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IMAR Test
2009 IMAR Test
2009 IMAR Test
Part of
IMAR Test
Subcontests
(4)
4
1
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exists a subsequence of consecutive terms, with product perfect square
Given any
n
n
n
positive integers, and a sequence of
2
n
2^n
2
n
integers (with terms among them), prove there exists a subsequence made of consecutive terms, such that the product of its terms is a perfect square. Also show that we cannot replace
2
n
2^n
2
n
with any lower value (therefore
2
n
2^n
2
n
is the threshold value for this property).
2
1
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7 vertices of a cube with value 0, eighth with value 1, increasing numbers
Of the vertices of a cube,
7
7
7
of them have assigned the value
0
0
0
, and the eighth the value
1
1
1
. A move is selecting an edge and increasing the numbers at its ends by an integer value
k
>
0
k > 0
k
>
0
. Prove that after any finite number of moves, the g.c.d. of the
8
8
8
numbers at vertices is equal to
1
1
1
.Russian M.O.
1
1
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diophantine system xy +zw = a, xz + yw = b has finitely solutions x,y,z,w
Given
a
a
a
and
b
b
b
distinct positive integers, show that the system of equations
x
y
+
z
w
=
a
x y +zw = a
x
y
+
z
w
=
a
x
z
+
y
w
=
b
xz + yw = b
x
z
+
y
w
=
b
has only finitely many solutions in integers
x
,
y
,
z
,
w
x, y, z,w
x
,
y
,
z
,
w
.
3
1
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convex ABCD, AB=CB, <ABC +2<CDA = \pi, AE=EC, prove <CDE=<BDA
Consider a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
with
A
B
=
C
B
AB=CB
A
B
=
CB
and
∠
A
B
C
+
2
∠
C
D
A
=
π
\angle ABC +2 \angle CDA = \pi
∠
A
BC
+
2∠
C
D
A
=
π
and let
E
E
E
be the midpoint of
A
C
AC
A
C
. Show that
∠
C
D
E
=
∠
B
D
A
\angle CDE =\angle BDA
∠
C
D
E
=
∠
B
D
A
.Paolo Leonetti