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Problem 1.
Let
n
be a positive integer and let
a
1
,...,a
k
(
k
≥
2
) be distinct integers in the set
{
1
,...,n
}
such that
n
divides
a
i
(
a
i
+1

1)
for
i
= 1
,...,k

1
. Prove that
n
does not divide
a
k
(
a
1

1)
.
Problem 2.
Let
ABC
be a triangle with circumcentre
O
. The points
P
and
Q
are interior points
of the sides
CA
and
AB
, respectively. Let
K
,
L
and
M
be the midpoints of the segments
BP
,
CQ
and
PQ
, respectively, and let
Γ
be the circle passing through
K
,
L
and
M
. Suppose that the line
PQ
is tangent to the circle
Γ
. Prove that
OP
=
OQ
.
Problem 3.
Suppose that
s
1
,s
2
,s
3
,...
is a strictly increasing sequence of positive integers such
that the subsequences
s
s
1
,s
s
2
,s
s
3
,...
and
s
s
1
+1
,s
s
2
+1
,s
s
3
+1
,...
are both arithmetic progressions. Prove that the sequence
s
1
,s
2
,s
3
,...
is itself an arithmetic pro
gression.
Language: English
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 Fall '11
 T.S.

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