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Given a group of prime order p = 7, prove that the group must be cyclic. | tiger_lab_math |
If we have a set of numbers {2, 4, 6, 8, 10}, and the predicate "is even" denoted by E(x), write down a predicate logic statement to represent the fact that all the numbers in the set are even. | tiger_lab_math |
Determine all integer solutions of the Diophantine equation 9x + 15y = 12. | tiger_lab_math |
Let M be a 2-dimensional manifold with a Riemannian metric g, and let ω be a 1-form on M such that ω = dx + dy, where x and y are local coordinates on M. Compute the Hodge star of ω with respect to g at a point p ∈ M, where g is given locally by g = 4(dx^2 + dy^2). | tiger_lab_math |
The point with coordinates $(6,-10)$ is the midpoint of the segment with one endpoint at $(8,0)$. Find the sum of the coordinates of the other endpoint. Let's write a Python program. | tiger_lab_math |
A square and a triangle have equal perimeters. The lengths of the three sides of the triangle are $6.1$ cm, $8.2$ cm and $9.7$ cm. What is the area of the square in square centimeters? Let's write a Python program. | tiger_lab_math |
a fellow borrowed a certain sum of money at 5 % per annum at simple interest and in 5 years the interest amounted to rs . 750 less than the sum lent . what was the sum lent ? Let's program in Python in the response. | tiger_lab_math |
Let U be the universal set and A be a subset of U, such that U contains exactly 100 elements and A contains exactly 60 elements. What is the cardinality of the complement of set A? Verify if the union of set A and its complement is equal to the universal set U. | tiger_lab_math |
Consider a network with 4 nodes and 5 edges depicted in the following graph:
```
(1)
/ \
10 / \ 20
(2) (3)
|\ /|\
30 | \ / | \
\|/ 5 \|/ \ 10
(4)-----(5)
```
Find the maximum flow that can be sent from node 1 to node 5, assuming that the capacity of each edge is as sh... | tiger_lab_math |
3 friends a , b , c went for week end party to mcdonald ’ s restaurant and there they measure there weights in some order in 7 rounds . a , b , c , ab , bc , ac , abc . final round measure is 175 kg then find the average weight of all the 7 rounds ? Let's write a program. | tiger_lab_math |
Find the smallest positive integer $x$ which is greater than $1$ and relatively prime to $120$ (recall that relatively prime means that the GCD of $x$ and $120$ is $1$) Please write a program to solve it | tiger_lab_math |
the mean of 50 observations was 36 . it was found later that an observation 44 was wrongly taken as 23 . the corrected new mean is ? Let's write a program. | tiger_lab_math |
A small square is entirely contained in a larger square, as shown. The side length of the small square is 3 units and the side length of the larger square is 7 units. What is the number of square units in the area of the black region?
[asy]
fill((0,0)--(21,0)--(21,21)--(0,21)--cycle,black);
fill((9,4)--(9,13)--(18,13)... | tiger_lab_math |
if 3 < x < 6 < y < 11 , then what is the greatest possible positive integer difference of x and y ? Let's program in Python in the response. | tiger_lab_math |
Find the value of the Euler's totient function of the number 25. | tiger_lab_math |
If a, b, c are consecutive positive integers and a<b<c, which of the following must be true?
I. a-c=-2
II. abc is an even integer
III. (a+b+c)/3 is an integer
Answer Choices: (A) All are correct (B) II (C) I and II (D) II and III (E) None are correct | tiger_lab_math |
A metal block has a square base with sides of 6 cm and a height of 10 cm. If the block is melted and then poured into a cylindrical container, find the radius of the container if the height of the cylindrical container is 12 cm. Use integration to find the volume of the metal block and the volume of the cylindrical con... | tiger_lab_math |
a particular store purchased a stock of turtleneck sweaters and marked up its cost by 20 % . during the new year season , it further marked up its prices by 25 % of the original retail price . in february , the store then offered a discount of 9 % . what was its profit on the items sold in february ? Let's write a prog... | tiger_lab_math |
Find all values of the real number $a$ so that the four complex roots of
\[z^4 - 6z^3 + 11az^2 - 3(2a^2 + 3a - 3) z + 1 = 0\]form the vertices of a parallelogram in the complex plane. Enter all the values, separated by commas. | tiger_lab_math |
in an electric circuit , two resistors with resistances x and y are connected in parallel . if r is the combined resistance of these two resistors , then the reciprocal of r is equal to the sum of the reciprocals of x and y . what is r if x is 4 ohms and y is 5 ohms ? Let's write a Python program. | tiger_lab_math |
[asy] unitsize(27); defaultpen(linewidth(.8pt)+fontsize(10pt)); pair A,B,C,D,E,F,X,Y,Z; A=(3,3); B=(0,0); C=(6,0); D=(4,0); E=(4,2); F=(1,1); draw(A--B--C--cycle); draw(A--D); draw(B--E); draw(C--F); X=intersectionpoint(A--D,C--F); Y=intersectionpoint(B--E,A--D); Z=intersectionpoint(B--E,C--F); label("$A$",A,N); label(... | tiger_lab_math |
How many of the natural numbers from 1 to 600, inclusive, contain the digit 5 at least once? (The numbers 152 and 553 are two natural numbers that contain the digit 5 at least once, but 430 is not.) Let's write a program. | tiger_lab_math |
a can contains a mixture of two liquids a and b in the ratio 7 : 5 . when 9 litres of mixture are drawn off and the can is filled with b , the ratio of a and b becomes 7 : 9 . how many litres of liquid a was contained by the can initially ? Let's write a Python program. | tiger_lab_math |
95 white and black tiles will be used to form a 10 x 10 square pattern . if there must be at least one black tile in every row and at least one white tile in every column , what is the maximum difference between the number of black and white tiles that can be used ? Please respond by writing a program in Python. | tiger_lab_math |
in a group of ducks and cows , the total number of legs are 36 more than twice the number of heads . find the total number of cows . Let's write a program. | tiger_lab_math |
A standard deck of cards has 52 cards divided into 4 suits, each of which has 13 cards. Two of the suits ($\heartsuit$ and $\diamondsuit$, called 'hearts' and 'diamonds') are red, the other two ($\spadesuit$ and $\clubsuit$, called 'spades' and 'clubs') are black. The cards in the deck are placed in random order (usu... | tiger_lab_math |
34 . the side surface of a cylinder can is rolled with a rectangular plate . if the height of a cylinder tank is 8 feet and the perimeter of the circular base is 6 feet , what is the diagonal of the rectangular plate ? Let's program in Python in the response. | tiger_lab_math |
on the first day of her vacation , louisa traveled 200 miles . on the second day , traveling at the same average speed , she traveled 350 miles . if the 200 - mile trip took 3 hours less than the 350 - mile trip , what was the average speed , in miles per hour ? Let's write a Python program. | tiger_lab_math |
the inner circumference of a circular race track , 25 m wide , is 880 m . find radius of the outer circle ? Let's write a Python program to solve it. | tiger_lab_math |
There are two ${\bf positive}$ integers $c$ for which the equation $$5x^2+11x+c=0$$has rational solutions. What is the product of those two values of $c$? | tiger_lab_math |
find the area , diameter = 10 m . Let's write a Python program. | tiger_lab_math |
ramesh purchased a refrigerator for rs . 12500 after getting a discount of 20 % on the labelled price . he spent rs . 125 on transport and rs . 250 on installation . at what price should it be sold so that the profit earned would be 12 % if no discount was offered ? Let's program in Python in the response. | tiger_lab_math |
For all positive integers n and m, the function A(n) equals the following product:
(1 + 1/2 + 1/2^2)(1 + 1/3 + 3^2)(1 + 1/5 + 5^2)…(1 + 1/p_n + 1/p_n^2), where p_n is the nth smallest prime number, while B(m) equals the sum of the reciprocals of all the positive integers from 1 through m, inclusive. The largest recipro... | tiger_lab_math |
The mean of four integers E will not change if all the integers are multiplied by any constant. What is always true about this set of numbers?
I. The mean of the set is 0
II. The sum of the largest member and the smallest member of the set is 0
III. The set contains both positive and negative integers
Answer Choices: (... | tiger_lab_math |
the sides of a square region , measured to the nearest centimeter , are 10 centimeters long . the least possible value of the actual area of the square region is ? Please respond by writing a program in Python. | tiger_lab_math |
Suppose that $f(z)$ is an entire function that satisfies the following conditions:
\begin{align*}
\mathrm{Re}f(z) &\leq \sqrt{\mathrm{Re}z}, \quad \mathrm{for \ all \ }z\in\mathbb{C},\\
\mathrm{Im}f(z) &\leq \sqrt{\mathrm{Im}z}, \quad \mathrm{for \ all \ }z\in\mathbb{C}.
\end{align*}
If $f(0)=0$ and $f(i)=2$, then what... | tiger_lab_math |
The line $y = 3$ intersects the graph of $y = 4x^2 + x - 1$ at the points $A$ and $B$. The distance between $A$ and $B$ can be written as $\frac{\sqrt{m}}{n}$, where $m$ and $n$ are positive integers that do not share any factors other than one. Find the value of $m - n$. | tiger_lab_math |
a tank holds x gallons of a saltwater solution that is 20 % salt by volume . one fourth of the water is evaporated , leaving all of the salt . when 5 gallons of water and 10 gallons of salt are added , the resulting mixture is 33 1 / 3 % salt by volume . what is the value of x ? Let's write a program. | tiger_lab_math |
Let $M$ be a smooth manifold of dimension $n$ and let $f:M\rightarrow \mathbb{R}$ be a smooth function. Define the set $N=\{(x,f(x))\in \mathbb{R}^{n+1}\mid x\in M\}$.
Prove that $N$ is a smooth submanifold of $\mathbb{R}^{n+1}$ of dimension $n+1$. | tiger_lab_math |
a invested $ 150 in a business after 6 months b invested $ 200 in the business . end of the year if they got $ 100 as profit . find a shares ? Please respond by writing a program in Python. | tiger_lab_math |
Ben's hockey team is 60% boys and the rest are girls. Half the girls are juniors and the other half are seniors. If the team has 50 players, how many junior girls are there? | tiger_lab_math |
a full stationary oil tank that is a right circular cylinder has a radius of 100 feet and a height of 25 feet . oil is pumped from the stationary tank to an oil truck that has a tank that is a right circular cylinder until the truck ' s tank is completely filled . if the truck ' s tank has a radius of 5 feet and a heig... | tiger_lab_math |
a gardener wants to plant trees in his garden in such a way that the number of trees in each row should be the same . if there are 4 rows or 5 rows or 6 rows , then no tree will be left . find the least number of trees required ? Let's write a Python program to solve it. | tiger_lab_math |
Let A = {2, 3} , B = {4, 5}, and C = {6, 7}. Find (A x B) x C and A x (B x C). Verify that the Cartesian product of two sets is associative. | tiger_lab_math |
A person stands at point A and observes a building at point B from a distance of 60 meters. The angle of elevation from the person's eyes to the top of the building is 45 degrees. If the person's eyes are 1.6 meters above the ground, find the height of the building in meters. | tiger_lab_math |
which no . need to add to 859622 to get a no . exactly divisible by 456 ? Let's write a Python program. | tiger_lab_math |
the length of a rectangular floor is more than its breadth by 200 % . if rs . 100 is required to paint the floor at the rate of rs . 2 per sq m , then what would be the length of the floor ? Please respond by writing a program in Python. | tiger_lab_math |
How many even three digit numbers are there such that the sum of the tens and units digits is 10? Please write a program to solve it | tiger_lab_math |
The average of 13 numbers is 10.9. If the average of the first seven numbers is 10.5 and that of the last seven numbers is 11.4, then the middle number is
Answer Choices: (A) 15.6 (B) 11.6 (C) 12.6 (D) 13.3 (E) 14.4 | tiger_lab_math |
a towel , when bleached , was found to have lost 20 % of its length and 10 % of its breadth . the percentage of decrease in area is ? Let's program in Python in the response. | tiger_lab_math |
In a community of 500 people, 320 are adults and 180 are children. If you choose 3 people at random from this community, what is the probability that all 3 people are children? | tiger_lab_math |
A group of people contains men, women, and children. The number of men is twice the number of women, and the number of women is 3 times the number of children. If the number of children is 30, how many people does this group contain? | tiger_lab_math |
There are six different models that are to appear in a fashion show. Two are from Europe, two are from South America, and two are from North America. If all the models from the same continent are to stand next to each other, how many ways can the fashion show organizer arrange the models?
Answer Choices: (A) 72 (B) 48 ... | tiger_lab_math |
a merchant purchased a jacket for $ 48 and then determined a selling price that equalled the purchase price of the jacket plus a markup that was 40 percent of the selling price . during a sale , the merchant discounted the selling price by 20 percent and sold the jacket . what was the merchant ’ s gross profit on this ... | tiger_lab_math |
The sum of three numbers $x$ ,$y$, $z$ is 165. When the smallest number $x$ is multiplied by 7, the result is $n$. The value $n$ is obtained by subtracting 9 from the largest number $y$. This number $n$ also results by adding 9 to the third number $z$. What is the product of the three numbers? | tiger_lab_math |
Prove that there are no non-isomorphic groups of order 12 other than cyclic group C12, alternating group A4, or dihedral group D6. | tiger_lab_math |
Six green balls and four red balls are in a bag. A ball is taken from the bag, its color recorded, then placed back in the bag. A second ball is taken and its color recorded. What is the probability the two balls are the same color? | tiger_lab_math |
the marked price of a book is 20 % more than the cost price . after the book is sold , the vendor realizes that he had wrongly raised the cost price by a margin of 25 % . if the marked price of the book is rs . 30 , what is the original cost price of the book ? Let's program in Python in the response. | tiger_lab_math |
the sum of the even numbers between 1 and n is 95 * 96 , where n is an odd number , then n = ? Let's write a Python program to solve it. | tiger_lab_math |
fox jeans regularly sell for $ 15 a pair and pony jeans regularly sell for $ 18 a pair . during a sale these regular unit prices are discounted at different rates so that a total of $ 9 is saved by purchasing 5 pairs of jeans : 3 pairs of fox jeans and 2 pairs of pony jeans . if the sum of the two discounts rates is 50... | tiger_lab_math |
Let set A = {1, 2} and set B = {3, 4}. Find the Cartesian product of A and B, then find the union of the resulting set. Next, find the Cartesian product of B and A, and find the union of that resulting set as well. Show that the union of the first resulting set is not equal to the union of the second resulting set, thu... | tiger_lab_math |
a textile manufacturing firm employees 125 looms . it makes fabrics for a branded company . the aggregate sales value of the output of the 125 looms is rs 5 , 00,000 and the monthly manufacturing expenses is rs 1 , 50,000 . assume that each loom contributes equally to the sales and manufacturing expenses are evenly spr... | tiger_lab_math |
Cory made a complete list of the prime numbers between 1 and 25. What is the sum of the smallest prime number and the largest prime number on his list? | tiger_lab_math |
if all the 6 are replaced by 9 , then the algebraic sum of all the numbers from 1 to 100 ( both inclusive ) varies by how much ? Please respond by writing a program in Python. | tiger_lab_math |
The sum of two numbers is 12 and the product of the numbers is 35. Find sum of the squares of that numbers.
Answer Choices: (A) 84 (B) 87 (C) 96 (D) 79 (E) 74 | tiger_lab_math |
in a hostel , the number of students decreased by 10 % and the price of food increased by 20 % over the previous year . if each student consumes the same amount of food then by how much should the consumption of food be cut short by every student , so that the total cost of the food remains the same as that of the prev... | tiger_lab_math |
A point $(x,y)$ is randomly picked from inside the rectangle with vertices $(0,0)$, $(4,0)$, $(4,1)$, and $(0,1)$. What is the probability that $x < y$? | tiger_lab_math |
the length of a rectangular plot is thrice its breadth . if the area of the rectangular plot is 972 sq m , then what is the breadth of the rectangular plot ? Let's program in Python in the response. | tiger_lab_math |
Prove that there are only two non-isomorphic groups of order 4. | tiger_lab_math |
Let $M$ be a compact smooth manifold of dimension $n$ without boundary, and let $f:M\rightarrow \mathbb{R}$ be a Morse function with $k$ critical points. Prove that the first $i$ Morse inequalities holds for $i=0,1,\ldots,n$, where the $i^{th}$ Morse inequalities is given by:
$$\sum_{j=0}^{i}(-1)^j\mathrm{rank}(H_j(M;... | tiger_lab_math |
x and y are two towns . ganesh covers the distance from x to y at an average speed of 54 km / hr . however , he covers the distance from y to x at an average speed of 36 km / hr . his average speed during the whole journey in km / hr . is : ? Let's write a Python program. | tiger_lab_math |
A meteorologist records that a particular region has a 30% chance of experiencing a thunderstorm in the afternoon. What is the probability that in a given week, the region will have thunderstorms on at least 3 afternoons? | tiger_lab_math |
a train can travel 50 % faster than a car . both start from point a at the same time and reach point b 75 kms away from a at the same time . on the way , however , the train lost about 12.5 minutes while stopping at the stations . the speed of the car is : ? Please respond by writing a program in Python. | tiger_lab_math |
Consider the following algebraic curve in the complex plane:
$$C : x^3 + y^3 - 3xy = 0$$
a) Show that the group $(\mathbb{Z}/3\mathbb{Z})^2$ acts on $C$ as automorphisms.
b) Determine the algebra of invariants of this group action.
c) Find an equation in the invariants that defines the quotient $C /(\mathbb{Z}/3\m... | tiger_lab_math |
the diagonals of a rhombus are 17 cm and 20 cm . find its area ? Let's write a Python program. | tiger_lab_math |
bill made a profit of 10 % by selling a product . if he had purchased that product for 10 % less and sold it at a profit of 30 % , he would have received $ 70 more . what was his original selling price ? Let's write a program. | tiger_lab_math |
the average of 11 results is 20 . the average of first 5 of them is 15 and that of last 5 is 22 . find the 6 th result ? Let's program in Python in the response. | tiger_lab_math |
Let X be a topological space, and suppose that there exist disjoint non-empty subsets U and V in X, such that U is open and V is closed. Show that there exists a continuous function f: X → [0,1] which separates U and V, that is f(U)={0} and f(V)={1}.
Then, consider the space X=[0,1] ∪ {2}, where {2} has open-generati... | tiger_lab_math |
after successive discounts of 20 % , 10 % and 5 % a certain good is sold for rs . 6600 . find the actual price of the good . Please write a program to solve it | tiger_lab_math |
Consider the cyclic group G = <𝑎> = {1, 𝑎, 𝑎^2, …, 𝑎^(𝑛−1)} with operation of multiplication modulo 𝑛.
Let H be a subgroup of G. Let 𝑏 be the smallest positive integer such that 𝑏∈H. Prove that H = <𝑏>.
Now, let 𝑛 = 14, and let H be the subgroup of G generated by 𝑎^3. Find the elements of H, and express ... | tiger_lab_math |
A road has four intersections. At the first intersection, there are 40 road signs showing different types of information mounted on the street poles. At the second intersection, 1/4 times more road signs than the first intersection are mounted on the street poles. The number of road signs at the third intersection is t... | tiger_lab_math |
The average of 11 numbers is 10.8. If the average of the first six numbers is 10.5 and that of the last six numbers is 11.4, then the middle number is
Answer Choices: (A) 71.5 (B) 11.5 (C) 81.5 (D) 11.5 (E) 12.6 | tiger_lab_math |
A regular octahedron is formed by joining the centers of adjoining faces of a cube. The ratio of the volume of the octahedron to the volume of the cube is
$\mathrm{(A) \frac{\sqrt{3}}{12} } \qquad \mathrm{(B) \frac{\sqrt{6}}{16} } \qquad \mathrm{(C) \frac{1}{6} } \qquad \mathrm{(D) \frac{\sqrt{2}}{8} } \qquad \mathrm{(... | tiger_lab_math |
a customer bought a product at the shop . however , the shopkeeper increased the price of the product by 20 % so that the customer could not buy the required amount of the product . the customer managed to buy only 70 % of the required amount . what is the difference in the amount of money that the customer paid for th... | tiger_lab_math |
Find the number of functions $f : \mathbb{R} \to \mathbb{R}$ such that
\[f(x + f(y)) = x + y\]for all real numbers $x$ and $y.$ | tiger_lab_math |
The dimensions of a triangle are tripled to form a new triangle. If the area of the new triangle is 54 square feet, how many square feet were in the area of the original triangle? | tiger_lab_math |
"statement 1": Bookman purchased 40 copies of a new book released recently , 10 of which are hardback and sold for $ 20 each , and rest are paperback and sold for $ 10 each, "statement 2" :Bookman purchased 20 copies of a new book released recently , 10 of which are hardback and sold for $ 20 each , and rest are paper... | tiger_lab_math |
If the parameter values of a Lorenz attractor are equal to $\sigma=10, \rho=28,$ and $\beta=8/3$, compute the maximum Lyapunov exponent of the system. | tiger_lab_math |
A company has 10 servers, each with a failure rate of 2%. Determine the probability that at least one of the servers fails within a month. | tiger_lab_math |
Jo adds up all the positive integers from 1 to 100. Kate does a similar thing with the first 100 positive integers; however, she first rounds every integer to its nearest multiple of 10 (rounding 5s up) and then adds the 100 values. What is the positive difference between Jo's sum and Kate's sum? | tiger_lab_math |
in an election , candidate a got 75 % of the total valid votes . if 15 % of the total votes were declared invalid and the total numbers of votes is 560000 , find the number of valid vote polled in favor of candidate . Let's write a program. | tiger_lab_math |
the width of a rectangular hall is ½ of its length . if the area of the hall is 200 sq . m , what is the difference between its length and breadth ? Let's write a Python program. | tiger_lab_math |
Find all entire functions $f(z)$ such that $|f(z)| = 1$ whenever $|z| = 1$. | tiger_lab_math |
a taxi leaves point a 3 hours after a bus left the same spot . the bus is traveling 30 mph slower than the taxi . find the speed of the taxi , if it overtakes the bus in three hours . Let's write a program. | tiger_lab_math |
what is the minimum percentage increase in the mean of set f { - 4 , - 1 , 0 , 6 , 9 } if its two smallest elements are replaced with two different primes ? Please write a program to solve it | tiger_lab_math |
Determine if the Fourier series of the function f(x) = x on the interval [-π, π] converges at x = 0 and find its sum if it does converge. | tiger_lab_math |
Four horizontal lines and four vertical lines are drawn in a plane. In how many ways can four lines be chosen such that a rectangular region is enclosed? | tiger_lab_math |
3 friends neha , sabi and mahi divide $ 1100 amongs them in such a way that if $ 5 , $ 8 and $ 4 are removed from the sums that neha , sabi and mahi received respectively , then the share of the sums that they got will be in the ratio of 2 : 8 : 6 . how much did mahi receive ? Let's write a program. | tiger_lab_math |
Consider the heat equation in 2D given by:
$\frac{\partial u}{\partial t} = k \left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}\right), \qquad 0 < x < L_x, \qquad 0 < y < L_y, \qquad t > 0$
Subject to the following boundary conditions:
$u(0,y,t) = 100, \qquad u(L_x,y,t) = 50, \qquad u(x,0,t... | tiger_lab_math |
f and e are midpoints of ab and ad respectively . if the side of square abcd is 8 , what is the area of triangle cef ? Let's program in Python in the response. | tiger_lab_math |
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