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National and Regional Contests
Romania Contests
IMAR Test
2014 IMAR Test
2014 IMAR Test
Part of
IMAR Test
Subcontests
(4)
4
1
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Steiner tree!
Let
n
n
n
be a positive integer. A Steiner tree associated with a finite set
S
S
S
of points in the Euclidean
n
n
n
-space is a finite collection
T
T
T
of straight-line segments in that space such that any two points in
S
S
S
are joined by a unique path in
T
T
T
, and its length is the sum of the segment lengths. Show that there exists a Steiner tree of length
1
+
(
2
n
−
1
−
1
)
3
1+(2^{n-1}-1)\sqrt{3}
1
+
(
2
n
−
1
−
1
)
3
associated with the vertex set of a unit
n
n
n
-cube.
3
1
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Polynomial expressible in a special form !!
Let
f
f
f
be a primitive polynomial with integral coefficients (their highest common factor is
1
1
1
) such that
f
f
f
is irreducible in
Q
[
X
]
\mathbb{Q}[X]
Q
[
X
]
, and
f
(
X
2
)
f(X^2)
f
(
X
2
)
is reducible in
Q
[
X
]
\mathbb{Q}[X]
Q
[
X
]
. Show that
f
=
±
(
u
2
−
X
v
2
)
f= \pm(u^2-Xv^2)
f
=
±
(
u
2
−
X
v
2
)
for some polynomials
u
u
u
and
v
v
v
with integral coefficients.
2
1
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Six consecutive squarish numbers!
Let
ϵ
\epsilon
ϵ
be a positive real number. A positive integer will be called
ϵ
\epsilon
ϵ
-squarish if it is the product of two integers
a
a
a
and
b
b
b
such that 1 < a < b < (1 +\epsilon )a. Prove that there are infinitely many occurrences of six consecutive
ϵ
\epsilon
ϵ
-squarish integers.
1
1
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Perpendicular bisector bisects segment !!
Let
A
B
C
ABC
A
BC
be a triangle and let
M
M
M
be the midpoint of the side
B
C
BC
BC
. The circle with radius
M
A
MA
M
A
centered in
M
M
M
meets the lines
A
B
AB
A
B
and
A
C
AC
A
C
again at
B
′
B^{'}
B
′
and
C
′
C^{'}
C
′
, respectively , and the tangents to this circle at
B
′
B^{'}
B
′
and
C
′
C^{'}
C
′
meet at
D
D
D
. Show that the perpendicular bisector of the segment
B
C
BC
BC
bisects the segment
A
D
AD
A
D
.