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Problems
Contests
National and Regional Contests
Serbia Contests
Serbia Team Selection Test
1992 Yugoslav Team Selection Test
1992 Yugoslav Team Selection Test
Part of
Serbia Team Selection Test
Subcontests
(3)
Problem 3
1
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permutation of [1992] with conditions
Does it exist a permutation of the numbers
1
,
2
,
…
,
1992
1,2,\ldots,1992
1
,
2
,
…
,
1992
such that the arithmetic mean of arbitrary two of the numbers is not equal to any of the numbers which is placed between these two numbers in the permutation?
Problem 2
1
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4-sequence recurrence, periodic
Periodic sequences
(
a
n
)
,
(
b
n
)
,
(
c
n
)
(a_n),(b_n),(c_n)
(
a
n
)
,
(
b
n
)
,
(
c
n
)
and
(
d
n
)
(d_n)
(
d
n
)
satisfy the following conditions:
a
n
+
1
=
a
n
+
b
n
,
b
n
+
1
=
b
n
+
c
n
,
a_{n+1}=a_n+b_n,\enspace\enspace b_{n+1}=b_n+c_n,
a
n
+
1
=
a
n
+
b
n
,
b
n
+
1
=
b
n
+
c
n
,
c
n
+
1
=
c
n
+
d
n
,
d
n
+
1
=
d
n
+
a
n
,
c_{n+1}=c_n+d_n,\enspace\enspace d_{n+1}=d_n+a_n,
c
n
+
1
=
c
n
+
d
n
,
d
n
+
1
=
d
n
+
a
n
,
for
n
=
1
,
2
,
…
n=1,2,\ldots
n
=
1
,
2
,
…
. Prove that
a
2
=
b
2
=
c
2
=
d
2
=
0
a_2=b_2=c_2=d_2=0
a
2
=
b
2
=
c
2
=
d
2
=
0
.
Problem 1
1
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squares on sides of triangle
Three squares
B
C
D
E
,
C
A
F
G
BCDE,CAFG
BC
D
E
,
C
A
FG
and
A
B
H
I
ABHI
A
B
H
I
are constructed outside the triangle
A
B
C
ABC
A
BC
. Let
G
C
D
Q
GCDQ
GC
D
Q
and
E
B
H
P
EBHP
EB
H
P
be parallelograms. Prove that
A
P
Q
APQ
A
PQ
is an isosceles right triangle.