North Dakota Mathematics Talent Search 2004-2005
PROBLEM SET 1
(Due 12/01/2004)
- Two men meet on the street. They haven't seen each other for many
years. They talk about various things, and then after some time
one of them says: "Since you're a professor in mathematics, I'd
like to give you a problem to solve. You know, today's a very
special day for me: All three of my sons celebrate their birthday
this very day! So, can you tell me how old each of them is?"
"Sure," answers the mathematician, "but you'll have to tell me
something about them."
"OK, I'll give you some hints," replies the father of the three
sons, "The product of the ages of my sons is 36."
"That's fine," says the mathematician, "but I'll need more than
just this."
"The sum of their ages is equal to the number of windows in that
building," says the father pointing at a structure next to them.
The mathematician thinks for some time and replies, "Still, I need
an additional hint to solve your puzzle."
"My oldest son has blue eyes," says the other man.
"Oh, this is sufficient!" exclaims the mathematician, and he gives
the father the correct answer: the ages of his three sons.
Your challenge now is to do the same: to follow the reasoning of
the mathematician and solve the puzzle.
- Mr. Smith and his wife invited four other couples for a party.
When everyone arrived, some of the people in the room shook hands
with some of the others. Of course, nobody shook hands with their
spouse and nobody shook hands with the same person twice.
After that, Mr. Smith asked everyone how many times they shook
someone's hand. He received different answers from everybody.
How many times did Mrs. Smith shake someone's hand?
- You drive a car at a constant speed of 40 miles/hour from
Washington D.C. to New York City and return immediately, but at
a higher speed of 60 miles/hour. What was your average speed for
the whole trip?
- Suppose that it takes four hours to fill a pool using a large pipe.
On the other hand, it takes six hours to fill the pool using a
small pipe. Prove that the time required to fill the pool when
using both pipes is less than 2 hours and 25 minutes.
- Prove that any polyhedron must have at least two faces with the
same number of edges.
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