Math 289 โ Winter 2006 โ Problem Set 7
Due March 8, 2006
1.(10) A mysterious letter was found next to the grave of Ludwig Wittgenstein. The letter
contained the following 2006 sentences:
There is exactly one false statement in this letter.
There are exactly two false statements in this letter.
There are exactly three false statements in this letter.
.
.
.
There are exactly 2005 false statements in this letter.
There are exactly 2006 false statements in this letter.
Find out how many of the statements (sentences) in the letter are true?
2.(10) Which digit is on the 7000th place of the decimal expansion of a number 1/7000?
3.(10) The vertices of a regular 100-gon all lie on a perimeter of a circle. Some 50 of
them are colored red, the 50 others are colored blue. Prove that the number of right-angle
triangles with all three vertices red is the same as the number of right-angle triangles with
all three vertices blue.
4.(10) Two parallelograms ABCD and EF GH are given such that
D=H, E lies on the side AB, C lies on the side F G.
Prove that the areas of both parallelograms are the same.
5.(15) Two Math 289 students play the following game. They start with two integers
written on a blackboard, e.g. 144 and 15. Then they alternate their turns.
In each turn a player who has the turn chooses two different integers already written
on the blackboard and writes a new one, which is an absolute value of the difference of
the two chosen numbers. The new number must be different from all the other numbers on
the blackboard, otherwise the turn must be repeated and the new number erased (no other
number is erased). This way both players take thier turns until one of them is unable to
make his turn. Then he looses and has to pay for dinner in Potbelly. Give an advice how
to play to win to a player Ludwig who has an initial turn if there are originally numbers
a) 17 and 4; b) 102 and 201 on the blackboard.
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