | tiger_math |
Three companies, A, B, and C, each purchased 10 ad spaces on a newspaper, with each ad space having a size of 12 foot by 5-foot rectangle. If each square foot ad was charged at $60, how much money did the three pay for the ads combined? | tiger_lab_math |
Find the smallest positive integer $y$ that satisfies $y + 6723 \equiv 1045 \pmod{15}$. | nvidia_math |
A vine had 800 grapes before the harvest season. In the first week of the harvest, the vine shed 3/8 of its grapes due to strong winds. In the second week, it shed 25% of the remaining grapes because of a fungal infection. In the third week, the vine shed 2/3 times as many grapes as it shed on the second week due to a ... | nvidia_math |
Maria rents a car for $30 a day and $0.30 a mile. If she rents it for 3 days and drives it 240 miles, how many dollars does she pay? | nvidia_math |
Given that $O$ is the center of the circle with $AB$ as diameter and $D$ a point between $A$ and $B$, if $OD \perp AB$ and $\angle AOB$ measures $120^\circ$, what is the ratio of $OA$ to $AD$? [asy]
import olympiad; import geometry; size(150); defaultpen(linewidth(0.8));
draw(origin--(4,4), black); draw(origin--(4,-4),... | nvidia_math |
Given the transformations \( x = r \cos(\theta) \) and \( y = r \sin(\theta) \), express the product of the partial derivatives \(\frac{\partial f}{\partial x} \times \frac{\partial f}{\partial y}\) in terms of \(\theta\) and \(r\) for a scalar function \(f\). What is this expression in terms of \( \theta \) and \( r \... | automath |
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