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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.
Solution
to May 2003 problem
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.
-
At least one of the last two
statements on this list is true.
-
Either this is the first true
statement, or it is the first false statement on the list.
-
This list contains at least
three consecutive false statements.
-
The difference between the
number of the last true statement and the
number of the first true statement is a divisor of n.
-
The sum of the numbers of the
true statements equals n.
-
This is not the last true statement.
-
The number of each of the
true statements is a divisor of n.
-
Exactly n% of the
statements on this list are true.
-
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.
-
This list does not contain
three consecutive true statements.
The problem for April: Find n
Solution
to April 2003 problem
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
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each committee have an EVEN number
of members,
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each pair of committees share
an EVEN number of members, and
-
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:
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each committee has an ODD number
of members, and
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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?
Solution
to March 2003 problem
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:
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Show that no power of 2 can be
written as the sum of two or more consecutive positive integers.
-
Show that any integer that is not
a power of 2 can be written as the
sum of consecutive integers.
Solution
to February 2003 problem
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,
| sequence |
1, |
2,2,1,1, |
2,1, |
2,2, |
1, |
2,2, |
... |
| run length |
1 |
22 |
11 |
2 |
1 |
2 |
|
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
- the distance between any two of them is an integer, and
- 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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