North Dakota Mathematics Talent Search 2004-2005
PROBLEM SET 2
(Due 02/01/2005)
- There is a chain of stores called 7-11. One day a customer
arrived in one of these 7-11 shops and selected four items.
He then approached the counter to pay for these items. The
salesman took his calculator, pressed a few buttons and said,
"The total price is $7.11."
The customer tried a joke, "Why? Do I have to pay $7.11
because the name of your shop is 7-11?"
The salesman didn't get the joke and answered, "Of course not!
I have multiplied the prices of these four items and I have just
given you the result!"
The customer was very surprised. "Why did you multiply these
numbers? You should have added them to get the total price!"
The salesman said, "Oh, yes, I'm sorry, I have a terrible
headache and I pressed the wrong button!"
Then the salesman repeated all the calculations, i.e. he added
prices of these four items, but to his and customer's great
surprise, the total was still $7.11.
Your task is to find the prices of these four items!
- In the times of Ancient Greece, Zeus commissioned a blacksmith
to make an iron ring that would go around the Earth, and the
blacksmith was asked to make the diameter of the ring match the
diameter of the Earth exactly. (Assuming here that the Earth is
a perfect sphere!) The poor blacksmith, however, made a mistake.
He made a ring that was just one meter longer in circumference
than it was supposed to be. Nevertheless, Zeus placed the ring
"around" the Earth, and it was made to touch the Earth at one
point. The question is, how much did the ring "stick out" on
the other side of the Earth? What kind of animal would be able
to squeeze under the gap between the Earth and the ring?
An ant? A mouse? A cat? A horse?
- A tall man walks into a jewelry store and places a bag full of
old mint coins on the counter. The jeweller picks the bag and
empties the coins out in front of them.
"How much will you give me for these coins?" asks the man.
"Ah, 1907 Indian head pennies. And they all look to be in very
good condition," says the jeweller as he examines them with an
eye piece, "I'd say each was worth $100. You've got 12 coins
here, right?"
"Well, yes, but there's a problem you should know about. You see,
when I bought these coins, the guy who sold them to me said that
one of them was counterfeit," admits the man.
"How about that? An honest thief?" exclaims the jeweller.
A moment of silence passes as the tall man tries to discern if
the jeweller was talking about him or the guy who sold him the
coins.
"Right, well, how can we go about finding the counterfeit?" asks
the tall man, now fidgeting slightly.
"Hmmm. I've got an idea. Most counterfeit coins are made with
cheaper materials and therefore don't weigh the same amount as the
real thing. I've got a balancing scale right here. Now all we
have to do is figure out how to weigh the coins so that we can
identify the fake."
The tall man smiles "Oh, this is easy. We can do it in just three
weighings. Not only that, but we can tell whether or not the fake
coin is lighter or heavier than the rest too."
"Really? How do you know that?" asks the jeweller, surprised at
the apparent talent of this man.
The two men slump down over the scale and start fiddling with the
coins. Can you figure out how to find the fake?
- Find all possible positive integers
A, B, C, D
such that simultaneously
AB = 2(C+D),
CD = 2(A+B).
- There are two rolls of toilet paper - A and B.
Roll A has inner diameter 4cm and outer diameter 15cm.
Roll B has inner diameter 6cm and outer diameter 16cm.
Each roll has 480 tissues, but the length of a single tissue
in roll A is 16cm, whereas the length of a tissue in
roll B is 17cm. The very practical and important question
is which roll has thicker tissues?
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