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Find the number of moles of Hydrochrolic acid formed on combining 3 moles of Methane and 3 moles of Chlorine Please respond by writing a program in Python. | tiger_lab_math |
Emily has a collection of 20 stamps. She knows that 4 of these stamps have a combined value of $8. If all the stamps have the same value, what is the total value of Emily's stamp collection? | tiger_math |
A train 605 m long is running with a speed of 60 km/hr. In what time will it pass a man who is running at 6 km/hr in the direction opposite to that in which the train is going?
Answer Choices: (A) 81 (B) 16 (C) 33 (D) 54 (E) 12 | tiger_lab_math |
Jenny wants to read 3 books in the next 10 days. She can read 100 words per hour. If the first book has 200 words, the second book has 400 words, and the third book has 300 words, how many minutes per day, on average, should she spend reading? | tiger_math |
A noodle merchant sells a mixture of two types of noodles, where the first type costs twice as much as the second type. The mixture is sold for Rs 17.50 per kilogram, resulting in a profit of 25%. If the ratio of the amounts of the first noodle to the second noodle in the mixture is 2:3, what are the respective costs o... | automath |
In an amusement park, there is a ride which consists of a large circle that rotates around its center. The circle is divided into sections, and each section is painted either red, blue, or green. Each color appears exactly one-third of the total number of sections around the circle. Now, the game organizer asks the par... | tiger_math |
A wheel rotates 10 times every minute and moves 30 cm during each rotation. How many metres does
the wheel move in two hour?
Answer Choices: (A) 6 metre (B) 12 metre (C) 120 metre (D) 876 metre (E) 360 metre | tiger_lab_math |
The square root of $2x$ is greater than 3 and less than 4. How many integer values of $x$ satisfy this condition? | tiger_math |
Tony has x. He needs to buy some cheese, which costs $7 a pound and a pound of beef that costs $5 a pound. After buying the beef and his cheese, he has $61 left. How many pounds of cheese did he buy?
If we know the answer to the above question is 3, what is the value of unknown variable x? | tiger_math |
A certain basketball team that has played 2/3 of its games has a record of 14 wins and 6 losses. What is the greatest number of the remaining games that the team can lose and still win at least 3/4 of all of its games?
Answer Choices: (A) 1 (B) 6 (C) 5 (D) 4 (E) 3 Let's program in Python in the response. | tiger_lab_math |
What is the $x$-coordinate of the point on the $x$-axis that is equidistant from points $A( -2, 0)$ and $B(0,4)$? | nvidia_math |
For positive integers $n$, define $S_n$ to be the minimum value of the sum
\[\sum_{k=1}^n \sqrt{(2k-1)^2+a_k^2},\]where $a_1,a_2,\ldots,a_n$ are positive real numbers whose sum is $17$. Find the unique positive integer $n$ for which $S_n$ is also an integer. | tiger_lab_math |
Jerry is cutting up wood for his wood-burning stove. Each pine tree makes 80 logs, each maple tree makes 60 logs, and each walnut tree makes 100 logs. If Jerry cuts up 8 pine trees, 3 maple trees, and 4 walnut trees, how many logs does he get? | nvidia_math |
Let $P(z)$ and $Q(z)$ be monic polynomials, each of degree $4$. The polynomial $P(z)$ has integer coefficients, while the polynomial $Q(z)$ has complex coefficients. The product $P(z)\cdot Q(z)$ is given by
$$P(z)\cdot Q(z)=z^8 + 4z^7 + 5z^6 + 4z^5 + 2z^4 + 5z^3 + 3z^2 + 2z + 2.$$
The constant term of $P(z)$ is a divis... | nvidia_math |
A and B starts a business with Rs.8000 each, and after 9 months, B withdraws half of his capital . How should they share the profits at the end of the 18 months?
Answer Choices: (A) 4:3 (B) 18:5 (C) 14:10 (D) 18:14 (E) 14:11 | tiger_lab_math |
Tom has $2,500 in a savings account that earns a 3% annual interest rate, compounded annually. How much interest will he have earned after 8 years? (Give your answer to the nearest dollar.) | nvidia_math |
An industrial loom weaves 0.128 metres of cloth every second. It takes a certain amount of time for the loom to weave a certain amount of cloth. If it takes approximately 195.3125 seconds, how much cloth was woven? | tiger_math |
Three plastic chairs cost as much as a portable table. Five plastic chairs cost $55. If Manny wants to buy one portable table and two chairs, how much will be left in his $100? | tiger_lab_math |
What is the maximum value of the expression $\frac{x^2 + 2x + 5}{x-3}$ for integer values of $x$? | tiger_math |
Given a point \( (x_0, y_0) = (0, 0) \) and a heading of \( 315^\circ \), calculate the coordinates of a point \( (x_1, y_1) \) on the line that extends from \( (x_0, y_0) \) in the direction of the heading such that \( t = 1 \). Use this information to find the slope-intercept equation of the line through these two po... | automath |
Let $ABCD$ be an isosceles trapezoid with bases $AB=92$ and $CD=19$. Suppose $AD=BC=x$ and a circle with center on $\overline{AB}$ is tangent to segments $\overline{AD}$ and $\overline{BC}$. If $m$ is the smallest possible value of $x$, then $m^2$=
$\text{(A) } 1369\quad \text{(B) } 1679\quad \text{(C) } 1748\quad \tex... | nvidia_math |
A bag contains 4 balls, with each ball either being white or not white. If two balls are drawn at random (without replacement) and both are found to be white, what is the probability that all 4 balls in the bag are white? Please provide your answer in its simplest form. | automath |
How many positive integers $x$ satisfy the inequality $100 \leq x^2 \leq 200$? | tiger_math |
How many integers lie between -219 and +2190? | automath |
Let A be an invertible n * n matrix and v and eigenvector of both A and B, is v necesarily an eigenvector of A + B? | tiger_lab_math |
In a certain programming language, a numeric string is considered "ternary-balanced" if the sum of its digits is divisible by 3. For example, the strings "27" and "108" are ternary-balanced, while "19" and "300" are not. How many of the integers from 1 to 1000, inclusive, have a ternary-balanced representation in this ... | nvidia_math |
There are 200 snakes in a park. There are three times as many pythons as boa constrictors. If there 40 boa constrictors and the rest of the snakes are rattlesnakes, calculate the total number of rattlesnakes in the park. Let's write a Python program to solve it. | tiger_lab_math |
In a leftist heap, the complexity of the merge operation is stated to be \(O(\lg(n_1 + n_2))\) for two heaps with sizes \(n_1\) and \(n_2\). If we assume that a leftist heap achieves its properties by maintaining a leftmost path with a length bound by \(O(\lg n)\), what is the maximum number of nodes that can be in the... | automath |
What is the equation of the curve generated by the points of intersection of the two lines given by \( y = \frac{k-1}{k}(x-k) \) and \( y = \frac{k+e-1}{k+e}(x-(k+e)) \) as \( e \to 0 \), and how does this relate to the expression \( y = (\sqrt{x}-1)^2 \)? | automath |
Sarah wants to build a fence around a rectangular area with a minimum area of 100 sq. ft. She wants to use the least amount of material possible for the fence. The length of the rectangular area should be 15 ft longer than the width. What should the width, in ft, be? | tiger_math |
Zena the zookeeper has 70 animals. She notices that if she brings in 3 more elephants, the number of monkeys will be exactly twice the number of elephants. Currently, there are 4 more monkeys than elephants. How many elephants does Zena currently have? | tiger_math |
A spherical capacitor has an inner sphere of radius 0.12 m and an outer sphere of radius 0.13 m. The outer sphere is earthed and the inner sphere is given a charge of 2.5 µC. The space between the spheres is filled with a liquid of dielectric constant 32.
(a) Calculate the capacitance of the capacitor in farads, and ... | automath |
What is the cardinality of the set of all compact subsets of the rational numbers \(\mathbb{Q}\)? | automath |
Vivian is responsible for making sure her students get 2 15-minute recess breaks a day, a 30-minute lunch and another 20-minute recess break. How much time do her students spend outside of class? | tiger_lab_math |
If the sets \( A \) and \( B \) have sizes \( |A| = 40 \) and \( |B| = 30 \), and their intersection has size \( |A \cap B| = 10 \), what is the size of the union \( |A \cup B| \)? | automath |
The Lee family visits a bakery to buy pastries. A small muffin costs $2, a medium muffin costs $3, and a large muffin costs $5. Mr. and Mrs. Lee each buy a medium muffin. Their two younger kids each get a small muffin. Their four older kids each purchase a large muffin. How much money does Mr. Lee pay the baker? | nvidia_math |
Given that the force \( F \) acting on a mop head is in equilibrium under the influence of static friction with coefficient \( \mu_k \), and considering the equation \( F = \frac{\mu_k mg}{\sin \theta - \mu_k \cos \theta} \), what is the minimum angle \( \theta_0 \) such that the force \( F \) remains positive?
Expre... | automath |
Simplify $\frac{2x^2 + 5x - 3}{x^2 - 4}$ for $x = -2$. | nvidia_math |
A school band found they could arrange themselves in rows of 6, 7, or 8 with no one left over. What is the minimum number of students in the band? | tiger_math |
What is an example of a polynomial function \( f(x_1, x_2, \ldots, x_n) \) such that a lower bound for its computation in Boolean circuits implies a corresponding lower bound for its computation in arithmetic circuits over \( \mathbb{Z} \), but the same is not true for arithmetic circuits over \( \mathbb{Q} \)? | automath |
A company wants to design a cylindrical tank with a volume of 500 cubic meters. The company wants to use the least amount of material possible to construct the tank. What are the dimensions of the tank that will minimize the cost of material? The material cost is directly proportional to the surface area of the tank. T... | tiger_lab_math |
Stan drove 300 miles in 5 hours, 20 minutes. Next, he drove X miles in 6 hours, 40 minutes. What was Stan's average speed in miles per hour for the total trip?
If we know the answer to the above question is 55, what is the value of unknown variable X? | tiger_math |
If $y$ varies directly as the square root of $x$, and $y = 60$ when $x = 16$, what is the value of $y$ when $x = 49$? | nvidia_math |
What conditions on the real numbers \( a \), \( b \), \( c \), and \( d \) ensure that the system of equations
\[
\begin{cases}
ax - by = c \\
bx + ay = d
\end{cases}
\]
has no solutions? Specifically, express these conditions in terms of \( a \), \( b \), \( c \), and \( d \). | automath |
What is the rational representation of the repeating decimal \(0.7777...\)? | automath |
In a 3D programming context, if a camera is positioned at the coordinate (0, 0, 4) and moves backward at a speed of 2 units per second for 3 seconds, what will be its new position in Cartesian coordinates? | automath |
Given that a fire hose ejects water at a velocity of \( v \) m/s and has a cross-sectional area of \( A \) m² at the nozzle, calculate the thrust force \( F \) experienced by the firefighter due to the water being expelled, considering the density of water is \( \rho \) kg/m³. What is the formula for the thrust force a... | automath |
Five shirts together cost $85. Of the 5 shirts, there are 3 shirts that cost $15 each. If the remaining shirts are each equal in value, what is the cost, in dollars, of one of the remaining shirts? | tiger_math |
What is the area of the red region, as a fraction of the area of the entire circle, given that the blue and white circles combined have half the area of the larger circle and that the arrangement forms an equilateral triangle? | automath |
A superhero can use superhuman speed to run 10 miles in 4 minutes. The supervillain has an Evil-Mobile that drives 100 miles per hour. How many miles farther can the superhero run in an hour than the supervillain can drive? Let's program in Python in the response. | tiger_lab_math |
What is the largest integer that is a divisor of \[
(n+1)(n+3)(n+5)(n+7)(n+9)
\]for all positive even integers $n$? | nvidia_math |
the length of the bridge , which a train 475 metres long and travelling at 90 km / hr can cross in 30 seconds , is : ? Let's program in Python in the response. | tiger_lab_math |
Suppose you have a sequence of random variables \(X_1, X_2, \ldots, X_n\) that are identically distributed and independent. Let \(P(X_i = 1)\) be the probability that any random variable \(X_i\) equals 1. If \(P(X_1 = 1) = p\), what is the probability \(P(X_1 = 1, X_2 = 1)\) in terms of \(p\)? | automath |
A woodworker is crafting enough furniture legs for their projects. They have made a total of 40 furniture legs so far, and this is the exact amount they needed for everything they’re building. If the woodworker is using these legs for their tables and chairs and they have built 6 chairs, how many tables have they made? | tiger_math |
What is the period of oscillation of the Rössler system given by the equations below?
x'=-y-z
y'=x+ay
z'=b+z(x-c)
where a=0.2, b=0.2, and c=5.7.
Assume that the initial conditions are x(0)=1, y(0)=0, and z(0)=0. | tiger_lab_math |
The linear transformations
\[A(\mathbf{x})=\begin{pmatrix}x+y+z\\ x-z\\ x+2y+3z\end{pmatrix} \quad \text{and} \quad B(\mathbf{x})=\begin{pmatrix}x+y-z\\ x+2y+z\\ x+y+2z\end{pmatrix}\]
are inverses. Find the sum of all the coefficients in the matrix representation of $B$. | nvidia_math |
A rectangular courtyard is surrounded by a walkway that is 5 meters wide. The length of the courtyard is twice its width. If the area of the walkway is 200 square meters, what is the total area of the courtyard and the walkway combined? | nvidia_math |
In the Science City, Kolkata, the rate of the ticket is increased by 50% to increase the revenue, but simultaneously some percentage of the visitors decreased. The percentage change in the revenue is 20%. It is known that the Science City collects revenue only from the visitors and it has no other financial supports. W... | tiger_math |
In a PCA analysis up to the 25th dimension, the first principal component (PC1) explains 40.6% of the total variance. After extrapolating eigenvalues to the 50th dimension, this percentage drops to 37.5%, and further extrapolation to the 1000th results in only 12.2% variance explained by PC1. What percentage of total v... | automath |
The length of the bridge, which a train 130 metres long and travelling at 36 km/hr can cross in 45 seconds, is:
Answer Choices: (A) 320 m (B) 225 m (C) 245 m (D) 250 m (E) 240 m Let's write a Python program to solve it. | tiger_lab_math |
The parabola $y^2=8x$ and the circle $x^2+y^2-2x-4y=0$ intersect at two points $A$ and $B$. Find the distance between $A$ and $B$. | nvidia_math |
Let $b_n = 2^{2n} + 1$. Define $e_n$ to be the greatest common divisor of $b_n$ and $b_{n+1}$. Determine the minimum possible value that $e_n$ can take on, and find the value of $n$ for which this minimum is attained. | nvidia_math |
If eight liters is a certain percentage of the capacity of a container filled with water, the total capacity of 40 such containers filled with water is 1600 liters. What percentage of the container's capacity is filled with water? | tiger_math |
Find the sum of the absolute values of all roots of the equation:
x = 3 + \frac{1}{x + \frac{1}{x + \frac{1}{x +...}}}
where the pattern of nested fractions continues indefinitely. | nvidia_math |
A box contains 5 white balls and 3 black balls. I draw them out of the box, one at a time. What is the probability that all of my draws alternate colors? | tiger_math |
Let $p(x)$ be a polynomial with integer coefficients, and let $a$ and $b$ be distinct integers such that $p(a) = p(b) = 1$. What is the smallest possible value of $|a - b|$? | nvidia_math |
The length of the bridge, which a train 130 metres long and travelling at 45 km/hr can cross in 30 seconds, is:
Answer Choices: (A) 200 meters (B) 225 meters (C) 245 meters (D) 250 meters (E) 300 meters | tiger_lab_math |
Which of the following triples of numbers have the same standard deviation as the numbers u, v and w?
I. u-2, v-2, w-2
II. 0, u-v, w-v
III. u-4, v+5, w-1
Answer Choices: (A) I only (B) II only (C) I and III only (D) II and I only (E) I, II, and III | tiger_lab_math |
What is the value of $a$ if the lines $2y - 2a = 6x$ and $y + 1 = (a + 6)x$ are parallel? | nvidia_math |
The sum of an infinite geometric series is X times the series that results if the first three terms of the original series are removed. What is the value of the series' common ratio?
If we know the answer to the above question is \frac{1}{3}, what is the value of unknown variable X? | tiger_math |
If the polynomial $(ax^3+bx^2+c)(dx^4+ex+f)(gx^2+h)$ has a degree of 9, where $a$ through $h$ are nonzero constants, what is the sum of the coefficients of the factors? | tiger_math |
The remainder when $5^{137}$ is divided by 8 can be found using the following problem:
Consider the set of integers $a$ for which there exists an integer $b$ such that the remainder when $b^n$ is divided by $a$ is equal to 1 for all positive integers $n$. What is the largest possible value of $a$? | nvidia_math |
Janet makes $20 per hour at work. She works 52 hours per week. Anything over 40 hours per week is considered overtime and is paid at 1.5 times the normal rate. She wants to purchase a $4640 car. How many weeks does she need to work to purchase the car? | tiger_lab_math |
Let $a$, $b$, and $c$ be positive real numbers such that $ax^2+bx+c=0$ and $cx^2+bx+a=0$ both have real roots. Find the largest possible value of $a/c+b$.
Note: The original problem focused on minimizing $a+b$ for two quadratic equations with real roots. In this new problem, we shift the focus to maximizing $a/c+b$ fo... | nvidia_math |
Solve for $y$: $2(y^3) = 54$. | nvidia_math |
A trader bought a car at 10% discount on its original price. He sold it at a 40% increase on the price he bought it. What percent of profit did he make on the original price?
Answer Choices: (A) 118 (B) 110 (C) 112 (D) 126 (E) 115 | tiger_lab_math |
Evaluate \( \cos(2 \arctan(1/3)) \). | automath |
What is the total number of bits required to store all possible 30-second long MP3 files encoded at 128 kbps, and how does this compare to the estimated number of atoms in the observable universe, approximately \(10^{80}\)? | automath |
A five-digit integer is divisible by 4 if and only if the last two digits form a number divisible by 4. What is the largest five-digit integer that contains no zero and is divisible by 4? | nvidia_math |
Given the coordinates of N stakes and M lambs on a 2D plane, determine the minimum length of rope needed to encircle all the lambs such that no part of the rope crosses over any lamb. Each stake can be used as an anchor point for the rope. What is the length of the rope needed in the following test case?
Test case:
``... | automath |
If there are \( N \) nodes in a Slotted Aloha network, and each node transmits in a slot with probability \( p \), what is the probability that exactly one node successfully transmits in a given slot? | automath |
On her birthday, Avianna bought red candles and blue candles in the ratio of 5:3. If Avianna had 45 red candles on her birthday, how many blue candles did she have? | tiger_lab_math |
The music class consists of 4 girls and 5 boys. How many ways can a group of 3 be formed if it has to include at least one boy?
Answer Choices: (A) 80 (B) 158 (C) 161 (D) 165 (E) 172 | tiger_lab_math |
The Lucas numbers $L_n$ are given by the recursion $L_0 = 2, L_1 = 1$, and $L_n = L_{n-1} + L_{n-2}$. Find the units digit of $L_{L_{10}}$. | nvidia_math |
The expression $6j^2 - 4j + 12$ can be rewritten in the form $c(j + p)^2 + q$, where $c$, $p$, and $q$ are constants. We are asked to find $\frac{q}{p}$.
$6j^2 - 4j + 12 = 6(j - \frac{1}{3})^2 + 11\frac{2}{3}$
$c = 6, p = -\frac{1}{3}, q = 11\frac{2}{3}$
Therefore, $\frac{q}{p} = \frac{11\frac{2}{3}}{-\frac{1}{3}} =... | nvidia_math |
A hexagon with center $O$ is located on the coordinate plane such that one of its vertices is at $(-1,-1)$. If $O$ has coordinates $(2,3)$ and the distance between $O$ and any vertex of the hexagon is $\sqrt{26}$, what are the coordinates of the other vertices of the hexagon? | nvidia_math |
A certain industrial loom weaves 0.128 meters of cloth every second. Approximately how many seconds will it take for the loom to weave 15 meters of cloth?
Answer Choices: (A) 114 (B) 115 (C) 116 (D) 117 (E) 118 Let's program in Python in the response. | tiger_lab_math |
The surface area of a sphere is 4πr^2, where r is the radius of the sphere. If the area of the base of a hemisphere is 3, what is the surface area of that hemisphere? | tiger_math |
A certain bakery sells a total of 480 muffins and cakes on a certain day. The number of muffins sold was 3/5 of the number of cakes sold. How many muffins were sold that day? | nvidia_math |
How many total days were there in the years 2005 through 2008? | nvidia_math |
What is the integral of the function \( \int x e^x \, dx \) evaluated using integration by parts, and what are the steps involved? | automath |
Given two bodies with masses \( m_1 = 10^{26} \) kg and \( m_2 = 10^{26} \) kg initially positioned at \( r_1(0) = (0, 0, 0) \) m and \( r_2(0) = (3000 \times 10^3, 0, 0) \) m, and with initial velocities \( v_1 = (10^4, 2 \times 10^4, 3 \times 10^4) \) m/s and \( v_2 = (0, 4 \times 10^4, 0) \) m/s, what is the analyti... | automath |
The sum of five numbers is 400. The average of the first two numbers is 50 and the third number is 60. Find the average of the two numbers?
Answer Choices: (A) 70 (B) 65 (C) 85 (D) 80 (E) 90 | tiger_lab_math |
How many natural numbers between 200 and 400 are divisible by 12? | nvidia_math |
At peak season, 6 packs of tuna fish are sold per hour, while in a low season 4 tuna packs are sold per hour. If each tuna pack is sold at $60, How much more money is made in a day during a high season than a low season if the fish are sold for 15 hours? | tiger_lab_math |
On an \( N \times N \) chessboard, a knight starts at the \( r \)-th row and \( c \)-th column. If \( N = 8 \), \( r = 1 \), \( c = 1 \) (representing the top-left corner), and the knight makes exactly \( K = 3 \) moves, what is the probability that the knight will still be on the chessboard after these moves? | automath |
Find the sum of all real values of x that satisfy the equation (x + 2)/(2x + 2) = (4x + 3)/(7x + 3). | tiger_math |
What is the sum of the roots of the quadratic equation \(3x^2 - 4x - 4 = 0\)? | automath |
A park is 1000 feet long and 2000 feet wide. If there is 1 tree per 20 square feet, how many trees are there in the park? | tiger_math |
Given the properties of the functions \(\sin\) and \(\cos\) defined in the axiomatic construction, how can we verify that
\[
\lim_{x\to 0} \frac{\sin x}{x} = 1?
\]
Provide a rigorous argument based on the conditions supplied, particularly focusing on the implications of the periodicity, continuity, and the increasin... | automath |
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