Some Fruit Punch

There is a party tonight at the Puzzlaria Tavern and Ross asked me to get some orange juice and pineapple juice for the Fruit Punch he is preparing. Being already giddy with all the drinks I had had, I tried some mixing of my own. Grabbing the large orange juice jar (lets call it jar O), I added enough juice into the second jar containing pineapple juice (jar P) till its contents were doubled. Then, I poured some of the mixture back into jar O to double its contents.

Fairly sloshed now, I decide a good way to settle matters would be to again pour from Jar O into Jar P to double P’s contents and proceed to do so. At this point I realize there is an equal quantity of pineapple juice in each jar. If Jar P now has one more litre of orange juice than pineapple juice, can you tell this poor muddle-headed soul, how much more orange juice than pineapple juice there is in Jar O?

Help me! The party is in full swing and Ross is getting impatient !!

Serial Killer

Provided by Cam.

Given the following series: 56__194__442__848__1484__2446__3854__?

What is the number corresponding to the 8th term, the 42nd term and the 200th term ?

Fermat upside-down

The world-famous Fermat's Last Theorem states that the equation aⁿ+bⁿ=cⁿ has no integer solutions where n > 2. It was proposed by Pierre de Fermat in 1637, and was not proven until 1995, 358 years later, when Andrew Wiles proved it. It took over 100 pages for him to do this, and even then, he relied on others' work in elliptic curve theory over the previous forty years.

I have a much simpler theorem I want proved. Prove that the equation n^a+n^b=n^c has no integer solutions where n > 2.

33^m - 7^n

If m and n are positive integers, what is the smallest absolute difference of 33^m - 7^n ?

Going home

A chauffeur always arrives at the train station at exactly five o'clock to pick up his boss and drive her home. One day his boss arrives an hour early, starts walking home, and is picked up by the chauffeur on the way out to the train station. They arrive at home twenty minutes earlier than usual. How long did she walk before she met her chauffeur?

How odd!

Is the the number 124 (in base 5) odd or even? More generally, how can you tell if a number in any base, is odd or even? (Don't worry about the obvious issues for bases > 10).

32 card trick

Alfred, Brian, Christopher and Damon play with a deck of 32 cards. Damon deals them out unequally, then says: "If you want us to have the same number of cards, do exactly as I say. You, Alfred, divide half of your cards between Brian and Christopher. Then, Brian, you do the same with Christopher and Alfred. Finally, Christopher, you follow suit with Alfred and Brian." How did Damon distribute the cards?

1,000,000,000!

How many 0s (zeroes) are there at the right hand end of 1,000,000,000! ?

That's one billion factorial ≈ 1.7 x 10^(20,723,265,847).

Sharing marbles

A group of children share marbles from a bag. The first child takes one marble and a tenth of the remainder. The second child takes two marbles and a tenth of the remainder. The third child takes three marbles and a tenth of the remainder. And so on until the last child takes whatever is left. Knowing that all the children end up with the same number of marble, how many children were there and how many marbles did each one get?

Family planning

My mother dreamed of having nineteen children. Her dream didn't come true. But I have three times as many sisters as first cousins, and I have half as many brothers as sisters. How many children did my mother have?

It's all not true.

Table mat

Feb 12, 7:05 PM. I've rephrased the question completely.

A table mat is made from red and white beads. There is an inner rectangle made entirely using white beads. It has a border (on all four sides) made from a single layer of red beads. Which mats can have as many white beads as red beads?

Next

What's the next term in the sequence: 77, 49, 36, 18, ... ?

Generations apart

A woman and her grandson have the same birthday. For six consecutive birthdays, she is an integral multiple of his age. How old is the grandmother at the sixth of these birthdays?

3D diagonal

What is the length of the diagonal of a 3D rectangular prism (a cuboid) whose surface area is 94, if the sum of the lengths of the sides is 48?

Weighty problem

You have goods to sell whose weights are whole numbers of kilograms, between 1 and 80. Using only 4 weights, what do they need to be?

Back to your roots

Evaluate rt(1 + rt(7 + rt( 1 + rt(7+......)))), where rt means square root. I have designed this so that the answer is an integer.

Prove that...

ok, here are some proving questions:

1) if H is the set to infinity of all positive integers, none of which have prime factors greater than 3, the PROVE THAT: the sum of the reciprocals of elements of H are smaller than 3

2) PROVE THAT any prime number (2^(2^n) + 1) cannot be represented as a difference of 2 fifth powers of integers.

NOTE: just for confirmation, this 2^2^n is the same like my last question, it is a continuation of it that i just found.

3) a₁ , a₂ , a₃ , ..... , a2009 are real numbers.

a₁ + a₂ + a₃ + ..... + a2009 >= 2009²

and, a₁² + a₂² + a₃² + ..... + a2009² <= 2009³ + 1

then, PROVE THAT 2008 <= a k <= 2010 for all values of k belonging to the set (1,2,3,4,.....,2009)

NOTE: here k is a subscript

Senior class

It appears that an ingenious or eccentric teacher being desirous of bringing together a number of older pupils into a class he was forming, offered to give a prize each day to the side of boys or girls whose combined ages would prove to be the greatest. Well, on the first day there was only one boy and one girl in attendance, and, as the boy's age was just twice that of the girl's, the first day's prize went to the boy. The next day the girl brought her sister to school, and it was found that their combined ages were just twice that of the boy, so the two girls divided the prize. When school opened the next day, however, the boy had recruited one of his brothers, and it was found that the combined ages of the two boys were exactly twice as much as the ages of the two girls, so the boys carried off the honors of that day and divided the prizes between them. The battle waxed warm and on the fourth day the two girls appeared accompanied by their elder sister; so it was then the combined ages of the three girls against the two boys, and the girls won off course, once more bringing their ages up to just twice that of the boys'. The struggle went on until the class was filled up, but as our problem does not need to go further than this point, to tell the age of that first boy, provided that the last young lady joined the class on her twenty-first birthday. What is the first boy's age (at the start).

3 Feb 2010, 6:10 AM. Express your answer in days.

Badly made wagons

When they started off on the great annual picnic every wagon in town was pressed into service. Half way to the picnic ground ten wagons broke down, so it was necessary for each of the remaining wagons to carry one more person. When they started for home it was discovered that fifteen more wagons were out of commission, so on the return trip there were three persons more in each wagon than when they started out in the morning. Now who can tell how many people attended the great annual picnic?


I should have called this one "Buggy wagons" - too late.

Ball breaking

Your job is to determine the highest floor of a 100 floor building from which a snooker ball may be dropped without breaking. You have two identical snooker balls. If a ball doesn't break, it is completely unharmed. If both balls break before you have determined the highest floor, or you drop the balls more times than is necessary, then you'll need to to seek alternative employment. What is the maximum number of times you have to drop the balls?

For consistency, assume the floors are numbered 1 to 100 in your explanations. Also assume, that there is no need to drop a ball from floor 1 (the ground floor).

Doubling up

Five schoolchildren were weighed in pairs. Their (combined) weights were:
129, 125, 124, 123, 122,121, 120, 118, 116 and 114 lbs. What were their individual weights?

What's the Next One?

Look at this series:

1 -- 6 -- 2 -- 7 -- 3 -- 8 --

What are the next 6 numbers in the series?

Finger counting

Start counting on the fingers on one hand thusly: 1 thumb, 2 index finger, 3 middle finger, 4 ring finger, 5 little finger, 6 ring finger, 7middle finger, 8 index finger, 9 thumb and so on.

Which finger would you end up on if you were counting to 123456789? That over 123 million in case you think it's a trick question. Assume that you aren't carted of to the funny farm while doing it.

6174

Take any 4 digit number which has at least 2 digits that are different (e.g. 1112 is OK).

Rearrange the digits to produce the largest possible number and the smallest possible number. Subtract the smaller from the larger one. After at most seven steps, you'll end up with 6174, and then you'll only get 6174 after that. Prove it.

e.g. for 6174 you'd do 7641 - 1467 = 6174

Sadly, this one's easily to find on the Internet, so please don't cheat. I've only looked enough to confirm it's there.

Rational or not

If x and y are irrational, prove that x^y can be rational.

You only need to find one example to make the proof.

Solve this

WWWDOT - GOOGLE = DOTCOM

Realizing ofcourse that values of M and E could be interchanged.
No leading zero's are allowed.
it is => wwwdot minus google equal to dotcom

each letter is a number

A Poorly Designed Clock

The hour and minute hands of a clock are indistinguishable. How many moments are there in a day when it is not possible to tell from this clock what time it is?

Sequencing the Digits

How many ways are there to write the numbers 0 through 9 in a row, such that each number, other than the left-most, is within one of some number to the left of it.

The Gambler's Ruin

At the beginning of play, gamblers A and B have m and n chips, respectively. Let their probabilities of winning be p for A, and q=1-p for B. After each play, the winner gets a chip from the loser, and play continues until one of the players is ruined.

What is the probability of A being ruined?

The Wrong Letter

An individual has written k letters to each of k different friends, and addressed the k corresponding envelopes.

How many different ways are there to place every letter into a wrong envelope?

The Monkey and the Coconuts

Five men and a monkey were shipwrecked on a desert island, and they spent the first day gathering coconuts for food. They piled them all together and went to sleep for the night.

One awakened and, wanting to avoid a fuss in the morning, decided to take his share. So he divided the coconuts into five piles. Having one coconut too many to make five equal piles, he gave the extra coconut to the monkey. After hiding his share, he made a single pile of the remaining coconuts and went back to sleep. (smart monkey - he also hid his coconut)

Each of the other four men did the same procedure, as the first man, at different times during the night. Each hid his share, gave one to the monkey to even the shares, and went back to sleep.

In the morning, each man awakened to find a much smaller pile of coconuts. No one commented on the reduced quantity, because each knew he was guilty.

How many coconuts were in the original pile? ( smallest quantity to satisfy the conditions)

Four Rowers and a Swan

Four rowers wish to cross the river by means of a boat that can only hold two men. The rower R1 needs one minute to cross the river alone, and the rowers R2, R3, and R4 need 2, 6, and 9 minutes respectively. Since the boat will hold only two men at a time, the rowers planned an optimum strategy for crossing the river. At the moment the rowers implement their plan, a swan starts to swim across the river at a speed of 60 feet per minute. The swan reaches the other side precisely at the moment the rowers complete their plan of getting all across the river.

What was the plan? Who rows with who?
How long did it take to accomplish the plan?
How wide is the river?

Two Cars and a Bird

Two cars, setting out from points A and B are 140 miles apart. They move toward each other on the same perfectly straight road, until they collide at C. Their speeds are 30 mph and 40 mph. At the very instant they start, a bird takes flight from point A heading straight toward the car which has left point B. As soon as the bird reaches the other car, it turns and changes direction. The bird flies back and forth between the two cars at a speed of 50 mph until the cars collide.

How long is the birds total flight path?

Gold Bars in a Pile

Three robbers A, B, and C each place stolen gold bars in a common pile, their shares being 1/2, 1/3, and 1/6 of the total amount, respectively. Next, each man takes the bars from the pile until none remain. Now A returns 1/2 of what he took, B 1/3, and C 1/6. When the last pile of gold is divided equally by the three men, it turns out that each has the amount he had at the beginning.

What was the total number of gold bars? (look for the minimum number of gold bars to satisfy the conditions)

A Particular Square Number

Find a square number such that, when five is added or subtracted, the result is again a square number!

Rabbits and More Rabbits

A man bought a pair of rabbits. How many pairs of rabbits can be produced from the original pair, in a year, if it is assumed that every month each pair begets a new pair that can reproduce after 2 months?

Probably relatively prime

What is the probability that two randomly chosen positive integers are relatively prime?

Two numbers are relatively prime if they have no common factors.

Number Logic

Two perfect logicians S and P are told that integers x and y are such that:
1 < x < y and that x + y < 100.

S and P are then given the values x+y and x*y respectively and privately. S and P know they have each been given the sum and product respectively. They then have the following conversation:

P: I cannot determine the two numbers.
S: I knew that.
P: Now I can determine them.
S: So can I.

Given that they spoke the truth, what are the two numbers?

Dec 24, 5: 10 AM. I made a significant modification to the conditionals for x and y. The place I found this sneakily used computers in deriving the solution. Unfortunately, my modifications altered the situation too much and invalidated the conversation between S and P.

In fact x < 10 and y < 20; but S and P don't know that. Unfortunately, I doubt that extra info is going to usefully simplify the problem solution. So apologies in advance if this one has a high labour requirement.

Nearly one

Prove that 0.9999999999..... = 1.

Absolute maximum

Define the functions Abs(x) and Max(a,b) as:

Abs(x) = (x if x>=0) or (-x if x < 0) e.g. Abs(-2) = 2
Max(a,b) = (a if a >= b) or (b if b > a) e.g. Max (2,-3) = 2

You might think these functions are independent, but they're not, because it is possible to express one in terms of the other. What are the equivalence formulae?

Average but special numbers

481 = the average of 148, 184, 418, 481, 814 and 841.

What are all the other such 3 digit numbers?

Insects on scale

There is a 100 cm long scale with 100 insects on it. At time t = 0, each starts moving either towards 0 or towards 100 at a speed of 1 cm/s. They fall off the scale when they reach any one of the ends.

You have to tell me this: what is the minimum time at which I can be sure that the scale has no insects left.

Spaceship Enterprise

Spaceship Enterprise starts at 3 am from planet Offensive to target planet Defensive.
It travels for one hour and then reduces its speed to 2/3rd for the next hour. The total distance at the end of the first hour is a four-digit square and at the end of the second hour a four-digit cube having no three digits identical. Defensive is informed by Informative about all this and it fires a missile Destroyer at 4 am itself. If the separation between the two planets is twice the initial speed of Speedster, at what speed must Destroyer travel to destroy Enterprise midway at 5 am?

* Speedster is the Enterprise

Boat Speed

Three boats, X, Y and Z are there to let the touristas tour a lake. They leave at different times and take 30 minutes to tour the lake. The time difference between X and Z is 20 minutes and that between Y and Z is 15 minutes. Given that the sum of time (in minutes) is 40 for all three to leave, after starting the countdown, at what time did each boat leave?

Peer to Peer

Suppose you are just starting to download a 4.2 GB file on a peer to peer network. There are 20 other people also up and/or downloading this same file. Nineteen have 20% and one has 100%, and you of course have 0%. (Assume each 20% is of random sections... none are the exact same sections.)

Everyone can download up to 500 k/sec and will upload 10 k/sec. When they get 100% each will begin uploading at 50k/sec.

You must understand that on peer to peer networks you request a random section of data you need and the request will be put on hold by others who have the data. Later someone will ask if you still need it and if you answer yes, it will be sent.

Assuming everyone stays active till everone reaches 100%, how long will it take for you to reach 100%?

Circles

You have 2009 concentric circles, with radii 1 to 2009.
a point is taken on the largest circle (the one with radius 2009)
from here, tangents are drawn to all the other 2008 circles.

How many of these tangents have a length that is a whole number?

Compute area between two curves

Given two simple equations y=sqrt(x) and y = -(x^2)/5

What is the area between the curves from x=0 to x=6?

Collatz Series 2

If you don't know what the Collatz series is, see the previous post.

We learned lots of the initial numbers give a series that contain 40.

Give two initial numbers between 100 and 200 that are ODD and don't contain a 40 in their series.

Water!

720!

The value of 720! is a large number.
Just how many digits does it have?
How many of the digits are 0?
How many of the other digits from 1 to 9?

I guess this is to hard.
Here's a link that will make it easy.



http://www.numberempire.com/factorialcalculator.php

1 to 9

The sum of the nine single digits 1,2,3,4,5,6,7,8,9 equals 45
and the product of the nine digits is 362,880.

Find a different string of nine single digits that have the same sum (45) and product (362,880) as the "123456789" string.

(To be different, some digits will be missing and some will occur more than once)

For example, a string of nine digits could be "112233445" but it would not sum to 45, and the product would not be 362,880, so it meets neither condition.

To easily identify a string, put the digits in increasing order.

Sum of digits

No new post? Ok let's do a replay, since Chris taught us how to do this.

The sum of the digits of 173^371 is A. The sum of the digits of A is B. The sum of the digits of B is C. What is the value of C?

fffffour

The sum of the digits of 4444^4444 is A. The sum of the digits of A is B. The sum of the digits of B is C. What is the value of C?


For info on modular arithmetic try:
http://www.cut-the-knot.org/blue/Modulo.shtml

Chessboard Steps

Starting at the bottom left-hand corner of a chessboard, how many different ways are there of moving to the top right-hand comer if

(a) You can move only one square at a time and
(b) You can move bottom to top or left to right or a diagonal combination of the two?

Tom's sweet tooth

Tom bought 1 lb of jellybeans and 2 lb of chocolate for $16. A week later, he bought 4 lb of caramels and a pound of jellybeans for $24. The next week, he bought 3 lb of liquorice, 1 lb pound of jellybeans and 1 lb of caramels for $12.

How much would he have to pay for a pound of each of the four kinds of candy?

Curious Thing

Can you prove that a triangle with sides that can be written in the form n²+1, n²-1 and 2n ( where n>1) is right angled ?

Cross that bridge

Three people (A, B and C) need to cross a bridge. A can cross the bridge in 10 minutes, B can cross in 5 minutes, and C can cross in 2 minutes. They have a bicycle and any person can cross the bridge in 1 minute using it. All three men and the bicycle start together on one side of the bridge. Only one person can use the bicycle at a time. The bicycle can be mounted and dismounted anywhere. All three men can be crossing together. What's the shortest time that all three men can cross the bridge in?

Prime time

Show that any odd prime can be written as the difference of the squares of two integers.

Paint balling

You have a bag with four balls, each is a different colour. You draw two balls from the bag, one at a time. You paint the first ball to match the colour of the second ball, you then put both balls back in the bag. What is the expected number of drawings before all four balls are the same colour?

Assume very quick drying paint.

Prime Density

Primes numbers become less frequent as the numbers get larger. For example between 0 and 100 there are 25 primes. From 100 to 200 there are only 21 prime numbers.

If we use this step size of 100 (numbers that end with a double zero), there's a step where there is only one prime number in that range of 100 numbers. Actually there's an infinite number. But what step is the lowest that has only 1 prime?

Sample answer = “1000 to 1100" (that step has 16 primes)

Smallest Integer #223

What is the smallest integer greater than 0 that can be written entirely with 0s and 1s and is exactly divisible by 223?


I don't know how difficult this will be for others, but it took me 4 seconds.

No binary numbers

Smallest integer

What is the smallest integer greater than 0 that can be written entirely with 0s and 1s and is exactly divisible by 225?

This one's fairly easy.

Elementary Arithmetic

In the introduction to an early literary work on elementary arithmetic, the author looking back over his life, muses: "I was once x years old in the year x cube. I am now x square years of age and in another x years my only son will be y years of age in the year y square."

In what year was the author of the manuscript born?

Wow, DVD Film

Solve: WOW/DVD=.FILMFILMFILMFILM ...

Each letter is a unique digit.

Humphrey Thou Sand

A camel has to transport as many as possible of an available 3000 bananas. For every comlete mile it walks, it must consume 1 banana (regardless of its load). It can only carry up to 1000 bananas at a time. What is the maximum number of bananas it can transport to a place a 1000 miles away?

Any way

Solve for A, B ,C ,D and E, the palindromic number equation:
AA + BCB = DEED

Literary Digest 2

A bookworm eats it way through a ten volume encyclopedia set on a bookshelf. Including the covers each of which are 2 mm thick, each volume is 5 cm thick. The bookworm starts at page 1 of volume 1 and makes its way through to the higher numbered volumes until it emerges from volume 10. Whilst partaking of this edible edification, how far did the bookworm travel?

Assume the bookworm goes in a straight line perpendicular to the covers and pages. Don't sweat it as to how the bookworm got to the starting point.

6

Ok this one's a bit tougher than the "17" problem.

Given an integer that ends with 6 (units column), when you move the 6 from the end to the beginning, the new number is 4 times the original number. What is the smallest number that does the trick?

Pretending that the first number is 3216, we want 4*3216 = 6321.
Obviously this example fails.

17

How many three digit numbers are divisible by 17?

Carry on towering

What is the range of convergence of x^(x^(x^(x^(x... ?
What are the corresponding minimum and maximum values of the expression?

You will find the following very useful:
http://en.wikipedia.org/wiki/E_%28mathematical_constant%29
Go half way down the page to x^(1/x). Have a glance, then follow the tetration link.

This is slightly modified from the original post in view of the info provided.

Only basic differential calculus is required.

Twoering inferno

What is the limit value of:

sqrt(2)^(sqrt(2)^(sqrt(2)^(sqrt(2)^(... ?

That's an infinite power tower of sqrt(2)s.

xth root of x

Just how big can x^(1/x) be? Assume x is real number.

Who squares?

How many squares are there on a chessboard?

Assume a simple chess board with no margin.

Just making a point

Prove that the three perpendicular bisectors of a triangle always meet at a point.

Here's a link so you can see what I'm blathering on about:
http://www.analyzemath.com/Geometry/Circumcircle/Circumcircle.html

On that page, click the "click here to start button". Ignore the circle that you see. Drag the triangle around and see what happens.

Classical geometry on allows the use of a straightedge and compass. Rulers and protractors are not allowed. Trigonometry is not allowed.

However for this problem any reasonable proofs are OK with me.

Happy Birthday

I expect everyone knows this one, but it's better than nothing.

In a random group of 23 people, there is a slightly better than a 50% chance that at least two of them share the same birthday. How is that possible?

How big would the group need to be to get the chance of a shared birthday up to at least 95%?

Discard all leap year complications. Assume each birthday is equally likely. Only consider the day and month, not the year of birth. It is not a trick problem.

Treasure Revisited

The solution to the original post (yesterday) was to quick.
But we can hopefully find the numeric solution with some given data.

Let's say the Elm tree is at (30,35) and the Oak is at (80,44).
These are x,y coordinates.

Find the x,y of the treasure within a 1/2 unit.

Treasure hunt

A man has a treasure map. It shows an island with a gallows, an elm tree and an oak tree. The map has instructions. They say go to the gallows and walk to the elm tree, counting your paces. Then turn 90 degrees right and walk the same number of paces as you had just counted. Mark that spot. Go back to the gallows. Now walk to the oak tree, counting your paces. Turn 90 degrees left and walk the same number of paces as you had just counted. Mark that spot. The treasure is buried halfway between the two marked spots.

When he got to the island, he could find the trees, but not the gallows. How can he find the treasure?

There are quite a few ways to answer this problem. One is partly outside the box. It is a famous problem.

You're so special

Numbers like pi and e are usually regarded as being special.
Are there any numbers that are not special?
Prove your assertion.

Three Magical Numbers

There are three simple numbers which has a mysterious property. Sum of any two numbers is a perfect square. Can you tell the numbers ?

Do you know Square Root

Leibniz , a German mathematician stated that

SQRT(1 + SQRT(-3)) + SQRT(1 - SQRT(-3)) = SQRT(6)

What do you think ?

Hole in a Cube

Is it possible to cut a hole in a cube in such a way that the larger cube can be passed through the hole ?

Do you know Addition

Which one is bigger ?

99⁵⁰ + 100⁵⁰ or 101⁵⁰

How ?

aye aye

If i is the usual unit imaginary number (Sqrt(-1)), what is i^i ?

Patient, Pills, and Probability

A Patient is taking one each of 5 different types of pills every day but he don’t like having to open and close 5 different bottles, so at the beginning of each (30-day) month he put 30 of each type of pill into one big bottle. When it is time to take your pills, he draw them out of the big bottle one at a time until he have (at least) one of each type.

On the last day of the month he will draw exactly 5 pills and they will all be different (because that’s all that’s left in the bottle), but on other days he will generally have to draw more than 5 pills in order to have (at least) one of each type. So, the question is: On the first day of each month (when there are 150 pills in the bottle), how many pills, on average, must he draw from the bottle in order to have (at least) one of each?

Magic !

A magician has one hundred cards numbered 1 to 100. He puts them into three boxes, a red one, a white one and a blue one, so that each box contains at least one card. A member of the audience selects two of the three boxes, chooses one card from each and announces the sum of the numbers on the chosen cards. Given this sum, the magician identifies the box from which no card has been chosen.

How many ways are there to put all the cards into the boxes so that this trick always works? (Two ways are considered different if at least one card is put into a different box.)

Trigonometry

Tan20.Tan40.Tan60.Tan80=?

This seems like a textbook question, but i want able to find it in the 6 books that I own.

Little Two

A number ends with the digit 2. If we move this 2 from the last position to the first, the new number is twice the original. What's the number ?

So You Know Maths

Can you think of a non prime number so that all its divisors is visible in the number itself ?

For example divisor of 13 is 1 and 13 so it satisfy the first condition but its a prime so not the second. You have to find a non prime. Your time starts now...

Hoover Dam Bridge 2

This is a continuation of the previous item.

A steady convoy of cement trucks are dumping cement into two
hoppers that feed two pumps. The pumps put the wet cement flow
into an 8 inch diameter pipe. The pipe runs almost straight
down 300 foot to the arch footing. The pumps run at 200 psi.

How much cement (in cubic yards) will be pumped into the reinforced
form in the 12 hour period from dusk to dawn?

Octagon Problem

Archimedes looked into the Octagon and immediately told that it is possible to create a perfect square out of it, if you cut it into five pieces and re join it.

Can you do that ?

Irrational Roots

If

P(x) = x⁴ - 2x³ - x² - 9x + 2 has only irrational roots, of which their real roots are between -1 and 3.

How does one find x?


-- abdeali kothari

Triangle Trio

There are three Triangles which has integer sides and whose perimeter is equal to the area. The only other information about them is they are not right triangles.

Can you find them ?

The Value

1¹⁰+2¹⁰ +3¹⁰ +4¹⁰.... +1000¹⁰ = ?

Believe it or not famous Mathematician Jakob Bernoulli determined the value in less than 10 minutes.

How long will you take ?

Areas of Circles

There are 3 congruent circles inscribed into an equilateral triangle. If one of the sides of the triangle is 6+2 long, what’s the sum of the areas of the 3 circles? Express in terms of pi.




-- James Yu

Vertical Velocity

A stone is thrown vertically upwards so as to reach 10 meters height in 10 seconds.

Now the question is not so simple, think! the velocity, with which the stone should be thrown to attain the same height in half the time will be more than the first velocity or less than the first velocity? What will be the required velocity?

Fancy Window

Ok, how many began figuring out the 68 inch string problem? Ok, it was a joke. Now for the real problem. (change the circle to a semi-circle with a diameter equal to one side of a rectangle)

Joan's house has a window in the shape of a rectangle surmounted with a semicircle. For a given total perimeter of 268 inches, what are the dimensions of the window if it allows the maximum amount of light?

Hint: Remember the 68 inch string solution?

68 inches of string

Divide the 68 inch string into two parts. With one
part make a square figure, the other make a circle.

What are the length of two parts of string that will
maximize the sum of the figure's area.

Warning: This one is pretty messy.

Plane and Space

Visualize !
One plane divides space into two parts. Two planes passing through a point divide space into four parts. See it with your two palms as two planes.

The question is to find out, How many parts three planes passing through a point divides the space ? What about four planes ?

Minimize triangle area

A right triangle, shown below,the right angle at origin. What are the dimensions of x and y such that the hypotenuse passes thru point (2,5) and the triangle area is minimized.

Another Image (click)

Power of Power

Can you give a method and Calculate


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