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Trapezoid JKLM in the x-y plane has coordinates J = (-3, -4), K = (-3, 1), L = (5, 7), and M = (5, -4). What is its perimeter? | tiger_math |
The average of two numbers is P. If one number is Q, then the other number is
Answer Choices: (A) 2Q (B) 2P (C) P - Q (D) 2P-Q (E) None of these | tiger_lab_math |
Donna worked 2 hours every morning walking dogs in her neighborhood for $10.00 an hour. 5 days a week, after school, she worked at a card shop for 2 hours and made a certain amount per hour. On the weekends, she usually made $10.00 an hour babysitting and was guaranteed 4 hours every Saturday from her neighbor. She mad... | tiger_math |
What is the expected value of a random variable \(X\) that has a normal distribution with mean \(m = 5\) and variance \(\sigma^2 = 4\)? | automath |
Compute the mean translational kinetic energy of a mole of ideal gas in J, both at room temperature 293 K. | tiger_lab_math |
Yan is somewhere between his home and the stadium. To get to the stadium, he can walk directly to the stadium, or else he can walk home and then ride his bicycle to the stadium. He rides 5 times as fast as he walks, and both choices require the same amount of time. What is the ratio of Yan's distance from his home to h... | tiger_math |
Find the number of distinct positive divisors of 81. | nvidia_math |
Passage: Conlan umpired in the National League from 1941 to 1965, officiating in five World Series (1945 World Series, 1950 World Series, 1954 World Series, 1957 World Series and 1961 World Series) and six Major League Baseball All-Star Game (1943, 1947, 1950, 1953, 1958 and the first 1962 contest). He also umpired in ... | tiger_lab_math |
The quadratic $x^2+300x+450$ can be written in the form $(x+d)^2+e$, where $d$ and $e$ are constants. What is $\frac{e}{d}$? | nvidia_math |
Each of the numbers $a_1,$ $a_2,$ $\dots,$ $a_{95}$ is $\pm 1.$ Find the smallest possible positive value of
\[\sum_{1 \le i < j \le 95} a_i a_j.\] | nvidia_math |
If Amy has 5 chocolate bars, 7 times as many M&Ms as chocolate bars, and 6 times as many marshmallows as M&Ms, and she plans to fill each basket with 10 candies, how many baskets will she be able to fill with the candies she has collected? | tiger_math |
A company has 150 employees, and 30% of them are managers. If 20% of the managers are in charge of international projects, how many managers are in charge of domestic projects? | nvidia_math |
Ben works 8-hour shifts in a furniture shop. It takes him 5 hours to build 1 rocking chair. How many chairs can he build in 10 days? | tiger_lab_math |
1/2, 1/4, 1/8, 1/16, 1/32, 1/64, ....
In the sequence above each term after after the first one-half the previous term. If x is the tenth term of the sequence, then x satisfies which of the following inequalities?
Answer Choices: (A) 0.1 < x < 1 (B) 0.01 < x < 0.1 (C) 0.001 < x < 0.01 (D) 0.0001 < x < 0.001 (E) 0.00001... | tiger_lab_math |
Find the difference between $3.75 and $2.01$. | tiger_math |
In the town of Mathville, the local baker and the local florist had a promotional idea. They decided to provide gift packs for a special event. Each pack contains either cookies or flowers. For each gift pack with cookies, the baker earns $4 and the florist $1. For each gift pack with flowers, the baker earns $1 and th... | tiger_math |
It starts raining at 7:00 and pours heavily until its stops at 17:00 on a particular day. On the second day, the rain takes 2 more hours than it took on the first day to stop. On the third day, the rain pours for twice the amount of time it took on the second day. Calculate the total time it was raining in the three d... | tiger_lab_math |
If $n = 2^{10} \cdot 3^{14} \cdot 5^{8}$, how many of the natural-number factors of $n$ are multiples of 150? | tiger_lab_math |
the h . c . f of two numbers is 23 and the other two factors of their l . c . m are 16 and 17 . the larger of the two numbers is : | tiger_math |
The length of a rectangle is given as $4x-15$ meters and its width as $20-3x$ meters. If the length and width are equal, what is the area of the rectangle in square meters? | tiger_math |
Our water polo team has 15 members. I want to choose a starting team consisting of 7 players, one of whom will be the goalie (the other six positions are interchangeable, so the order in which they are chosen doesn't matter). In how many ways can I choose my starting team? | nvidia_math |
Consider the following set \( A \subset \mathbb{R} \) defined as \( A = \{ x \in \mathbb{R} : \text{ there exists a continuous function } f : [0, 1] \to \mathbb{R} \text{ such that } f(t) = x \text{ for some } t \in [0, 1] \} \).
If \( A \) is compact, what is the dimension of the cone \( CA = A \times [0, 1]/(A \tim... | automath |
Let $a,$ $b,$ $c$ be distinct complex numbers such that
\[\frac{a^2}{b - c} + \frac{b^2}{c - a} + \frac{c^2}{a - b} = 0.\]Find all possible values of
\[\frac{a^3}{(b - c)^2} + \frac{b^3}{(c - a)^2} + \frac{c^3}{(a - b)^2}.\]Enter all the possible values, separated by commas. | nvidia_math |
The coefficients of the polynomial
\[a_{10} x^{10} + a_9 x^9 + a_8 x^8 + \dots + a_2 x^2 + a_1 x + a_0 = 0\]are all integers, and its roots $r_1,$ $r_2,$ $\dots,$ $r_{10}$ are all integers. Furthermore, the roots of the polynomial
\[a_0 x^{10} + a_1 x^9 + a_2 x^8 + \dots + a_8 x^2 + a_9 x + a_{10} = 0\]are also $r_1,$... | nvidia_math |
If 5 women or 8 girls can do a work in 84 days.In how many days can 10 women and 5 girls can do the same work?
Answer Choices: (A) 32 (B) 33 (C) 34 (D) 35 (E) cannot be determined | tiger_lab_math |
Given a cubic polynomial \( p(x) = ax^3 + bx^2 + cx + d \), where \( a, b, c, d \) are real coefficients, what is the condition on the coefficients that ensures the function \( p \colon \mathbb{R} \to \mathbb{R} \) is injective? What value must the discriminant of the derivative polynomial \( p'(x) = 3ax^2 + 2bx + c \)... | automath |
Given the complex number \( z = a + bi \), where \( a \) and \( b \) are real numbers, determine the number of distinct square roots of \( z \) in the complex plane. If \( a \) and \( b \) are allowed to take any real values, express your answer in the format of a definitive quantity. | automath |
a train crosses a platform of 160 m in 15 sec , same train crosses another platform of length 250 m in 20 sec . then find the length of the train ? Let's write a program. | tiger_lab_math |
Calculate the mean and standard deviation of a binomial probability distribution with parameters \( n = 10 \) and \( p = \frac{2}{5} \). Provide your answers in fractional form. | automath |
In a Poisson process with jump intensity \( h \), the probability of at least one jump occurring in the interval \([t, t + \Delta t]\) is approximated by \( h \Delta t + h^3 \Delta t^3 \). If we consider a very small interval where \( \Delta t \) approaches 0, what is the limiting form of the probability density, and h... | automath |
At a shop in Japan, women's T-shirts are sold every 30 minutes for $18, and men's T-shirts are sold every 40 minutes for $15. This shop is open from 10 am until 10 pm. How much does the shop earn selling T-shirts per week? | tiger_lab_math |
Danny loves collecting cards from a popular game. He trades them regularly with his friends to try and complete his collection. This week, Danny traded with three different friends. From the first friend, he received 4 cards for every 3 cards he gave away. From the second friend, he received 1 card for every 2 cards he... | tiger_math |
Debby and Maggie agreed to share a sum of money so that Debby takes x% and Maggie takes the rest. If Maggie's share is $4,500, They shared 6000. What is the value of unknown variable x? What is the value of unknown variable x? | tiger_math |
Consider two samples drawn from a uniform distribution on the integers from 1 to 10, with each sample containing 1,000 observations. Sample A results in the following counts for each number:
- 1: 98
- 2: 105
- 3: 97
- 4: 102
- 5: 94
- 6: 101
- 7: 99
- 8: 100
- 9: 99
- 10: 106
Sample B results in the following counts ... | automath |
Two complex numbers are chosen at random from the set $\{1+2i, -1+i, 2-i, -2+3i\}$. What is the probability that the distance between the chosen points is $\sqrt{5}$? | tiger_math |
A car is 15 minutes late by running at a speed of 4/5th of its actual speed. The actual time taken by the car to cover that distance is:
Answer Choices: (A) 30 minutes (B) 1 hour (C) 2 hours (D) 3 hours (E) None of these | tiger_lab_math |
Viola is planning to distribute apples, bananas, and carrots to children at a neighborhood party. She has arranged the fruits and vegetables such that there are 10 more apples than bananas, and the number of carrots is three times the number of apples. If the total number of apples, bananas, and carrots Viola has is 12... | tiger_math |
Consider the following undirected graph:
```
A --- B --- F G
| | |
C --- D --- E H
```
Identify if there are any cut vertices and bridges in this graph. | tiger_lab_math |
Express as a common fraction: $\sqrt{0.\overline5}+\sqrt{0.\overline1}-\sqrt{0.\overline3}$ | nvidia_math |
Lena creates her own granola mix with oats and nuts in a specific ratio. She uses three ounces of nuts for every five ounces of oats. Her latest batch of granola weighed 24 ounces. How many ounces of nuts did she use? | nvidia_math |
Let $a$ and $b$ be positive real numbers such that each of the equations $x^2 + ax + 2b = 0$ and $x^2 + 2bx + a = 0$ has real roots. Find the smallest possible value of $a + b.$ | tiger_lab_math |
Is there an isomorphism between the semigroups \( S(4) \), which is the set of all functions from the set \( \{1, 2, 3, 4\} \) to itself under composition, and \( \mathbb{Z}_{256} \), the set of integers from \( 0 \) to \( 255 \) under multiplication modulo \( 256 \)? Justify your answer by considering the properties o... | automath |
A building has 10 floors. It takes 15 seconds to go up the stairs to the even-numbered floors and 9 seconds to go up to the odd-numbered floors. This includes getting to the first floor. How many minutes does it take to get to the 10th floor? Please write a program to solve it | tiger_lab_math |
If the areas of three different faces of a rectangular solid are 6, 8, and 12 square units, what is the volume of the solid in cubic units? | automath |
A pizza parlor offers six toppings. What is the greatest number of four-topping pizzas that can be made such that no two pizzas have the same topping combination? | nvidia_math |
John has $2500 in his bank account that earns 10% interest every year. How much will be in John's bank account after one year?
Answer Choices: (A) $2600 (B) $3000 (C) $2250 (D) $2750 (E) $3500 | tiger_lab_math |
"statement 1": Sandy can do a job in 21 days and Molly can do the same job in 42 days, "statement 2" :Sandy can do a job in less than 41 days and Molly can do the same job in 42 days, "options: " Entailment or contradiction or neutral? Let's write a program. | tiger_lab_math |
In a standard deck of 52 playing cards, if 2 red cards are drawn, what is the probability that the next 3 cards drawn are also red? Express your answer as a simplified fraction. | automath |
Given the polynomial \( p(x) = 4x^5 + x^3 + 2x^2 \), what is the leading coefficient of \( p(x) \)? | automath |
Every day, Billie bakes 3 pumpkin pies for 11 days and puts them in the refrigerator. It takes 2 cans of whipped cream to cover 1 pie. If Tiffany comes over and eats x pies, how many cans of whipped cream does Billie need to buy to cover the remaining pies?
If we know the answer to the above question is 58, what is t... | tiger_math |
A bakery sells a total of 250 loaves of bread per day. They offer a 10% discount for customers who buy 10 or more loaves at once. If the average customer buys 2 loaves, and 10% of customers take advantage of the discount, how many loaves of bread are sold at the discounted price? | nvidia_math |
How many positive integers less than 200 are both a perfect square and a perfect fourth power? | nvidia_math |
A tetrahedron has three of its edges of lengths 4, 5, and 6 cm. Determine the length of the fourth edge, if it is known that the opposite faces are mutually perpendicular to each other. | nvidia_math |
If Teresa has 14 colored pencils and 35 black pencils, and her dad asks her to share all her pencils equally among her three younger siblings, while allowing her to keep 10 pencils for herself, how many pencils does each sibling receive? | tiger_math |
What is the arithmetic mean of the integers from -4 through 5, inclusive? Express your answer as a decimal to the nearest tenth. | tiger_math |
What can be concluded about the behavior of a non-negative harmonic function \( u: \mathbb{R}^n \to \mathbb{R} \) as its value at the origin approaches zero? Specifically, if \( u(0) = 0 \), what is the limit of \( u(x) \) as \( \|x\| \to \infty \)? | automath |
A rectangular prism has its length, width, and height all multiplied by a factor of 4. If the volume of the original prism was 24 cubic meters, what is the volume of the new prism, in cubic meters? | tiger_math |
Let $x_1 > y_1 > 0$ be two positive real numbers. Define the sequences $(x_n)$ and $(y_n)$ recursively by
$$x_{n+1} = \frac{x_n + y_n}{2}, \quad y_{n+1} = \frac{2x_ny_n}{x_n + y_n}.$$
What is the limit of the sequences $(x_n)$ and $(y_n)$ as $n \to \infty$ in terms of $x_1$ and $y_1$? | automath |
Margo walks to her friend's house in 10 minutes. Using the same route, it takes Margo 15 minutes to get back home. If her average walking rate for the entire trip is 3 miles per hour, how many total miles did she walk? | tiger_math |
If the sum of five numbers is 20.5, what is the average of the five numbers? | tiger_math |
What is the length of a side of an equilateral triangle if its altitude measures 24? Additionally, what would be the altitude of an equilateral triangle with a perimeter of 45? | automath |
What is the minimum possible cost of a path that visits each vertex in a weighted complete graph \( K_n \) exactly once, considering both the direct edges between adjacent vertices and the edges connecting a vertex to the vertex two places ahead in the path, for a given set of edge weights \( w_{ij} \)? | automath |
Consider the following parametric equations:
\[x(t) = 2t^2\]
\[y(t) = t^3 - 1\]
Which of the following curves is described by these equations?
(A) Circle
(B) Parabola
(C) Ellipse
(D) Hyperbola
(E) Cubic curve | nvidia_math |
If James trains twice a day for 4 hours each time, excluding 2 days per week, how many hours does he train in a year? | tiger_math |
If Rs.450 amount to Rs.540 in 4 years, what will it amount to in 6 years at the same rate % per annum?
Answer Choices: (A) 227 (B) 299 (C) 270 (D) 585 (E) 271 | tiger_lab_math |
What is the exact area of the triangular region in the first quadrant that is bounded above by the curve \( y = e^{2x} \), below by \( y = e^x \), and on the right by the line \( x = \ln(3) \)? | automath |
If it takes Rob 1 hour to get to the national park and it takes Mark three times as long, at what time should Mark leave his home if Rob leaves his home at 11 a.m. so that they both arrive at the national park at the same time? | tiger_math |
The greatest possible length which can be used to measure exactly the length 7m ,3m 85cm, 12m 95 cm is?
Answer Choices: (A) 25cm (B) 101cm (C) 56cm (D) 42cm (E) 35cm | tiger_lab_math |
John and Jane went out for a dinner and they ordered the same dish. Both used a 10% discount coupon. John paid a 15% tip over the original price of the dish, while Jane paid the tip over the discounted price for the coupon. If John paid $0.63 more than Jane, what was the original price of the dish?
Answer Choices: (A) ... | tiger_lab_math |
Let $x$ and $y$ be positive real numbers such that $\frac{1}{x} + \frac{1}{y} = 2$. Find the maximum value of $x + y$. | nvidia_math |
When Rahul was born, his father was 35 years older than his brother and his mother was 25 years older than his sister. If Rahul's brother is 5 years older than him and his mother is 3 years younger than his father, how old was Rahul's sister when he was born ?
Answer Choices: (A) 11 years (B) 10 years (C) 12 years (D) ... | tiger_lab_math |
What is the molecular weight of 4 moles of a certain compound if the molecular weight is 1188? | tiger_math |
What is the diameter of a complete graph K8?
(Note: A complete graph K8 is a graph with 8 vertices, where each vertex is connected to every other vertex.) | tiger_lab_math |
the cost price of 20 articles is the same as the selling price of x articles . if the profit is 25 % , find out the value of x ? Let's write a Python program to solve it. | tiger_lab_math |
The Lebesgue measure of the set of all real numbers $x$ such that $x^2 < \infty$ can be represented as an infinite series. Find the sum of the series $1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots$. Express your answer as a real number. | nvidia_math |
A bookstore has 5 shelves, each containing 3 books of fiction and 2 books of non-fiction. If the store owner wants to display a selection of 3 fiction books and 2 non-fiction books in the store window, how many different selections can be made? | nvidia_math |
Given the metrics for the flat spacetime described by \(\eta_{\mu\nu} = \mathop{\mathrm{diag}}(1, -1, -1, -1)\) and the definition of the D'Alembert operator \(\Box = -\eta^{\mu\nu} \partial_\mu \partial_\nu\), determine the explicit form of the D'Alembert operator in Cartesian coordinates. Specifically, what is the va... | automath |
If a rancher initially had 340 head of cattle and planned to sell them all for $204,000, but 172 of them fell sick and died, causing his customers to lose confidence in his cattle and forcing him to lower the price by $150 per head for the remaining cattle, what is the total amount of money the rancher would lose by se... | tiger_math |
Given the hyperbola defined by the equation \( 7x^2 - 3y^2 - 25 = 0 \), find the parametric equations that represent one of its branches in the form \( x(t) \) and \( y(t) \). Additionally, what are the specific values of the parameters that constitute each branch of the hyperbola? | automath |
The longest side of a right triangle is 8 meters and the shortest side is 6 meters. What is the area of the triangle in square meters? | nvidia_math |
What is the value of the expression $$\frac{2}{3}\times9\times\frac{3}{8}\times24\times\frac{4}{27}\times81\times\frac{5}{72}\times216\times\frac{6}{216}\times648?$$ | tiger_math |
A leprechaun is hiding at the end of a rainbow that is 100 meters away from a pot of gold. The leprechaun can move 20% of the remaining distance towards the pot each minute. However, every third minute, a mischievous fairy pushes the leprechaun back 10 meters. How far is the leprechaun from the pot of gold after 10 min... | nvidia_math |
Each of the nine dots in this figure is to be colored red, white or blue. No two dots connected by a segment (with no other dots between) may be the same color. If the dots on the boundary of the figure are required to be red, how many ways are there to color the remaining dots?
[asy]
draw((-75,0)--(-45,0)--(-60,26)--... | tiger_math |
Let $B$ be the set of all numbers which can be represented as the sum of five consecutive positive odd integers. What is the least common multiple of all numbers in $B$? | nvidia_math |
For a certain exam,a score of 58 was 2 standard deviations below mean and a score of 98 was 3 standard deviations above mean.What was the mean score Q for the exam?
Answer Choices: (A) 74 (B) 76 (C) 78 (D) 80 (E) 82 Let's write a Python program. | tiger_lab_math |
What are the integer coefficients \( a_n \) and \( b_n \) in the Machin-like formula for \( \frac{\pi}{4} \) represented by \( \frac{\pi}{4} = 12 \arctan\left(\frac{1}{49}\right) + 32 \arctan\left(\frac{1}{57}\right) - 5 \arctan\left(\frac{1}{239}\right) + 12 \arctan\left(\frac{1}{110443}\right) \)? | automath |
What is the largest integer $x$ such that $\frac{x}{4}+\frac{3}{5} < \frac{7}{4}$? | tiger_math |
In a box containing 4 red balls and 3 yellow balls, three balls are drawn without replacement. What is the exact probability that exactly one ball of each color is selected? | automath |
In the context of Hall's theorem, consider \( n = 5 \) sets \( A_1, A_2, A_3, A_4, A_5 \) where each set \( A_i \) has been truncated to subsets \( B_i \) such that \( |B_i| = 2 \) for all \( i \), and it is no longer possible to remove elements from these subsets without violating the Hall condition.
If we denote th... | automath |
A rectangular lawn of dimensions 70 m * 60 m has two roads each 10 m wide running in the middle of the lawn, one parallel to the length and the other parallel to the breadth. What is the cost of traveling the two roads at Rs.3 per sq m?
Answer Choices: (A) 2288 (B) 2779 (C) 2779 (D) 3900 (E) 3600 Let's write a program. | tiger_lab_math |
Given a twice differentiable function \( f: \mathbb{R}^n \rightarrow \mathbb{R} \) that is \( l \)-smooth and \( \alpha \)-strongly convex, derive the regularity condition \( \langle \nabla f(x), x - x^* \rangle \geq \frac{\alpha}{2} \| x - x^* \|^2 + \frac{1}{2\beta} \|\nabla f(x)\|^2 \) for any \( x \in \mathbb{R}^n ... | automath |
A sphere has a radius of 5 cm. Find the volume of the sphere. | tiger_lab_math |
What is the remainder when the sum of all the terms in a $\frac{1}{2008}$ -array, which is a structured, infinite collection of numbers, is divided by 2008? The first entry of each row is $\frac{1}{2p}$ times the first entry of the previous row, and each succeeding term in a row is $\frac{1}{p}$ times the previous term... | tiger_math |
The total marks obtained by a student in Mathematics and Physics is 60 and his score in Chemistry is 20 marks more than that in Physics. Find the average marks scored in Mathamatics and Chemistry together.
Answer Choices: (A) 40 (B) 99 (C) 77 (D) 66 (E) 12 | tiger_lab_math |
Let the roots of the cubic equation $2x^3 - x^2 + 5x + 1 = 0$ be $r_1, r_2,$ and $r_3$. Find the sum of the reciprocals of these roots, $\frac{1}{r_1} + \frac{1}{r_2} + \frac{1}{r_3}$. | nvidia_math |
If Julia wants to purchase 4 rock and roll CDs priced at $5 each, 4 pop CDs priced at $10 each, 4 dance CDs priced at $3 each, and 4 country CDs priced at $7 each, but she only has $75, how much money does she still need to buy all the CDs? | tiger_math |
A card shop contains 5 birthday cards, 4 holiday cards, and 3 graduation cards. If three cards are purchased at random from the shop, what is the probability that the three cards will be graduation cards?
Answer Choices: (A) 1/33 (B) 6/264 (C) 1/44 (D) 1/264 (E) 1/1320 | tiger_lab_math |
Given the point \( P(2at, at^2) \) on the parabola defined by the equation \( x^2 = 4ay \), where the normal at \( P \) is given by \( x + ty = at^3 + 2at \), find the coordinates of the point \( Q \) on the same parabola such that the normal at \( Q \) is perpendicular to the normal at \( P \). What are the coordinate... | automath |
Little John had $5.10. He spent $1.05 on sweets and gave to his two friends $1.00 each. How much money was left?
Answer Choices: (A) $2.15 (B) $2.05 (C) $2.45 (D) $2.25 (E) $2.85 | tiger_lab_math |
Sarah is planning to do some baking. She buys 5 pounds of rye flour, some whole-wheat bread flour, and 3 pounds of chickpea flour. Sarah already had 2 pounds of whole-wheat pastry flour at home. She now has 20 pounds of flour. How many pounds of whole-wheat bread flour did she buy? | tiger_math |
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