CMC
Charlotte Mathematics Club
Problem Page
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Pi Mu Epsilon Problem of the Week
Here are some Team Problems from this year (and last year).
October 28, 2000
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- Four suspects, one of whom was known to have committed a murder,
made the following statements when questioned by police.
If one and only one of these is telling the truth, who did it?
Arby: Becky did it.
Becky: Ducky did it.
Cindy: I didn't do it.
Ducky: Becky is lying.
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Numbers are written in all 16 squares of a 4-by-4 table
so that the sum of neighbors of each number is equal to 1
(neighboring squares are those with a common side).
Find the sum of all of the numbers in the table.
Call a number n fortunate if it can be written with four equal digits in some base
b in Z+ (that is, b is a positive whole number).
For example, the four-digit base 10 numbers 1111, 2222, ..., ane 9999 are all fortunate.
Also, the four-digit base 5 number 22225 represents
(2 * 53) + (2 * 52) + (2 * 5) + 2 = 312, so 312 is fortunate, too.
- What is the smallest fortunate number?
- Why is 80 fortunate?
- Why is 2000 fortunate?
- What is the largest fortunate number less than 2000?
- What is the smallest fortunate number greater than 2000?
- Are any perfect squares fortunate?
- Are any prime numbers fortunate?
- Are there any doubly fortunate numbers (fortunate in two different bases)?
November 15, 1997
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Suppose that u and d are numbers satisfying 1 < u < d.
Explain why (u+1)/(u-1) > (d+1)/(d-1).
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Cinderella can walk up one floor on the central staircase at the castle in
20 seconds, but only takes 15 seconds coming down.
In a modernization move, the stairs are replaced by up and down escalators,
each of which take 30 seconds to transport a standing person a distance of
one floor.
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How long would it take a walking Cinderella to walk up one floor
on an up-escalator?
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How long would it take a walking Cinderella to walk down one floor
on an up-escalator?
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How long would it take a walking Cinderella to walk up one floor
on a down-escalator?
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How long would it take a walking Cinderella to walk down one floor
on a down-escalator?
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Suppose that the escalator moves at a rate of e floors per second,
and Cindefella walks up stairs at a rate of u floors per second,
while he walks down stairs at a rate of d floors per second.
Assume that e < u < d.
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How long would it take a walking Cindefella to walk one round trip,
up one floor, then down one floor, on an up-escalator?
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How long would it take a walking Cindefella to walk one round trip,
up one floor, then down one floor, on a down-escalator?
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Which is faster:
the round trip up and down the up-escalator,
or the round trip up and down the down-escalator?
Explain.
October 18, 1997
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A treasure is located at a point along a straight road with towns A,
B, C, and D on it in that order.
A map gives the following instructions for locating the treasure:
- Start at town A and go half of the way to C.
- Then go one third of the way to D.
- Then go one fourth of the way to B, and dig for the treasure.
If AB = 6 miles, BC = 8 miles, and the treasure is buried midway between
A and D, find the distance from C to D.
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A second treasure is buried somewhere along another straight road on which are
five towns.
An old manuscript contains directions to the treasure.
These directions are similar to those in the previous problem,
but the identities of the towns have been scratched out.
All that can be determined is that the treasure-hunter should start at one of the towns
(which town isn't clear) and travel half of the way to a second town.
From there, travel one third of the way to a third town.
From there, travel one fourth of the way to a fourth town.
Finally, from there, travel one fifth of the way to the fifth town.
Explain how to find the location of the treasure.
September 6, 1997
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Five pieces of cheese weigh 1, 3, 5, 7 and 9 ounces, respectively.
We want to cut one of the pieces into two parts so that the resulting
six pieces can be gathered into two packets where:
- each packet contains three pieces,
- the two packets have equal total weights, and
- the two parts of the cut piece are in different packets.
Which of the original five cheese pieces can be cut so that two packets
can be formed which satisfy these three properties.
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Answer the same question in the case that the weights of the five cheese pieces are
2, 3, 5, 8 and 13 ounces.
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Answer the same question in the case that the weights of the five cheese pieces are
2, 3, 5, 8 and 14 ounces.
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Suppose that our five cheese pieces each have a different weight.
Show that at least one piece can be cut so that two packets
can be formed which satisfy the three properties.
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Suppose that our five cheese pieces each have a different weight.
Suppose further that the smallest piece can be cut to accomplish this task.
Show that any of the five pieces can be cut so that two packets
can be formed which satisfy the three properties.
Oct 12, 1996
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Baby Sally tears a page from granddad's favorite book.
It's an odd--even page, as pages usually are.
Granddad realizes that the sum of the pages before the torn one
have the same sum as the page numbers of the pages after the torn one.
What page was torn and how long is the book?
(Assume the book is less than 1000 pages long; find all solutions.)
Sep 14, 1996
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A ball rolls around on an P x Q pool table.
The ball always starts at a lattice point.
When rolling in the interior of the table, it travels along
a straight line whose direction is at 45 degree angles from the sides.
When it strikes a side of the table, the ball reflects in the expected 45
degree angle; when it rolls into a corner of the table, the ball rebounds
out of the corner along the same route it entered.
A complete trajectory is accomplished when the ball returns to its
starting point travelling in the same direction as it started.
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Count the number of complete trajectories as a function of
P and Q.
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Determine the conditions under which there is a trajectory which
includes diagonally opposite corners of the table.