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A candy company sells premium chocolate at $5 per pound and regular chocolate at $4 per pound in increment of whole pound only.If Barrett buys a pack of candies which costs him $40, how many pounds of premium chocolates are in the box?
Answer Choices: (A) [A] 1 (B) [B] 2 (C) [C] 3 (D) [D] 4 (E) [E] 5 | tiger_lab_math |
Find the smallest positive four-digit number that is divisible by each of the four smallest prime numbers. | nvidia_math |
Find the remainder when $8730 + 8731 + 8732 + 8733 + 8734$ is divided by 11. | tiger_math |
What is the greatest three-digit multiple of 19? | tiger_math |
Given the runs scored by eleven cricket players P1, P2, P3, ..., P11 during their careers, the total runs scored is 33,120. If the contribution of each player to the total runs is as follows:
- P1: 2,600
- P2: 1,070
- P3: 5,350
- P4: 7,215
- P5: 2,400
- P6: 1,440
- P7: 2,110
- P8: 1,840
- P9: 1,900
- P10: 3,880
- P11:... | automath |
In triangle $ABC,$ $b = 5,$ $c = 4,$ and $\cos (B - C) = \frac{31}{32}.$ Find $a.$
Note: $a$ is the side length opposite $\angle A,$ etc. | nvidia_math |
Determine the relationships among the sets \( A, B, C, \) and \( D \) defined as follows:
\[
A = \{ x : x^2 - 8x + 12 = 0 \}, \quad B = \{ 2, 4, 6 \}, \quad C = \{ 2, 4, 6, 8, \ldots \}, \quad D = \{ 6 \}
\]
Which of the sets are subsets of which? Provide a detailed explanation of your reasoning. | automath |
Let \( f: [3, \infty) \to [0, \infty) \) be defined by \( f(x) = \sqrt{x^2 - 9} \). What is the range of the function \( f(x) \) in interval notation? | automath |
Given the following stem and leaf plot:
```
Stem | Leaf
-------------
2 | 2 5 6
3 | 3 5 6 8
4 | 2 5 5 6 7 7 7 8 8 9
5 | 2 4 6 8 8
6 | 5 8 9
7 | 4 5
8 | 7
9 | 9
```
What is the mode of the dataset represented by this stem and leaf plot? | automath |
A Statistics student wants to find out the average daily allowance of the middle school students. According to his survey, 2/3 of the students receive an average of $6 allowance per day while the rest gets an average of $4 a day. If he surveyed 60 students, what is the total amount of money those 60 students get in a d... | tiger_lab_math |
Eight people fit in a row on an airplane, and there are 12 rows. Only 3/4 of the seats in each row are allowed to be seated. How many seats will not be occupied on that plane? Please write a program to solve it | tiger_lab_math |
I had some donuts when I left home this morning but only 9 when I came back home. 70% of the donuts are missing. How many donuts did I have initially? | tiger_math |
How many portions of 200 ml milk can Jasmine pour from a full 2-liter container of milk? | tiger_math |
The operation $\dagger$ is defined as $\frac{m}{n}\dagger\frac{p}{q} = (m)(p)(\frac{q}{n}).$ What is the simplified value of $\frac{7}{12}\dagger\frac{8}{3}$? Please respond by writing a program in Python. | tiger_lab_math |
Simplify $\displaystyle\frac{2-3i}{1+i}$, where $i^2 = -1.$ | nvidia_math |
What is the value of the number represented by the power circuit defined by the expression \( x = 2^{2^{2}} \) and \( y = x^{2} \) if the power circuit allows for exponentiation using previously computed values? | automath |
The American Academy of Pediatrics recommended no more than 2 hours of screen time each day for children. Mrs. Merril wants to follow a 2-hour screen time for her child. How many minutes will she allow her child to use his gadget this evening if he already used his gadget for 45 minutes in the morning? | nvidia_math |
If 2 tablespoons of popcorn kernels can make 4 cups of popcorn, how many tablespoons of popcorn kernels will be needed for Joanie's 3 cups, Mitchell's 4 cups, Miles and Davis's shared 6 cups, and Cliff's 3 cups for movie night? | tiger_math |
Given an array of integers \( nums \), determine if there exist indices \( i, j, k \) such that \( i < j < k \) and \( nums[i] < nums[j] < nums[k] \). If such indices exist, return the count of all unique triples \( (i, j, k) \) that satisfy this condition. For instance, if \( nums = [1, 2, 3, 4] \), the output should ... | automath |
a river 2 m deep and 45 m wide is flowing at the rate of 6 kmph the amount of water that runs into the sea per minute is ? Let's write a Python program. | tiger_lab_math |
What is the number of degrees in $x$? [asy]
import olympiad; import geometry; defaultpen(linewidth(0.8)); size(150);
pair A = origin, B = (Cos(50),Sin(50)), C = (Cos(50) + Sin(50)*sqrt(3),0);
draw(A--B--C--cycle);
label("$30^\circ$",C-(0.3,-.03),NW);
label("$2x$",B + (-.10,-0.1),SE);
label("$x$",A+(0.15,.03),NE);
[/asy... | nvidia_math |
Gumballs come in packages of 5. Nathan ate a certain number of gumballs and consumed 4 whole boxes. How many gumballs did Nathan eat? | tiger_math |
On a bookshelf, LIBRARY is spelled out with 7 letters, one letter per book. Two vowels and three consonants are removed and put away in a box. If the L's and I's are indistinguishable, how many distinct possible collections of letters could be put in the box? | nvidia_math |
Three cars all leave Denver at the same time and travel in different directions. One car travels 10 mi/h faster than the second car, and the second car travels 5 mi/h faster than the third car. After 7 hours, the combined distance between all three cars is 840 mi. How fast is each car traveling? | tiger_math |
What are the necessary conditions under which the equality
$$\int_a^b f(x) \, dx = \lim_{n\to \infty} \int_a^b f_n(x) \, dx$$
holds true for a sequence of functions \( f_n \) converging to \( f \)? Specifically, discuss the implications of uniform convergence and integrability in the context of Riemann and Lebesgue... | automath |
A student needs to calculate the scalar curvature of the two-dimensional surface with parametric equations:
x(u,v) = u^2 + v^2
y(u,v) = u^2 - v^2
z(u,v) = 2uv
over the point (1,1). | tiger_lab_math |
Find the number of positive divisors of 2002. Let's write a program. | tiger_lab_math |
An alien invades Earth. It abducts 200 people. He returns 80% of the people abducted. After that he takes 10 people to another planet. He took the rest to his home planet. How many people did he take to his home planet? | tiger_lab_math |
What is the maximum number of Voronoi cells that can be created when generating a Voronoi diagram with 100 radial points and 360 azimuthal points arranged in a circular pattern? Given the behavior observed, why does the actual count fall short by a specific number, and what conditions would lead to this discrepancy? | automath |
In a bag containing 8 balls—3 red, 1 green, and 4 blue—what is the probability of drawing 2 red balls, 1 green ball, and 2 blue balls in a sequence of 5 draws, assuming that the draws are made with replacement? | automath |
What is the greatest product obtainable from two integers whose sum is 246? | tiger_math |
Given the Taylor series expansions for sine and cosine, calculate the sine and cosine of \( \frac{\pi}{6} \) (30 degrees) using the first four terms of their respective series, and provide the approximate values. | automath |
Derrick measures the length of his yard. The length of Alex's yard is x% the size of Derrick's and the length of Brianne's yard is 6 times the size of Alex's. If Brianne's yard is 30 yards long, how long is Derrick's yard, In yards, the answer is 10. What is the value of unknown variable x? | tiger_math |
Problem:
Convert $\sqrt{3} e^{i \tan^{-1} 2}$ to rectangular form using only the properties of complex numbers and trigonometric identities. | nvidia_math |
A point P(-2,9) lies on the curve y = 2x^3 - 3x^2 + 1. Find the slope of the tangent line to the curve at point P. | tiger_lab_math |
Rachel is 4 years older than Leah. If the sum of their ages is 34, how old is Rachel? | tiger_lab_math |
Ross is 20 years older than his daughter Julie. In 5 years Ross will be twice as old as Julie. What is Julies current age.
Answer Choices: (A) 10 (B) 15 (C) 20 (D) 30 (E) 25 | tiger_lab_math |
The radius of a circle is increased by 1%. Find how much % does its area increases?
Answer Choices: (A) 2.06 (B) 2.09 (C) 2.03 (D) 2.01 (E) 2.02 | tiger_lab_math |
On a quiz, Martin answered three fewer questions correctly than Kelsey, Kelsey answered eight more questions correctly than Campbell, and Campbell answered twice as many questions correctly as Wendy. If Wendy answered 20 questions correctly, how many questions did Martin answer correctly? | tiger_math |
Bill, Ted, and Alice head to the river to throw both sticks and rocks into the water. Bill throws 6 more sticks into the river than Ted does, and Alice throws half as many sticks as Ted. However, Ted tosses twice as many rocks into the river as Bill, and Alice tosses 3 times as many rocks as Bill.
Each stick that Ted... | tiger_math |
Using all the letters of the word "WEDNESDAY", how many different words can be formed?
Answer Choices: (A) A) 7 (B) B) 9! (C) C) 8 (D) D) 7! (E) E) 3 | tiger_lab_math |
A swimming pool is being filled with water at a rate of 1 cubic foot per minute. The pool is in the shape of a rectangular prism with a length of 15 feet, a width of 8 feet, and a height of 4 feet. How many minutes will it take to fill the pool if it is currently empty? | nvidia_math |
Every some minutes during sunset, the sky changes to a new color. Over the two hours of a long summer sunset, the sky turned 12 colors. How many minutes does it take for the sky to change to a new color? | tiger_math |
If $\theta=\arctan(\sqrt{x})$, what is the value of $x$ when $\tan2\theta = 2$? | nvidia_math |
Let \( f(t) \) be an energy signal defined by the condition \( \int_{-\infty}^{\infty} |f(t)|^2 dt < \infty \). If we denote the DC component of \( f(t) \) as
\[
f_0 = \lim_{T \to \infty} \frac{1}{T} \int_{-T/2}^{T/2} f(t) \, dt,
\]
what is the value of \( f_0 \) for an energy signal, and explain why this value does... | automath |
Facebook decided to award a productivity bonus to all its female employees who are mothers. This productivity bonus will total 25% of Facebook's annual earnings, which was $5,000,000 for the year 2020. It is known that Facebook employs 3300 employees; one-third are men, and of the women, 1200 are not mothers. How much ... | tiger_lab_math |
Determine the number of ways to paint a set of 6 identical balls with 3 different colors, such that each color is used at least once. | nvidia_math |
Consider a holomorphic function \( f \in \mathscr{O}(\mathbb{C}^n) \) that is irreducible at the origin. Let \( D = \mathbb{C}^n \) and \( p = 0 \). Given the function \( f(y,z,x) = y^2 - xz^2 \), determine if it is true that there exists a neighborhood \( U \) of the origin such that \( f \) remains irreducible in \( ... | automath |
Find avrg speed if a man travels at speed of 16km/hr up and 30km/hr dawn at an altitude of 200m.
Answer Choices: (A) 22.8 (B) 21.8 (C) 20.8 (D) 28.8 (E) 29.8 | tiger_lab_math |
Given the equation \( ax^2 + bx + c = 0 \) where \( x \) has dimensions of length, determine the dimensional units for the coefficients \( a \), \( b \), and \( c \) in terms of length. Specifically, express the relationship between their dimensions in an equation format. | automath |
Let \( r_8 \) be the real root of the equation \( x + x^2 + \ldots + x^8 = 8 - 10x^9 \), and let \( r_{10} \) be the real root of the equation \( x + x^2 + \ldots + x^{10} = 8 - 10x^{11} \). Determine whether \( r_8 < r_{10} \) or \( r_8 > r_{10} \). Additionally, calculate the approximate numerical values of \( r_8 \)... | automath |
If five shirts were purchased, with three of them costing $15 each and the remaining shirts costing $20 each, what is the total cost of all five shirts in dollars? | tiger_math |
Albert is wondering how much pizza he can eat in one day. He buys 2 large pizzas and 2 small pizzas. A large pizza has 16 slices and a small pizza has 8 slices. If he eats it all, how many pieces does he eat that day? | tiger_math |
Pipe A can fill a tank in 4 hours, pipe B in 8 hours, and pipe C in 24 hours. If all the pipes are open, in how many hours will the tank be filled?
Answer Choices: (A) 2.4 hours (B) 2.8 hours (C) 3.2 hours (D) 4.8 hours (E) 5.2 hours | tiger_lab_math |
Twenty gremlins and fifteen imps are at the Annual Mischief Convention. The imps have had a lot of in-fighting lately and refuse to shake hands with each other, but they readily shake hands with all of the gremlins. Meanwhile, all the gremlins are quite friendly and shake hands with all of the other gremlins as well as... | nvidia_math |
Write $4.\overline{054}$ as a simplified fraction in lowest terms. | tiger_math |
Let $x_1,$ $x_2,$ $\dots,$ $x_n$ be nonnegative real numbers such that $x_1 + x_2 + \dots + x_n = 1$ and
\[x_1^2 + x_2^2 + \dots + x_n^2 \le \frac{1}{100}.\]Find the smallest possible value of $n.$ | tiger_math |
A particle is located on the coordinate plane at $(-3,0)$. Define a ''move'' for the particle as a counterclockwise rotation of $\frac{\pi}{2}$ radians about the origin followed by a translation of $5$ units in the positive $y$-direction. Find the particle's position after $100$ moves. | tiger_math |
A certain class of students is being divided into teams. The class can either be divided into 18 teams with an equal number of players on each team or 24 teams with an equal number of players on each team. What is the lowest possible number of students in the class?
Answer Choices: (A) 6 (B) 36 (C) 48 (D) 60 (E) 72 | tiger_lab_math |
What is the ratio of the area of a square inscribed in a circle to the area of the circle, and what is the ratio of the area of a circle inscribed in a square to the area of the square? Calculate these ratios and provide the numerical values. | automath |
Given a collection of \( m \) sparse matrices \( A_i \) of size \( n \times n \) and a vector \( b \) of size \( m \), if you manage to find a rank-1 matrix \( Z = xy^T \) that satisfies the bilinear equations \( x^T A_i y = b_i \) for \( i = 1, \ldots, m \), how can you determine the dimensions \( n \) and \( m \) suc... | automath |
Sure, here's a precise numerical problem for the student to solve:
A factory wants to minimize the cost of producing a certain item. The cost is a function of the production rate per hour (x) and the number of workers employed (y) and is defined as:
C(x,y) = 2x^2 + 4y^2 + 4xy - 32x - 48y + 200
Using simulated annea... | tiger_lab_math |
Apple sold 100 iPhones at their New York store today for an average cost of $1000. They also sold 20 iPads for an average cost of $900 and 80 Apple TVs for an average cost of $200. What was the average cost across all products sold today? | nvidia_math |
A sphere is inscribed in a cube with edge length 9 inches. Then a smaller cube is inscribed in the sphere. How many cubic inches are in the volume of the inscribed cube? Express your answer in simplest radical form. Let's write a Python program to solve it. | tiger_lab_math |
How many extra electrons must be added to the Earth and the Moon such that the electric repulsion between them completely offsets their gravitational attraction? Assume the number of electrons corresponds to the ratio of their radii, specifically, \( \frac{R_{earth}}{R_{moon}} = \frac{6.38}{1.74} \). Use the known valu... | automath |
What is the mathematical justification for using the negative logarithm of the predicted probabilities as the cost function in logistic regression, and how does this choice ensure convexity of the cost function? | automath |
Mr. Thompson previously spent 35% of his monthly income on transportation costs. After a promotion, his salary was increased by $800, and now his transportation costs only account for 20% of his monthly income. What was his previous monthly income? | nvidia_math |
If Steve found 100 gold bars while visiting Oregon and wants to distribute them evenly among his 4 friends, but lost 20 gold bars on the way back to San Diego, how many gold bars will each of his friends receive when he returns? | tiger_math |
In $\triangle PQR$, point $T$ is on side $QR$ such that $QT=6$ and $TR=10$. What is the ratio of the area of $\triangle PQT$ to the area of $\triangle PTR$? [asy]
size(6cm);
pair q = (0, 0); pair t = (6, 0); pair r = (16, 0);
pair p = (4, 8);
draw(p--q--r--cycle--t);
label("$P$", p, N);
label("$Q$", q, SW);
label("$T$"... | nvidia_math |
Consider a locally connected subset \( A \subset \mathbb{R}^2 \). Is it possible for \( A \) to be locally path disconnected? If so, provide a specific example of such a subset. | automath |
John tossed a fair coin 6 times. What is the probability that the coin landed heads up exactly twice?
Answer Choices: (A) 0.094 (B) 0.15 (C) 0.194 (D) 0.25 (E) 0.294 | tiger_lab_math |
Add 147.32 to 28.191 and round your answer to the nearest hundredth. | nvidia_math |
Let $(S, \preceq)$ be a totally ordered set with five elements: $a$, $b$, $c$, $d$, and $e$, where the relation $\preceq$ is defined by $a \preceq b$, $a \preceq c$, $b \preceq c$, $b \preceq d$, and $c \preceq d$. Identify all maximal elements in the subset $T = \{a, b, c, d\}$. How many maximal elements does $T$ have... | automath |
A shopkeeper loses 15%,if an article is sold for Rs. 136. What should be the selling price of the article to gain 20%?
Answer Choices: (A) s.147 (B) s.148 (C) s.144 (D) s.189 (E) s.192 | tiger_lab_math |
If the ratio of the amount of money that Cara, Janet, and Jerry have is 4:5:6, and the total amount of money they have is $75, what is the loss that Cara and Janet will incur if they combine their money to buy oranges and sell them at 80% of the buying price? | tiger_math |
The monthly incomes of A and B are in the ratio 5 : 2. B's monthly income is 12% more than C's monthly income. If C's monthly income is Rs. 16000, then find the annual income of A?
Answer Choices: (A) Rs. 537600 (B) Rs. 180000 (C) Rs. 201600 (D) Rs. 504000 (E) None of these | tiger_lab_math |
Two goods trains each 500 m long are running in opposite directions on parallel tracks. Their speeds are 45 km/hr and 30 km/hr respectively. Find the time taken by the slower train to pass the driver of the faster one?
Answer Choices: (A) 27 sec (B) 17 sec (C) 48 sec (D) 96 sec (E) 17 sec | tiger_lab_math |
Amelia has a flower shop. She sells 5 types of flowers and makes bouquets using a combination of all flower types in two different sizes: small and large. Each small bouquet costs her $15 to create, and each large bouquet costs $25. She sold twice as many large bouquets as small bouquets in a day. If Amelia made a tota... | tiger_math |
If $\det \mathbf{B} = -2$, then find $\det (\mathbf{B^2})$. | nvidia_math |
A mason is constructing a wall that requires 120 bricks. He already has 12 crates, each containing 8 bricks. How many more crates of bricks does the mason need to buy to complete the wall? | nvidia_math |
Given the functions \( a(x) = \left(\frac{x}{4}\right)^2 \) and \( b(x) = -\left(\frac{x}{4}\right)^2 - 1 \), calculate the value of \( F(5) \) using the function \( F(x) = 2\sqrt{2} \cdot \tan^{-1}\left(\frac{x}{2\sqrt{2}}\right) \) and determine the horizontal distribution normalization factor \( \frac{x_{\text{max}}... | automath |
Sarah bought a $\$2$ ice cream cone with 25 coins and received no change. The coins were either dimes or quarters. How many quarters did Sarah use? | tiger_math |
Archie is playing with his marbles outside. He loses 60% of them into the street. Of the remaining ones, he loses half down a sewer. If he has 20 left, how many did he start with? | tiger_lab_math |
In a study, the hours studied and the grades earned by a group of 10 students are recorded in the table below. Calculate the correlation coefficient between the hours studied and the grades earned, and interpret the result.
```
Hours Studied: 5 7 3 2 8 6 1 4 9 10
Grades Earned: 70 85 65 60 90 80 55 75 95 92
``` | tiger_lab_math |
John cuts his grass to 2 inches. It grows .5 inches per month. When it gets to 4 inches he cuts it back down to 2 inches. It cost $100 to get his grass cut. How much does he pay per year? | tiger_math |
The weights of three boys are in the ratio 4 : 5 : 6. If the sum of the weights of the heaviest and the lightest boy is 47 kg more than the weight of the third boy, what is the weight of the lightest boy?
Answer Choices: (A) 87.6 kg (B) 37.6 kg (C) 98.6 kg (D) 65.6 kg (E) 87 kg | tiger_lab_math |
Find the maximum value of
\[f(x) = \sin x - \cos x\]for $0 \le x \le \frac{5\pi}{4}.$ | nvidia_math |
Alex is snacking on cookies. Each cookie is 12 calories. A box contains 30 cookies and costs $3. If she wants to consume 720 calories, how much will she need to spend on cookies? | nvidia_math |
Calculate the greatest integer less than or equal to $\frac{2022^3}{2020 \cdot 2021} - \frac{2020^3}{2021 \cdot 2022}$. | tiger_math |
Consider the polynomial equation:
\[x^5 + ax^4 + bx^3 + cx^2 + dx + f = 0\]
where all coefficients are integers. Let $m$ be the exact number of rational roots of the polynomial, counting multiplicity. For example, the polynomial $(x + 2)^3 (x^2 + 5x + 13) = 0$ has three rational roots counting multiplicity, because t... | nvidia_math |
Given the equations \( eq_1 = a b \dot{x}^2 - a c x'' \) and \( eq_2 = x'' - b \dot{x} - c = 0 \), substitute \( x'' \) from \( eq_2 \) into \( eq_1 \) and simplify the expression. What is the resulting expression for \( eq_1 \) after the substitution? | automath |
Emily organized a charity event and needs to track the donations. The total pledged amount is $750, but she has only received $560 so far. There are four people who still need to pay: Jack, Rachel, Michael, and David. Jack and Rachel owe the same amount, Michael owes $50, and David owes twice as much as Michael. How mu... | nvidia_math |
Given a scheme \( X \) that is either normal or Cohen-Macaulay, if \( \delta : X \to \mathbf{Z} \) denotes a dimension function such that for every point \( x \in X \), there exists an integer \( n_x \) with \( \delta(\xi) = n_x \) for any generic point \( \xi \) of an irreducible component of \( X \) containing \( x \... | automath |
Compute the values of the hyperbolic cosine function \( \cosh(x) \) at \( x = -2 \), \( x = -\pi i \), \( x = \frac{\pi}{6} \), \( x = \frac{5\pi i}{7} \), and \( x = \frac{3\pi}{2} \). Provide both the decimal approximations and the exact symbolic results wherever applicable. | automath |
Find the reciprocal of the sum of the roots of the equation $18t^2 + 45t -500 =0$. | tiger_math |
at a loading dock , each worker on the night crew loaded 3 / 4 as many boxes as each worker on the day crew . if the night crew has 1 / 2 as many workers as the day crew , what fraction of all the boxes loaded by the two crews did the day crew load ? | tiger_math |
A certain car's price decreased by 2.5% (from the original price) each year from 1996 to 2002, during that time the owner of the car invested in a new carburetor and a new audio system for the car, which increased car's price by $1,500. If the price of the car in 1996 was $22,000, what is the car's price in 2002?
Answe... | tiger_lab_math |
There were 60 students on the field trip to the amusement park. A third of the students rode the roller coaster, and then a quarter of those students also rode the Ferris wheel. How many of the students who rode the Ferris wheel also rode the roller coaster? | nvidia_math |
Last Sunday a certain store sold copies of Newspaper A for $1.00 each and copies of Newspaper B for $1.25 each, and the store sold no other newspapers that day. If r percent of the store's revenues from newspapers sales was from Newspaper A and if q percent of the newspapers that the store sold were copies of Newspaper... | tiger_lab_math |
how many seconds will a 800 metre long train take to cross a man walking with a speed of 3 km / hr in the direction of the moving train if the speed of the train is 63 km / hr ? Please respond by writing a program in Python. | tiger_lab_math |
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