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How many natural numbers between 150 and 300 are divisible by 9? | tiger_math |
Let $f(x)$ and $g(x)$ be two functions defined on the interval $[0,2\pi]$. The graphs of $y=f(x)$ and $y=g(x)$ intersect at exactly two points: $(\pi/2,1)$ and $(3\pi/2,-1)$. Find the maximum possible value of $\int_0^{2\pi} |f(x)-g(x)| dx$. | nvidia_math |
Tom has $15$ boxes of pens to sell at the school store. Each box contains $20$ pens. He wants to package the pens in bags of $5$ pens each and sell them for $2$ dollars per bag. If he already has $30$ bags prepared, how many more bags can he make with the remaining pens? | nvidia_math |
Solve the equation $7^x + 7^{2x} + 7^{3x} + 7^{4x} + 7^{5x} + 7^{6x} = 7^{7x}$ for $x$, given that $\log_7 2$ is a solution. | nvidia_math |
A magic square is an array of numbers in which the sum of the numbers in each row, in each column, and along the two main diagonals are equal. The numbers in the magic square shown are not written in base 10. For what base will this be a magic square?
[asy]
unitsize(0.75cm);
for (int i=0; i<4; ++i) {
draw((0,i)--(3,i)... | nvidia_math |
Let $P$ be the parabola with equation $y^2 + 2x - 6y - 8 = 0$. If $(a, b)$ is the vertex of $P$ and $f$ is the focal length, what is the value of $a + b + f$? | nvidia_math |
Praveen starts business with Rs.3640 and after 5 months, Hari joins with Praveen as his partner. After a year, the profit is divided in the ratio 2 : 3. What is Hari’s contribution in the capital?
Answer Choices: (A) s.7500 (B) s.8000 (C) s.8500 (D) s.9000 (E) s.9360 | tiger_lab_math |
Jenna collects stamps. She puts the same number of stamps on each page and then inserts each page into one of her two stamp books. One of her stamp books has a total of 840 stamps. The other has 1008. What is the largest number of stamps that Jenna could be putting on each page? | tiger_lab_math |
one fourth of one third of two fifth of a number is 10 . what will be 40 % of that number | tiger_math |
What is the period of $y = \tan x + \cot x$? | nvidia_math |
Given the series \( S(x) = 2x - 4x^3 + 6x^5 - 8x^7 + \cdots \) for \( |x| < 1 \), find a closed form for this series. Evaluate your answer for \( x = \frac{1}{2} \) and provide the numerical value. | automath |
What is the value of \( t_{5} \) in the Thue-Morse sequence, and how is it determined based on the binary representation of 5? | automath |
What is the expected number of hits when rolling 70 six-sided dice where a roll of 4 or 5 counts as 1 hit and a roll of 6 counts as 2 hits? Additionally, how does this expected value change if a roll of 3 also counts as 1 hit? | automath |
In triangle \( ABC \), let \( D \), \( E \), and \( F \) be the mid-points of the sides \( BC \), \( CA \), and \( AB \) respectively. If the lengths of the sides are given as \( AB = 6 \text{ cm} \), \( BC = 4.8 \text{ cm} \), and \( CA = 5.6 \text{ cm} \), calculate the perimeter of triangle \( DEF \). | automath |
The price of buying a wooden toy at the new Craftee And Best store is $20, and the cost of buying a hat is $10. If Kendra went to the shop with a $100 bill and bought x wooden toys and three hats, calculate the change she received. The answer is 30. What is the value of unknown variable x? | tiger_math |
The function
$$f(x)=\frac{x^2+ax+b}{x^2-4x+3}$$
can be written as
$$f(x)=\frac{x^2+ax+b}{(x-1)(x-3)}=\frac{A}{x-1}+\frac{B}{x-3}$$
for some constants $A$ and $B$. If $f(x)$ has a horizontal asymptote $y=1$, and $f(2)=4$, determine $A+B$. | nvidia_math |
Find the value of $(32)^{3/5}$. | tiger_math |
Evaluate the limit:
\[
\lim_{x \to 0} \frac{-2x}{\sin x}
\]
and explain the reasoning behind your solution without using L'Hôpital's theorem. | automath |
Solving a linear equation with several occurrences of the variable, solve for w. Simplify answer as much as possible.
(7w + 6)/6 + (9w +8)/2 = 22
Answer Choices: (A) 1 (B) 2 (C) 3 (D) 4 (E) 5 | tiger_lab_math |
In a certain triangle, the medians intersect at a point that divides each median into two segments, one of which is twice as long as the other. The sum of the lengths of the longer segments of the medians is 120 cm. Find the perimeter of the triangle. | nvidia_math |
Emily wants to find all the five-letter words that have the same letter in the second and fourth positions. How many combinations of letters satisfy this property? | nvidia_math |
If a graph \( G \) has \( n \) vertices, with \( n-1 \) of them having odd degree and 1 vertex having even degree, how many vertices in the complement graph \( \overline{G} \) have odd degree? | automath |
A semi truck travels on a paved road for 2 hours at an average speed of 20 miles per hour faster than it travels on a dirt road. The time spent on the dirt is 3 hours. The entire trip is 200 miles. How fast is the truck traveling on the dirt road? | tiger_math |
Consider a tridiagonal matrix \( A \) of size \( n \times n \). Given that such a matrix can have non-zero entries only along its main diagonal and the diagonals immediately above and below it, determine the maximum number of non-zero entries in \( A \). Express your answer as a function of \( n \). | automath |
At a reception, one-third of the guests departed at a certain time. Later two-fifths of the guests departed. Even later two-thirds of the remaining guests departed. If six people were left, how many were originally present at the party?
Answer Choices: (A) 28 (B) 37 (C) 20 (D) 45 people (E) 27 | tiger_lab_math |
A cylindrical can has a surface area of 360 square inches. What is the maximum volume of liquid that can be stored in the can? | nvidia_math |
Given that $a + b = c$, $b + c = 5$, and $c = 3$, what is the value of $a$? | tiger_math |
A mass \( m \) is attached to a string of length \( r \) and swings in a vertical circle under the influence of gravity. If the angle \( \alpha \) that the string makes with the vertical is given by \( \tan \alpha = \frac{r}{h} \), where \( h \) is the height above the lowest point, derive an expression for the angular... | automath |
If the difference between two numbers is 12 and their product is 105, what is the sum of the two numbers? | tiger_math |
Circle $T$ has its center at point $T(-2,6)$. Circle $T$ is reflected across the $y$-axis and then translated 8 units down. What are the coordinates of the image of the center of circle $T$? | tiger_math |
Given three urns: Urn 1 contains 2 white and 8 red balls, Urn 2 contains 8 white and 2 red balls, and Urn 3 contains all 10 white balls. Each urn is selected with equal probability. If an urn is selected and a ball is drawn and replaced, followed by another ball drawn from the same urn, resulting in both balls being wh... | automath |
Given a normal distribution \( N(\mu, \sigma^2) \) and a sample of size \( n \), suppose the sample mean \( \bar{X} \) is used as an estimator for the population mean \( \mu \). If the variance of the population is known to be \( \sigma^2 \), derive the Cramér-Rao lower bound for the variance of any unbiased estimator ... | automath |
Given a cube with side length 6, points M, N, and P are located on its surface, one point per face, and not along any edge or vertex of the cube. The line segment MP is parallel to an edge of the cube, and the triangle MNP has right angles at both M and N. Find the area of triangle MNP. | nvidia_math |
What is the remainder when $15^{43}$ is divided by 4? | tiger_math |
If Mary chops up 18 chairs, 6 tables, and 4 stools to make firewood, and each chair produces 6 sticks of wood, each table produces 9 sticks of wood, and each stool produces 2 sticks of wood, how many hours can she stay warm if she needs to burn 5 sticks of wood per hour? | tiger_math |
Ali, Bella, and Cindy went out to collect berries. They agreed that Bella would collect two times the number of berries Ali collects, and Cindy would collect three times Bella’s quantity. After a full day of collecting, the total amount of berries they collected was 360. How many berries did each person collect? | tiger_math |
How many distinct solutions are there to the equation \( x^3 \equiv -1 \pmod{365} \)? | automath |
In a zoo, a hare is 14 inches tall, while a camel is 24 times taller than the hare. How tall is the camel in feet? | tiger_lab_math |
Given an isosceles triangle with a perimeter of 71 centimeters and one side measuring 22 centimeters, what are the possible lengths of the other two sides? Present your answer including a clear verification of the triangle inequality for each case. | automath |
Roxanne bought 2 cups of lemonade for $2 each and 2 sandwiches for $2.50 each. How much change must she get from a $20 bill? | nvidia_math |
Determine the number of real solutions for $x$ in the equation $$(x - 5x + 12)^2 + 1 = -|x|$$ | tiger_math |
Karl was attempting to calculate economic figures. He found the following equation to be true:\[fp-w=10000\]If $f=5$ and $w=5+125i$, what is $p$? | tiger_math |
The base-10 numbers 217 and 45 are multiplied. The product is then written in base-6. What is the units digit of the base-6 representation? | nvidia_math |
What is the next term in the geometric sequence $$2, 6x, 18x^2, 54x^3, \ldots ?$$ Express your answer in terms of $x$. | nvidia_math |
If Jerome has 20 classmates on his cell phone contact list and half as many out-of-school friends as classmates, as well as his two parents and his sister, how many people are on Jerome's contact list in total? | tiger_math |
Find the last three digits of the following sum:
$$7! + 14! + 21! + \cdots + 120!$$ | nvidia_math |
If Leicester City had a 5000-1 chance of winning the Premier League at the start of the 2015-2016 season, and 12 people placed a £10 bet each, what would be the total amount of money bet on Leicester City if they did win, and how much profit would each of those bettors receive? | automath |
If $4:x^2 = x:16$, what is the value of $x$? | tiger_math |
Let \( B_t \) be a standard Brownian motion, and let \( \rho \) be a correlation coefficient such that between \( B_t \) and another independent Brownian motion \( W_t \) we know that \( [B,W]_t = \rho t \) for all \( t \geq 0 \). What is the value of \( \rho \) when \( B_t \) and \( W_t \) are independent Brownian mot... | automath |
Given the two particle states represented by Dirac kets as $$\lvert\textbf{r}_1\rangle \otimes \lvert\textbf{r}_2 \rangle,$$ how would you express the bra vector corresponding to this state? Specifically, is it correct to state that $$\langle\textbf{r}_1\rvert \otimes \langle\textbf{r}_2\rvert$$ is the proper notation,... | automath |
What are the three distinct cube roots of \(-8\) in the complex number system, and can you provide them in trigonometric form \(r(\cos \theta + i \sin \theta)\)? | automath |
A man purchased 3 blankets @ Rs.100 each, 1 blankets @ Rs.150 each and two blankets at a certain rate which is now slipped off from his memory. But he remembers that the average price of the blankets was Rs.150. Find the unknown rate of two blankets?
Answer Choices: (A) 1050 (B) 2770 (C) 1550 (D) 5102 (E) 5000 | tiger_lab_math |
Using the secant method, perform 3 iterations to approximate a root of the function \( f(x) = x^3 - 4 \) starting with the initial points \( x_0 = 1 \) and \( x_1 = 3 \). What is the value of \( x_2 \) after the third iteration? | automath |
If $x=2$, what is the value of $\frac{x^1\cdot x^2\cdot x^3\cdots x^{15}}{x^2\cdot x^4 \cdot x^6 \cdots x^{20}}$? | tiger_math |
Is
\[f(x) = \frac{1}{2^x - 1} + \frac{1}{2}\]an even function, odd function, or neither?
Enter "odd", "even", or "neither". | nvidia_math |
The average weight of A, B and C is 45 kg. If the average weight of A and B be 40 kg and that of B and C be 43 kg, then the weight of B is?
Answer Choices: (A) 65 kg (B) 26 kg (C) 16 kg (D) 31 kg (E) 18 kg | tiger_lab_math |
what is the remainder if 7 ^ 8 is divided by 100 ? Let's write a Python program. | tiger_lab_math |
Solve for $y$:
$$y^2 - 2y - 5 = (y - 3)(y + 2).$$ | nvidia_math |
What is the time complexity of calculating the n-th Fibonacci number using the naive recursive method, and how does the recursion tree structure relate to this complexity? | automath |
Don and Rachel have a different bottle of milk that is being filled at a rate of 1/10 gallons per minute. However, there is a leak in the bottle that causes it to lose 1/20 gallons per minute. If they start with an empty bottle, how many minutes will it take to fill the bottle with 1/2 gallon of milk? | nvidia_math |
Given the function \( f(x) = -3x + \frac{1}{x} - 3 \), simplify the expression
\[
\frac{f(4+h) - f(4)}{h}
\]
and evaluate the limit as \( h \) approaches 0. What is the value of this limit? | automath |
Compute the exact value of the expression $\left| 3\pi - | 3\pi - 11 | \right|$. Write your answer using only integers and $\pi$, without any absolute value signs. | nvidia_math |
If $x^2 + bx + c$ is the square of a binomial for some constants $b$ and $c$, and $b = 10$, what is the value of $c$? | tiger_math |
For integers \(a\), \(b\), and \(c\) with absolute values not exceeding \(10\), determine the number of ordered triples \((a,b,c)\) such that there exists an integer \(d\) making the polynomial \(g(x) = x^2 + cx + d\) have \(g(f(2)) = g(f(4)) = 0\), where \(f(x) = x^2 + ax + b\). | automath |
Molly had 14 candles on her birthday cake. She grew older and got 6 more on her birthday cake. Molly is _____ years old now. | tiger_math |
In a system where two elastic balls collide, the potential energy \( V(r) \) is related to the separation \( r \) between their centers. Given that the force \( F \) acting between the balls is defined as \( F = -\frac{dV}{dr} \), describe the relationship between potential energy and separation during the collision. S... | automath |
Find $f(x)$, with terms in order of increasing degree, if $x^2 + 2x - 3 = f(x) + 2x^3 - x + 4$. | nvidia_math |
Express 60 mps in kmph. | tiger_math |
Let $p(x)$ be a monic polynomial of degree 4, such that $p(1) = 18,$ $p(2) = 34,$ and $p(3) = 52.$ Find $p(0) + p(4).$ | tiger_math |
If the total number of employees in a company is 450 and each employee earns $2000 per month, and the company laid off 1/3 of its employees due to tough economic times, what is the total amount of money that the company paid to the remaining employees? | tiger_math |
Kaitlyn, Sophia, and Ava participated in a math competition. Kaitlyn scored 1.5 times as many points as Sophia, and Sophia scored 2/3 times as many points as Ava. If Ava scored 120 points, calculate the average points scored by the three. | nvidia_math |
Given the recursive function defined by:
\[
f_n =
\begin{cases}
0 & \text{if } n \leq 1 \\
1 & \text{if } n = 2 \\
f_{n-1} + 8n - 18 & \text{if } n > 2
\end{cases}
\]
what is the closed formula for \( f_n \) in terms of \( n \)? Specifically, can you find \( f_{10} \)? | automath |
Consider a polynomial function $g(x)$ defined as $g(x) = x^5 - 3x^4 + 2x^3 - x^2 + 4x - 1$. If $g(5) = 641$, find the value of $g(-3)$. | nvidia_math |
Find the phase shift of the graph of $y = 3 \sin \left( x - \frac{\pi}{5} \right).$ | nvidia_math |
In the arithmetic sequence $3$, $7$, $11$, Y$, $19$, $\ldots$, what is the value of the unknown variable Y? | tiger_math |
Four car rental agencies A, B, C and D rented a plot for parking their cars during the night. A parked 15 cars for 12 days, B parked 12 cars for 20 days, C parked 18 cars for 18 days and D parked 16 cars for 15 days. If A paid Rs. 1395 as rent for parking his cars, what is the total rent paid by all the four agencies?
... | tiger_lab_math |
What is the topological space formed by the product of the two topological spaces \( \mathbb{R}^2 \) and the discrete space \( \{1, 2\} \), and how can we describe its open sets? | automath |
Given a pendulum with a length \( l \) and experiencing a gravitational acceleration \( g \), if the initial angle \( \theta_0 \) is \( \frac{\pi}{4} \) radians and the initial angular velocity \( \dot{\theta}_0 \) is \( 0 \) radians per second, what is the total mechanical energy \( E_0 \) of the pendulum? Use the for... | automath |
Given a quadratic Bézier curve defined by control points \( P_0, P_1, P_2 \) and the De Casteljau algorithm, if we modify the algorithm to use different weights for the linear interpolations (for instance, \( t = 0.3 \) for \( P_0 \) and \( P_1 \), \( t = 0.6 \) for \( P_1 \) and \( P_2 \), and \( t = 0.4 \) for the fi... | automath |
Let $\mathbf{a} = \begin{pmatrix} 3 \\ 2 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 1 \\ 3 \end{pmatrix}.$ Find the area of the triangle with vertices $\mathbf{0},$ $\mathbf{a},$ and $\mathbf{2b}.$ | tiger_math |
Consider a graph G with 7 vertices and the following adjacency matrix:
$$
\begin{bmatrix}
0 & 1 & 0 & 1 & 0 & 1 & 1 \\
1 & 0 & 0 & 1 & 1 & 0 & 0 \\
0 & 0 & 0 & 1 & 1 & 1 & 0 \\
1 & 1 & 1 & 0 & 0 & 0 & 1 \\
0 & 1 & 1 & 0 & 0 & 1 & 1 \\
1 & 0 & 1 & 0 & 1 & 0 & 0 \\
1 & 0 & 0 & 1 & 1 & 0 & 0 \\
\end{bmatrix}
$$
What is ... | tiger_lab_math |
Let $b_1,b_2,\dots,b_{100}$ be integers such that for any $i$ we have $|b_i|\leq 2$, and for any $i\neq j$ we have $|b_i+b_j|\leq 2$. Find the largest possible value of $$\sum_{1\leq i<j\leq 100} b_ib_j.$$ | nvidia_math |
Given a covering of a space \( X \) by \( n \) open subsets \( \{ U_i \}_{i=1}^n \), suppose we have \( n \) objects \( E_i \) in a derived category with the property that their restrictions satisfy cocycle conditions. If the sheaves \( \mathop{\mathcal{E}\! \mathit{xt}}\nolimits^p_{\mathcal{O}_{U_i}}(E_i, E_i) = 0 \) ... | automath |
The difference of and its reciprocal is equal to :
Answer Choices: (A) 105/302 (B) 305/309 (C) 105/312 (D) 105/304 (E) 105/309 | tiger_lab_math |
A cube of edge 12 cm is immersed completely in a rectangular vessel containing water. If the dimensions of the base of the vessel are 20 cm * 15 cm, find the rise in water level? | tiger_math |
How many distinct values can be obtained by changing the order of exponentiation in the expression \( 2^{2^{2^{2}}} \)? | automath |
Find the volume of the solid of revolution generated by rotating the region enclosed by the equation $x^2 + y^2 + 10x + 24y = 0$ about the y-axis. | tiger_math |
A regular hexagon has six diagonals, and a regular octagon has twenty diagonals. How many diagonals does a regular dodecagon have? | tiger_math |
John bought 2 shares and sold them for $75 each. If he had a profit of 20% on the sale of one of the shares but a loss of 20% on the sale of the other share, then on the sale of both shares John had
Answer Choices: (A) a profit of $10 (B) a profit of $6.25 (C) a loss of $6.25 (D) a loss of $10 (E) neither a profit nor ... | tiger_lab_math |
This week, the fishmonger sold 50 kg of salmon. He sold three times more the following week. What was the total amount of fish sold in two weeks? | tiger_math |
"statement 1": Molly can do a task in 18 days while Sandy can do the task in 12 days, "statement 2" :Molly can do a task in less than 68 days while Sandy can do the task in 12 days, "options: " Entailment or contradiction or neutral? Please write a program to solve it | tiger_lab_math |
The figure shows a regular hexagon inscribed in a circle of radius $10$ units. The shaded region is formed by removing six congruent isosceles triangles from the hexagon. If the height of each triangle is $2$ units, what is the area of the shaded region in square units? Express your answer in terms of $\pi$.
[asy]
imp... | nvidia_math |
Find the modular inverse of 7 modulo 42, within the range of 0 to 41. | tiger_math |
In a circuit with resistors of 2 Ω each and a 10 V battery with an internal resistance of 1/6 Ω, what is the equivalent resistance of the network between points A and C? | automath |
To fill an order, the factory dyed 72,315 yards of silk green, 54,215 yards pink, 63,480 yards blue, and 81,990 yards yellow. The order required that 40% of the silk should be green, 30% pink, 20% blue, and 10% yellow. However, during the dyeing process, there was 7% wastage for green silk, 5% wastage for pink silk, 3%... | tiger_math |
A man can row 7.5 kmph in still water and he finds that it takes him twice as long to row up as to row down the river. Find the rate of stream.
Answer Choices: (A) 2.5 km/hr (B) 12.5 km/hr (C) 1 2.6 km/hr (D) 22.5 km/hr (E) 22.6 km/hr | tiger_lab_math |
Maria can make three scarves out of ten balls of yarn. How many scarves can she make out of forty balls of yarn? | nvidia_math |
What is the limit of the Lebesgue measure of the set \( A_n \) as \( n \) approaches infinity? | automath |
What is the total of all the positive divisors of 18? | tiger_math |
If \( f(x) = 2x + 3 \), what is the value of \( e^{\ln(f(1))} \)? | automath |
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