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National and Regional Contests
Lithuania Contests
Lithuania Team Selection Test
2005 Lithuania Team Selection Test
2005 Lithuania Team Selection Test
Part of
Lithuania Team Selection Test
Subcontests
(3)
1
1
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Division of square nxn into 40x40 and 49x49 squares
Find the smallest integer
n
n
n
such that an
n
×
n
n\times n
n
×
n
square can be partitioned into
40
×
40
40\times 40
40
×
40
and
49
×
49
49\times 49
49
×
49
squares, with both types of squares present in the partition, if a)
40
∣
n
40|n
40∣
n
; b)
49
∣
n
49|n
49∣
n
; c)
n
∈
N
n\in \mathbb N
n
∈
N
.
2
1
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A convex quadrilateral
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral, and write
α
=
∠
D
A
B
\alpha=\angle DAB
α
=
∠
D
A
B
;
β
=
∠
A
D
B
\beta=\angle ADB
β
=
∠
A
D
B
;
γ
=
∠
A
C
B
\gamma=\angle ACB
γ
=
∠
A
CB
;
δ
=
∠
D
B
C
\delta= \angle DBC
δ
=
∠
D
BC
; and
ϵ
=
∠
D
B
A
\epsilon=\angle DBA
ϵ
=
∠
D
B
A
. Assuming that
α
<
π
/
2
\alpha<\pi/2
α
<
π
/2
,
β
+
γ
=
π
/
2
\beta+\gamma=\pi /2
β
+
γ
=
π
/2
, and
δ
+
2
ϵ
=
π
\delta+2\epsilon=\pi
δ
+
2
ϵ
=
π
, prove that
(
D
B
+
B
C
)
2
=
A
D
2
+
A
C
2
(DB+BC)^2=AD^2+AC^2
(
D
B
+
BC
)
2
=
A
D
2
+
A
C
2
[Moderator edit: Also discussed at http://www.mathlinks.ro/Forum/viewtopic.php?t=30569 .]
3
1
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A sequence of 2000 reals
The sequence
a
1
,
a
2
,
.
.
.
,
a
2000
a_1, a_2,..., a_{2000}
a
1
,
a
2
,
...
,
a
2000
of real numbers satisfies the condition
a
1
3
+
a
2
3
+
.
.
.
+
a
n
3
=
(
a
1
+
a
2
+
.
.
.
+
a
n
)
2
a_1^3+a_2^3+...+a_n^3=(a_1+a_2+...+a_n)^2
a
1
3
+
a
2
3
+
...
+
a
n
3
=
(
a
1
+
a
2
+
...
+
a
n
)
2
for all
n
n
n
,
1
≤
n
≤
2000
1\leq n \leq 2000
1
≤
n
≤
2000
. Prove that every element of the sequence is an integer.