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Problems
Contests
National and Regional Contests
Romania Contests
IMAR Test
2010 IMAR Test
2010 IMAR Test
Part of
IMAR Test
Subcontests
(3)
3
1
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the upper bound of number of triangles
Given an integer
n
≥
2
n\ge 2
n
≥
2
, given
n
+
1
n+1
n
+
1
distinct points
X
0
,
X
1
,
…
,
X
n
X_0,X_1,\ldots,X_n
X
0
,
X
1
,
…
,
X
n
in the plane, and a positive real number
A
A
A
, show that the number of triangles
X
0
X
i
X
j
X_0X_iX_j
X
0
X
i
X
j
of area
A
A
A
does not exceed
4
n
n
4n\sqrt n
4
n
n
.
2
1
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8th IMAR,problem 2
Given a triangle
A
B
C
ABC
A
BC
, let
D
D
D
be the point where the incircle of the triangle
A
B
C
ABC
A
BC
touches the side
B
C
BC
BC
. A circle through the vertices
B
B
B
and
C
C
C
is tangent to the incircle of triangle
A
B
C
ABC
A
BC
at the point
E
E
E
. Show that the line
D
E
DE
D
E
passes through the excentre of triangle
A
B
C
ABC
A
BC
corresponding to vertex
A
A
A
.
1
1
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periodic with period $2^k$
Show that a sequence
(
a
n
)
(a_n)
(
a
n
)
of
+
1
+1
+
1
and
−
1
-1
−
1
is periodic with period a power of
2
2
2
if and only if
a
n
=
(
−
1
)
P
(
n
)
a_n=(-1)^{P(n)}
a
n
=
(
−
1
)
P
(
n
)
, where
P
P
P
is an integer-valued polynomial with rational coefficients.