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What is half of the absolute value of the difference of the squares of 18 and 16? | tiger_lab_math |
Determine the number of ways to arrange the letters of the word ALABAMA. | tiger_lab_math |
How many positive integers less than $500$ can be written as the sum of two positive perfect cubes? Let's write a program. | tiger_lab_math |
An organization trying to save endangered species of turtles has taken twice more hawksbill turtles than the number of green turtles into a conservation center. If the number of green turtles is 800, calculate the number of turtles that have been taken into the conservation center. | tiger_lab_math |
4 balls of different colors are to be placed in 3 different boxes such that any box contains at least 1 ball . what is the maximum number of different ways in which this can be done ? Let's write a Python program to solve it. | tiger_lab_math |
Using the Riemann mapping theorem, find a conformal map that transforms the unit disk $\mathbb{D}$ onto a vertical slit domain from $-1$ to $1$ that excludes the point $0$. What is the image of the point $i$ under this map? | tiger_lab_math |
Consider an undirected graph G with 8 vertices and 12 edges. Determine the minimum number of edges that need to be removed from G in order to disconnect the graph into two components. | tiger_lab_math |
The function $y=\frac{x^3+8x^2+21x+18}{x+2}$ can be simplified into the function $y=Ax^2+Bx+C$, defined everywhere except at $x=D$. What is the sum of the values of $A$, $B$, $C$, and $D$? | tiger_lab_math |
Suppose there are three mathematicians, A, B, and C, and one of them knows the answer to a math problem while the other two don't. Each of the mathematicians knows whether or not they themselves know the answer. If A knows the answer, then B and C both know that A knows. Likewise, if B knows the answer, then A and C bo... | tiger_lab_math |
Consider the projective variety defined by the homogeneous polynomial equation $f(x,y,z) = x^3 - 2y^3 + 3z^3 = 0$. Find the points on this variety in the projective plane $\mathbb{P}^2$ over the field $\mathbb{Q}$ with coordinates $(x:y:z)$, where $x$, $y$, and $z$ are not all zero, and each of them has absolute value ... | tiger_lab_math |
Let X be a smooth projective curve of genus g, and let C be a smooth projective curve of genus h, such that C intersects X transversely at n distinct points. Compute the dimension of the vector space of global sections of the sheaf Intersection Cohomology complex of X with coefficients in the sheaf of rational function... | tiger_lab_math |
A quantum system is described by the wave function:
$\psi(x) = \begin{cases} 0, & \text{if } -L<x<0\\
a(1-|x|/L), & \text{if } 0\leq|x|\leq L\\
0, & \text{otherwise}
\end{cases} $
where $a$ is a constant with units $\text{m}^{-1/2}$ and $L$ is a constant with units of meters.
Find the Fourier transform of the wave... | tiger_lab_math |
$A, B, C$ and $D$ are distinct positive integers such that the product $AB = 60$, the product $CD = 60$ and $A - B = C + D$ . What is the value of $A$? | tiger_lab_math |
A line is parameterized by a parameter $t,$ so that the vector on the line at $t = -1$ is $\begin{pmatrix} 1 \\ 3 \\ 8 \end{pmatrix},$ and the vector on the line at $t = 2$ is $\begin{pmatrix} 0 \\ -2 \\ -4 \end{pmatrix}.$ Find the vector on the line at $t = 3.$ | tiger_lab_math |
A square is inscribed in the ellipse
\[\frac{x^2}{3} + \frac{y^2}{6} = 1,\]so that its sides are parallel to the coordinate axes. Find the area of the square.
[asy]
unitsize(1 cm);
draw(xscale(sqrt(3))*yscale(sqrt(6))*Circle((0,0),1));
draw((sqrt(2),sqrt(2))--(-sqrt(2),sqrt(2))--(-sqrt(2),-sqrt(2))--(sqrt(2),-sqrt(2... | tiger_lab_math |
For each permutation $a_1,a_2,a_3,\cdots,a_{10}$ of the integers $1,2,3,\cdots,10$, form the sum
\[|a_1-a_2|+|a_3-a_4|+|a_5-a_6|+|a_7-a_8|+|a_9-a_{10}|.\]
The average value of all such sums can be written in the form $\dfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.
| tiger_lab_math |
Suppose you are given a box containing 100 balls, out of which 60 are blue and 40 are red. You randomly select 10 balls from the box without replacement, and find that 6 of them are blue. What is the probability that the next ball you select from the box will be red, given that 6 of the first 10 balls you selected were... | tiger_lab_math |
an engineer undertakes a project to build a road 15 km long in 300 days and employs 30 men for the purpose . after 100 days , he finds only 2.5 km of the road has been completed . find the ( approximate ) number of extra men he must employ to finish the work in time . Please write a program to solve it | tiger_lab_math |
A computer program calculated the sum of the first 100 counting numbers as 5,050. Using inductive logic, how can you prove that this is the correct sum? | tiger_lab_math |
Facebook decided to award a productivity bonus to all its female employees who are mothers. This productivity bonus will total 25% of Facebook's annual earnings, which was $5,000,000 for the year 2020. It is known that Facebook employs 3300 employees; one-third are men, and of the women, 1200 are not mothers. How much ... | tiger_lab_math |
working alone , a can complete a certain kind of job in 12 hours . a and d , working together at their respective rates , can complete one of these jobs in 4 hours . in how many hours can d , working alone , complete one of these jobs ? Let's write a Python program. | tiger_lab_math |
Suppose we have a graph G with 8 vertices. What is the minimum number of colors needed to color the vertices of G such that no two adjacent vertices have the same color? | tiger_lab_math |
a man sitting in a train which is traveling at 56 kmph observes that a goods train , traveling in opposite direction , takes 15 seconds to pass him . if the goods train is 410 m long , find its speed ? Let's write a Python program. | tiger_lab_math |
Given the function $f(z)=\sqrt{z^2+1}$ defined on the open upper half-plane $\mathbb{H}$, find the analytic continuation of $f(z)$ across the real axis onto the open lower half-plane, $\mathbb{H'}$. | tiger_lab_math |
Three is the largest number that can be divided evenly into 27 and the positive integer x, while 10 is the largest number that can be divided evenly into both 100 and x. Which of the following is the largest possible number that could be divided into x and 2400
Answer Choices: (A) 30 (B) 70 (C) 210 (D) 240 (E) 700 | tiger_lab_math |
Consider the sphere $S^2$ and a point $p \in S^2$. Assume that you have a continuous function $f_t: S^2 \rightarrow S^2$ for $t \in [0,1]$ that satisfies $f_0=id$ and $f_1(p)=p$, where $id$ is the identity map.
What is the value of the homotopy group $\pi_1(S^2 \setminus \{p\})$? | tiger_lab_math |
cubes with each side one inch long are glued together to form a larger cube . the larger cube ' s face is painted with red color and the entire assembly is taken apart . 24 small cubes are found with no paints on them . how many of unit cubes have at least one face that is painted red ? Please respond by writing a prog... | tiger_lab_math |
7 does not occur in 1000 . so we have to count the number of times it appears between 1 and 999 . any number between 1 and 999 can be expressed in the form of xyz where 0 < x , y , z < 9 . Let's write a program. | tiger_lab_math |
Find the smallest positive integer $n$ such that
\[\begin{pmatrix} \cos 170^\circ & -\sin 170^\circ \\ \sin 170^\circ & \cos 170^\circ \end{pmatrix}^n = \mathbf{I}.\] Please write a program to solve it | tiger_lab_math |
Let $z$ be a complex number such that $|z| = 13.$ Find $z \times \overline{z}.$ | tiger_lab_math |
Suppose there are 15 books, each of which is written by one of the following 5 authors: A, B, C, D, and E. In how many ways can these books be arranged on a shelf such that there are exactly 3 books by each author next to each other? | tiger_lab_math |
lagaan is levied on the 60 percent of the cultivated land . the revenue department collected total rs . 3 , 44,000 through the lagaan from the village of mutter . mutter , a very rich farmer , paid only rs . 480 as lagaan . the percentage of total land of mutter over the total taxable land of the village is : ? Let's w... | tiger_lab_math |
It rained 0.9 inches on Monday . On Tuesday , it rained 0.7 inches less than on Monday. It rained _____ inches on Tuesday . Let's write a Python program to solve it. | tiger_lab_math |
A Ferris wheel has a radius of 30 meters, and completes one revolution in 2 minutes. Find the maximum height of a passenger if the passenger gets onto the wheel at ground level. (Assume the center of the wheel is at ground level) | tiger_lab_math |
Suppose two burger shops, Burger A and Burger B, are located across the street from each other. If both shops sell identical burgers for $5, what is the probability that a customer will choose to buy from Burger A instead of Burger B if Burger A sets its price at $4.50 and Burger B maintains its price at $5? Assume tha... | tiger_lab_math |
the sum of number of boys and girls in a school is 150 . if the number of boys is x , then the number of girls becomes x % of the total number of students . the number of boys is ? Please respond by writing a program in Python. | tiger_lab_math |
What is the sum of all the positive two-digit integers divisible by both the sum and product of their digits? | tiger_lab_math |
In how many ways can you partition the number 7 into distinct positive integers? | tiger_lab_math |
A right-angled triangle has two legs of length 5 cm and 12 cm. Find the length of the hypotenuse by using trigonometry and then find the perimeter of the triangle. | tiger_lab_math |
the ages of two person differ by 20 years . if 8 years ago , the elder one be 5 times as old as the younger one , their present ages ( in years ) are respectively ? Let's write a Python program to solve it. | tiger_lab_math |
A rectangular piece of land measures 80 meters by 100 meters, and a rectangular building is to be constructed within the land. What are the dimensions of the building with the maximum area that can be constructed? | tiger_lab_math |
Consider the unit circle, centered at the origin in the complex plane. Find a bijective mapping that maps the inside of the unit circle to the upper half plane. | tiger_lab_math |
the breadth of a rectangular field is 60 % of its length . if the perimeter of the field is 800 m , find out the area of the field . Let's write a program. | tiger_lab_math |
b completes a work in 7 days . a alone can do it in 10 days . if both work together , the work can be completed in how many days ? Let's write a Python program. | tiger_lab_math |
Consider a simple graph G with 7 vertices and 9 edges. Find the minimum number of colors needed to color the edges of the graph such that no adjacent edges have the same color. | tiger_lab_math |
If p,q ,and r are positive real numbers such that p(q+r) = 152, q(r+p) =162 , and r(p+q) = 170, then pqr is
Answer Choices: (A) a) 672 (B) b) 688 (C) c) 704 (D) d) 720 (E) e) 750 | tiger_lab_math |
roses can be purchased individually for $ 7.30 , one dozen for $ 36 , or two dozen for $ 50 . what is the greatest number of roses that can be purchased for $ 680 ? Let's write a program. | tiger_lab_math |
A two-digit integer $AB$ equals $\frac{1}{9}$ of the three-digit integer $AAB$, where $A$ and $B$ represent distinct digits from 1 to 9. What is the smallest possible value of the three-digit integer $AAB$? | tiger_lab_math |
if x is an integer such that 1 < x < 9 , 2 < x < 15 , 7 > x > – 1 , 4 > x > 0 , and x + 1 < 5 , then x is ? Let's program in Python in the response. | tiger_lab_math |
Let A = {1, 2, 3} and B = {2, 4, 6}. Determine the union of set A and set B, and confirm that set A and set B are subsets of the union. | tiger_lab_math |
a owes b rs . 50 . he agrees to pay b over a number of consecutive days on a monday , paying single note or rs . 10 or rs . 20 on each day . in how many different ways can a repay b . Let's write a Python program. | tiger_lab_math |
a trader bought a car at 30 % discount on its original price . he sold it at a 80 % increase on the price he bought it . what percent of profit did he make on the original price ? Let's write a program. | tiger_lab_math |
the monthly incomes of a and b are in the ratio 5 : 2 . b ' s monthly income is 12 % more than c ' s monthly income . if c ' s monthly income is rs . 13000 , then find the annual income of a ? Please write a program to solve it | tiger_lab_math |
in a bus left side are 15 seats available , 3 few seats in right side because in rear exit door . each seat hold 3 people . in addition , there is a seat back can sit 7 people all together . how many people can sit in a bus ? Let's write a Python program to solve it. | tiger_lab_math |
4 12-sided dice are rolled. What is the probability that the number of dice showing a two digit number is equal to the number of dice showing a one digit number? Express your answer as a common fraction. (Assume that the numbers on the 12 sides are the numbers from 1 to 12 expressed in decimal.) | tiger_lab_math |
A rectangular cardboard box with an open top must have a volume of 6,000 cubic centimeters. The length of the base is twice the width of the base. The cost of the box material is 10 cents per square centimeter. What are the dimensions of the box that will minimize the cost of the box? | tiger_lab_math |
linda spent 5 / 6 of her savings on furniture and the rest on a tv . if the tv cost her $ 500 , what were her original savings ? Let's program in Python in the response. | tiger_lab_math |
Here is a numerical problem for Diophantine equations that a math student can solve:
Find all integer solutions for the equation:
$$4x + 7y = 15$$ | tiger_lab_math |
Consider the algebraic surface defined by the equation:
x^2 + y^2 - z^2 = 1
Find the coordinates of all the points on the surface that have x = 2. | tiger_lab_math |
Let $M$ be a smooth, compact manifold without boundary with dimension $n$. Suppose that the De Rham cohomology groups of $M$ satisfy $H^i(M,\mathbb{R})=0$ for $i\neq k$ and $H^k(M,\mathbb{R})=1$.
(a) Show that $k\leq n$.
(b) Suppose that $k=n$. Prove that $M$ is diffeomorphic to the sphere $S^n$.
(c) Suppose that $... | tiger_lab_math |
Let f(z) be an entire function such that |f(z)| ≤ 7|z|^3 for all z ∈ ℂ. Prove that f(z) is a polynomial and find its degree. | tiger_lab_math |
Find the degree of the Fano variety that is the intersection of two quadrics in projective 4-space with equations:
$x_0^2+x_1^2+x_2^2=0$
$x_0x_2+x_1x_3=0$ | tiger_lab_math |
For some positive integer $n$, the number $110n^3$ has $110$ positive integer divisors, including $1$ and the number $110n^3$. How many positive integer divisors does the number $81n^4$ have?
$\textbf{(A) }110\qquad\textbf{(B) }191\qquad\textbf{(C) }261\qquad\textbf{(D) }325\qquad\textbf{(E) }425$
| tiger_lab_math |
A cube has a side length of 10 cm. Find the volume of the sphere that is inscribed inside the cube. | tiger_lab_math |
income and expenditure of a person are in the ratio 5 : 4 . if the income of the person is rs . 14000 , then find his savings ? Let's write a Python program to solve it. | tiger_lab_math |
a student chose a number , multiplied it by 2 , then subtracted 138 from the result and got 104 . what was the number he chose ? Let's write a program. | tiger_lab_math |
Find the general solution of the differential equation:
$$
y'' - 6y' + 9y = 0,
$$
where $y$ is a complex-valued function. | tiger_lab_math |
Compute the number of real solutions $(x,y,z,w)$ to the system of equations:
\begin{align*}
x &= z+w+zwx, \\
y &= w+x+wxy, \\
z &= x+y+xyz, \\
w &= y+z+yzw.
\end{align*} | tiger_lab_math |
the average mark of the students of a class in a particular exam is 80 . if 5 students whose average mark in that exam is 50 are excluded , the average mark of the remaining will be 90 . find the number of students who wrote the exam . Please respond by writing a program in Python. | tiger_lab_math |
The average (arithmetic mean)E of the four distinct positive integers p, q, s and t is x. If p and q are odd, and s and t are even, which of the following must be true?
I. x is an integer
II. (x−0.5) is an integer.
III. 2x is an odd integer.
Answer Choices: (A) I only (B) II only (C) III only (D) II and III only (E) No... | tiger_lab_math |
The quotient of two positive integers is $\frac{5}{2}$ and their product is 160. What is the value of the larger of the two integers? | tiger_lab_math |
if each participant of a chess tournament plays exactly one game with each of the remaining participants , then 171 games will be played during the tournament . find the number of participants . Please write a program to solve it | tiger_lab_math |
a grocer has a sale of rs . 5420 , rs . 5660 , rs . 6200 , rs . 6350 and rs . 6500 for 5 consecutive months . find the sale he should have in the sixth month , so that he gets an average sale of rs . 6100 ? Let's write a Python program. | tiger_lab_math |
Suppose there are 5 contestants in a beauty contest game, and each contestant must choose a number between 0 and 100. The winner of the contest will be the contestant whose number is closest to two-thirds of the average of all the numbers chosen. What number do you choose to maximize your chances of winning the game, a... | tiger_lab_math |
of the 160 people at a party , 70 were women , and 50 women tried the appetizer . if 50 people did not try the appetizer , what is the total number of men who tried the appetizer ? Let's write a program. | tiger_lab_math |
100 people are attending a newspaper conference . 35 of them are writers and more than 38 are editors . of the people at the conference , x are both writers and editors and 2 x are neither . what is the largest possible number of people who are both writers and editors ? Let's write a program. | tiger_lab_math |
a report consists of 20 sheets each of 55 lines and each such line consist of 65 characters . this report is retyped into sheets each of 65 lines such that each line consists of 70 characters . the percentage reduction in number of sheets is closest to ? Please respond by writing a program in Python. | tiger_lab_math |
Consider a square ABCD with sides of length 6 units. This square can be divided into four congruent squares as shown below. If we continue this process of dividing each square into four equal squares and then shading the first obtained square red, what will be the total area of red squares for the fourth subdivision?
... | tiger_lab_math |
The volume of a certain rectangular solidis $216\text{ cm}^3$, its total surface area is $288\text{ cm}^2$, and its three dimensions are in geometric progression. Find the sum of the lengths in cm of all the edges of this solid.
| tiger_lab_math |
Let $a$ and $b$ be positive real numbers such that $a + 2b = 1.$ Find the minimum value of
\[\frac{1}{a} + \frac{1}{b}.\] | tiger_lab_math |
Let X be a topological space and let A be a connected subset of X. Suppose that A and X/A are both path-connected, where X/A denotes the quotient space formed by identifying all points of A to a single point. Prove that X is path-connected.
Question for the math student:
Let X be a topological space and let A be a co... | tiger_lab_math |
a certain automobile company ’ s best - selling model is the speedster . the speedster , like all of their other models , comes in coupe and convertible styles . 3 / 4 of the current inventory is speedsters , of which 3 / 5 are convertibles . if there are 30 vehicles that are not speedsters , how many speedster convert... | tiger_lab_math |
Using type theory, prove that there exists a natural number that cannot be expressed as the sum of three cubes of integers. | tiger_lab_math |
Let X be a set of real numbers and let P(x) be the statement "x is positive". Using higher-order logic, express the statement "every positive real number has a square root" as a mathematical formula. | tiger_lab_math |
Let $N$ denote the number of permutations of the $15$-character string $AAAABBBBBCCCCCC$ such that
None of the first four letters is an $A$.
None of the next five letters is a $B$.
None of the last six letters is a $C$.
Find the remainder when $N$ is divided by $1000$.
| tiger_lab_math |
The number of arrangements that can be made with the letters of the word MEADOWS so that the vowels occupy the even places?
Answer Choices: (A) 14 (B) 144 (C) 777 (D) 992 (E) 221 | tiger_lab_math |
the width of a rectangular hall is ½ of its length . if the area of the hall is 288 sq . m , what is the difference between its length and breadth ? Please write a program to solve it | tiger_lab_math |
Triangle $ABC$ has side lengths $AB=4$, $BC=5$, and $CA=6$. Points $D$ and $E$ are on ray $AB$ with $AB<AD<AE$. The point $F \neq C$ is a point of intersection of the circumcircles of $\triangle ACD$ and $\triangle EBC$ satisfying $DF=2$ and $EF=7$. Then $BE$ can be expressed as $\tfrac{a+b\sqrt{c}}{d}$, where $a$, $b$... | tiger_lab_math |
the average age of 15 students of a class is 15 years . out of these , the average age of 5 students is 14 years and that of the other 9 students is 16 years . the age of the 15 th student is ? Let's program in Python in the response. | tiger_lab_math |
What is the maximum number of chaotic attractors that can be observed in the Rikitake system when the following parameters are used?
$a=-0.15$, $b=0.2$, $c=10$, $d=0.2$, $l=2.5$, $m=0.7$ and $n=1.05$. | tiger_lab_math |
the length of rectangle is thrice its breadth and its perimeter is 64 m , find the area of the rectangle ? Let's write a Python program. | tiger_lab_math |
a and b began business with rs . 3000 and rs . 4000 after 8 months , a withdraws rs . 1000 and b advances rs . 1000 more . at the end of the year , their profits amounted to rs . 756 find the share of a . Please respond by writing a program in Python. | tiger_lab_math |
Suppose that we have a smooth plane curve of degree 4 given by the equation $C: f(x,y) = x^4+y^4+3x^2y^2-4x^3=0$ and a divisor $D=(p)-(q)$, where $p=(-1,0)$ and $q=(1,0)$. Find the dimension of the space $L(D)$ of sections of the line bundle associated to $D$ on $C$, that is, find $\text{dim }L(D)$. | tiger_lab_math |
If $a$ is a nonzero integer and $b$ is a positive number such that $ab^2=\log_{10} b$, what is the median of the set $\{0, 1, a, b,
1/b\}$? | tiger_lab_math |
on sunday , bill ran 4 more miles than he ran on saturday . julia did not run on saturday , but she ran twice the number of miles on sunday that bill ran on sunday . if bill and julia ran a total of 28 miles on saturday and sunday , how many miles did bill run on sunday ? Please respond by writing a program in Python. | tiger_lab_math |
What is the smallest even number that can be expressed as the difference between two consecutive prime numbers? | tiger_lab_math |
Find the number of ordered pairs of positive integers $(a,b)$ such that $a+b=1000$ and neither $a$ nor $b$ has a zero digit.
| tiger_lab_math |
Find the number of ways of arranging the letters of the word "MATERIAL" such that all the vowels in the word are to come together?
Answer Choices: (A) 1840 ways (B) 1440 ways (C) 1490 ways (D) 1447 ways (E) 1240 ways | tiger_lab_math |
Determine the Lyapunov exponent of the Rikitake system with parameters a=1.2 and b=2.8 using the forward Euler method with a time step of 0.01. | tiger_lab_math |
Let G be a group of order 27, and let H be a subgroup of G with order 9. Prove that H is a normal subgroup of G. | tiger_lab_math |
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