Numbers and Sets  exercises for enthusiasts 1.
W. T. G.
1.
Let
A
be the sum of the digits of 4444
4444
, and let
B
be the sum of the digits of
A
.
What is the sum of the digits of
B
?
2.
Let
x
1
,. .. ,x
n
be real numbers such that
∑
n
i
=1
x
i
= 0 and
∑
n
i
=1
x
2
i
= 1. How large can
x
1
x
2
+
x
2
x
3
+
.. .
+
x
n

1
x
n
+
x
n
x
1
be? (If you cannot solve this problem, try it for small
values of
n
, or just experiment to see how large you can make the sum.)
3.
Let
R
be a rectangle which can be divided into smaller rectangles, each of which has
at least one side of integer length. Prove that
R
has at least one side of integer length.
4.
Does there exist a real number
c >
0 such that for every positive integer
n
it is possible
to choose points (
x
1
,y
1
)
,. .. ,
(
x
n
,y
n
) in the unit square [0
,
1]
2
with the property that

x
i

x
j

y
i

y
j

>
cn

1
whenever
i
6
=
j
?
5.
Does there exist a cycle in
Z
3
(i.e., a path consisting of line segments going from
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 Fall '10
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 Real Numbers, Sets, positive integer, lattice point, integer length, W. T. G

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