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Let $g(x) = x^5 + x^4 + x^3 + x^2 + x + 1.$ What is the remainder when the polynomial $g(x^{12})$ is divided by the polynomial $g(x)$? Let's write a Python program to solve it. | tiger_lab_math |
There are 3 eggs in each box. So , there are _____ eggs in 2 boxes. Please respond by writing a program in Python. | tiger_lab_math |
Let \( u = 5(2x^2 + 3)^3 \) and \( v = 3(4x^3 - 1)^2 \). If \( y = \frac{u}{v} \), find the derivative \( y' \) using both the quotient rule and the chain rule, and simplify your answer. What is the value of \( y' \) at \( x = 1 \)? | automath |
Tom has a quarter as much money as Nataly. Nataly has three times as much money as Raquel. How much money do Tom, Raquel, and Nataly have combined if Raquel has $40? | nvidia_math |
Determine whether the series
\[
\sum_{n=1}^{\infty} \frac{1}{n^2 + n}
\]
converges or diverges, and explain your reasoning using a comparison test with a known benchmark series. | automath |
Henry bought 2 new fireworks and his friend bought some. They already had 6 saved from last year and now they have 11 fireworks. How many fireworks did Henry's friend buy? | tiger_math |
If a shop is offering a discount on pens where you can buy 10 pens at the regular price and get the next 10 for half off, and a customer purchased 20 pens for $30, what is the regular price of one pen in dollars? | tiger_math |
In an arithmetic sequence, the first three terms are 1, 10, and 19. What is the value of the 21st term? | tiger_math |
Tom can type x words a minute. A page is 450 words. It would take him 50 to type out 10 pages. What is the value of unknown variable x? What is the value of unknown variable x? | tiger_math |
Janice can type 6 sentences per minute. Today at work, Janice continued working on a paper she started typing yesterday. She typed for 20 minutes, took a break, and typed 15 minutes longer. She then had to erase 40 sentences she had typed incorrectly. After a meeting, she typed for 18 minutes more. In all, the paper ha... | tiger_lab_math |
What is the unbiased correction factor \( C_n \) for estimating the population standard deviation \( \sigma \) using the sample standard deviation \( s \) when the sample size \( n \) is 20? | automath |
A person was asked to state his age in years.His reply was,”Take my age three years hence,multiply it by 3 and then subtract three times my age three years ago and you will know how old I am”.What was the age of the person ?
Answer Choices: (A) 18 years (B) 20 years (C) 24 years (D) 32 years (E) 43 years | tiger_lab_math |
Emily has 24 rare stamps and wants to know their total value. She finds out that 8 of these stamps are worth $36 in total. Assuming each stamp has the same value, what is the total value of her entire collection? | nvidia_math |
Alex earns $30 per hour by tutoring students. He wants to save at least $1800 for a summer vacation. He can tutor for 2 hours every weekday and 4 hours on both Saturday and Sunday. How many weeks does it take for Alex to earn enough to cover his vacation expenses? | nvidia_math |
In Rodrigo's classroom in the morning there are red chairs, yellow chairs, and blue chairs. There are 4 red chairs. There are 2 times as many yellow chairs as red chairs, and there are 2 fewer blue chairs than yellow chairs. In the afternoon, Lisa borrows 3 chairs. How many chairs are left in Rodrigo's classroom? | tiger_math |
An isosceles triangle $ABC$ has base $\overline{BC}$, whose midpoint is the origin of the coordinate plane, and the altitude from $A$ is perpendicular to $\overline{BC}$. Given that the vertices $B$ and $C$ have integer coordinates, find the largest possible number of triangles with different areas that can be construc... | nvidia_math |
Suppose you are studying the Rikitake system with the following differential equations:
dx/dt = y(z - 1 + a(x^2 - 1))
dy/dt = x(3z + 1 - a(x^2 - 1))
dz/dt = -2z/tau - xy
where a = 0.2 and tau = 10.
Given initial conditions x(0) = 1.5, y(0) = -0.5, z(0) = 2.5, solve the Rikitake system numerically using a step siz... | tiger_lab_math |
On a monthly basis, Esperanza pays $600 in rent, 3/5 as much money on food as she pays on rent and her mortgage bill costs three times the amount of money she spends on food. If she saves $2000 and also pays the equivalent of 2/5 of her savings in taxes, calculate her gross monthly salary. | tiger_lab_math |
The average of five numbers is 24. Three of the numbers are 20, 25, and 30, and the other two numbers are in the ratio of 2:3. What is the sum of the squares of the two numbers that are in the ratio? | nvidia_math |
What is the Euclidean norm of the complex vector \(\boldsymbol{v} = \begin{pmatrix} 2 + 3i \\ 4i \end{pmatrix}\)? | automath |
What is the cardinality of the continuum if strong reflection principles hold, and how does this relate to the continuum hypothesis? | automath |
If the apple tree on Magdalena's farm produced 200 apples this year and she picked 1/5 of the fruits on the first day, twice that number on the second day, and 20 more apples than the first day on the third day, what is the total number of apples remaining on the tree? | tiger_math |
Evaluate the double integral of the function \( f(x, y) = \frac{1}{(x+y)^2} \) over the region defined by \( 2 \leq x \leq 4 \) and \( 1 \leq y \leq 6 \). That is, compute:
\[
\iint_{R} \frac{1}{(x+y)^2} \, dy \, dx
\] | automath |
Given the function
\[g(x) =
\begin{cases}
-x^2+2&\text{if }x<1,\\
x^2+2&\text{if }x\geq 1,
\end{cases}
\]
find the domain of the function $h(x) = \frac{1}{g(x)}$. | nvidia_math |
If two vectors \( \mathbf{u} \) and \( \mathbf{v} \) in three-dimensional space satisfy the conditions \( \mathbf{u} \cdot \mathbf{v} = -6 \), \( \mathbf{v} \cdot \mathbf{w} = 18 \), and \( \mathbf{u} \cdot \mathbf{w} = -81 \), determine whether the vectors \( \mathbf{u} \), \( \mathbf{v} \), and \( \mathbf{w} \) are o... | automath |
What is the maximum number of balls \( n \) that can be tested using a three-pan balance to find one heavier ball, given that you can perform at most \( k \) weighings? Express your answer in terms of \( k \). | automath |
Determine the value of $\dfrac{3 \times 4}{6}$. | tiger_math |
The matrices
\[\begin{pmatrix} a & 2 \\ 1 & 4 \end{pmatrix} \quad \text{and} \quad \begin{pmatrix} -\frac{2}{7} & \frac{1}{7} \\ b & \frac{3}{14} \end{pmatrix}\]are inverses. Enter the ordered pair $(a,b).$
A new problem:
Let $A$ and $B$ be two $2\times2$ matrices such that $AB=BA$ and $A^2=B^2$. If the trace of $A$... | nvidia_math |
Given a backpacker's knapsack with a volume of \( V \) cubic inches and a weight capacity of \( W \) pounds, suppose there are 7 items with specific volume (\( a_i \)), weight (\( w_i \)), and value (\( c_i \)) considerations. Items 1, 2, and 3 (canned goods) provide no value unless Item 4 (a can opener) is included, a... | automath |
Rectangle A has sides a and b, and Rectangle B has sides c and d. The ratio a/c = b/d = 3/4. If Rectangle A has sides of length 3 and 6, what is the length of side d in Rectangle B? | tiger_math |
A bag contains 8 red and 4 white balls. One ball is drawn at random. What is the probability that the ball drawn is white?
Answer Choices: (A) 4/7 (B) 2/3 (C) 5/8 (D) 1/4 (E) 5/7 | tiger_lab_math |
Given three random variables \( x_1, x_2, x_3 \), consider the following relationships based on the conditional independence structures described:
1. If \( p(x_1, x_2, x_3) = p(x_1 \vert x_3)p(x_2 \vert x_3)p(x_3) \), what can be inferred about the dependency between \( x_1 \) and \( x_2 \) when conditioned on \( x_3 ... | automath |
If Hana sold 4/7 of her stamp collection for $28, what would be the total amount she would have earned if she sold the entire collection? | tiger_math |
The $\$4.55$ in Carol's piggy bank consists of quarters and nickels. There are seven more nickels than quarters. How many nickels does Carol have in her bank? | tiger_math |
In a set of five consecutive integers, the sum of the smallest and largest numbers is equal to twice the middle number. If the second smallest number is 17, find the product of the three largest numbers. | nvidia_math |
The salary of a typist was raised by a certain percentage and then the same was reduced by 5%. The typist presently draws Rs. 3135. The original salary was Rs. 3000. What was the percentage increase in the salary initially? | tiger_math |
Point $C$ is the midpoint of $\overline{AB}$, point $D$ is the midpoint of $\overline{AC}$, and point $E$ is the midpoint of $\overline{AD}$. If $AE=6$, what is the length of $\overline{AB}$? | tiger_math |
Alex consumes 3 16-oz bottles of juice and 48 ounces of milk daily. How many ounces of fluid does she drink in a 5-day period? | nvidia_math |
A fair 8-sided die is rolled. If the result is $n$, then I win $\frac{1}{n}$ dollars. What is the expected value of my win? Express your answer as a dollar value rounded to the nearest cent. | nvidia_math |
Let $a$, $b$, and $c$ be positive integers such that $a>b>c$. The system of equations
$$ab+c \equiv b(a+c) \pmod{c(a+b)}$$
and
$$c(a+b)+1 \equiv ac+1 \pmod{ab}$$
has a unique solution $(a,b,c)$. Find $c^{-1} \pmod{a}$. | nvidia_math |
We are given the positive numbers a,b,c, which form an arithmetic progresion in the given order. We know that a+b+c=9. The numbers a+1,b+1,c+3 form a geometric progression in the given order. Find c.
Answer Choices: (A) 5 (B) 6 (C) 7 (D) 8 (E) 9 | tiger_lab_math |
A box contains 5 green balls, 3 blue balls, and 2 red balls. If 2 balls are drawn at random without replacement, what is the probability that the first ball drawn is green and the second ball drawn is not green? | nvidia_math |
What is the ratio of the surface area of a sphere to the surface area of a cube, if the sphere is inscribed in the cube? Express your answer as a common fraction in terms of pi. | automath |
a car gets 20 kilometers per gallon of gasoline . how many gallons of gasoline would the car need to travel 190 kilometers ? Let's write a program. | tiger_lab_math |
Let $\mathbf{v}$ and $\mathbf{w}$ be vectors such that
\[\operatorname{proj}_{\mathbf{w}} \mathbf{v} = \begin{pmatrix} 1 \\ 0 \\ -3 \end{pmatrix}.\]Compute $\operatorname{proj}_{\mathbf{w}} (-2 \mathbf{v}).$ | nvidia_math |
The expression $3^x + 1$ is a perfect square for a certain positive integer $x$. What is the largest prime factor of this perfect square? | nvidia_math |
In four years, Suzy will be twice Mary's age then. If Suzy is 20 now, how old is Mary? | tiger_lab_math |
Emily walks down 12 levels of stairs. Each level is 12 feet. If each step is 15 inches, how many steps does she walk down? | nvidia_math |
In a town, there are four primary schools. Two of them can teach 400 students at a time, and the other two x students at a time. How many students at a time in total, These four primary schools can teach 1480. What is the value of unknown variable x? | tiger_math |
A fair six-sided die is rolled 10 times. What is the expected number of consecutive pairs of rolls where the sum of the numbers rolled is 7? Express your answer as a common fraction. | nvidia_math |
What is the degree of the polynomial \( P_n(x) = x^n - F_n x - F_{n-1} \), where \( F_n \) denotes the \( n \)-th Fibonacci number, and how many distinct roots does it have over the real numbers? | automath |
$n$ coins are simultaneously flipped. The probability that at most one of them shows tails is $\frac{3}{16}$. Find $n$. | nvidia_math |
If a trading algorithm has a win rate of 55% and the average gain per winning trade is 2%, while the average loss per losing trade is 1%, using a fixed dollar amount of $5000 per trade on an initial capital of $100,000, what is the expected number of trades required for the probability of losing the entire capital to e... | automath |
a group of 6 children and 10 adults are going to the zoo . child tickets cost $ 10 , and adult tickets cost $ 16 . how much will the zoo tickets cost in all ? | tiger_math |
The difference between Sarah's weight and Lisa's weight is Y pounds. If you add Sarah's weight to Lisa's weight, you get twice Lisa's weight. Lisa weighs 150 pounds. What is the value of the unknown variable Y? | tiger_math |
A right triangle has a hypotenuse of 12 cm. If one of the acute angles is $30^{\circ}$, find the area of the triangle in square centimeters. | tiger_math |
Consider a graph G with 8 vertices and 14 edges. Determine if G has a perfect matching, and if so, provide one such matching. | tiger_lab_math |
Given a set of tiles where each tile has a color (black, red, orange, or blue) and a number (from 1 to 15), with the constraint that no tile can appear more than twice, how many valid ways can you group all tiles into valid subsets, where a valid subset is defined as either a group (three or four tiles of the same numb... | automath |
Matthew, Patrick and Alvin are eating dinner together. Matthew eats three times as many egg rolls as Patrick eats. Patrick eats x% as many egg rolls as Alvin eats. If Alvin ate 4 egg rolls, Matthew ate 6. What is the value of unknown variable x? What is the value of unknown variable x? | tiger_math |
A polynomial $P(x)$ is called self-centered if it has integer coefficients and $P(100) = 100.$ Find the smallest positive integer constant $c$ such that $c$ is the leading coefficient of a self-centered polynomial. | nvidia_math |
A man sitting in a train which is traveling at 70 kmph observes that a goods train, traveling in opposite direction, takes 15 seconds to pass him. If the goods train is 450 m long, find its speed. | tiger_math |
A father said to his son, "I was as old as you are at present at the time of your birth." If the father's age is 26 years now, the son's age five years back was?
Answer Choices: (A) 8 years (B) 14 years (C) 18 years (D) 19 years (E) 15 years | tiger_lab_math |
What is the mean and variance of a half-Cauchy distribution with scale parameter \(\sigma = 25\)? | automath |
Let \( m \) be a positive integer. How many elements of the ring \( \mathbb{Z}_m \) (the integers modulo \( m \)) have a multiplicative inverse? Provide the answer in terms of \( m \). | automath |
What is the result of performing the long division of 2011 by 3, and how would you verify that your answer is correct using multiplication? | automath |
Tom has been tracking his daily expenses for a week. He spent a total of $120 on Monday and Tuesday. On Tuesday and Wednesday, his average daily expenditure was $50. On Thursday and Friday, he spent a total of $90. The total expenditure for Monday and Wednesday was $80. If Tom's expenses on Saturday and Sunday were 20%... | nvidia_math |
Find the second smallest solution of the equation $(y^2-16)(y^2-4)=0$? | tiger_math |
Tim's cat bit him. He decided to get himself and the cat checked out. His doctor's visits $300 and insurance covered 75%. His cat's visit cost $120 and his pet insurance covered $60. How much did he pay? | nvidia_math |
A window is made up of 8 glass panes. Each pane has a length of 12 inches and a width of 8 inches. What is the area of the window? | tiger_math |
A parametric curve is defined by the equations
\[x = 3 \cos t + 2 \sin t,\]
\[y = 5 \sin t - 2 \cos t.\]
Express the graph of the curve in the form
\[Ax^2 + Bxy + Cy^2 = 1,\]
and find the ordered triple $(A, B, C)$. | nvidia_math |
In a certain scenario, the probability of an event A occurring is given by \( P(A) = 0.69 \), and the probability of event A not occurring is \( P(A') = 0.31 \). If \( n \) independent trials are conducted, what is the probability that event A occurs at least 4 times if \( n = 5 \)? Specifically, calculate \( P(X \geq ... | automath |
In a set \( X \) containing 5 elements, consider a semiconnex relation \( R \). What is the maximum number of elements in \( X \) that can be in the range of \( R \)? | automath |
Given a smooth fiber bundle \( p: E \to M \) with fiber \( F \), and a smooth section \( s: M \to E \), can you determine the rank of the linear map \( T_xM \to V_{s(x)}F \) obtained by projecting the tangent map \( Ts: TM \to TF \) to the vertical subbundle \( VF = \ker(Tp) \) at a point \( x \in M \)? Specifically, e... | automath |
If the volume of two cubes are in the ratio 27:1, the ratio of their edges is:
Answer Choices: (A) 3:1 (B) 3:2 (C) 3:5 (D) 3:7 (E) 3:4 | tiger_lab_math |
Given the function \( f(x,y) = x^2y \) defined on the region \( \{(x,y) : x^2 + y^2 \leq 1\} \), determine the maximum and minimum values of \( f \) and identify their locations, considering both the interior and the boundary of the constraint set. What are the coordinates of the global maximum and minimum of \( f \) w... | automath |
What is the first term and the common difference of the arithmetic series defined by the summation \( S_m = \sum_{m=1}^{10}\left(-2 + \frac{4}{3}m\right) \)? | automath |
Given that $a,$ $b,$ and $c$ are the roots of $x^3 + px + q = 0$, and the determinant
\[\begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix} = 0,\]
find the value of $a^2 + b^2 + c^2$. | tiger_math |
What conditions must be satisfied for the weighted projective stack $\mathcal{P}_R(a_0,\dots, a_n)$ to be proper over $Spec R$? Specifically, if $R$ is a complete discrete valuation ring, what must be shown regarding the $R$-points and $K$-points of the stack? | automath |
if p and q are prime numbers , how many divisors does the product p ^ 6 * q ^ 7 have ? Let's write a program. | tiger_lab_math |
Ed booked a hotel while he was on vacation. Staying at the hotel cost was $x per hour every night and $2 per hour every morning. If Ed had $80 and he stayed in the hotel for 6 hours last night and 4 hours this morning, He was left with 63 money after paying for his stay at the hotel. What is the value of unknown variab... | tiger_math |
The cubic polynomial $p(x)$ satisfies $p(2) = 1,$ $p(7) = 19,$ $p(15) = 11,$ and $p(20) = 29.$ Find
\[p(1) + p(2) + p(3) + \dots + p(21).\] | nvidia_math |
the volume of the largest right circular cone that can be cut out of a cube of edge 7 cm is : ? Let's write a Python program. | tiger_lab_math |
Mary wants to bake 10 apple pies for a charity event. Each pie needs 8 apples and she already harvested 50 apples from the trees in her garden. How many more apples does she need to buy to make all 10 pies? | tiger_lab_math |
Consider the following periodic function:
$f(x) =
\begin{cases}
-1 & \text{if } -\pi<x<0 \\
2 & \text{if } 0<x<\pi
\end{cases}$
Find the Fourier series coefficients of the function $f(x)$ with $T=2\pi$. Specifically, find the expression for $a_n$ and $b_n$, where
$f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left(a_n... | tiger_lab_math |
A man has Rs. 630 in the denominations of one-rupee notes, five-rupee notes and ten-rupee notes. The number of notes of each denomination is equal. What is the total number of notes that he has ?
Answer Choices: (A) 45 (B) 60 (C) 75 (D) 90 (E) 118.125 | tiger_lab_math |
After spending Rs. 5000 on rent, Rs. 1500 on milk, Rs. 4500 on groceries, some amount on children's education, Rs. 2000 on petrol and Rs. 700 on miscellaneous expenses, Mr. Kishore saved 10% of his monthly salary. He saved Rs. 1800. How much did he spend on children's education? | tiger_math |
Antonov bought 60 candies. He gave a pack of candy to his sister. If a pack of candy has 20 pieces, how many packs does Antonov still have? | tiger_math |
A regular polygon has sides of length 5 units and an exterior angle of 120 degrees. What is the perimeter of the polygon, in units? | nvidia_math |
In an aligned mathematical environment where you want to keep the equations aligned neatly, if you are using the `\MoveEqLeft` command from the `mathtools` package and you want to shift the first equation three ems to the left, how many points (pt) should the adjustment be, given that 1 em is equal to 10 pt in a 10 pt ... | automath |
What is the radius of the circle inscribed in triangle $ABC$ if $AB = 5, AC=6, BC=7$? Express your answer in simplest radical form. | nvidia_math |
The areas of two squares are in the ratio $25:36$. What is the ratio of their perimeters? Express your answer in the form $a:b$. | nvidia_math |
Adam has $5.00 (USD) to buy an airplane that costs €3.80 (Euros) at a store. The exchange rate at the time of purchase is 1 USD = 0.82 EUR. The sales tax in his state is 7.5%, there is a 3.5% surcharge for using a credit card, and an additional $0.25 (USD) processing fee. Taking all these costs into account, as well as... | tiger_math |
In one day, Juan sends Keith five times as many messages as he sends Laurence, and Laurence sends Missy 4.5 times as many messages as he receives from Juan. If Missy received 18 messages from Laurence, how many messages does Keith receive from Juan?
Answer Choices: (A) 3 (B) 12 (C) 16 (D) 18 (E) 20 | tiger_lab_math |
How many units long is a segment whose endpoints are (2,3) and (7,15)? Please respond by writing a program in Python. | tiger_lab_math |
Tom decides to get a new floor for his room. It cost $x to remove the floor. The new floor costs $1.25 per square foot and Tom's room is 8*7 feet. How much did it cost to replace the floor?
If we know the answer to the above question is 120, what is the value of unknown variable x? | tiger_math |
A sequence of five integers starts with -2 and ends with 3. Each term is obtained by adding a fixed constant to the previous term. How many such sequences have the property that the middle term is the arithmetic mean of the first and last terms? | nvidia_math |
Aliana is planning her birthday party and wants to equally distribute 68 candies among her friends. If she decides to invite either 7 or 9 friends, calculate how many candies each friend would get in each scenario. | tiger_math |
A curtain requires 3.25 yards of fabric. Jasmine needs to make 5 curtains for the living room and she has 10 feet of fabric. How many feet of fabric does Jasmine still need for the curtains? | nvidia_math |
Let \( H \) be a finite group with a subgroup \( H_1 \). If \( H_1 \) is generated by \( k \) cyclic groups, how many distinct elements can be formed by taking products of these \( k \) cyclic groups taken \( n \) at a time, where \( n \leq k \)? Specifically, if each cyclic group has size \( m_i \) for \( i = 1, \ldot... | automath |
What is the period of the attractor for the Ikeda map equation given as:
x(n+1) = 1 + a(x(n)cos(t(n)) - y(n)sin(t(n)))
y(n+1) = b(x(n)sin(t(n)) + y(n)cos(t(n)))
assuming values of a=0.9184 and b=0.4, and initial conditions x(0) = 0.1 and y(0) = 0.1? | tiger_lab_math |
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