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the function f ( p ) represents the number of ways that prime numbers can be uniquely summed to form a certain number p such that p = a + b + c + d … where those summed variables are each prime and a ≤ b ≤ c ≤ d . . . for instance f ( 8 ) = 3 and the unique ways are 2 + 2 + 2 + 2 and 2 + 3 + 3 and 3 + 5 . what is f ( 1... | tiger_lab_math |
an seller earns an income of re 3 on the first day of his business . on every subsequent day , he earns an income which is just thrice of that made on the previous day . on the 15 th day of business , he earns an income of : ? Please write a program to solve it | tiger_lab_math |
Equilateral $\triangle ABC$ is inscribed in a circle of radius $2$. Extend $\overline{AB}$ through $B$ to point $D$ so that $AD=13,$ and extend $\overline{AC}$ through $C$ to point $E$ so that $AE = 11.$ Through $D,$ draw a line $l_1$ parallel to $\overline{AE},$ and through $E,$ draw a line $l_2$ parallel to $\overlin... | tiger_lab_math |
Consider the set of real numbers given by the interval [0,1]. Prove that this set is both connected and path-connected. Could you come up with a similar example to prove that the statement is not true for other sets of real numbers, say an open interval (0,1) or a closed interval [0,1)? | tiger_lab_math |
Consider a two-dimensional manifold with metric given by
$$
ds^2 = 2(dx^2 + dy^2) + 3dxdy.
$$
Find the nonzero components of the Riemann curvature tensor and evaluate the scalar curvature at the origin $(0,0)$. | tiger_lab_math |
For positive integers $N$ and $k$, define $N$ to be $k$-nice if there exists a positive integer $a$ such that $a^{k}$ has exactly $N$ positive divisors. Find the number of positive integers less than $1000$ that are neither $7$-nice nor $8$-nice.
| tiger_lab_math |
A mass of 0.5 kg is attached to a spring with a spring constant of 50 N/m. The mass-spring system is initially at rest but is then given an initial displacement of 0.1 m from its equilibrium position. Find the equation of motion for the mass-spring system and determine the amplitude, period, and frequency of the result... | tiger_lab_math |
A trapezoid has a length of 6 cm and a height of 4 cm. The length of the two parallel sides are 10 cm and 8 cm respectively. Find the area of the trapezoid. | tiger_lab_math |
Let A={1,2,3} and B={4,5}. Find the Cartesian product of A×B and B×A and then show that the two sets are not equal. | tiger_lab_math |
circles x and y are concentric . if the radius of circle x is 6 times that of circle y , what is the probability that a point selected inside circle x at random will be outside circle y ? Please write a program to solve it | tiger_lab_math |
a boat crossed a lake from north to east at the speed of 10 km / h , entered a river and covered twice as much distance going upstream at 9 km / h . it then turned around and stopped at the south shore of the lake . if it averaged 3.8 km / h that day , what was its approximate downstream speed ? Let's write a Python pr... | tiger_lab_math |
Let $A$ and $B$ be real numbers such that $\frac{A}{x-5}+B(x+1)=\frac{-3x^2+12x+22}{x-5}$. What is $A+B$? | tiger_lab_math |
a good train and a passenger train are running on parallel tracks in the same direction . the driver of the goods train observes that the passenger train coming from behind overtakes and crosses his train completely in 60 sec . whereas a passenger on the passenger train marks that he crosses the goods train in 40 sec .... | tiger_lab_math |
find the number which is nearest to 3105 and is exactly divisible by 21 . Let's write a Python program. | tiger_lab_math |
The area of the square formed on the diagonal of a rectangle as its side is 108 1/3 % more than the area of the rectangle. If the perimeter of the rectangle is 28 units, find the difference between the sides of the rectangle?
Answer Choices: (A) 6m (B) 9m (C) 3m (D) 2m (E) 1m | tiger_lab_math |
Carolyn is planning out her final essay. The introduction will be 450 words, the conclusion will be triple the length of the introduction, and each of the four body sections will be the same length. If her essay has to be 5000 words total, how long is each section? | tiger_lab_math |
A string of length 10m is fixed at both ends. It has a mass of 0.1kg and is under a tension of 100N. Find the wave speed of the transverse waves on the string if the frequency is 50Hz. | tiger_lab_math |
Dr. Math's four-digit house number $ABCD$ contains no zeroes and can be split into two different two-digit primes ``$AB$'' and ``$CD$'' where the digits $A$, $B$, $C$ and $D$ are not necessarily distinct. If each of the two-digit primes is less than 40, how many such house numbers are possible? | tiger_lab_math |
Here's a question based on Wilson's theorem:
What is the smallest positive integer greater than 1 that satisfies the following condition?
(x - 1)! + 1 is divisible by x
Note: This condition is equivalent to saying that x is a prime number. | tiger_lab_math |
a train running at the speed of 120 km / hr crosses a pole in 9 seconds . find the length of the train . Let's write a program. | tiger_lab_math |
A triangle ABC has a side AB of length 8 cm and a side AC of length 10 cm. The angle between sides AB and AC is 60 degrees. Find the perimeter of the triangle using trigonometry. | tiger_lab_math |
For any 7 given numbers, which of the following is always equivalent to adding the 3 numbers together and then dividing the sum by 3?
I. Ordering the 3 numbers numerically, from highest to lowest, and then selecting the middle number.
II. Dividing each of the numbers by 3 and then adding the results together.
III. Mult... | tiger_lab_math |
A point $(x,y)$ is randomly and uniformly chosen inside the square with vertices (0,0), (0,2), (2,2), and (2,0). What is the probability that $x+y < 3$? | tiger_lab_math |
a train passes a platform in 30 seconds . the same train passes a man standing on the platform in 20 seconds . if the speed of the train is 54 km / hr , the length of the platform is ? Let's program in Python in the response. | tiger_lab_math |
15 litres of mixture contains 20 % alcohol and the rest water . if 3 litres of water be mixed with it , the percentage of alcohol in the new mixture would be ? Let's write a program. | tiger_lab_math |
the average of 8 numbers is 21 . if each of the number is multiplied by 8 , the average of the new set of numbers is : ? Let's write a Python program to solve it. | tiger_lab_math |
each week , harry is paid x dollars per hour for the first 12 hours and 1.5 x dollars for each additional hour worked that week . each week , james is paid x dollars per per hour for the first 40 hours and 2 x dollars for each additional hour worked that week . last week james worked a total of 41 hours if harry and ja... | tiger_lab_math |
A certain function $f$ has the properties that $f(3x) = 3f(x)$ for all positive real values of $x$, and that $f(x) = 1 - |x - 2|$ for $1\leq x \leq 3$. Find the smallest $x$ for which $f(x) = f(2001)$. | tiger_lab_math |
Let $ABCDE$ be a pentagon inscribed in a circle such that $AB = CD = 3$, $BC = DE = 10$, and $AE= 14$. The sum of the lengths of all diagonals of $ABCDE$ is equal to $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$ ?
$\textbf{(A) }129\qquad \textbf{(B) }247\qquad \textbf{(C) }353\q... | tiger_lab_math |
hillary and eddy are climbing to the summit of mt . everest from a base camp 5000 ft from the summit . when they depart for the summit at 06 : 00 , hillary climbs at a rate of 800 ft / hr with eddy lagging behind at a slower rate of 500 ft / hr . if hillary stops 1000 ft short of the summit and then descends at a rate ... | tiger_lab_math |
selling an kite for rs . 30 , a shop keeper gains 25 % . during a clearance sale , the shopkeeper allows a discount of 10 % on the marked price . his gain percent during the sale is ? Please write a program to solve it | tiger_lab_math |
Let X be a smooth projective surface such that the canonical class KX is not nef. Assume that there is a contraction f : X → Y to a smooth projective curve Y such that −KX is f -ample. Compute the value of the self-intersection number −KX ⋅ C, where C is a curve contracted by f. | tiger_lab_math |
a shopkeeper bought 600 oranges and 400 bananas . he found 15 % of oranges and 8 % of bananas were rotten . find the percentage of fruits in good condition . Let's write a program. | tiger_lab_math |
at the end of the month , a certain ocean desalination plant ’ s reservoir contained 30 million gallons of water . this amount is twice the normal level . if this amount represents 75 % of the reservoir ’ s total capacity , how many million gallons short of total capacity is the normal level ? Let's write a program. | tiger_lab_math |
The points $(0,0)\,$, $(a,11)\,$, and $(b,37)\,$ are the vertices of an equilateral triangle. Find the value of $ab\,$.
| tiger_lab_math |
The letters of the alphabet are given numeric values based on the two conditions below.
$\bullet$ Only the numeric values of $-2,$ $-1,$ $0,$ $1$ and $2$ are used.
$\bullet$ Starting with A and going through Z, a numeric value is assigned to each letter according to the following pattern: $$
1, 2, 1, 0, -1, -2, -1,... | tiger_lab_math |
What is the smallest positive integer "n" such that (n-1)! + 1 is divisible by "n"? | tiger_lab_math |
Find the minimum value of
\[\sin^4 x + \frac{3}{2} \cos^4 x,\]as $x$ varies over all real numbers. | tiger_lab_math |
A certain collection of numbered indexed cards includes one card with a 1 written on it, two cards with a 2, and so forth up to $n$ cards showing an $n,$ for some positive integer $n$. Determine $n,$ if the average value of a card in this collection is 2017. | tiger_lab_math |
n a certain flower shop , which stocks 4 types of flowers , there are 1 / 3 as many violets as carnations , and 1 / 4 as many tulips as violets . if there are equal numbers of roses and tulips , what percent of the flowers in the shop are carnations ? Please write a program to solve it | tiger_lab_math |
A math student is given the following problem to solve from the subject of Topology:
Let X be a path-connected topological space, and let f and g be two continuous maps from S^1 to X such that f(1) = g(1). Prove that f and g are homotopic with respect to their basepoint f(1) if and only if their induced maps on the fu... | tiger_lab_math |
in a mixture 60 litres , the ratio of milk and water 2 : 1 . if this ratio is to be 1 : 2 , then the quanity of water to be further added is ? Let's write a Python program. | tiger_lab_math |
the average of 6 number is 3.95 . the average of two of them is 3.8 , while the average of the other two is 3.85 . what is the average of the remaining two number ? Let's program in Python in the response. | tiger_lab_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 . Let's write a Python program to solve it. | tiger_lab_math |
Alan likes to support music creators, so instead of downloading songs from the internet, he visits a record store once in a while. He decided now to buy 2 CDs by "The Dark", 1 CD by the creator "AVN", and 5 CDs that are mixes of songs from the 90s. The price of the "AVN" CD is $12 and it is half the price of one CD by ... | tiger_lab_math |
at a restaurant , glasses are stored in two different - sized boxes . one box contains 12 glasses , and the other contains 16 glasses . if the average number of glasses per box is 15 , and there are 16 more of the larger boxes , what is the total number of glasses e at the restaurant ? ( assume that all boxes are fille... | tiger_lab_math |
In the six-digit integer $3A6,\!792$, what is the largest digit $A$ so that the six-digit integer will be divisible by 3? | tiger_lab_math |
at a certain bowling alley , it costs $ 0.50 to rent bowling shoes for the day and $ 1.75 to bowl 1 game . if a person has $ 12.80 and must rent shoes , what is the greatest number of complete games that person can bowl in one day ? Please respond by writing a program in Python. | tiger_lab_math |
Let $\alpha \neq 1$ be a complex number such that the distance from $\alpha^2$ to 1 is twice the distance from $\alpha$ to 1, while the distance from $\alpha^4$ to 1 is four times the distance from $\alpha$ to 1. Enter all possible values of $\alpha,$ separated by commas. | tiger_lab_math |
A lattice point in the $x,y$-plane is a point both of whose coordinates are integers (not necessarily positive). How many lattice points lie on the graph of the equation $x^2-y^2=47$? Let's write a program. | tiger_lab_math |
When three positive integers are divided by $24$, the remainders are $10,$ $4,$ and $12,$ respectively.
When the sum of the three integers is divided by $24$, what is the remainder? Let's write a Python program. | tiger_lab_math |
What is the sum of all integer solutions to $1<(x-2)^2<25$? | tiger_lab_math |
What is the smallest positive integer that is divisible by 6, 9, and 15? | tiger_lab_math |
rajat , vikas and abhishek are submitting questions in the ratio 7 : 3 : 2 . if total number of questions submitted by them is 24 . find the number of questions submitted by vikas . Please write a program to solve it | tiger_lab_math |
a bag contains an equal number of one rupee , 50 paise and 25 paise coins respectively . if the total value is 210 , how many coins of each type are there ? Let's write a program. | tiger_lab_math |
a man can row a distance of 5 km in 60 min with the help of the tide . the direction of the tide reverses with the same speed . now he travels a further 30 km in 10 hours . how much time he would have saved if the direction of tide has not changed ? Let's program in Python in the response. | tiger_lab_math |
at what price must an article costing rs . 51.50 be marked in order that after deducting 5 % from the list price . it may be sold at a profit of 25 % on the cost price ? Let's program in Python in the response. | tiger_lab_math |
John buys a chair. He then buys a table that is 3 times the price of the chair. Then, he buys a couch that is 5 times the price of the table. If John paid $380 for all these items, what is the price of the couch? | tiger_lab_math |
In a two-digit, if it is known that its unit's digit exceeds its ten's digit by 2 and that the product of the given number and the sum of its digits is equal to 144, then the number is:
Answer Choices: (A) 24 (B) 26 (C) 42 (D) 46 (E) None of these | tiger_lab_math |
Let $\mathcal{P}$ be the parabola in the plane determined by the equation $y = x^2.$ Suppose a circle $\mathcal{C}$ intersects $\mathcal{P}$ at four distinct points. If three of these points are $(-28,784),$ $(-2,4),$ and $(13,169),$ find the sum of the distances from the focus of $\mathcal{P}$ to all four of the int... | tiger_lab_math |
during a thanksgiving weekend , a car rental company rented 6 - tenths of their vehicles , including two - fifths of the 4 wds that it had . if 40 % of the vehicles are 4 wds , then what percent of the vehicles that were not rented were not 4 wds ? Let's write a Python program to solve it. | tiger_lab_math |
the average age of a group of 10 persons was decreased by 3 years when one person , whose age was 44 years , was replaced by a new person . find the age of the new person ? Let's program in Python in the response. | tiger_lab_math |
the length of a train and that of a platform are equal . if with a speed of 90 k / hr , the train crosses the platform in one minute , then the length of the train ( in meters ) is ? Let's program in Python in the response. | tiger_lab_math |
soymilk are sold in packages of two or 3 only . if hannah bought 17 soymilk exactly , what could be the number of large packs hannah bought ? Let's write a Python program. | tiger_lab_math |
A high school has 500 students, out of which 300 are boys and 200 are girls. The probability of a boy graduating with honors is 0.6, whereas the probability of a girl graduating with honors is 0.4. What is the probability that a student chosen at random from the school will be a girl who does not graduate with honors? | tiger_lab_math |
In a party there are 5 couples. Out of them 5 people are chosen at random. Find the probability that there are at the least two couples?
Answer Choices: (A) 5/21 (B) 5/27 (C) 5/25 (D) 5/24 (E) 5/22 | tiger_lab_math |
For how many integer values of $n$ between 1 and 120 inclusive does the decimal representation of $\frac{n}{120}$ terminate? | tiger_lab_math |
In a substitution cipher, the letter 'C' is always replaced with the letter 'E', and the letter 'R' is always replaced with the letter 'T'. If the original message contains 5 instances of the letter 'C' and 7 instances of the letter 'R', how many instances of the letter 'E' and how many instances of the letter 'T' woul... | tiger_lab_math |
sheila works 8 hours per day on monday , wednesday and friday , and 6 hours per day on tuesday and thursday . she does not work on saturday and sunday . she earns $ 288 per week . how much does she earn in dollars per hour ? Let's write a Python program to solve it. | tiger_lab_math |
Let G be a group of order 27. Show that there exists an element a in G such that a has order 27, which means a generates the cyclic group C27. | tiger_lab_math |
vishal invested 10 % more than trishul . trishul invested 10 % less than raghu . if the total sum of their investments is rs . 6358 , how much amount did raghu invest ? Let's write a Python program. | tiger_lab_math |
A mass-spring system has a natural frequency of 2 radians per second. If the mass of the system is 0.5 kg and the spring constant is 4 N/m, find the equation of motion of the system if it is displaced 2 cm from its equilibrium position and released. (Assume no damping.) | tiger_lab_math |
Using second-order logic, determine if the statement "For all integers a and b, a+b = b+a" is true. | tiger_lab_math |
a plot abcd is as shown in figure , where af = 30 m , ce = 40 m , ed = 50 m , ae = 120 m . find the area of the plot abcd ? Let's write a Python program. | tiger_lab_math |
In a gaming competition, a total of 25 players are participating. The first prize will be given to the player who wins the most number of games, and a second prize will be given to the player who wins the second-most number of games. However, there is a condition that each player cannot play against any other player mo... | tiger_lab_math |
Let $A$ be a point on the circle $x^2 + y^2 - 12x + 31 = 0,$ and let $B$ be a point on the parabola $y^2 = 4x.$ Find the smallest possible distance $AB.$ | tiger_lab_math |
When three positive integers are divided by $12$, the remainders are $7,$ $9,$ and $10,$ respectively.
When the sum of the three integers is divided by $12$, what is the remainder? Let's program in Python in the response. | tiger_lab_math |
keats library purchases a number of new books , all in the category of biography , and the library does not acquire any other books . with the addition of the new biographies , the biography collection of the library amounts to 32 % of the total number of books in the library . if prior to the purchase , only 20 % of t... | tiger_lab_math |
in a party every person shakes hands with every other person . if there are 105 hands shakes , find the number of person in the party . Let's program in Python in the response. | tiger_lab_math |
if $ 10 be allowed as true discount on a bill of $ 110 due at the end of a certain time , then the discount allowed on the same sum due at the end of double the time is : ? Let's program in Python in the response. | tiger_lab_math |
Triangle $ABC$ is inscribed in circle $\omega$ with $AB=5$, $BC=7$, and $AC=3$. The bisector of angle $A$ meets side $\overline{BC}$ at $D$ and circle $\omega$ at a second point $E$. Let $\gamma$ be the circle with diameter $\overline{DE}$. Circles $\omega$ and $\gamma$ meet at $E$ and a second point $F$. Then $AF^2 = ... | tiger_lab_math |
Find all possible isomorphisms between the group G of order 20 and each of the following groups:
a) the cyclic group C20
b) the dihedral group D10
c) the non-abelian group of order 20. | tiger_lab_math |
in a group of ducks and cows , the total number of legs are 28 more than twice the no . of heads . find the total no . of buffaloes . Let's write a program. | tiger_lab_math |
a certain sum is invested at simple interest at 18 % p . a . for two years instead of investing at 12 % p . a . for the same time period . therefore the interest received is more by rs . 840 . find the sum ? Let's program in Python in the response. | tiger_lab_math |
A circle rests in the interior of the parabola with equation $y = x^2,$ so that it is tangent to the parabola at two points. How much higher is the center of the circle than the points of tangency? | tiger_lab_math |
cathy and jim begin running at the same time and they start at the same place on a straight path . cathy runs at 10 miles per hour , and jim runs at 6 miles per hour . after 18 minutes , cathy stops to stretch . if it takes cathy 27 minutes to stretch and jim continues to run during this time , how many minutes will it... | tiger_lab_math |
the pressure someone experiences as he or she dives deeper and deeper in the ocean increases linearly . on the surface , the pressure is close to 15 pounds per square inch . 33 feet below the surface , the pressure is 30 pounds . if 25000 pounds per sq inch can crush your bones , what depth is extremely dangerous for h... | tiger_lab_math |
Consider a game of Matching Pennies between two players, player A and player B. Player A has two pennies, one with heads and one with tails, while player B also has two pennies, one with heads and one with tails. Both players simultaneously place one of their pennies face-up on the table. Player A wins if both pennies ... | tiger_lab_math |
the distance from the x - axis to point p is half the distance from the y - axis to point p . if the coordinates of p are ( x , - 6 ) , how many units is p from the y - axis ? Let's write a Python program to solve it. | tiger_lab_math |
x is a positive integer less than 400 . when x is divided by 7 , the remainder is 1 ; when x is divided by 3 , the remainder is 2 . how many x are there ? Please respond by writing a program in Python. | tiger_lab_math |
if x is equal to the sum of the integers from 40 to 60 , inclusive , and y is the number of even integers from 40 to 60 , inclusive , what is the value of x + y ? Let's write a program. | tiger_lab_math |
The equation $x^3 - 4x^2 + 5x - \frac{19}{10} = 0$ has real roots $r,$ $s,$ and $t.$ Find the area of the triangle with sides $r,$ $s,$ and $t.$ | tiger_lab_math |
in the recent , climate conference in new york , out of 700 men , 500 women , 800 children present inside the building premises , 20 % of the men , 40 % of the women and 10 % of the children were indians . find the percentage of people who were not indian ? Please respond by writing a program in Python. | tiger_lab_math |
Consider the following graph represented by adjacency matrix:
$$
\begin{bmatrix}
0 & 1 & 0 & 1 & 0\\
1 & 0 & 1 & 0 & 1\\
0 & 1 & 0 & 1 & 0\\
1 & 0 & 1 & 0 & 1\\
0 & 1 & 0 & 1 & 0
\end{bmatrix}
$$
What is the maximum number of perfect matchings that can be identified in this graph? | tiger_lab_math |
For a certain weekend, the weatherman predicts that it will rain with a $40\%$ probability on Saturday and a $50\%$ probability on Sunday. Assuming these probabilities are independent, what is the probability that it rains over the weekend (that is, on at least one of the days)? Express your answer as a percentage. Let... | tiger_lab_math |
A is a set containing 8 different positive odd numbers. B is a set containing 7 different numbers, all of which are members of A. Which of the following statements CANNOT be true?
Answer Choices: (A) The range of B is even. (B) The mean of A is even. (C) The mean of B is even. (D) The range of A is equal to the range o... | tiger_lab_math |
Suppose that there is a system that continuously cycles between two states, A and B, with a cycle time of 10 seconds. Using temporal logic, determine if the system is currently in state A if it was in state B 25 seconds ago. | tiger_lab_math |
if 2 ^ 5 , 3 ^ 3 , and 13 ^ 2 are all factors of the product of 936 and w where w is a positive integer , what is the smallest possible value of w ? Let's program in Python in the response. | tiger_lab_math |
the ratio of three numbers is 1 : 2 : 3 and the sum of their squares is 2744 . the sum of the numbers is ? Please write a program to solve it | tiger_lab_math |
In how many ways can the number 10 be written as a sum of positive integers without regard to order? | tiger_lab_math |
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