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Determine all real numbers $a$ such that the function
$$f(x)=\left(\frac{x^2}{2a}\right)^{1-\frac{x}{a}}$$
is defined for any $x$, $0<x<a$, and satisfies
$$f(x)+f(a-x)=1$$
for all $x$ from the same interval. | nvidia_math |
Simplify $\sqrt{\frac{x^3+8}{x+2}}$. Express your answer in simplest radical form in terms of $x$. | nvidia_math |
Let \( S = \{1, e^x, e^{-x}, e^{2x}, e^{-2x}\} \) and \( B = \{1, \sinh(x), \cosh(x), \sinh(2x), \cosh(2x)\} \). A linear transformation \( T: V \to V \) is defined by \( T(y) = y'' - 3y' - 4y \).
1. What is the representation matrix of the transformation \( T \) with respect to the bases \( S \) and \( B \)?
2. Giv... | automath |
What is the greatest three-digit number which is a multiple of 13? | nvidia_math |
If a function \( f \) is continuous on the closed unit disk and holomorphic in its interior, and satisfies the conditions \( |f(z)| \leq 1 \) for \( \Re(z) \leq 0 \) and \( |f(z)| \leq 2 \) for \( \Re(z) > 0 \), what is the maximum possible value of \( |f(0)| \)? Is this bound tight? | automath |
Compute the first six terms of the simple continued fraction for \(\sqrt{3}\) and express the result as \( [a_0; a_1, a_2, a_3, a_4, a_5] \). What are the values of \(a_0, a_1, a_2, a_3, a_4, \text{ and } a_5\)? | automath |
In a rectangular coordinate system, if a line passes through the points (-17,-18), (13,22) and (x,2) then what is the value of x?
Answer Choices: (A) -2 (B) -1 (C) 0 (D) 1 (E) 2 | tiger_lab_math |
An amount of Rs.1560 was divided among A, B and C in the ratio 1/2:1/3:1/4. Find the share of C?
Answer Choices: (A) Rs.367 (B) Rs.386 (C) Rs.369 (D) Rs.360 (E) Rs.362 Please write a program to solve it | tiger_lab_math |
What is the value of $\frac{2023^3-2 \cdot 2023^2 \cdot 2024+3 \cdot 2023 \cdot 2024^2-2024^3+1}{2023 \cdot 2024}$? | tiger_math |
In the graph below, no axes or origin is shown. If point B's coordinates are (10,4), which of the following coordinates would most likely be A's?
Answer Choices: (A) (17, -2) (B) (10, 6) (C) (6, 8) (D) (-10, 3) (E) (-2, -17) Let's write a program. | tiger_lab_math |
If a and b are integers and (a*b)^5 = 48y, y could be:
Answer Choices: (A) 8 (B) 16 (C) 162 (D) 144 (E) 100 | tiger_lab_math |
Find all real numbers $a$ for which there exists a function $g : \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x$, $y \in \mathbb{R}$,
\[g(x) + g(y) = g(xy) + axy.\] | nvidia_math |
Given a positive integer \( k \geq 2 \) and a permutation \( \sigma \) of \( \{1, 2, \ldots, k\} \), does there exist a positive integer \( n \) such that the sequence of numbers of distinct prime factors \( \omega(n+\sigma(1)), \omega(n+\sigma(2)), \ldots, \omega(n+\sigma(k)) \) can be arranged in strictly increasing ... | automath |
A set of numbers has the property that for any number t in the set, t + 2 is in the set. If -1 is in the set, which of the following must also be in the set?
I. 1
II. -3
III. 5
Answer Choices: (A) I only (B) II only (C) I and III only (D) II and III only (E) I, II, and III | tiger_lab_math |
The arithmetic mean of four numbers is 15. Two of the numbers are 10 and 18 and the other two are equal. If the product of the two equal numbers is 256, what is the sum of the two equal numbers? | tiger_math |
What are the exact solutions for the system of nonlinear equations given by
\[
\begin{align*}
2 - a_0 &= 0, \\
a_1 &= 0, \\
a_2 x_s^2 + a_1 x_s + a_0 - 3 - x_s &= 0, \\
2 a_2 x_s + a_1 + 1 &= 0
\end{align*}
\]
in terms of \(a_0\), \(a_1\), \(a_2\), and \(x_s\)? | automath |
The lengths of the three sides of a right-angled triangle are consecutive terms of an arithmetic progression. Determine the product of the lengths of the three sides. | nvidia_math |
2 friends a and b running up hill and then to get down length of road - 440 yads a on his return journey met b going up at 20 yards from top if a has finished race 5 minute earlier than b then how much time a had taken to complete the race ? Please respond by writing a program in Python. | tiger_lab_math |
Let $\mathbf{F}$ be a matrix representing a reflection over the $y$-axis, and let $\mathbf{T}$ be a matrix representing a translation by a vector of $\begin{pmatrix} a \\ b \end{pmatrix}$. If
\[\mathbf{T} \mathbf{F} = \begin{pmatrix} -2 & 0 \\ 0 & 3 \end{pmatrix},\]then find $a + b$. | nvidia_math |
A certain population of rabbits and foxes have the following growth rates:
- The population of rabbits grows at a rate of 3 rabbits per month.
- The population of foxes grows at a rate of 2.5 foxes per month.
- For every fox, it kills 1.5 rabbits per month.
If there are initially 20 rabbits and 10 foxes, write a syst... | tiger_lab_math |
Solve for $c$: \[\frac{c-23}{2} = \frac{2c +5}{7}.\] | nvidia_math |
Larry is mixing oil, vinegar, and water to make salad dressing in a ratio of 3:2:1. If he has 8 cups of oil and 7 cups of vinegar, but can use any amount of water, what is the maximum total volume of salad dressing he can make, given that fractional cup measurements are allowed?
(A) 12
(B) 13
(C) 14
(D) 15
(E... | automath |
Suppose the weight of apples in a grocery store follows a normal distribution with a mean of 150 grams and a standard deviation of 20 grams. What is the probability that a randomly selected apple will weigh between 135 grams and 165 grams? | tiger_lab_math |
Let $A = (3, \theta_1)$ and $B = (9, \theta_2)$ in polar coordinates. If $\theta_1 + \theta_2 = \frac{3\pi}{2},$ then find the distance $AB.$ | tiger_math |
Find the fraction that equals $0.\overline{73}$. | nvidia_math |
Jenny is trying to convince her cat to walk on a leash. The cat spends twenty minutes resisting. Then Jenny coaxes the cat to walk 64 feet at a rate of 8 feet/minute. How many minutes does this whole process take? | tiger_math |
A polynomial $P(x)$ with real coefficients has the property that $P(x) = 0$ if and only if $P(1/x) = 0$. Furthermore, if $r$ is a root of $P$, then $|r| = 1$. Suppose that $P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0$, where $a_n \neq 0$. Determine $|a_{n-1}|$. | nvidia_math |
A stock currently trades at $100, with a volatility of 20% per year. The risk-free rate is 5% per year, and the option's time to maturity is 1 year. What is the price of a European call option on this stock with a strike price of $110 based on the Black-Scholes-Merton equation for option pricing with stochastic volatil... | tiger_lab_math |
What is the largest integer $n$ such that exactly half of the positive divisors of $3^n \cdot 5^n$ are multiples of 15? | nvidia_math |
Find 283/1983×100
Answer Choices: (A) 14.29 (B) 14.21 (C) 14.22 (D) 14.27 (E) 14.23 | tiger_lab_math |
In triangle PRQ, PR is a diameter of the semicircle, and it is given that \(\angle PRQ = 25^\circ\). What is the measure of angle PQR? | automath |
In a certain class of 25 students, 10 are right-handed, 18 enjoy reading fiction novels, and 3 are left-handed and do not read fiction novels. How many students in the class are right-handed and enjoy reading fiction novels? Assume that each student is either right-handed or left-handed, but not both. | nvidia_math |
an inspector rejects 0.08 % of the meters as defective , how many meters he examine to reject 2 meteres ? Let's write a Python program to solve it. | tiger_lab_math |
If \( a = \frac{1}{b} \), what is the simplified form of \( a^{\log_b n} \)? | automath |
Given the parameters of a Black-Scholes framework for a European call option with stock price \( S_0 = 100 \), strike price \( K = 130 \), volatility \( \sigma = 1.0 \), time to expiration \( t = 1 \) year, and risk-free interest rate \( r = 0 \), calculate the initial value of the option price \( C \) and the correspo... | automath |
For the function \( f(x) \) defined by its derivative
\[
\frac{d}{dx}f(x)=-e^{\frac{1}{2x+2}}\left(\operatorname{sgn}\left(x\right)+\frac{1-\left|x\right|}{2\left(x+1\right)^2}\right),
\]
determine the intervals on which \( f(x) \) is increasing and decreasing, and find the stationary points of \( f(x) \). Additiona... | automath |
Let X be a space obtained by gluing a square and a triangle along their corresponding edges. Compute the intersection number of the diagonal of the square with the boundary of the triangle using Poincaré duality. | tiger_lab_math |
Emma purchased a new laptop and paid $15\%$ of the cost upfront, which was $\$250$. What was the price of the laptop? | nvidia_math |
Find the sum of the cubes of the coefficients of the expression $2(x^3 - 2x^2 + 5)$ when expanded. | tiger_math |
The slant height of a cone is 15 cm. The volume of the cone is 225π cubic centimeters. What is the height of the cone from the vertex to the center of the base? | tiger_math |
After Nancy bought a pie and sliced it into 8 pieces, she gave 1/2 of the pie to Joe and Darcy and 1/4 of the pie to Carl. How many slices of pie are left? | tiger_math |
What is the probability that the sum of two different single-digit prime numbers selected from the set \{2, 3, 5, 7\} is not a prime number? Express your answer as a simplified fraction. | automath |
The numbers 2, 4, 6, and 8 are a set of four consecutive even numbers. Suppose the sum of five consecutive even numbers is 320. What is the smallest of the five numbers? | nvidia_math |
Giselle is in charge of the relay run on track and field day. Last year, the race was 300 meters. This year, it will be 4 times as long. Giselle needs to set up 6 tables for the relay run. The distance between the tables must be the same and the last table will be at the finish line. What is the distance between table ... | tiger_math |
Bryan has some 3 cent stamps and some 4 cent stamps. What is the least number of stamps he can combine so the value of the stamps is at least 33 cents? | tiger_math |
The sum of the squares of two consecutive integers is 17 times the square of the larger integer. Determine the value of the smaller integer. | nvidia_math |
What is the probability that a college student, who has a 1 in 3.8 chance of winning on each scratch-off ticket, will lose all four of their $2 scratchers? | automath |
Find the number of positive solutions to
$$\cos (\arctan( \sin (\arccos x) ) ) ) = x^2.$$
Note: This problem requires a different approach than the original problem, as it involves different trigonometric functions and a squared variable. The solution will likely involve identities and properties of trigonometric fun... | nvidia_math |
Given a rectangular box with measured dimensions of length \( L = 3 \) cm, width \( W = 4 \) cm, and height \( H = 5 \) cm, each with a maximum measurement error of \( 0.05 \) cm, use differentials to find the maximum possible error in the volume of the box, \( V = L \times W \times H \). What is the approximate value ... | automath |
What is the integrating factor for the differential equation \((x^2y - 2xy^2) \, dx + (x^3 - 3x^2y) \, dy = 0\)? | automath |
How many integers $n$ are there such that $2 \leq n \leq 15$ and $242_n$ is a perfect cube? | tiger_math |
At a birthday party, 30% of the guests are married, 50% are single, and the rest are children. If there are 1000 guests, how many more married people are there than children? | tiger_math |
A customer arrives at a service center at an average rate of one every two minutes, following a Poisson process. What is the probability that at least one customer arrives in a 5-minute interval? | automath |
What are the signs of the first two eigenvectors obtained from the PCA of the first two columns of the iris dataset using the `prcomp()` function, and how do they compare to the signs of the corresponding eigenvectors obtained from the covariance matrix using the `eigen()` function without the `symmetric = FALSE` flag? | automath |
Given the linear equation \( 3x + 4y = 12 \), find the \( x \)-intercept and the \( y \)-intercept. Then, determine the coordinates of an additional point on the line. What are the coordinates of the three points you identified? | automath |
Nancy had some files on her computer. She deleted thirty-one of them and put the rest into folders with six files in each one. Nancy ended up with 2 folders. How many files did Nancy have on her computer initially? | tiger_math |
After offering a group of items at a 30% discount, a merchant further reduces the sale prices by 20% and asserts that the final price is 50% off the original price. What is the positive difference, expressed as a percentage of the original price, between the actual discount and the merchant's claimed discount? | tiger_math |
If $A = 2023 \div 5$, $B = A \div 5$, and $Y = A + B$, then what is the value of $Y$? | tiger_math |
What is the remainder when $3^{200}$ is divided by 7? | nvidia_math |
The quantity $\sqrt{60} - 3\sqrt{5} + \frac{\sqrt{270}}{\sqrt{3}}$ can be expressed as $\sqrt{N}$, where $N$ is an integer. Find $N$. | tiger_math |
Solve the equation:
z^2 - (3 - 4i)z + 13i = 0
where z is a complex number. | tiger_lab_math |
How many 3-letter words can we make from the letters A, B, C, and D, if we are allowed to repeat letters, and we must use the letter A at least once? (Here, a word is an arbitrary sequence of letters.) | tiger_math |
A right circular cylinder has a height of 10 cm and a volume of $250\pi$ cubic centimeters. What is the surface area of the cylinder, in terms of $\pi$? | nvidia_math |
8th Dec 2007 was Saturday, what day of the week was it on 8th Dec, 2006?
Answer Choices: (A) Sunday (B) Tuesday (C) Friday (D) Tuesday (E) Monday | tiger_lab_math |
If the complement of an angle is 5 degrees more than four times the angle, what is the measure of the angle in degrees? | tiger_math |
In a box, there are tiles numbered from 1 to 50. The tiles marked with a number congruent to 2 modulo 5 are blue. If one tile is chosen randomly from the box, what is the probability that the chosen tile is blue? | tiger_math |
Find the sum of all solutions to the equation $(x-6)^2=25$. | nvidia_math |
When Carlos divides his marbles into groups of six, seven, or eight, he has one marble left over. Assuming Carlos has more than one marble, what is the smallest possible number of marbles in Carlos's collection? | tiger_lab_math |
In a certain city, the number of buses is 15 more than twice the number of trains. If there are currently 25 buses, how many trains were added to the city's transportation system yesterday? | nvidia_math |
A grocer purchased 80 kg of rice at Rs. 13.50 per kg and mixed it with 120 kg rice at Rs. 16 per kg. At what rate per kg should he sell the mixture to gain 16%?
Answer Choices: (A) Rs. 19 (B) Rs. 20.5 (C) Rs. 17.4 (D) Rs. 21.6 (E) Rs. 22.6 | tiger_lab_math |
In the new budget, the price of kerosene oil rose by 25%. By how much percent must a person reduce his consumption so that his expenditure on it does not increase? | tiger_math |
A train running at the speed of 90 km/hr crosses a pole in 6 seconds. What is the length of the train?
Answer Choices: (A) 150 (B) 2766 (C) 187 (D) 267 (E) 191 | tiger_lab_math |
Given a hyperbola defined by the equation \(\frac{x^2}{9} - \frac{y^2}{16} = 1\), find the eccentricity of the hyperbola and verify it is greater than 1. What are the coordinates of the foci? | automath |
John starts a TV show. He pays $1000 per episode for the first half of the season. The second half of the season had episodes that cost 120% more expensive. If there are 22 episodes how much did the entire season cost? | tiger_lab_math |
Of the 24 students in class, one-third are in the after-school chess program, and half of those students will be absent from school on Friday for the regional chess tournament. How many of the students in this class are going to that tournament? | tiger_math |
Evaluate the definite integral \( \int_0^1 \sin^2(x) \, dx \). What is the exact value of this integral? | automath |
A bag of dozen apples costs $14 and Brian has already spent $10 on kiwis and half that much on bananas. What's the maximum number of apples Brian can buy if he left his house with only $50 and needs to pay the $3.50 subway fare each way? | tiger_lab_math |
Determine the general term of the sequence defined by the terms \( \frac{2}{3}, \frac{1}{3}, \frac{2}{9}, \frac{1}{6} \). What type of sequence is it (arithmetic, geometric, or harmonic), and can you provide a formula for the \( n \)-th term? | automath |
What is the solution for t in the equation $3 \cdot 3^t + \sqrt{9 \cdot 9^t} = 18$? | tiger_math |
What is the value of \( R(1) \) when we substitute \( \varepsilon = 0.1 \) into the function \( R(x) = \frac{x}{2}\left( 1 + \frac{x}{\sqrt{x^2 + \varepsilon^2}} \right) \)? | automath |
Given the points \( P(2, 0, 2) \), \( Q(3, 1, 4) \), and \( R(1, 3, 2) \), calculate a normal vector to the plane defined by these three points. What are the coordinates of this normal vector? | automath |
Given that angle \( x \) is obtuse and \( \sin{x} = \frac{\sqrt{11}}{6} \), what is the value of \( \cos{x} \)? | automath |
Evaluate the limit:
\[
\lim_{h \to 0} \int_{-1}^{1} \frac{h}{h^2 + x^2} \, dx
\]
and provide the numerical value of the limit. | automath |
What is the number of distinct sequences of 0s and 1s of length 6 that contain exactly three 1s? | automath |
Find the remainder when the expanded form of $(x + 2)(3x^2 - x + 5)$ is divided by $x-1$. | tiger_math |
In the expansion of $(x + 1)^5$, find the ratio of the coefficient of the term with $x^3$ to the coefficient of the term with $x^2$. | nvidia_math |
Consider the linear logic connective ! (called "of course"). Let A be any formula. Find a formula B such that !A → B is true in linear logic. | tiger_lab_math |
Aria has fifteen vases in her dining room, three times the number of vases in the hallway. Her mother brings home 12 more vases she purchased from the antique store. What's the total number of vases in the house? | nvidia_math |
On a quiz, Martin answered x fewer questions correctly than Kelsey, and Kelsey answered eight more questions correctly than Campbell. If Campbell answered 35 questions correctly, how many did Martin answer correctly?
If we know the answer to the above question is 40, what is the value of unknown variable x? | tiger_math |
Find the absolute value of the difference of the solutions of $x^2-5x+5=0$. | tiger_lab_math |
If Greg's PPO algorithm achieved 90% of the maximum possible reward on the CoinRun environment, and the maximum reward for CoinRun is half of the maximum ProcGen reward of 240, what is the reward obtained by Greg's PPO algorithm? | tiger_math |
For given integers \( k, \ell \geq 2 \), determine the smallest integer \( n \) such that any set of at least \( n \) points in the plane contains either at least \( \ell \) collinear points or at least \( k \) pairwise visible points. | automath |
Given the equation of a circle:
$$x^2 - 6x + y^2 - 4y + c = 0$$
determine $c$ such that the radius of the circle is 4.
Write another problem inspired by this one:
Given the equation of a parabola:
$$y^2 - 2x = 0$$
determine the value of $k$ such that the parabola is tangent to the circle:
$$(x - 2)^2 + (y - 3)^2 = k^2... | nvidia_math |
A point (x, y) is reflected about the line y = x. If the image point has coordinates (8, 13), what is the equation of the line that contains the segment connecting the original point and its image? | nvidia_math |
How many ways can you form a board of 31 people from 40 men and 40 women such that there are more women than men on the board? | automath |
Let $x$ be a value such that $8x^2 + 7x - 1 = 0$ and $24x^2+53x-7 = 0.$ What is the value of $x$? Express your answer as a simplified common fraction. | nvidia_math |
How many negative integers $m$ satisfy $300 < m^2 < 1200$? | nvidia_math |
Emily purchased 8 books on Friday. On Saturday, she bought 2 times that quantity. On Sunday, she bought 5 times the quantity she bought on Saturday. What is the total number of books Emily bought over these three days? | nvidia_math |
Two boats A and B start from a buoy in the middle of a river, with boat A traveling downstream and boat B traveling across the river, both moving away for a distance \( d \) and then returning to the buoy. If each boat's speed with respect to still water is twice the stream velocity \( u \), what is the ratio of the to... | automath |
Two similar right triangles have areas of 6 square inches and 150 square inches. The length of the hypotenuse of the smaller triangle is 5 inches. What is the sum of the lengths of the legs of the larger triangle? | nvidia_math |
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