instruction_seed stringlengths 17 6.7k | source stringclasses 4
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Find the function $g(x)$ such that $g(x) + 2x^3 - 5x^2 + 1 = x^4 - 3x^2 - 2$, given that $g(x)$ has only even-degree terms. | nvidia_math |
Given points \( P \) and \( Q \) that are 4 units apart on a horizontal line, and a line \( L \) that touches a circle centered at the midpoint of \( P \) and \( Q \) with a radius of 1, determine the distance from \( P \) to line \( L \) if \( PM + 3QN = 4 \). What is the angle \( \theta \) formed between segment \( P... | automath |
To express 20 as a sum of different powers of 2, we would write $20 = 2^4 + 2^2$. The sum of the exponents of these powers is $4 + 2 = 6$. If 500 were expressed as a sum of at least two distinct powers of 2, what would be the least possible sum of the exponents of these powers? | tiger_math |
Determine the number of roots of unity that are also roots of $z^3 + az^2 + bz + c = 0$ for some integers $a$, $b$, and $c$. | nvidia_math |
What is the minimum number of theorem schemas listed in the text on normal modal logic that are needed to derive the schema \(\square(A \land B) \to \square A \land \square B\) using only the rules of modus ponens and necessitation? | automath |
A dog chases a cat 3 hours after the cat runs. dog takes 6 hours to reach the cat. If the average speed of the dog is 72 kmph, what s the average speed of the cat?
Answer Choices: (A) 48kmph (B) 47kmph (C) 46kmph (D) 38kmph (E) 37kmph | tiger_lab_math |
Consider the sequence defined by \( x_n = (-1)^n n \) for \( n \in \mathbb{N} \). Is this sequence bounded below? If so, find a lower bound \( m \) such that for all \( n \in \mathbb{N} \), \( m \leq x_n \). If not, explain why no such \( m \) exists. | automath |
The function $f(x) = \frac{ax^2 + bx + c}{dx^2 + ex + f}$ has a domain of all real numbers except $x = 2$. Find all values of $a, b, c, d, e,$ and $f$ such that the graph of $f(x)$ has a horizontal asymptote of $y = 3$ and a vertical asymptote of $x = 2$. | nvidia_math |
A manufacturing company wants to increase the production of a product, and the manager believes that a change in the raw material supplier will result in an increase in production. The company decides to switch suppliers and records the production for the next ten days. The results are as follows:
Day 1: 300 units
Day... | tiger_lab_math |
in a certain pond , 40 fish were caught , tagged , and returned to the pond . a few days later , 40 fish were caught again , of which 2 were found to have been tagged . if the percent of tagged fish in the second catch approximates the percent of tagged fish in the pond , what ` s the approximate number of fish in the ... | tiger_lab_math |
Find the remainder when $x^4 - 2x^2 + 3$ is divided by $x - 1$. | nvidia_math |
Given two continuous functions \( f(x) \) and \( g(x) \) defined on the interval \([a, b]\), how can we quantitatively determine which function is more similar to a reference function \( h(x) \) over the given interval? Specifically, if we define the similarity \( S \) as the area between the curves of the functions an... | automath |
A cinema has 18 rows of seats, with each row divided into three sections by two aisles. If the cinema management decides to reserve each section for a group of 4 people, how many people can be seated in the cinema? | nvidia_math |
Exactly two-fifths of NBA players who signed up for a test are aged between 25 and 35 years. If three-eighths of them are older than 35, and a total of 1000 players signed up, how many players are younger than 25 years? | tiger_lab_math |
Consider a group of 7 people. How many ways can they be seated in a row if there are 3 women and 4 men, and the women must sit together? | tiger_lab_math |
Let \( S \) be a subset of the reals, and let \( f: S \rightarrow \mathbb{R} \) and \( g: [0, \infty) \rightarrow [0, \infty) \) be functions such that \( g \) is continuous at \( 0 \), \( g(0) = 0 \), and for any \( x, y \in S \), it holds that \( |f(x) - f(y)| \leq g(|x - y|) \). If \( \epsilon > 0 \), can you determ... | automath |
When the expression $4(x^2-2x+2)-7(x^3-3x+1)$ is fully simplified, what is the sum of the squares of the coefficients of the terms? | nvidia_math |
In how many distinct ways can the letters of the word "LETTER" be arranged in a grid of 5 rows and 11 boxes, ensuring that no row is empty and each box contains at most one letter, considering that the letters L and E are identical? | automath |
Two cyclists, Maria and John, depart from the same location simultaneously. Maria rides west at 10 mph, while John rides north at 12 mph. How long will it take for them to be 80 miles apart? | tiger_math |
If a bank offers a nominal annual interest rate of 8% compounded quarterly, what is the effective annual rate (EAR) for this interest rate? | automath |
Given the recurrence relation \( T(p, q) = T(p-1, q) + T(p, q-1) \) with boundary conditions \( T(p, 0) = 1 \) and \( T(0, q) = 1 \) for all \( p, q \in \mathbb{N}^{*} \), find a closed formula for \( T(p, q) \) and verify that it simplifies to \( T(p, q) = {p+q \choose p} \). Additionally, compute \( T(5, 7) \). | automath |
Given a wonky pizza defined by the shape that extends \(\frac{\alpha \theta }{2\pi }+\beta\) units from the origin at an angle \(\theta\) radians, where \(0 \leq \alpha, \beta \leq 10\) and \(N\) is an integer with \(2 \leq N \leq 100\), how can you determine the angles for \(N-1\) cuts, starting from \(0\) radians, so... | automath |
James has a habit of being late to school and has received detention for it multiple times. For every instance of tardiness, he received 2 hours of detention. If he has spent twice as many hours in detention as there are days in a typical month, how many instances of tardiness is he responsible for? | nvidia_math |
The compound interest earned by Sunil on a certain amount at the end of two years at the rate of 8% p.a. was Rs.2828.80. Find the total amount that Sunil got back at the end of two years in the form of principal plus interest earned.
Answer Choices: (A) 19828.83 (B) 19828.8 (C) 19828.82 (D) 19828.89 (E) 19828.81 | tiger_lab_math |
The ratio of the volumes of two spheres is $\frac{125}{216}$. The ratio of their radii can be expressed in the simplified form $\frac{a}{b}$ where $a$ and $b$ are integers. What is the value of the sum $a+b$? | nvidia_math |
A math student needs to find the value of the 6th term in the 7th row of Pascal's Triangle. | tiger_lab_math |
In a geometric progression, it is given that \( a_4 \cdot a_5 = \frac{1}{64} \) and that \( a_5 - a_3 = 24 \). If \( a_3 \) and \( a_5 \) are both positive integers, what is the value of \( a_3 + a_5 \)? | automath |
Two of the altitudes of the scalene triangle $ABC$ have length $4$ and $12$. If the length of the third altitude is also an integer, what is the biggest it can be?
$\textbf{(A)}\ 4\qquad \textbf{(B)}\ 5\qquad \textbf{(C)}\ 6\qquad \textbf{(D)}\ 7\qquad \textbf{(E)}\ \text{none of these}$
| nvidia_math |
A train 100 meters long takes 6 seconds to cross a man walking at a certain speed in the direction opposite to that of the train. The speed of the train is 54.99520038396929 kmph. What is the speed of the man in kmph? | tiger_math |
What is the largest integer \( k \) such that there exist positive integers \( a \) and \( n \) with \( n \geq 2 \) for which the sequence \( a, an, an^2, an^3, \ldots, an^k \) consists entirely of palindromes in base 10? | automath |
If Emily has three times as much money in her wallet as David does, and David has one-quarter the amount of money that Rachel has in her wallet, and Rachel has $80 in her wallet, how much more money does Emily have in her wallet than David? | nvidia_math |
The Grey's bought several chickens at a sale. John took 5 more of the chickens than Mary took. Ray took 6 chickens less than Mary. If Ray took 10 chickens, how many more chickens did John take than Ray? | tiger_math |
The average (arithmetic mean) of 27, 32, and 64 is 6 less than the average of 29, 42, and x. What is x?
Answer Choices: (A) 62 (B) 64 (C) 66 (D) 68 (E) 70 | tiger_lab_math |
Let \( f: \mathbb{R} \to \mathbb{R} \) be defined as follows:
\[
f(x) =
\begin{cases}
x^2 \sin(1/x) & \text{if } x \neq 0 \\
0 & \text{if } x = 0
\end{cases}
\]
For the function \( f(x) \), determine if the following statements are true or false:
1. \( f \) is continuous at \( x = 0 \).
2. \( f \) is differentia... | automath |
Compute the integral
\[
\int \frac{dx}{(1 - \sin x)^2}.
\]
Additionally, evaluate the integral
\[
\int \sin x \sin 2x \sin 3x \, dx.
\]
Provide your answers in a simplified form. | automath |
Find the largest constant $m,$ so that for any positive real numbers $a,$ $b,$ $c,$ and $d,$
\[\sqrt{\frac{a}{b + c + d}} + \sqrt{\frac{b}{a + c + d}} + \sqrt{\frac{c}{a + b + d}} + \sqrt{\frac{d}{a + b + c}} > m.\] | tiger_lab_math |
What is the number of units in the area of the circle with center at $P$ and passing through $Q$? Express your answer in terms of $\pi$.
[asy]
size(150); pair P = (-3,4), Q=(9,-3); string stringpair(pair p){return "$("+string(p.x)+", "+string(p.y)+"$)";}
draw((-15,0)--(15,0),Arrows(4)); draw((0,-15)--(0,15),Arrows(4))... | tiger_lab_math |
Given the data set \( e2.7 \) consisting of 100 values, calculate the mean, standard deviation, and coefficient of variation without groupings. Then, determine the mean again using the grouping method with 10 equally spaced bins. Provide all numerical results, including each mean value, standard deviation, and the coef... | automath |
Consider a graph with 6 vertices and 7 edges. Is this graph connected or disconnected? If it is connected, which vertices are cut vertices? If it is disconnected, how many components does it have? | tiger_lab_math |
In a card game, you are awarded 10 points if you win one round. While playing, Jane ends up with x points. If she lost 20 points, then how many rounds did she play?
If we know the answer to the above question is 8, what is the value of unknown variable x? | tiger_math |
Jamie ordered a set of various shaped cookies to use in her math class for a lesson on geometry. The set includes 10 square cookies, 15 circular cookies, and some triangular and hexagonal cookies. The total number of cookies she received was 60. If the number of triangular cookies is four times the number of hexagonal ... | tiger_math |
Given the functions \( x = f(t) = (t-1)^3 \) and \( y = g(t) = (t-1)^2 \), calculate the value of \( g'(t) \) at \( t = 1 \) and determine whether \( g'(1) \) is greater than, less than, or equal to 1. What is the conclusion based on this calculation? | automath |
Medians $\overline{AD}$ and $\overline{BE}$ of $\triangle ABC$ are perpendicular. If $AD= 15$ and $BE = 20$, then what is ${AB}$? | nvidia_math |
Farmer Tilly has 22 hectares of land on which she wants to plant three types of crops: corn, wheat, and soybeans. However, each crop has different spacing requirements due to their root systems. Corn needs at least 4 hectares per field to grow properly, wheat needs at least 3 hectares per field, and soybeans need at le... | tiger_math |
If an object starts from rest and experiences a constant acceleration of \( a \) metres per second squared, how long will it take for the object to reach a speed of \( 30 \) m/s in terms of \( a \)? Provide your answer in seconds. | automath |
Let \( f(z) = \frac{1}{z^2 - z} \). Calculate the residues of \( f(z) \) at its poles. Then, find the sum of these residues. What is the final result? | automath |
Two positive integers, $m$ and $n$, are such that $m$ is divisible by $n$, and the sum of their squares is 2025. How many possible values of $m$ are there? | nvidia_math |
Given four collinear points $A$, $B$, $C$, $D$ so that $AB = x$, $BC = 3x$, and $CD = 8x$. What is the ratio of $AC$ to $BD$ if point $A$ is between points $B$ and $C$, point $C$ is between points $A$ and $D$, and $AB + BC + CD = AD$? Express your answer as a common fraction. | nvidia_math |
Evaluate
\[\begin{vmatrix} 0 & \sin \alpha & -\cos \alpha \\ -\sin \alpha & 0 & \sin \beta \\ \cos \alpha & -\sin \beta & 0 \end{vmatrix}.\] | nvidia_math |
What is the area of the shaded region in the figure below? The figure is not drawn to scale. Round your answer to the nearest square centimeter.
[asy]
draw((0,0) -- (3,0) -- (3,3) -- (0,3)--cycle) ; draw((3,0)-- (12,0) -- (12,9) -- (3, 9)--cycle);
label ( "3 cm", (0,1.5), W); label ( "3 cm", (1.5 ,0), S);
label ( "9 ... | tiger_math |
What is the definite integral of \( x^3 e^x \) from \( x = 0 \) to \( x = 1 \)? | automath |
The area of a square equals the square of a length of the side of the square. The perimeter of a square equals the sum of the lengths of all four sides. The sum of the areas of two squares is X, while the difference in their areas is 33. Find the sum of their perimeters. The answer is 44. What is the value of unknown ... | tiger_math |
Let $f(x) = ax^3 + bx^2 - cx + d$ and $g(x) = -ax^3 + bx^2 + cx + d$. If $f(2) = 7$ and $g(2) = -1$, find $f(-2) + g(-2)$. | nvidia_math |
Mary is baking a cake . She already put in 8 cups of flour . She added 2 cups more of flour. The recipe wants _____ cups of flour. | tiger_math |
Compute the integral \( I = \int_{-6}^6 \frac{(4e^{2x} + 2)^2}{e^{2x}} \, \mathrm{d}x \) and find the exact numerical value of \( I \). | automath |
What is the 75th digit after the decimal point when 5/37 is expressed as a decimal? | nvidia_math |
A 270 metres long train running at the speed of 120 kmph crosses another train running in opposite direction at the speed of 80 kmph in 9 seconds. What is the length of the other train?
Answer Choices: (A) 220 meter (B) 225 meter (C) 230 meter (D) 235 meter (E) 240 meter | tiger_lab_math |
A list of measurements in increasing order is 4, 5, 6, 8, 10 and x. If the median of these measurements is 3/4 times their arithmetic mean, what is the value of x?
Answer Choices: (A) 17 (B) 19 (C) 21 (D) 23 (E) 25 | tiger_lab_math |
A cube is painted red on all faces and then cut into some equal smaller cubes. There are 12 smaller cubes painted on only 2 faces. How many smaller cubes were created in total? | tiger_math |
The scores on Trisha's first three tests were 88, 73 and 70. After two more tests, the mean score for all five tests was 81. Each test score was less than X, and all of Trisha's test scores were different integer values. List Trisha's five test scores from the greatest to the least, separated by commas. The answer is 7... | tiger_math |
How many people need to be in a room for there to be at least a 50% chance that at least two of them share the same birthday, assuming there are 365 days in a year? | automath |
What is the single digit result obtained by repeatedly applying the sum of digits function to the factorial of \(10\) (i.e., \(10!\))? | automath |
Evaluate the expression $\frac{\sqrt{338}}{\sqrt{288}}+\frac{\sqrt{150}}{\sqrt{96}}$ and simplify the result as a fraction. | tiger_math |
Let \( X \sim \text{Geom}(p) \) be a random variable representing the number of failures before the first success in a series of independent Bernoulli trials with probability of success \( p \). What is the expected value \( \mathbb{E}(X) \) in terms of \( p \)? | automath |
In a grocery store, Julia bought 2 pieces of Snickers and 3 packs of M&M's. If each piece of Snickers costs $1.5 and a pack of M&M's has the same cost as 2 Snickers, how much is Julia's change if she gave the cashier 2 $10 bills? | tiger_math |
Which type of conic section is described by the equation
\[\frac{x^2}{9} + \frac{y^2}{16} - \frac{2xy}{12} = 1?\]
Enter "C" for circle, "P" for parabola, "E" for ellipse, "H" for hyperbola, and "N" for none of the above. | nvidia_math |
Mike's pickle seems to roll around at the speed of 10.5 cm/min on his ceramic plate and at a speed of 79 mm/min on his paper plate. This means his pickle heats up more rolling around on the? "Option 1": paper plate, "Option 2" :ceramic plate Let's write a Python program. | tiger_lab_math |
A block of mass \( m = 4 \) kg is placed on a horizontal surface with a coefficient of friction \( \mu = 0.20 \). If a force \( F \) is applied horizontally to the block, what is the maximum acceleration \( a_{\text{max}} \) that the block can achieve without sliding? Please provide the answer in meters per second squa... | automath |
If a/b = 1/7, b/c = 3, c/d = 2/3, d/e = 2 and e/f = 1/2, then what is the value of abc/def ?
Answer Choices: (A) 72/21 (B) 27/85 (C) 34/47 (D) 13/81 (E) 16/21 | tiger_lab_math |
Three of the vertices of a regular octahedron are to be chosen at random. What is the probability that they will be the vertices of a triangle that is a face of the octahedron? Express your answer as a common fraction. | tiger_math |
Let the probability density function \( f(x) = kx^2 \) for \( x \in [0, 1] \) and \( f(x) = k(2-x) \) for \( x \in [1, 2] \). Find the value of the constant \( k \) such that \( f(x) \) is a valid probability density function over the interval \( [0, 2] \).
Calculate \( k \). | automath |
After harvesting 400 kg of tomatoes on Wednesday and half as much on Thursday, Tedra harvested a total of 2000 kg of tomatoes on Wednesday, Thursday, and Friday. If he gives away 700 kg of tomatoes to his friends, how many tomatoes does he have left from the number he harvested on Friday? | tiger_math |
Given the confidence of an association rule is 90.35% and the support for the consequent is calculated as 846/2000, what is the lift of the rule, and what does it signify about the relationship between the antecedent and the consequent? | automath |
Given the function $f(x) = \frac{x^2 + 4x + 4}{x - 1}$, find the sum of the coefficients of the simplified function $f(x) = Ax + B + \frac{C}{x-1}$, where A, B, and C are constants, and the function is defined everywhere except at $x = D$. | nvidia_math |
Amanda is preparing packaging boxes for her cookie shop. She has three sizes of boxes: Small, Medium, and Large. Each Small box can hold 2 cookies, each Medium box can hold 5 cookies, and each Large box can hold 10 cookies. If Amanda has exactly 47 cookies to package and she must use at least one box of each size, what... | tiger_math |
A company has 50 employees, of which 22 studied accounting, 15 studied finance and 14 studied marketing. Eight employees have done just two courses and only one made the three courses. How many employees have no training?
Answer Choices: (A) A) 0 (B) B) 9 (C) C) 10 (D) D) 11 (E) E) 26 | tiger_lab_math |
A train of length 180 metres takes 54 seconds to cross a tunnel of length 270 metres. What is the speed of the train in km/hr?
Answer Choices: (A) 50 km/hr. (B) 20 km/hr. (C) 30 km/hr. (D) 40 km/hr. (E) None | tiger_lab_math |
Passage: Canada, the United States and Mexico host communities of both Jews (6 million or about 1.8%), Buddhists (3.8 million or 1.1%) and Muslims (3.4 million or 1.0%). The biggest number of Jewish individuals can be found in the United States (5.4 million), Canada (375,000) and Mexico (67,476). The United States host... | tiger_lab_math |
What is the area of the region in the complex plane defined by the inequality \( |z^2 - z| < 1 \)? | automath |
The product of two positive integers plus their sum is 103. The integers are relatively prime, and each is less than 20. What is the sum of the two integers? Please write a program to solve it | tiger_lab_math |
What is the limit of the sequence formed by the partial sums of the harmonic series \( H_n = \sum_{k=1}^{n} \frac{1}{k} \) as \( n \) approaches infinity? | automath |
In an orchestra, Sebastian plays drums for percussion. He sits next to the brass section, which consists of four trombone players, two trumpet players, and one French horn player. In front of them is the strings section, which includes three violinists, a cellist, and a contrabassist. Next to the strings are a trio of ... | tiger_math |
Consider the function \( f(x) = \sin\left(\frac{\pi}{5} x\right) \). Determine which of the following functions, when graphed, represents \( \sin(2x - \frac{\pi}{2}) \):
A) \( f\left(2(x - \frac{\pi}{4})\right) \)
B) \( f\left(\frac{x}{2} + \frac{\pi}{10}\right) \)
C) \( f\left(\frac{1}{2}x + \frac{\pi}{5}\right) \)... | automath |
A bakery sells a special type of bread in boxes of 24 loaves each. The bakery's owner, Maria, wants to offer a discount to customers who buy multiple boxes at once. She decides to create a special deal where customers can buy a set of boxes that is divisible by 12, and she will give them a free box of bread. What is th... | nvidia_math |
Evaluate the determinant of the matrix:
\[\begin{bmatrix} 1 & 0 & 2 \\ 3 & 2 & -1 \\ 1 & 1 & 3 \end{bmatrix}.\] | tiger_math |
If you want to make a 2-liter bottle of Coca-Cola using a syrup that has a dilution ratio of 5.4:1 with carbonated water, and assuming the syrup costs $65 for 5 gallons (approximately 18.925 liters), while carbon dioxide costs $0.21 per liter, how much will it cost in total to make one 2-liter bottle of Coca-Cola? | automath |
The average score of a cricketer in 2 matches is 20 and in other 3 matches is 30. Then find the average score in all the 5 matches?
Answer Choices: (A) 26 (B) 27 (C) 30 (D) 35 (E) 42 | tiger_lab_math |
Given the polynomial \( x^2 - 3xy + 3y^2 - xy \), factor it completely and express the factorization in terms of \( (x - ay)(x - by) \), where \( a \) and \( b \) are real numbers. What are the values of \( a \) and \( b \)? | automath |
Let \( P(x) \) be a polynomial of degree \( n \) with real coefficients. What is the maximum number of distinct real roots that \( P(x) \) can have, and under what conditions does this maximum occur? | automath |
If x, y, and z are positive integers and x^2 = y^2 + z^2, which of the following must be true?
I. x > z
II. x = y + z
III. y^2 + z^2 is a positive integer
Answer Choices: (A) I only (B) II only (C) III only (D) I and II only (E) I and III only | tiger_lab_math |
Find the sum of the roots of the equation $$\sqrt{x+2\sqrt{x-1}}+\sqrt{x-2\sqrt{x-1}}=2.$$ | tiger_math |
If $\frac{1}{4}$ of a number is equal to $\frac{1}{5}$ of $\frac{1}{3}$ of 60, what is the number? | tiger_math |
Given a non-zero column matrix \( A \) of size \( m \times 1 \) and a non-zero row matrix \( B \) of size \( 1 \times n \), what is the rank of the resulting matrix \( X = AB \) in terms of \( m \) and \( n \)? Additionally, if \( m = 3 \) and \( n = 4 \), what is the specific rank of \( X \)? | automath |
A plane needs to fly due north for 400 miles in 2.5 hours, with a wind blowing from 328° at 11 mph. If the plane's current airspeed is 160 mph, what bearing and airspeed should the plane maintain to reach its destination? | automath |
Passage: In a 15-year career with the Boston Red Sox, New York Yankees, Cincinnati Reds, and New York Giants (NL), Mays compiled a 207-126 record with 29 shutouts in baseballs, 862 strikeouts and a 2.92 earned run average when the league average was 3.48. He won twenty or more games five times. He was also noted for hi... | tiger_lab_math |
Collin learned that he could earn different amounts per aluminum can based on their weight, ranging from €0.15 to €0.35. He found 12 lightweight cans at home worth €0.15 each, three times as many medium weight cans at his grandparents' house worth €0.25 each, his neighbor gave him 46 heavyweight cans worth €0.35 each, ... | tiger_math |
Find the integer $m$ that satisfies $0 \le m < 7$ and $$2345 \equiv m \pmod 7.$$ | nvidia_math |
A local non-profit organization offers a $1000 grant to the 15% of the lowest-income families in a town with 800 residents. It has been estimated that this type of grant generates 4 times as much money in local economic growth, in terms of increased sales tax revenue, as it costs. What is the non-profit organization's ... | nvidia_math |
What is the sum of the five largest distinct positive integer multiples of 7 that are less than 100? | tiger_math |
Find the value of $a$ so that the lines with the given equations are perpendicular. \begin{align*}
y &= 2x+13 \\
6y+ax &= 6.
\end{align*} | nvidia_math |
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