Matematika :

May 3, 2011

Problem of The Month From Math Central Uregina (3)

MP33: May 2003
Given a triangle and an infinite blue and white checkerboard, show that the triangle can be placed on the board with all its vertices strictly in the blue.


MP32: April 2003
Below is a list of ten statements, numbered from 1 to 10, that enable you to determine a number n. Unfortunately, some of these statements happen to be false.
  1. At least one of the last two statements on this list is true.
  2. Either this is the first true statement, or it is the first false statement on the list.
  3. This list contains at least three consecutive false statements.
  4. The difference between the number of the last true statement and the
    number of the first true statement is a divisor of n.
  5. The sum of the numbers of the true statements equals n.
  6. This is not the last true statement.
  7. The number of each of the true statements is a divisor of n.
  8. Exactly n% of the statements on this list are true.
  9. The number of divisors of n that are bigger than 1 and less than n is
    greater than the sum of the numbers of the true statements.
  10. This list does not contain three consecutive true statements.
The problem for April: Find n

MP31: March 2003
The faculty of the Even University of Regina considers serving on committees to be their main duty. Their faculty association passed a rule requiring that
  1. each committee have an EVEN number of members,
  2. each pair of committees share an EVEN number of members, and
  3. no two committees are allowed to have identical membership.

Their cross-town rival is the Odd University of Regina whose faculty likewise spends most of their day serving on committees. Their rules are a bit different:
  1. each committee has an ODD number of members, and
  2. each pair of committees share an EVEN number of members.

The problem for March: Even University has only 20 faculty members, while Odd University has 1000 members. Odd University tries to form as many committees as it can following its own rule, but still comes short of the number of committees at Even University. How is this possible?

MP30: February 2003
Note that 3 = 1+2, 5 = 2+3, 6 = 1+2+3, 7 = 3+4, but neither 4 nor 8 can be written as the sum of two or more CONSECUTIVE positive integers. Your task for February is to show that this pattern continues forever:
  1. Show that no power of 2 can be written as the sum of two or more consecutive positive integers.
  2. Show that any integer that is not a power of 2 can be written as the
    sum of consecutive integers.

MP29: January 2003
An L-shaped tromino is a piece consisting of three squares, two of which are attached to adjacent sides of the third.
You are given a 2n by 2n grid of squares, and uncle Albert places a penny on one of them.
Your job is to cover the remaining 22n – 1 squares with (22n – 1)/3 trominoes. Can you do it?

MP28: December 2002
Complete the following sentence by filling in each blank with a numeral of one or more digits.
In this sentence, the number of occurrences of 0 is ____, of 1 is ____, of 2 is ____, of 3 is ____, of 4 is ____, of 5 is ____, of 6 is ____, of 7 is ____ of 8 is ____, and of 9 is ____.
The sentence was devised by Raphael Robinson, who proved that there are exactly two correct ways to fill in the blanks.

MP27: November 2002
The Kolakoski sequence 1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, ... is an example of what people call a self-reading sequence: {an} is defined to be the sequence of 1's and 2's whose first term is 1, and each subsequent term an is the length of the nth run (of ones or twos). In more detail,
run length12211212 
Start with a1 = 1. The rule says that the first run (which is a single 1) has length 1.
Thus a2 must be different, so that a2 = 2. The second run therefore has length 2, which forces the third term, a3, to be a two also. This completes the second run, so the third run begins with a 1; since its length is 2 we have a4 = 1 and a5 = 1. The fourth and fifth runs are consequently the singletons 2 then 1. And so on.
Problem for November: Prove that 0.122112122122112... is irrational. (Recall that a number in decimal form is irrational if its digits are nonrepeating.)

MP26: October 2002
Given a set of points in the plane such that
  1. the distance between any two of them is an integer, and
  2. infinitely many of them lie on the same line,
prove that all points in the set lie on that line.

What happens if we ask only that 5 points lie on the same line?

MP25: September 2002
Each person in a group of seven people speaks at most two languages, while among every three of them there are at least two who can communicate. Prove that there are three people who speak a common language.

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