instruction_seed stringlengths 17 6.7k | source stringclasses 4
values |
|---|---|
What is the sum of the greatest common factor of 24 and 64 and the least common multiple of 20 and 15? | tiger_math |
Given a list of distinct integers, how many swaps are required to sort the list in ascending order? For example, if the input list is \([5, 3, 2, 1, 4]\), how many swaps are needed to transform it into \([1, 2, 3, 4, 5]\)? Additionally, determine if the number of swaps is odd or even, and explain the significance of th... | automath |
The square shown is divided into 4 congruent rectangles. If the perimeter of the square is 144 units, what is the perimeter of one of the four congruent rectangles? [asy]
size(75); defaultpen(linewidth(0.8));
draw(unitsquare);
draw((0,0.25)--(1,0.25)); draw((0,0.5)--(1,0.5)); draw((0,0.75)--(1,0.75));
[/asy] | tiger_lab_math |
Find the sum of the following series in simplified fraction form:
$$\frac{1}{2\times3} + \frac{1}{3\times4} + \frac{1}{4\times5} + \cdots + \frac{1}{19\times20}$$ | tiger_math |
Melanie baked 10 batches of brownies, with 20 brownies in each batch. She set aside 3/4 of the brownies in each batch for a bake sale, put 3/5 of the remaining in a container, and gave out the rest. How many brownies are given out? | tiger_math |
Farmer Randy has 1700 acres of cotton he needs to have planted in 5 days. With a crew of 2 tractors working for 2 days and then a crew of 7 tractors working for another 3 days, how many acres of cotton per day does each tractor need to plant to meet their planting deadline? | tiger_math |
Express as a fraction in lowest terms: $0.\overline{1} + 0.\overline{01}$ | tiger_lab_math |
Consider two sets \( A \) and \( B \) defined as follows:
\( A = \{2, 3, 5, 7, 11, 13\} \) (the set of prime numbers less than 15) and
\( B = \{2, 4, 6, 8, 10, 12, 14\} \) (the set of even numbers less than 15).
What is the intersection \( A \cap B \), and how many elements does it contain? | automath |
"statement 1": Carl can wash all the windows of his house in more than 1 hours, "statement 2" :Carl can wash all the windows of his house in 6 hours, "options: " Entailment or contradiction or neutral? Let's write a Python program to solve it. | tiger_lab_math |
Ms. Apple plants trees in her orchard in a grid pattern. Each row contains exactly $n$ trees, and there are a total of $n$ rows. Ms. Apple notices that if she counts the trees along the diagonals from the top-left corner to the bottom-right corner, each diagonal contains $p$ trees, with $p = 1, 2, 3, ..., n$. Curious a... | tiger_math |
A goods train runs at the speed of 72 kmph and crosses a platform of a certain length in 26 seconds. The length of the goods train is 260.0416 meters. What is the length of the platform in meters? | tiger_math |
A historian discovered an ancient artifact with a cryptic inscription indicating that a certain event occurred 3174 years ago. However, the historian suspects that the ancient civilization used a non-standard calendar system where each year was divided into 15 months, with each month having exactly 20 days. How many st... | nvidia_math |
A right circular cylinder has a radius of 5 cm and a height of 16 cm. Using integration, what is the volume of the cylindrical shape? | tiger_lab_math |
If $7^{500}\equiv 1\pmod{1000}$, what are the last three digits of $7^{15000}$? | tiger_math |
A certain car can travel 32 kilometers on a liter of fuel. The fuel tank's contents decrease by 3.9 gallons as the car moves at a constant speed of approximately 52 miles per hour. How long did it take for the fuel tank's contents to decrease by 3.9 gallons? (1 gallon = 3.8 liters; 1 mile = 1.6 kilometers) | tiger_math |
Matthias has 40 soccer balls and 15 basketballs. 30 soccer balls and 7 basketballs have a hole in them. How many balls in total does Matthias have without holes in them? | tiger_lab_math |
When Jeff asks a Magic 8 Ball a question, there is a 2/5 probability of receiving a positive answer. If he asks 5 questions, what is the probability of getting exactly 2 positive answers? | tiger_math |
Lucas went to the store with $20 and needed to buy 3 avocados that cost $2 each. How much change does he bring home? | tiger_math |
Hamza has several empty buckets of different sizes, holding either 3, 5, or 6 liters. She fills the 5-liter bucket and pours as much as she can into the 3-liter bucket. Then, she pours the remainder into the 6-liter bucket. How much more water, in liters, can she put into the 6-liter bucket, without overflowing? | tiger_math |
In a classroom, 10 children have 7 books each. Their teacher brings another x books to the classroom. There are 78 books in the classroom altogether. What is the value of unknown variable x? | tiger_math |
What is the value of \( 27^{\frac{4}{3}} \) expressed as a whole number? | automath |
In a survey of 1200 people, 756 reported that they like lemons, indicating a sample proportion of \( \hat{p} = 0.63 \). Using the normal approximation, compute the 95% confidence interval for the true population proportion \( p \). What are the lower and upper bounds of this confidence interval? | automath |
a rectangular photograph is surrounded by a border that is 1 inch wide on each side . the total area of the photograph and the border is m square inches . if the border had been 3 inches wide on each side , the total area would have been ( m + 52 ) square inches . what is the perimeter of the photograph , in inches ? L... | tiger_lab_math |
A company manufacturing t-shirts aims to maximize their profit. The cost of producing each t-shirt is $6 and they sell each t-shirt for $15. The company also incurs a fixed cost of $5000 for rent, salaries, and other overhead expenses. Using whale optimization algorithm, what is the maximum number of t-shirts they need... | tiger_lab_math |
A regular polygon has exterior angles each measuring 18 degrees. How many sides does the polygon have? | tiger_math |
What is the minimum value of the function f(x) = x^2 - 6x + 8 using Whale Optimization Algorithm? | tiger_lab_math |
Let $f(x)=-3x^2+x-4$, $g(x)=-5x^2+3x-8$, and $h(x)=5x^2+5x+1$. Express $f(x)+g(x)+h(x)$ as a single polynomial, with the terms in order by decreasing degree. | nvidia_math |
Given a directed graph \( G \) with edge weights representing the number of cars on each road, consider the following two edges: \( e_{1,2} \) with a weight of 10 cars traveling from vertex 1 to vertex 2, and \( e_{2,4} \) with an unknown weight \( x \) cars traveling from vertex 2 to vertex 4.
If the net flow at eac... | automath |
Given a 4x4 matrix \( A \) with singular values \(\text{diag}(12, 10, 0.1, 0.05)\), compute the Moore-Penrose pseudoinverse \( A^+ \). What is the resulting matrix \( A^+ \)? | automath |
Lisa, Jack, and Tommy earned $60 from washing cars all week. However, half of the $60 was earned by Lisa. Tommy earned half of what Lisa earned. How much more money did Lisa earn than Tommy? | tiger_lab_math |
What percent is 320 gm of 1 kg?
Answer Choices: (A) 25% (B) 40% (C) 10% (D) 8% (E) 32% Please write a program to solve it | tiger_lab_math |
Emily consumes 4 meals a day. Breakfast is 400 calories. Her mid-morning snack has 10% more calories than that. Her lunch is 1.5 times as many calories as the mid-morning snack, and her dinner is 2.5 times as many calories as lunch. She also has 2 energy bars that are each 250 calories. How many calories does she get i... | nvidia_math |
How many $x$-intercepts does the graph of the parabola $x = -2y^2 + y + 1$ have? | nvidia_math |
What is the volume of the solid generated by rotating the region \( R \), bounded by the curves \( y = x \), \( x = 2 - y^2 \), and \( y = 0 \), around the x-axis? | automath |
If 4 men can color a 48 m long cloth in 2 days, then a certain number of men can color a 36 m long cloth in 1 day. How many men are in the second group? | tiger_math |
At a company, the average wage of male employees is $4 per hour, and the average wage of female employees is $8 per hour. If the average wage of all employees is $7 per hour, what is the ratio of the number of male employees to the number of female employees?
Answer Choices: (A) 1 to 3 (B) 2 to 3 (C) 3 to 2 (D) 5 to 3 ... | tiger_lab_math |
Given a matrix \( X \) defined as
\[
X = \begin{bmatrix} A & C \\ 0 & B \end{bmatrix}
\]
where \( A \) and \( B \) are invertible matrices of dimensions \( n \times n \) and \( m \times m \) respectively, and \( 0 \) is an \( m \times n \) zero matrix, what is the inverse of \( X \) expressed in terms of \( A \), \(... | automath |
Working alone at their respective constant rates, A can complete a task in ‘a’ days and B in ‘b’ days. They take turns in doing the task with each working 2 days at a time. If A starts they finish the task in exact 8 days. If B starts, they take a day more. How long does it take to complete the task if they both work t... | tiger_lab_math |
Sum of 53 odd numbers is?
Answer Choices: (A) 2709 (B) 2809 (C) 2909 (D) 3009 (E) 3109 | tiger_lab_math |
The average of first 35 prime numbers is?
Answer Choices: (A) 14.54 (B) 15.6 (C) 15.8 (D) 14.7 (E) 14.8 | tiger_lab_math |
Find $AX$ in the diagram.
[asy]
import markers;
real t=.56;
pair A=(0,0);
pair B=(3,2);
pair C=(.5,1.5);
pair X=t*A+(1-t)*B;
draw(C--A--B--C--X);
label("$A$",A,SW);
label("$B$",B,E);
label("$C$",C,N);
label("$X$",X,SE);
markangle(n=1,radius=15,A,C,X,marker(markinterval(stickframe(n=1),true)));
markangle(n... | nvidia_math |
What is that largest positive integer $n$ for which $n^3+200$ is divisible by $n+15$? | tiger_math |
A car is moving due east at a speed of 50.0 km/h, and raindrops are falling vertically at a constant speed. If the traces of the rain on the car's windows form an angle of 45.0° with the vertical, what are the following velocities?
(a) What is the speed of the rain with respect to the car in m/s?
(b) What is the spee... | automath |
Emily is thinking of a three-digit number. She gives the following clues: "My number is divisible by 7. The sum of its digits is 18. When you reverse the digits, the new number is 99 more than the original number." What is Emily's number? | nvidia_math |
A deck of forty cards consists of four $1$'s, four $2$'s,..., and four $10$'s. A matching pair (two cards with the same number) is removed from the deck. Given that these cards are not returned to the deck, let $m/n$ be the probability that two randomly selected cards also form a pair, where $m$ and $n$ are relatively ... | nvidia_math |
When Erick went to the market to sell his fruits, he realized that the price of lemons had risen by $4 for each lemon. The price of grapes had also increased by half the price that the price of lemon increased by per grape. If he had planned to sell the lemons at $8 and the grapes at $7, and he had 80 lemons and x grap... | tiger_math |
The division of a whole number N by 13 gives a quotient of 15 and a remainder of 2. Find N.
Answer Choices: (A) 797 (B) 197 (C) 597 (D) 997 (E) 297 | tiger_lab_math |
In a two-period difference-in-differences (DiD) model where treatment is applied in the second period, I estimate the coefficient \(\beta_3\) from the model including a control variable \(x_{it}\) and find that the fixed effects (FE) estimator and the ordinary least squares (OLS) estimator yield different results. If I... | automath |
Todd borrowed $100 from his brother to start a snow-cone stand. He has to repay him $110 as soon as he can. Todd spent $75 on ingredients for the snow cones. He sells 200 of them for $.75 each. After he pays back his brother, how much money does Todd have? | tiger_math |
The students in Mrs. Reed's English class are reading the same $760$-page novel. Three friends, Alice, Bob and Chandra, are in the class. Alice reads a page in $20$ seconds, Bob reads a page in $45$ seconds and Chandra reads a page in $30$ seconds. Chandra and Bob, who each have a copy of the book, decide that they c... | nvidia_math |
A line segment begins at (2, 5). It is 8 units long and ends at the point (7, y) and y < 0. What is the value of y? | nvidia_math |
Round $54.\overline{54}$ to the nearest hundredth. | tiger_math |
Given the function \( P(t) = 690\left(1 + \frac{6t}{68 + t^2}\right) \) that describes the population of a bacteria culture over time \( t \), calculate the rate of change of the population \( P'(t) \) when \( t = 7 \) hours, and provide the answer rounded to two decimal places. | automath |
45 x ? = 35% of 900
Answer Choices: (A) 16.2 (B) 7 (C) 5 (D) 500 (E) None | tiger_lab_math |
Let $\mathbf{M}$ be a matrix such that
\[\mathbf{M} \begin{pmatrix} 1 \\ 2 \end{pmatrix} = \begin{pmatrix} 4 \\ 1 \end{pmatrix} \quad \text{and} \quad \mathbf{M} \begin{pmatrix} -2 \\ 3 \end{pmatrix} = \begin{pmatrix} -3 \\ -2 \end{pmatrix}.\]Compute $\mathbf{M} \begin{pmatrix} 7 \\ 0 \end{pmatrix}.$ | tiger_math |
Two particles, one red and one blue, start at the origin and move along the positive $x$-axis. At each time step $n$, the red particle moves to the right by a distance of $1/n$ and the blue particle moves to the right by a distance of $1/n^2$. Find the smallest value of $n$ such that the blue particle is closer to the ... | nvidia_math |
You're at an airport and need to reach a flight gate that is a distance of \( D \) kilometers away. You can run at a speed of \( 10 \) km/h, and you also have access to moving walkways that move at a speed of \( v \) km/h. You must rest for a total of \( 1 \) minute during your journey, and while resting, you can walk ... | automath |
A can contains a mixture of liquids A and B is the ratio 7:5. When 9 litres of mixture are drawn off and the can is filled with B, the ratio of A and B becomes 7:9. How many liter W of liquid A was contained by the can initially?
Answer Choices: (A) 10 (B) 20 (C) 21 (D) 25 (E) 27 | tiger_lab_math |
A musician is composing a song that consists of a repeating pattern of notes. The pattern starts with a 2-second note, followed by a 3-second note, and then a 5-second note. If the song is designed to last exactly 20 minutes, how many times will the 5-second note be played? | nvidia_math |
If the odds for pulling a prize out of the box are $3:4$, what is the probability of not pulling the prize out of the box? Express your answer as a common fraction. | tiger_math |
42, 40, 35, 33, 28, 26, ?
Answer Choices: (A) 21 (B) 23 (C) 28 (D) 25 (E) 28 | tiger_lab_math |
What is the value of the integral \( I = \int \frac{x^2}{(x-1)^{3/2}(x+3)^{1/2}} \, dx \) after applying the appropriate substitutions and simplifications? | automath |
Let \( \triangle ABC \) be a triangle with circumcenter \( O \). Define points \( A_1, B_1, C_1 \) as the projections of some point \( P \) onto the sides \( BC, AC, AB \), respectively. Let \( M_a, M_b, M_c \) be the midpoints of sides \( BC, AC, AB \). Consider the reflections \( A_2, B_2, C_2 \) of points \( A_1, B_... | automath |
What is the expected value of the quadratic form \( E[(\mathbf{x}^\top A \mathbf{x})^2] \) when \( \mathbf{x} \) is uniformly distributed on the unit sphere in \( \mathbb{R}^d \), expressed in terms of the trace of \( A \) and \( A^2 \)? | automath |
From a circular piece of paper with radius $OE$, Rachel removes the unshaded sector shown. Using the larger shaded sector, she joins edge $OE$ to edge $OD$ (without overlap) to form a cone of radius 9 centimeters and of volume $243\pi$ cubic centimeters. What is the number of degrees in the measure of angle $EOD$ of ... | nvidia_math |
If Gerald wants to purchase a meat pie for 2 pfennigs and he currently has 54 farthings, with 6 farthings equivalent to 1 pfennig, how many pfennigs will Gerald have remaining after buying the pie? | tiger_math |
If $x$ and $y$ are positive integers less than $20$ for which $x + y + xy = 76$, what is the value of $x + y$? | nvidia_math |
What is the expected value of \( R^2 \) when considering a regression model with \( m \) uncorrelated predictors and a sample size of \( n \)? Can you derive this value using the properties of the Beta distribution? | automath |
A train covers a distance of 12 km in 10 min. If it takes 6 sec to pass a telegraph post, then the length of the train is?
Answer Choices: (A) 999 (B) 279 (C) 120 (D) 99 (E) 21 | tiger_lab_math |
One leg of a right triangle is 12 inches, and the measure of the angle opposite that leg is $30^\circ$. What is the number of inches in the hypotenuse of the triangle? Let's write a Python program to solve it. | tiger_lab_math |
In the parametric definition of a torus, if the radius \( R \) is set to 10 and the tube radius \( r \) is set to 5, what are the coordinates of the points on the torus for \( u = 0 \) and \( v = \frac{\pi}{2} \)? | automath |
For the "Prime Factor Look-and-Say" sequence starting at a positive integer \( n \), is there a proven upper bound on the length of the sequence before it terminates, and if so, what is that bound? If no such bound exists, provide a counterexample that demonstrates a potentially long sequence. | automath |
Calculate the modular inverse of 3 modulo 257, expressed as a residue modulo 257. | tiger_math |
How many of the positive factors of 10 , 16 and how many common factors are there in numbers?
Answer Choices: (A) 1 (B) 2 (C) 3 (D) 4 (E) 5 | tiger_lab_math |
Given the following payoff matrix for a two-player game:
\[
\begin{array}{c|c|c}
& \text{Left} & \text{Right} \\
\hline
\text{Up} & 3,0 & 0,1 \\
\text{Mid} & 1,1 & 2,0 \\
\text{Down} & 0,0 & 1,2 \\
\end{array}
\]
What are the rationalizable strategies for Player 1 (the row player) and Player 2 (the column player)? S... | automath |
Grandma Molly created statues of turtles for her front lawn. The first year, she created 4 statues and placed them on her lawn. The second year, she quadrupled the number of statues on her front lawn. In the third year, she added another 12 statues to the front lawn, but a hail storm broke 3 of the statues, which sh... | tiger_lab_math |
How many different problems did students present from the beginning of Week 1 on January 10 to the end of Week 15 on April 29, inclusive? | automath |
Briar is attending a one-week community empowerment event and has to take a cab ride to the event and back home every day. A cab ride costs $2.5 per mile. If the event is 200 miles away from Briar's home, calculate the total amount of money the cab rides cost would be at the end of the event? | tiger_lab_math |
Imagine you are a treasure hunter tracking a long-lost golden artifact. According to ancient clues, the artifact is buried in a rectangular field. The field's width measures 24 feet, and the diagonal from one corner to the opposite corner slightly sunk into the ground measures 40 feet. What is the length of the field w... | tiger_math |
If the current estimated age of the universe is 13.7 billion years, and a previous estimate was 20 billion years, what is the minimum number of significant figures that the original estimate could have been measured with, given that significant figures can cause such a large change in the estimate? | automath |
The Tlingit language of the Pacific Northwest Coast of North America has 14 letters in its alphabet: A, Ch, E, G, H, I, K, L, S, T, U, W, X, and Y. Suppose identification cards of six letters utilize only the letters in the Tlingit alphabet. How many identification cards of six letters are possible that begin with eith... | nvidia_math |
Given a boolean function \( f: \{-1, 1\}^n \to \{-1, 1\} \) that is \( \epsilon \)-close to the set of dictators or the constant function \( 1 \), express the probability that the test defined by picking a random \( y \) and accepting iff \( f(y) \neq f(-y) \) passes in terms of the Fourier coefficients of \( f \). Wha... | automath |
Given that \( x, y > 0 \) and the constraint \( 9x + 4y = 2005 \), what is the minimum value of the expression \( \frac{1}{x} + \frac{1}{y} \)? | automath |
A math field day competition is held in a room with many tables, and there are 5 stools at each table. Each stool has 4 legs, and each table has 3 legs. If there is a total of 365 legs on all the tables and stools in the room, how many tables are in the room? | tiger_math |
Find the exact value of $\frac{1}{9}$ as a decimal, where the decimal expansion is obtained by dividing 1 by 9. | nvidia_math |
If positive integer k is divided by 14, the quotient is m and the remainder is 2, then which of the following must be true?
I. k + m is even
II. k cannot be a prime number
III. k is divisible by 4
Answer Choices: (A) None (B) I (C) II (D) III (E) I and II only | tiger_lab_math |
Find the smallest positive integer $n$ such that
\[\begin{pmatrix} \cos 200^\circ & -\sin 200^\circ \\ \sin 200^\circ & \cos 200^\circ \end{pmatrix}^n = \mathbf{I}.\] | tiger_math |
Point $C$ is the midpoint of $\overline{AB}$, point $D$ is the midpoint of $\overline{AC}$, point $E$ is the midpoint of $\overline{AD}$, and point $F$ is the midpoint of $\overline{AE}$. If $AF=3$, what is the number of units in the length of $\overline{AB}$? | nvidia_math |
Given a directed graph where the objects correspond to vertices and morphisms correspond to directed edges, under what conditions can this directed graph be classified as a category? Specifically, demonstrate that if there exist directed edges \( A \to B \) and \( B \to C \), how would you express the requirement for a... | automath |
Find the smallest prime factor of 2307. | nvidia_math |
Given two non-stationary time series \(X(t)\) and \(Y(t)\) where both are integrated of order 1, if we find that the first differences \(\Delta X(t) = X(t) - X(t-1)\) and \(\Delta Y(t) = Y(t) - Y(t-1)\) are stationary and cointegrated, what is the implication for the long-term relationship between \(X(t)\) and \(Y(t)\)... | automath |
Under what conditions do the sheaves of real analytic functions on a real analytic manifold exhibit coherence, and can you provide a specific example of a real analytic manifold of a certain dimension where this coherence holds? | automath |
Let \( A = \mathbb{Z}_2 \) and \( B = \mathbb{Z}_2 \times \mathbb{Z}_2 \). Construct two distinct non-isomorphic group extensions \( G_1 \) and \( G_2 \) of \( B \) by \( A \), where both extensions yield the abelian group \( G = \mathbb{Z}_4 \times \mathbb{Z}_2 \). Provide a detailed explanation of the injections from... | automath |
Kevin has a tree growing in his garden that is currently 180 inches tall. That is 50% taller than it was when he planted it there. How tall was the tree, in feet, then? | tiger_lab_math |
Find the smallest positive integer that leaves a remainder of 1 when divided by 5, a remainder of 2 when divided by 7, and a remainder of 3 when divided by 11. | nvidia_math |
After breaking a mirror into 60 pieces, Anne sweeps up half of them. Her cat steals 3 pieces and her boyfriend picks up 1/3 of the remaining pieces. How many pieces does her boyfriend pick up? | tiger_math |
Let $g(x)$ be a function defined for all integers $x \ge 0,$ with $g(1) = 2.$ For all integers $a,$ $b \ge 0,$ $g(a + b) = g(a) \cdot g(b) - g(ab).$ Find $g(2024).$ | nvidia_math |
What digit $B$ will make the number $45B2$ divisible by $3$? | nvidia_math |
Dave bought 8 books about animals, some books about outer space, and 3 books about trains to keep him busy over the holidays. Each book cost $6. Dave spent $102 on the books. How many books about outer space did he buy? | tiger_math |
Given the vectors \(\vec{A} = [4, 4, 2]\) and \(\vec{B} = [4, 3, 12]\), calculate the angle \(\theta\) formed when the vectors are placed tail to tail. Then, determine the components of the vector that results from projecting \(\vec{B}\) onto \(\vec{A}\).
What is the value of \(\theta\) in degrees, and what are the c... | automath |
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