Matematika :

May 3, 2011

Problem of The Month From Math Central Uregina (2)

MP24: August 2002
A problem from Denmark's "Georg Mohr Konkurrencen I Matematik 1996" was generalized by Pierre Bornsztein [CRUX MATHEMATICORUM 2001, page 240]. He showed that if (a) P is a permutation of the set {1, 2, ..., n}, and (b) n is congruent to 2 or 3 modulo 4, then the numbers |k - P(k)| cannot be distinct. Our problem for August is to say what happens when n = 20:
Is there a permutation P for which the numbers |k - P(k)| take on all values from 0 to 19?

MP23: July 2002
For July we look at the problem of fitting an equilateral triangle inside a triangle T labeled with sides a >= b >= c. Note that the angles opposite these sides are ordered A >= B >= C. It is not hard to show that when the middle angle B is at most 60 degrees, the largest equilateral triangle that fits in T rests on the longest side a. When B >= 60 there is a condition involving a, c, and B that tells whether the largest equilateral triangle in T will rest on the longest side a or on the shortest side c. That formula can be found in "Equilateral Triangles in Triangles," a paper by Richard P. Jerrard and John E. Wetzel that will appear in the American Mathematical Monthly later this year.

Describe the unique nonequilateral triangle for which the three largest inscribed equilateral triangles, one on each side, are equal.

MP22: June 2002
The Seven Dwarfs are having breakfast, and Snow White has just poured them some milk. Before drinking, the dwarfs have a ritual. First, Dwarf #1 splits his milk equally among his brothers' mugs (leaving himself with nothing). Then Dwarf #2 does the same with his milk, etc. The process continues around the table, until Dwarf #7 has distributed his milk in this way. (Note that Dwarf #7 is named Dopey!) At the end, each dwarf has exactly the same amount of milk as he started with! If the total amount of milk was 42 ounces, how much milk did each dwarf have at the beginning? Is this the only possible distribution of milk, or does the problem admit multiple solutions?
The source of this month's problem is the US-Canada Mathcamp 2002 Application Quiz.

MP21: May 2002
The source of this month's problem is the fifth annual Team Competition organized by the North Central Section of the Math Association of America. It was held November 10, 2001.
We define the fractional part of r to be
F(r) = r - (the largest integer that divides evenly into r).Some examples:F(12.34) = .34, F( 11/2) = 1/2, F( 8/3) = 2/3.Problem for May:Find a positive number r such that F(r) + F( 1/r) = 1.

MP20: April 2002
To mark the arrival of spring, our monthly problem pays tribute to Canadian cuisine.
  1. In a large bowl, mix together one egg, one cup of milk, one cup of flour, a teaspoon of baking powder, and a pinch of salt. You get something called pancake batter.

    Drop one big spoonful of the batter into a hot greased pan and cook on both sides for approximately 17 minutes. You get something called a burnt pancake.

    Put the burnt pancake on a sheet of paper and trace its contour with a pencil. You get something called a simple closed curve.
  2. Prove that your simple closed curve contains the three vertices of an equilateral triangle.
Step 4. above will require some thought. While you are at it you may as well turn the rest of the batter into pancakes and eat them with a generous portion of maple syrup.

MP19: March 2002
Your enemy gets to choose 2000 of the counting numbers from 1 to 3000. Your job is to find a subsequence of 1000 of your enemy's numbers so that they alternate odd, even, odd, even, ... when listed in their natural order. Prove that no matter how clever your enemy might be, you can always succeed.

MP18: February 2002
A regular tetrahedron is a pyramid that consists of four equilateral triangles, each joined to the other three along their sides. The resulting six edges, consequently, all have equal lengths. Imagine that on each of the four faces, a car is traveling in a clockwise direction at a constant speed along the edges bounding that face. Each of the four cars may be traveling at a different speed and may start anywhere on the boundary of its face. Can the cars always keep from crashing, or is there destined to be a collision along some edge?

MP17: January 2002
For each real number r, let Tr be the transformation of the plane that takes the point (x, y) into the point (10r x, y + r). Find the equation of the continuous curve y = f(x) that contains the image of the point (2002, 2002) for every transformation Tr.
MP16: December 2001
There is only one integer n for which the expression

is an integer. Find this value of n and show that there are no others.

MP15: November 2001
The Efron dice are laid out before you:
Die A has four faces showing 4, and two faces showing 0,
die B has all six faces showing 3,
die C has two faces showing 6 and four faces showing 2, and
die D has three faces showing 5 and three faces showing 1.
The game is played by two people. The first chooses one of the four dice to roll, and the opponent chooses from the three that remain. The player who rolls the higher number wins. Your opponent graciously invites you to pick a die first. What should you do?

MP14: October 2001
The theatre charges one dollar for its Sunday afternoon show. One Sunday the cashier finds that he has no change. Eight people arrive at the theatre; four have only a one-dollar coin (a loonie) and four have only a two-dollar coin (a toonie). Depending on how the people line up, the cashier may or may not be able to make change for every person in the line as they buy their tickets one at a time. Suppose that the eight people form a line in random order, without knowing who has a loonie and who has a toonie. What is the probability that the cashier will be able to make change for every person in the line?
This month's problem is taken from the 2001 British Columbia Colleges High School Mathematics Contest.

MP13: September 2001
Six professors in the Mathematics Department visited the coffee room on the day that the coffee fund was stolen. Each entered the room only once, stayed for some time, and then left. Whenever any two of them were in the room at the same time, at least one of them noticed the other was there. (Professors of mathematics do not always notice one another.) When the Department secretary quizzed the six professors about their visits to the coffee room the following information was given:
ProfessorClaimed to have seen
AbelBernoulli and Erdos
BernoulliAbel and Fermat
CauchyDescartes and Fermat
DescartesAbel and Fermat
ErdosBernoulli and Cauchy
FermatCauchy and Erdos

If exactly one of the professors was lying to frame someone else (where here "lying" means giving false information, not omitting information), who is the culprit?

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