Parity Questions (First Meeting) Math Worksheet

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Parity Questions (First Meeting):
1. In a 6
6 chart all but one corner blue square are painted white. You are
allowed to repaint any column or any row in the chart (i.e., you can select
any row or column and flip the color of all squares within that line). Is
it possible to attain an entirely white chart by using only the permitted
operations?
Example operation:
2. John and Pete have three pieces of paper. Each of the boys picks one
piece, tears it up, and puts the smaller pieces back. John only tears a
piece of paper into 3 smaller pieces while Pete only tears a piece of paper
into 5 smaller pieces. After a few minutes can there be exactly 100 pieces
of paper?
3. All natural numbers from 1 to 101 are written in a row. Can the signs
“+” and “ ” be placed between them so that the value of the resulting
expression is 0?
4. Of 101 coins, 50 are counterfeit, and they differ from the genuine coins
in weight by 1 gram. Peter has a scale in the form of a balance which
shows the difference in weight between the objects placed on each pan.
He chooses one coin, and wants to find out whether it is counterfeit. Can
he do this in one weighing?
5. Suppose a, b and c are integers such that the equation ax + bx + c = 0
has a rational solution. Prove that at least one of the integers a, b and c
must be even.
6. Can a convex nonagon (a polygon with 9 sides) be cut into parallelograms?
7. Consider a football conference with 13 teams. Is it possible to schedule
games so that each team plays exactly 9 games within the conference?
8. (2002, A3) Let n
2 be an integer and T be the number of nonempty
subsets S of 1, 2, 3, . . . , n with the property that the average of the
elements of S is an integer. Prove that T
n is always even.
Note: Ask me if you don’t know what a set is!
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