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Problems
Contests
National and Regional Contests
Serbia Contests
Serbia Team Selection Test
1980 Yugoslav Team Selection Test
1980 Yugoslav Team Selection Test
Part of
Serbia Team Selection Test
Subcontests
(3)
Problem 2
1
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a modular polynomial
Let
a
,
b
,
c
,
m
a,b,c,m
a
,
b
,
c
,
m
be integers, where
m
>
1
m>1
m
>
1
. Prove that if
a
n
+
b
n
+
c
≡
0
(
m
o
d
m
)
a^n+bn+c\equiv0\pmod m
a
n
+
bn
+
c
≡
0
(
mod
m
)
for each natural number
n
n
n
, then
b
2
≡
0
(
m
o
d
m
)
b^2\equiv0\pmod m
b
2
≡
0
(
mod
m
)
. Must
b
≡
0
(
m
o
d
m
)
b\equiv0\pmod m
b
≡
0
(
mod
m
)
also hold?
Problem 1
1
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geo config of venn diagram
Circles
k
k
k
and
l
l
l
intersect at points
P
P
P
and
Q
Q
Q
. Let
A
A
A
be an arbitrary point on
k
k
k
distinct from
P
P
P
and
Q
Q
Q
. Lines
A
P
AP
A
P
and
A
Q
AQ
A
Q
meet
l
l
l
again at
B
B
B
and
C
C
C
. Prove that the altitude from
A
A
A
in triangle
A
B
C
ABC
A
BC
passes through a point that does not depend on
A
A
A
.
Problem 3
1
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fractional recurrence, output is integer
A sequence
(
x
n
)
(x_n)
(
x
n
)
satisfies
x
n
+
1
=
x
n
2
+
a
x
n
−
1
x_{n+1}=\frac{x_n^2+a}{x_{n-1}}
x
n
+
1
=
x
n
−
1
x
n
2
+
a
for all
n
∈
N
n\in\mathbb N
n
∈
N
. Prove that if
x
0
,
x
1
x_0,x_1
x
0
,
x
1
, and
x
0
2
+
x
1
2
+
a
x
0
x
1
\frac{x_0^2+x_1^2+a}{x_0x_1}
x
0
x
1
x
0
2
+
x
1
2
+
a
are integers, then all the terms of sequence
(
x
n
)
(x_n)
(
x
n
)
are integers.