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The set of all solutions of the system $$
\begin{cases}
x+y\leq 3 \\
2x+y\geq 2 \\
x\geq 0 \\
y\geq 0
\end{cases}
$$ is a quadrilateral region. Find the number of units in the length of the longest side. Express your answer in simplest radical form. | tiger_lab_math |
What is the sum of the lengths of all the line segments removed in the construction of the Cantor set up to the nth iteration, where n is a positive integer greater than or equal to 1? | tiger_lab_math |
Let X be the metric space with the standard Euclidean topology and suppose Y is a nonempty subset of X such that every continuous function from Y to the discrete space with two points is constant. Prove that Y is both connected and path-connected.
Question: Let Y be a subset of R^2 defined by Y = {(x, y) : x^2 + y^2 ≤... | tiger_lab_math |
A cylindrical water tank has a radius of 5 meters and a height of 20 meters. Determine the mean curvature of the tank's surface at the top and at the bottom. | tiger_lab_math |
According to Goldbach's Conjecture, every even integer greater than 2 can be expressed as the sum of two prime numbers. Can you find the prime numbers that add up to 126? | tiger_lab_math |
A game show offers a prize of $10,000 to the contestant who has the lowest unique bid among all the other contestants. Suppose there are 10 contestants in total, and each contestant can only see the bids of the other contestants after they have placed their own bid. However, each contestant knows their own valuation fo... | tiger_lab_math |
Which of the options is a two-digit number, which has a unit's digit that exceeds its ten's digit by 4 and the product of the given number and the sum of its digits is equal to 208.
Answer Choices: (A) 15 (B) 37 (C) 55 (D) 26 (E) 73 | tiger_lab_math |
All the numbers 2, 3, 4, 5, 6, 7 are assigned to the six faces of a cube, one number to each face. For each of the eight vertices of the cube, a product of three numbers is computed, where the three numbers are the numbers assigned to the three faces that include that vertex. What is the greatest possible value of the ... | tiger_lab_math |
Suppose that a group of students take a test that consists of 10 questions. The test is designed so that each question is worth a certain amount of points, ranging from 1 to 5 points per question. You decide to perform a factor analysis on the test data to determine whether there are any underlying factors that influen... | tiger_lab_math |
workers decided to raise rs . 3 lacs by equal contribution from each . had they contributed rs . 50 eachextra , the contribution would have been rs . 3.60 lacs . how many workers were they ? Let's write a Python program. | tiger_lab_math |
excluding stoppages , the speed of the bus is 54 kms / hr and with stoppages , it is 45 kmph . for how many minutes does the bus stop per hour ? Let's write a Python program. | tiger_lab_math |
Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that
\[f(f(x) + y) = f(x^2 - y) + 4f(x) y\]for all real numbers $x$ and $y.$
Let $n$ be the number of possible values of $f(3),$ and let $s$ be the sum of all possible values of $f(3).$ Find $n \times s.$ | tiger_lab_math |
in a rectangular coordinate system , what is the area of a triangle whose vertices have the coordinates ( 3 , 0 ) , ( 6 , 3 ) , and ( 6 , - 3 ) ? Please write a program to solve it | tiger_lab_math |
find the compound interest on $ 1200 for 5 years at 20 % p . a . if ci is component yearly ? Please write a program to solve it | tiger_lab_math |
if there are only 2 wheelers and 4 wheelers parked in a school located at the heart of the city , find the number of 4 wheelers parked there if the total number of wheels is 58 ? Let's write a Python program. | tiger_lab_math |
Lagoon island is populated by alligators. Half the alligators are male. The rest are female. Among the females, 40% are juveniles. There are 15 adult females. How many male alligators are there? | tiger_lab_math |
Let A = {1,2} and B = {3,4}. Find the Cartesian product of (A union B) and the set {5,6}. Then find the Cartesian product of A with {5,6} and union it with the Cartesian product of B with {5,6}. Show that the two products are equal, proving that the Cartesian product of two sets is distributive over union. | tiger_lab_math |
Let $M$ be a smooth manifold that can be covered by two open sets $U$ and $V$ such that $U \cap V$ is non-empty and contractible. The De Rham cohomology group $H^k(M)$ is isomorphic to the direct sum of the De Rham cohomology groups $H^k(U)$ and $H^k(V)$. Suppose that $H^1(U) \cong \mathbb{R}^2$ and $H^1(V) \cong \math... | tiger_lab_math |
Let G be a group of order 4. Prove that G is isomorphic to either the cyclic group C4 or the Klein four-group V4.
Hint: Consider the possible values of the orders of elements in G. | tiger_lab_math |
triangle xyz is an isosceles right triangle . if side xy is longer than side yz , and the area of the triangle is 36 , what is the measure of side xy ? Let's program in Python in the response. | tiger_lab_math |
Kevin bought 2 watermelons . The first watermelon was 9.91 pounds , and the second watermelon was 4.11 pounds. Kevin baught _____ pounds of watermelon . Let's write a Python program. | tiger_lab_math |
Ryan works in an office that has an even number of men and women working there. Ryan participates in a meeting composed of 4 men and 6 women who are pulled from the office floor. This reduces the number of women working on the office floor by 20%. How many people work at Ryan's office? | tiger_lab_math |
if each participant of a chess tournament plays exactly one game with each of the remaining participants , then 120 games will be played during the tournament . find the number of participants . Let's write a program. | tiger_lab_math |
Let $A,$ $R,$ $M,$ and $L$ be positive real numbers such that
\begin{align*}
\log_{10} (AL) + \log_{10} (AM) &= 2, \\
\log_{10} (ML) + \log_{10} (MR) &= 3, \\
\log_{10} (RA) + \log_{10} (RL) &= 4.
\end{align*}Compute the value of the product $ARML.$ | tiger_lab_math |
when positive integer x is divided by positive integer y , the remainder is 8 . if x / y = 76.4 , what is the value of y ? Let's write a Python program. | tiger_lab_math |
find out the c . i on rs . 5000 at 4 % p . a . compound half - yearly for 1 1 / 2 year ? Let's write a Python program to solve it. | tiger_lab_math |
An integer $n$ is said to be square-free if the only perfect square that divides $n$ is $1^2$. How many positive odd integers greater than 1 and less than $100$ are square-free? Please respond by writing a program in Python. | tiger_lab_math |
A stream cipher generates a random key stream that is combined with the plaintext to produce the ciphertext. Let's say you are using a stream cipher that generates a key stream of the form 10110101 repeatedly. If you encrypt the plaintext "10101010" using this cipher, what would be the resulting ciphertext? | tiger_lab_math |
A student wants to find the number of intersection points of two given algebraic curves from the equations:
$$
x^3 + y^3 = 16 \\
x^2 + y^2 = 8
$$
Using Bezout's theorem, how many intersection points do these two curves have? | tiger_lab_math |
Matt's four cousins are coming to visit. There are four identical rooms that they can stay in. If any number of the cousins can stay in one room, how many different ways are there to put the cousins in the rooms? | tiger_lab_math |
on a race track a maximum of 5 horses can race together at a time . there are a total of 25 horses . there is no way of timing the races . what is the minimum number r of races we need to conduct to get the top 3 fastest horses ? Let's write a Python program. | tiger_lab_math |
if 28 % of a number exceeds 18 % of it by 7.2 , then find the number ? Let's write a Python program. | tiger_lab_math |
shekhar started a business investing rs . 25,000 in 1999 . in 2000 , he invested an additional amount of rs . 10,000 and rajeev joined him with an amount of rs . 35,000 . in 2001 , shekhar invested another additional amount of rs . 10,000 and jatin joined them with an amount of rs . 35,000 . what will be rajeev ’ s sha... | tiger_lab_math |
two employees x and y are paid a total of rs . 590 per week by their employer . if x is paid 120 percent of the sum paid to y , how much is y paid per week ? Let's program in Python in the response. | tiger_lab_math |
What is the smallest even number that cannot be expressed as the sum of two prime numbers according to Goldbach's conjecture? | tiger_lab_math |
A portion of the graph of a quadratic function $f(x)$ is shown below.
Let $g(x)=-f(x)$ and $h(x)=f(-x)$. If $a$ is the number of points where the graphs of $y=f(x)$ and $y=g(x)$ intersect, and $b$ is the number of points where the graphs of $y=f(x)$ and $y=h(x)$ intersect, then what is $10a+b$?
[asy]
size(150);
real ... | tiger_lab_math |
Consider a group G of order 35. Prove that the group is either cyclic or isomorphic to the direct product of two cyclic groups with orders 5 and 7, respectively. | tiger_lab_math |
Find the sum of all real values of $x$ that satisfy
\[x = 1 - x + x^2 - x^3 + x^4 - x^5 + \dotsb.\] | tiger_lab_math |
A group of 6 friends are at a party. Each pair of friends is either friends or not friends with each other. What is the minimum number of friendships that must exist among these 6 friends to guarantee that there are either 3 mutual friends or 3 mutual strangers in the group? | tiger_lab_math |
a man ' s regular pay is $ 3 per hour up to 40 hours . overtime is twice the payment for regular time . if he was paid $ 174 , how many hours overtime did he work ? Let's program in Python in the response. | tiger_lab_math |
3 houses are available in a new flat . 3 persons apply for the houses . each applies for 1 house without consulting others . the probability that all the 3 apply for the same house is ? Let's write a program. | tiger_lab_math |
the angle of elevation of a ladder leaning against a wall is 60 º and the foot of the ladder is 4.6 m away from the wall . the length of the ladder is : ? Let's write a program. | tiger_lab_math |
the sides of a square region , measured to the nearest centimeter , are 4 centimeters long . the least possible value of the actual area of the square region is ? Let's write a Python program. | tiger_lab_math |
What is the sum of the two smallest prime numbers that can be added up to give an even number greater than or equal to 10, according to Goldbach's conjecture? | tiger_lab_math |
Determine the number of ways to arrange the letters of the word RADII. | tiger_lab_math |
the length of a room is 5.5 m and width is 3.75 m . what is the cost of paying the floor by slabs at the rate of $ 500 per sq . metre . Let's write a program. | tiger_lab_math |
you collect pens . suppose you start out with 7 . mike gives you another 22 pens . since her father makes pens , cindy decides to double your pens . since you ' re nice , you give sharon 19 pens . how many pens do you have at the end ? Please respond by writing a program in Python. | tiger_lab_math |
there are 6 working days in a regular week and for each day , the working hours are 10 . a man earns rs . 2.10 per hour for regular work and rs . 4.20 per hour for overtime . if he earns rs . 525 in 4 weeks , how many hours did he work ? Please write a program to solve it | tiger_lab_math |
For any positive integer n, the sum of the first n positive integers equals [n(n+1)]/2.
What is the sum of all the even integers between 99 and 181?
Answer Choices: (A) 4360 (B) 4890 (C) 5280 (D) 5740 (E) 6320 | tiger_lab_math |
A hyperbolic plane has a curvature of -1.5. If a hyperbolic triangle has angles of 60°, 70°, and 50°, find the area of the hyperbolic triangle. | tiger_lab_math |
At a university there are 1800 students. 30% of all the students are from other countries. Starting next semester, 200 new foreign students will begin studying at this University. How many foreign students will then study at the University if all other students continue to study? | tiger_lab_math |
two integers are in the ratio of 1 to 4 . if 12 is added to the smaller number , the ratio becomes 1 to 2 . find the larger integer . Let's program in Python in the response. | tiger_lab_math |
A block of wood has the shape of a right circular cylinder with radius $6$ and height $8$, and its entire surface has been painted blue. Points $A$ and $B$ are chosen on the edge of one of the circular faces of the cylinder so that $\overarc{AB}$ on that face measures $120^\text{o}$. The block is then sliced in half al... | tiger_lab_math |
What is the smallest four-digit positive integer that is divisible by 47? | tiger_lab_math |
a sum was put a simple interest at a certain rate for 2 years . had it been put at 4 % higher rate , it would have fetched rs . 60 more . the sum is : ? Please respond by writing a program in Python. | tiger_lab_math |
Let $a$ and $b$ be positive real numbers such that $a + 2b = 1.$ Find the minimum value of
\[\frac{1}{a} + \frac{2}{b}.\] | tiger_lab_math |
There are 12 points in a given plane, no three of which are on the same line. If one of the points is represented as 'A', then how many triangles can be determined with the 12 points that contain the point A?
Answer Choices: (A) 55 (B) 105 (C) 182 (D) 210 (E) 455 | tiger_lab_math |
rs . 500 amounts to rs . 670 in 5 years at simple interest . if the interest is increased by 2 % , it would amount to how much ? Let's write a Python program. | tiger_lab_math |
3 buffaloes eat as much as 4 cows or 2 oxen . at a farm , there are 15 buffaloes , 8 oxen , and 24 cows . the fodder for these cattle is available for 24 days . if 60 more cows and 30 more buffaloes are brought in , how many days will the fodder last ? Please respond by writing a program in Python. | tiger_lab_math |
the fuel indicator in a car shows 1 / 5 th of the fuel tank as full . when 22 more liters of fuel are poured in to the tank , the indicator rests at the 3 / 4 of the full mark . find the capacity of the tank . Let's write a Python program. | tiger_lab_math |
the circumferences of two circles are 268 meters and 380 meters . find the difference between the areas of the larger and the smaller circles . Let's write a Python program. | tiger_lab_math |
Here's a precise numerical problem related to Heat transfer and Diffusion Equations for the math student to solve:
A metal plate of dimensions 8cm x 12cm and thickness 0.5cm is initially at a temperature of 60°C. The temperature at x=0 and x=8cm is fixed at 100°C, while the temperature at y=0 and y=12cm is fixed at 2... | tiger_lab_math |
in a family 15 people eat only vegetarian , 8 people eat only non veg . , 11 people eat both veg and non veg . . how many people eat veg in the family ? Please respond by writing a program in Python. | tiger_lab_math |
"statement 1": The average number of shirts with Salman , Ambani and Dalmiya is more than 24 , if all of them reached a shopping mall in Delhi and purchased 8 shirts each of them then average number of shirt each of them now has, "statement 2" :The average number of shirts with Salman , Ambani and Dalmiya is 54 , if a... | tiger_lab_math |
For any positive integer n, the sum of the first n positive integers equals n(n+1)/2. What is the sum of all the even integers between 99 and 201?
Answer Choices: (A) 7650 (B) 9250 (C) 12,650 (D) 14,250 (E) 15,150 | tiger_lab_math |
a certain sky blue paint contains 10 percent blue pigment and 90 percent red pigment by weight . a certain green paint contains 70 percent blue pigment and 30 percent yellow pigment . when these paints are mixed to produce a brown paint , the brown paint contains 40 percent blue pigment . if the brown paint weighs 10 g... | tiger_lab_math |
What is the smallest positive integer x that satisfies the equation 5^x ≡ 3 (mod 7)? | tiger_lab_math |
Consider the following modal logic formula:
(p → q) ⊢ ◇p → ◇q
Prove that the formula is valid using a natural deduction proof. | tiger_lab_math |
a crate measures 6 feet by 8 feet by 12 feet on the inside . a stone pillar in the shape of a right circular cylinder must fit into the crate for shipping so that it rests upright when the crate sits on at least one of its six sides . what is the radius , in feet , of the pillar with the largest volume that could still... | tiger_lab_math |
the number n is 76 h 4 , where h represents the 10 ' s digit . if n is divisible by 9 , what is the value of h ? Let's write a Python program. | tiger_lab_math |
Consider the sphere $S^2$ and a unit circle $S^1$ lying on it. Let $f:S^1\to S^1$ be a continuous map such that $f(-x)=-f(x)$ for all $x\in S^1$. Define $g:S^2\to S^2$ by $g(x,y,z)=(\cos 2\pi f(\theta)\cdot x,\sin 2\pi f(\theta)\cdot y,z)$ where $(x,y,z)$ are the spherical coordinates of a point in $S^2$ and $f(\theta)... | tiger_lab_math |
The age of a lioness in a park is twice the age of a hyena in the same park. The lioness is 12 years old. If the babies of the two animals are half the age of their mothers, calculate the sum of the age of the babies in five years. | tiger_lab_math |
A city is located in a region that has a 0.05 probability of being hit by a hurricane every year. What is the probability that in the next 10 years, the city will be hit by at least 2 hurricanes? | tiger_lab_math |
a computer manufacturer produces a certain electronic component at a cost of $ 80 per component . shipping costs for delivering the components are $ 6 per unit . further , the manufacturer has costs of $ 16,500 a month related to the electronic component regardless of how many it produces . if the manufacturer produces... | tiger_lab_math |
All the stocks on the over-the-counter market are designated by either a 7 letter or 6 letter codes that is created by using the 26 letters of the alphabet. Which of the following gives the maximum number of different stocks that can be designated with these codes?
Answer Choices: (A) 2 (26^5) (B) 26 (26^4) (C) 27 (2... | 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 |
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 |
Find all integer solutions to the Diophantine equation 2x + 3y = 7. | tiger_lab_math |
the difference between a two - digit number and the number obtained by interchanging the positions of its digits is 45 . what is the difference between the two digits of that number ? Please write a program to solve it | tiger_lab_math |
if difference between compound interest and simple interest on a sum at 10 % p . a . for 2 years is rs . 61 then sum is ? Let's write a Python program. | tiger_lab_math |
What is the number of non-isomorphic groups of order 60? Prove that every group of order 60 is isomorphic to one of the three groups mentioned in the subtopic, using the given information and by constructing the necessary isomorphisms. | tiger_lab_math |
in a company , 50 percent of the employees are men . if 60 percent of the employees are unionized and 70 percent of these are men , what percent of the non - union employees are women ? Please respond by writing a program in Python. | tiger_lab_math |
What is the largest three-digit integer $n$ that satisfies $$55n\equiv 165\pmod{260}~?$$ | tiger_lab_math |
a textile manufacturing firm employees 80 looms . it makes fabrics for a branded company . the aggregate sales value of the output of the 80 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 sprea... | tiger_lab_math |
Raj is working on a set of Data Sufficiency problems for his December GMAT: a geometry problem, an algebra problem, and a data interpretation problem. He has determined that statement 1 of the geometry problem is insufficient on its own, that both statement 1 and 2 of the algebra problem are insufficient on their own, ... | tiger_lab_math |
if x is an integer such that 5 < x < 21 , 7 < x < 18 , 13 > x > 2 , 12 > x > 9 , and x + 1 < 13 , then x is ? Please respond by writing a program in Python. | tiger_lab_math |
an equal number of desks and bookcases are to be placed along a library wall that is 15 meters long . each desk is 2 meters long , and each bookshelf is 1.5 meters long . if the maximum possible number of desks and bookcases are to be placed along the wall , then the space along the wall that is left over will be how m... | tiger_lab_math |
if 5 % more is gained by selling an book for $ 350 than by selling it for $ 340 , the cost of the book is : ? Let's write a Python program. | tiger_lab_math |
Consider the relevance logic formula R := A ⊃ B, where A and B are propositions. Let A be the proposition "x is an even number" and B be the proposition "x is a number greater than 10."
Provide an example of a natural number x that satisfies the formula R, and prove that it satisfies the formula. | tiger_lab_math |
Given a graph with 7 vertices and 9 edges, what is the size of the maximum independent set that can be identified in the graph? | tiger_lab_math |
Let $f$ be a non-constant polynomial such that
\[f(x - 1) + f(x) + f(x + 1) = \frac{[f(x)]^2}{2013x}\]for all nonzero real numbers $x.$ Find the sum of all possible values of $f(1).$ | tiger_lab_math |
if the average of 5 positive integers is 45 and the difference between the largest and the smallest of these 5 numbers is 10 , what is the maximum value possible for the largest of these 5 integers ? Please write a program to solve it | tiger_lab_math |
Consider the circle with radius 5 centered at the point (2,3) in the x-y plane. Using homotopy and homotopy groups, calculate the fundamental group of the space obtained by removing the point (2,6) from this circle. | tiger_lab_math |
Let X be the set of real numbers. Define a predicate P on X such that P(x) is true if and only if x is a positive integer. Write a second-order sentence that expresses the statement "for every subset S of X, there exists a least element of S that satisfies P". | tiger_lab_math |
Three circles, each of radius $3$, are drawn with centers at $(14, 92)$, $(17, 76)$, and $(19, 84)$. A line passing through $(17,76)$ is such that the total area of the parts of the three circles to one side of the line is equal to the total area of the parts of the three circles to the other side of it. What is the ab... | tiger_lab_math |
If the digit in the unit’s place of a two-digit number is halved and the digit in the ten’s place is doubled, the number thus, obtained is equal to the number obtained by interchanging the digits. Which of the following is definitely true?
Answer Choices: (A) Digits in the unit’s place and the ten’s place are equal. (B... | tiger_lab_math |
Let $x$ and $y$ be positive real numbers such that $x + y = 10.$ Find the minimum value of $\frac{1}{x} + \frac{1}{y}.$ | tiger_lab_math |
the avg weight of a , b & c is 84 kg . if d joins the group , the avg weight of the group becomes 80 kg . if another man e who weights is 7 kg more than d replaces a , then the avgof b , c , d & e becomes 79 kg . what is the weight of a ? Let's write a Python program to solve it. | tiger_lab_math |
the cost price of a radio is rs . 1500 and it was sold for rs . 1245 , find the loss % ? Let's write a program. | tiger_lab_math |
If the point $(3,6)$ is on the graph of $y=g(x)$, and $h(x)=(g(x))^2$ for all $x$, then there is one point that must be on the graph of $y=h(x)$. What is the sum of the coordinates of that point? Please write a program to solve it | tiger_lab_math |
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