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The two integers closest to $\sqrt{200}$ are relatively prime. Find the smaller one. | nvidia_math |
A rectangular pyramid has a base with dimensions 6 inches by 4 inches and a height of 12 inches. If a sphere has the same volume as the pyramid, what is the radius of the sphere, in inches? | nvidia_math |
What are the number of ways to select 3 men and 2 women such that one man and one woman are always selected?
Answer Choices: (A) 26 ways (B) 19 ways (C) 30 ways (D) 76 ways (E) 15 ways | tiger_lab_math |
a hall 36 m long and 15 m broad is to be paved with stones , each measuring 8 dm by 5 dm . the number of stones required is : ? Let's program in Python in the response. | tiger_lab_math |
Two lines in space are represented by the equations \(\frac{x - 1}{2} = \frac{y + 3}{4} = \frac{z - 5}{1}\) and \(\frac{x - 2}{1} = \frac{y + 1}{-1} = \frac{z - 3}{3}\). What is the acute angle between these two lines? Provide your answer in radians. | automath |
The vertical drop of a roller coaster is the largest difference in height between any high point and the next low point. The vertical drops of five roller coasters at Mandelbrot Amusement Park are shown in the table. \begin{tabular}{|l|c|} \hline
The Parabola & 165 feet \\ \hline
The G Force & 119 feet \\ \hline
The Me... | tiger_math |
The coordinates of the vertices of a hyperbola are $(3, 5)$ and $(7, 5).$ One of the asymptotes of the hyperbola has equation $y = 2x.$ Find the equation of the other asymptote of the hyperbola, giving your answer in the form "$y = mx + b$". | nvidia_math |
Given the functions \( f(n) = 3n^2 + 5n + 2 \) and \( g(n) = n^3 \), determine whether \( f(n) \in \mathcal{O}(g(n)) \). If so, find suitable constants \( c \) and \( n_0 \) that satisfy the definition of Big-O notation. If not, justify your answer. | automath |
Let $A$, $B$, $C$, and $D$ be points on a circle such that $AB = 11$ and $CD = 19.$ Point $P$ is on segment $AB$ with $AP = 6$, and $Q$ is on segment $CD$ with $CQ = 7$. The line through $P$ and $Q$ intersects the circle at $X$ and $Y$. If $PQ = 27$, find $XY$. | nvidia_math |
Natasha has 3 times as much money as Carla, and Carla has twice as much money as Cosima. If Natasha has $60, and the three decide to buy goods worth the total amount they have and sell the goods at 7/5 of the buying price, how much profit would they make? | nvidia_math |
Let $r$ be the common ratio of a geometric sequence whose terms are all positive integers. If $a, b, c$ are three consecutive terms of the sequence such that $\sqrt[3]{a}\cdot\sqrt{b}\cdot\sqrt[4]{c}$ is an integer, determine the smallest possible value of $r$. | nvidia_math |
Maria is a decorator hired to wallpaper a client's office building. The building has five conference rooms, and three times as many offices as conference rooms. The client wants each conference room wallpapered with a unique design, and the offices wallpapered with a standard pattern. Each room requires 3 rolls of wall... | nvidia_math |
By selling 150 mangoes, a fruit-seller gains the selling price of 30 mangoes. Find the gain percent?
Answer Choices: (A) 28% (B) 25% (C) 85% (D) 24% (E) 22% | tiger_lab_math |
Reese had a sum of money in her savings account. The company she used to work with was not doing well which is why she lost her job. So she spent 20% of her savings for her expenses in February, 40% of it in March, and $1500 in April. She had 11000 in her savings account if she still has $x after spending those amounts... | tiger_math |
What is the value of $\log_{10}{4}+2\log_{10}{5}+3\log_{10}{2}+6\log_{10}{5}+\log_{10}{8}$? Let's program in Python in the response. | tiger_lab_math |
Doris works at the Widget Factory in the packing department. She puts 3 widgets in each carton, which are 4 inches wide, 4 inches long, and 5 inches tall. She then packs those cartons into a shipping box before sending it to the loading bay. The shipping boxes are 20 inches wide, 20 inches long, and 20 inches high. How... | tiger_lab_math |
Kristen notices an identical number of three types of insects in her neighborhood: butterflies, fireflies, and ladybugs. She always seems to observe butterflies in groups of 44, fireflies in groups of 17, and ladybugs in groups of 25. What is the smallest number of butterflies that she could have seen such that the tot... | tiger_math |
Max has forgotten his bike lock combination. It is a sequence of four numbers, each in the range from 1 to 25, inclusive. He knows that the first number is a multiple of 5, the second number is prime, the third number is even, and the fourth number is a multiple of 4. How many combinations could possibly be Max's? | nvidia_math |
If the clown initially has 3 dozen balloons on a string in his hand and 3 boys and 12 girls each buy a balloon, how many balloons is the clown still holding? | tiger_math |
A Lamplighter monkey has long arms and can swing from branch to branch at a speed of 10 feet per second. But when the monkey is frightened, it will run along the branches at a speed of 15 feet per second. If a Lamplighter monkey runs for x seconds to evade a predator, then swings for another 10 seconds to add distanc... | tiger_math |
Find the solution to the non-homogeneous system of linear equations:
x + 2y - z = 11
3x - 4y + z = -1
2x + y + 2z = 3 | tiger_lab_math |
Given the function \( f(z) = \tan^{-1}(z \cdot (0.058659782 - 0.35747651i)) \), determine the values of \( f(-20 + 3.282i) \) and \( f(-20 + 3.281i) \), and verify why these values have a significant difference. Specifically, calculate the real and imaginary parts of these two values, and explain the behavior of the fu... | automath |
How many 4-critical graphs exist on 7 vertices, and can you provide an example of one? | automath |
Let $f(x) = 2x + 1$ and $g(f(x)) = x^3 - 2x + 1$. Find $g(-1)$.
We have $g(f(x))=g(2x+1)=x^3-2x+1$.
So $$g(2x+1)=x^3-2x+1$$ and $$g(2(x-1)+1)=(x-1)^3-2(x-1)+1=x^3-3x^2+5x-2$$So $x^3-2x+1=(x^3-3x^2+5x-2)+3x^2-3x+1$ and the corresponding solution is $$g(x)=x^3-3x^2+5x-2$$ wherefrom $g(-1)=(-1)^3-3(-1)^2+5(-1)-2=-11$.
T... | nvidia_math |
An integer n between 1 and 100, inclusive, is to be chosen at random. What is the probability that n(n + 1) will be divisible by a certain number? The answer is 0.5. | tiger_math |
What is the diameter in centimeters of a circle whose area is $100\pi \text{cm}^2$? | tiger_math |
Carrie wants to buy a new iPhone. The new iPhone costs $800. She can trade in her Samsung Galaxy for $240. She can make $80 per week babysitting. How many weeks does she have to work before she can purchase the iPhone? | tiger_lab_math |
Given the lattice \(\mathbf{Z}^4 \subset \mathbf{R}^4\), calculate \(N_m\) for \(m = 9\), which represents the number of ways of writing the number \(9\) as a sum of four integral squares. | automath |
A biologist wants to calculate the number of fish in a lake. On May 1 she catches a random sample of 50 fish, tags them, and releases them. On September 1 she catches a random sample of 80 fish and finds that 5 of them are tagged. To calculate the number of fish in the lake on May 1, she assumes that 20% of these fish ... | tiger_math |
The teacher divided the students into four groups. One group had 5 students, another 8 students, and the third 7 students. If there were x total students, There were 4 students in the fourth group. What is the value of unknown variable x? | tiger_math |
What is the length of the diagonal of a rectangle with a perimeter of 40 cm and a area of 84 cm²? | nvidia_math |
Determine the value of the expression
\[\frac{6}{7} + \frac{\frac{6}{7}}{6 + \frac{\frac{6}{7}}{6 + \frac{\frac{6}{7}}{6 + \cdots}}}.\]
Assume it is positive. | nvidia_math |
In a simple linear regression model given by the equation \( y_j = \beta_0 + \beta_1 x_j + \epsilon_j \), if it is known that the parameter \( \beta_1 \) is restricted to the value of 1 during the estimation process, how many independent parameters are estimated in the model using the method of least squares? | automath |
Emily purchases a sandwich for $6.50 every day for lunch, 5 days a week. On Fridays, she also buys a pastry for $3.00. Her goal for the next 6 months is to reduce her lunch spending by 30%. How much will she save? | nvidia_math |
The age difference between Asaf and Alexander's age is half the total number of pencils Asaf has. The sum of their ages is 140, and Asaf is 50 years old. If Alexander has 60 more pencils than Asaf, calculate the total number of pencils they have together. | tiger_lab_math |
If Maria wants to paint a picture and needs to purchase tools, including a set of brushes for $20 and canvas that costs three times more than the brushes, as well as paint that costs $8 per liter, with a minimum requirement of 5 liters. If she is able to sell the painting for $200, how much profit will she earn from th... | tiger_math |
The set X = {11,12,13,14,15}. The set Y = {2,3,4,5}. We will randomly choose one element x from set X and one element y from set Y. What is the probability that x / y will be an integer?
Answer Choices: (A) 11/20 (B) 7/20 (C) 3/20 (D) 7/10 (E) 3/10 | tiger_lab_math |
If a student is required to maintain a grade point average (GPA) of at least 3.5 while also carrying a course load of at least 12 credit hours to qualify for the honor roll, what set of values does the GPA represent, expressed in interval notation? | automath |
Let \( f: [0, 1] \to \mathbb{R} \) be a continuously differentiable function such that
\[
\left| \frac{f(0) + f(1)}{2} - \int_0^1 f(x) \, dx \right| \leq \alpha \max_{x \in [0,1]} |f'(x)|.
\]
What is the minimum possible value of \( \alpha \) such that the inequality holds for all continuously differentiable functio... | automath |
Given a set of points \((x_i, y_i)\) for \(i = 1, \ldots, N\) that are purported to lie on a Lissajous curve defined by the equations \(x = A \sin(at + \delta)\) and \(y = B \cos(bt)\), describe a method to determine the parameters \(A\), \(B\), \(a\), \(b\), and \(\delta\) that best fit these points using the method o... | automath |
A merchant has selected two items to be placed on sale, one of which currently sells for 35 percent less than the other. If he wishes to raise the price of the cheaper item so that the two items are equally priced, by what percentage must he raise the price of the less expensive item?
Answer Choices: (A) 30% (B) 40% (C... | tiger_lab_math |
When Greenville State University decided to move its fine arts collection to a new library, it had to package the collection in 20-inch by 20-inch by 12-inch boxes. The university pays a certain amount for every box, and it needs 2.4 million cubic inches to package the collection. The minimum amount the university must... | tiger_math |
A petri dish originally contained 600 bacteria. A scientist let the bacteria grow and now there are a certain number of them. There are now 8317 more bacteria. How many bacteria are there now? | tiger_math |
What is a general formula for calculating the area of a cyclic n-gon given its side lengths? | automath |
A sum of money is to be distributed among A, B, C, D in the proportion of 5 : 2 : 4 : 3. If C gets Rs. 1000 more than D, what is B's share?
Answer Choices: (A) 4000 (B) 5000 (C) 2000 (D) 1000 (E) 6000 | tiger_lab_math |
a man can row 8 kmph in still water . when the river is running at 1.8 kmph , it takes him 2 hour to row to a place and back . what is the total distance traveled by the man ? Let's write a program. | tiger_lab_math |
Let $a,$ $b,$ $c$ be a three-term arithmetic series where all the terms are positive, such that $abc = 64.$ Find the smallest possible value of $b.$ | nvidia_math |
Let \( n \) be a positive integer. If \( x \) and \( y \) are square roots of \( 51733469 \) modulo \( n \) such that \( x \not\equiv \pm y \mod n \), find the necessary condition on \( n \) for \( n \mid (x+y)(x-y) \) to hold true, and state how many distinct pairs \( (x, y) \) exist under these conditions if \( n \) ... | automath |
Given an initial amount of $875,000 in a 401(k) account, with a monthly withdrawal of $5,000 and an annual interest rate of 5% (compounded monthly), how many months will it take for the account to be depleted, accounting for taxes of 17.5% on each withdrawal? | automath |
Let $C = (2,3)$ and $D = (6,1).$ Let $Q$ be a point on the circle $x^2 + y^2 = 4.$ Find the largest possible value of $\angle CQD.$ | nvidia_math |
If Jeff picks two letters randomly from the alphabet without replacement, what is the probability that both letters are in the word `PROBABILITY'? | tiger_math |
A particle moves in a circular path with a radius of \( r \) meters and experiences a centripetal acceleration of \( 4 \, \text{m/s}^2 \). If the particle's mass is \( 2 \, \text{kg} \), what is the net force acting on the particle? | automath |
Given the following data on surgery success rates for two hospitals, Mercy and Hope:
Mercy Hospital:
- Type A: Successful 292 out of 359
- Type B: Successful 1449 out of 1836
- Type C: Successful 179 out of 299
- Type D: Successful 434 out of 2086
- Type E: Successful 13 out of 149
- Total: 2366 successful out of 4729... | automath |
James streams on twitch. He had 150 subscribers and then someone gifted 50 subscribers. If he gets $9 a month per subscriber how much money does he make a month? | nvidia_math |
If there are sixty-five dolphins in the ocean, and three times that number joins in from a river which has its mouth at the ocean, calculate the total number of dolphins in the ocean. | tiger_math |
What positive value of $t$ makes $|{-4+ti}|$ equal to $2\sqrt{13}$? | tiger_math |
If we define the matrices \( C = \begin{pmatrix} 1 & 4 \\ 3 & 2 \end{pmatrix} \) and \( D = \begin{pmatrix} 10 & 4 \\ 5 & 2 \end{pmatrix} \), what is the determinant of the matrix obtained by multiplying \( C \) and \( D \), i.e., what is \( \text{det}(C \times D) \)? | automath |
The cost price of 12 articles is equal to the selling price of 10 articles. What is the profit percent?
Answer Choices: (A) 10% (B) 20% (C) 30% (D) 40% (E) 50% | tiger_lab_math |
Given the sequence defined by
\[
x_n = \begin{cases}
\frac{n+1}{n} & \text{if } n \text{ is odd,} \\
0 & \text{if } n \text{ is even.}
\end{cases}
\]
calculate both \(\liminf_{n \to \infty} x_n\) and \(\limsup_{n \to \infty} x_n\). What are the final values of these limits? | automath |
If n is a positive integer and n^2 is divisible by 450, then what is the largest positive integer that must divide n?
Answer Choices: (A) 10 (B) 20 (C) 30 (D) 40 (E) 50 | tiger_lab_math |
If the price of 23 balls is 276, then what will the price of 12 balls?
Answer Choices: (A) 144 (B) 148 (C) 156 (D) 186 (E) None of these Let's write a program. | tiger_lab_math |
Consider a chain complex \( K \) consisting of objects \( K_n \) in an Abelian category, with a differential \( d : K \rightarrow K \) of degree \(-1\). If the complex satisfies \( d^2 = 0 \) and the homology groups \( H_n(K) = Z_n(K) / B_n(K) \) are all zero, what can be concluded about the complex \( K \)?
1. Is the... | automath |
Two different solutions of alcohol are mixed: the first solution has a proportion of alcohol to water of 3:1, and the second solution has a proportion of 2:3. If the first solution is four times the volume of the second solution, what is the concentration of alcohol in the resulting mixture expressed as a percentage? | automath |
Compute $\binom{17}{9}$. You are told that $\binom{15}{6} = 5005$ and $\binom{15}{8} = 6435$. | nvidia_math |
Find the area of a triangle with side lengths 8, 9, and 9. | nvidia_math |
Compute $\sqrt{54}\cdot\sqrt{32}\cdot \sqrt{6}$. | nvidia_math |
Jess and her family play Jenga, a game made up of 54 stacked blocks in which each player removes one block in turns until the stack falls. The 5 players, including Jess, play 5 rounds in which each player removes one block. In the sixth round, Jess's father goes first. He removes a block, causing the tower to almost fa... | tiger_lab_math |
What is the final numerical result obtained by substituting the values \(\alpha_{c,c} = 30\), \(f_{c,k} = 1\), and \(\gamma_c = 1.5\) into the formula \(f_{c,d} = \frac{\alpha_{c,c} f_{c,k}}{\gamma_c}\)? | automath |
A firefighter's hose can deliver 20 gallons of water per minute. For a building fire that requires 4000 gallons of water to put out, how long will it take, in minutes, for a team of 5 firefighters, each with their own hose, to put out the fire? | tiger_math |
A cyclist travels uphill at a constant speed of $v$ kph for $t$ hours, covering a distance of $d$ kilometers. Then, they turn around and travel downhill at a constant speed of $2v$ kph for the same amount of time $t$. What is the average speed of the cyclist for the entire round trip? | nvidia_math |
On Rudolph's car trip across town, he traveled 2 more than 5 miles and encountered 3 less than x stop signs. Rudolph encountered 2 stop signs per mile on his trip across town. What is the value of unknown variable x? What is the value of unknown variable x? | tiger_math |
There are 25 roses in a garden. There are 40 tulips. There are 35 daisies. What percentage of flowers are not roses? Let's program in Python in the response. | tiger_lab_math |
In a binary tree with \( n \) nodes, where \( n \) is greater than 1, is there always a node whose subtree contains at least \( \left\lfloor \frac{n}{3} \right\rfloor \) and at most \( \left\lfloor \frac{2n}{3} \right\rfloor \) nodes? If so, how can this be proven? | automath |
A gambler began playing blackjack with $110 in chips. After exactly 12 hands, he left the table with $320 in chips, having won some hands and lost others. Each win earned $100 and each loss cost $10. How many possible outcomes were there for the first 5 hands he played? (For example, won the first hand, lost the second... | tiger_lab_math |
The operation $\heartsuit$ is defined as $a\,\heartsuit\,b = \sqrt{a^2 + b^2}$ for any real numbers $a$ and $b$. What is the value of $(3\,\heartsuit\,5)\,\heartsuit\,(2\,\heartsuit\,7)$? | nvidia_math |
A combination lock consists of 5 buttons, each with a different color (red, green, blue, yellow, and purple). The lock can be opened by pressing any 3 buttons in a specific order, with the condition that exactly two of the pressed buttons must be of primary colors (red, green, or blue). How many different combinations ... | nvidia_math |
Given the points (2, 1) and (0, -3), what is the slope-intercept form of the equation of the line that passes through these two points? Please provide the equation in the form \( y = mx + b \). | automath |
Let \( N \) be the sum of five consecutive integers. If the first three of these integers, when arranged in ascending order, sum to zero, determine the product of the last three integers when they are arranged in descending order. What is the product?
A. -1
B. 0
C. 1
D. 2
E. 6 | automath |
Given the polynomial \( f(x) = x^3 - 4x^2 + 9x - 10 \), find all the real and complex zeros of \( f \) and express the polynomial in its completely factored form over the complex numbers. | automath |
Let \( \{X_n\}_{n \geq 1} \) be a sequence of independent and identically distributed random variables, each following a normal distribution with mean \( \mu \) and variance \( \sigma^2 \). According to the Central Limit Theorem, what is the limiting distribution of the normalized sum \( S_n = \frac{1}{\sigma \sqrt{n}}... | automath |
In a regular pentagon $ABCDE$, the apothem (the distance from the center of the pentagon to one of its vertices) is 12 units. Find the side length of the pentagon to the nearest tenth. | nvidia_math |
What is the Lyapunov exponent for a weather system, given a set of initial conditions, where the values for the system parameters are: the Rayleigh number, R = 200; the Prandtl number, Pr = 6.95; and the aspect ratio, A = 1.23? | tiger_lab_math |
The number of students in Teresa's graduating class is more than 50 and fewer than 100 and is 1 less than a multiple of 3, X less than a multiple of 4, and 3 less than a multiple of 5. There are 62 students in Teresa's graduating class. What is the value of unknown variable X? | tiger_math |
Given a strictly increasing array of positive floating-point numbers \( X_0, X_1, \ldots, X_n \), what is the minimum value of \( H \) such that the following condition holds for all \( i = 0, 1, \ldots, n-1 \):
\[
\lfloor H X_{i+1} \rfloor > \lfloor H X_i \rfloor
\]
Moreover, express \( H \) in terms of the elements... | automath |
What is the greatest common divisor of 84 and 108? | tiger_lab_math |
if 9 / ( 5 + 3 / x ) = 1 , then x = ? Let's write a program. | tiger_lab_math |
If two numbers are in the ratio 1:2. If 10 is added to both of the numbers then the ratio becomes 2:3 then find the smallest number?
Answer Choices: (A) 10 (B) 20 (C) 30 (D) 40 (E) 50 | tiger_lab_math |
Find the inverse of the matrix
\[\begin{pmatrix} 1 & 2 \\ 2 & 4 \end{pmatrix}.\]If the inverse does not exist, then enter the zero matrix. | tiger_math |
From a theatre troupe of eight members, including Lou, one person will be randomly chosen to play Abel, another person will be randomly chosen to play Barry, and a third person will be randomly chosen to play Caine. What is the probability that Lou will NOT be chosen to play either Abel or Caine?
Answer Choices: (A) 1/... | tiger_lab_math |
What is the covariance \( \text{Cov}(e^{\int_{t}^{T} W^1_s ds}, \int_{t}^{t+1} W^2_s ds) \) given that \( W^1_t \) and \( W^2_t \) are two correlated Brownian motions with a correlation structure such that \( dW^1_t dW^2_t = \rho dt \)? Please provide a detailed derivation of your answer. | automath |
At A.T. Cross Luxury Pens, a pencil sells at twenty-five cents while a pen sells at fifteen cents. Bowen buys 40 pens and 2/5 times more pencils than pens from A.T. Cross Luxury Pens. Calculate the total amount of money Bowen spends. | tiger_lab_math |
If x+y=5, and x=15/y, what is the value of x^2+y^2?
Answer Choices: (A) 12 (B) 14 (C) 16 (D) 18 (E) 10 | tiger_lab_math |
If you have a positive integer \( n \) with the digits \( a_k, a_{k-1}, \ldots, a_1, a_0 \) (where \( a_k \) is the most significant digit), describe an algorithm using a do-while loop to compute the sum of its digits, and determine the final value of this sum when \( n = 5634 \). | automath |
the distance from the x - axis to point p is half the distance from the y - axis to point p . if the coordinates of p are ( x , - 8 ) , how many units is p from the y - axis ? Please write a program to solve it | tiger_lab_math |
Given an object’s positions at two time points, \( (1,3) \) at \( t=6 \) and \( (3,3) \) at \( t=7 \), determine its future position at \( t=8 \). If the object moves from \( (5,3) \) at \( t=6 \) to \( (3,3) \) at \( t=7 \), what is its position at \( t=8 \)? Calculate both future positions. | automath |
A psychiatrist has 4 patients that need 25 sessions in total. One of the patients needs 6 sessions. Another patient needs 5 more than that. How many sessions would the remaining patients need? | tiger_math |
A small landholding is divided into three parts, and on each part a different crop is planted: oats, wheat and rye.
The yield of the three crops is proportional to the area sown.
If the yield of oats is six quintals and that of wheat is half as much again,
what is the yield of rye if it is known that the total yield fo... | nvidia_math |
Philip is a painter. He makes 2 paintings per day. If he already has 20 paintings, He will have 80 paintings in total after x days. What is the value of unknown variable x? What is the value of unknown variable x? | tiger_math |
The average of a set of Y numbers is 92. If two new numbers, 80 and 88, are added to the set, the average of the new set of numbers becomes 90. Find the value of Y. | tiger_math |
All the numbers 2, 3, 4, 5, 6, 7 are assigned to the six faces of a cube, one number to each face. For each of the eight vertices of the cube, a product of three numbers is computed, where the three numbers are the numbers assigned to the three faces that include that vertex. What is the greatest possible value of the ... | nvidia_math |
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