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Contests
National and Regional Contests
Serbia Contests
Serbia National Math Olympiad
2011 Serbia National Math Olympiad
2011 Serbia National Math Olympiad
Part of
Serbia National Math Olympiad
Subcontests
(3)
3
2
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collinear iff X is circumcenter Serbia Math Olyimpiad 2011
Let
H
H
H
be orthocenter and
O
O
O
circumcenter of an acuted angled triangle
A
B
C
ABC
A
BC
.
D
D
D
and
E
E
E
are feets of perpendiculars from
A
A
A
and
B
B
B
on
B
C
BC
BC
and
A
C
AC
A
C
respectively. Let
O
D
OD
O
D
and
O
E
OE
OE
intersect
B
E
BE
BE
and
A
D
AD
A
D
in
K
K
K
and
L
L
L
, respectively. Let
X
X
X
be intersection of circumcircles of
H
K
D
HKD
HKD
and
H
L
E
HLE
H
L
E
different than
H
H
H
, and
M
M
M
is midpoint of
A
B
AB
A
B
. Prove that
K
,
L
,
M
K, L, M
K
,
L
,
M
are collinear iff
X
X
X
is circumcenter of
E
O
D
EOD
EO
D
.
T with 66 points
Set
T
T
T
consists of
66
66
66
points in plane, and
P
P
P
consists of
16
16
16
lines in plane. Pair
(
A
,
l
)
(A,l)
(
A
,
l
)
is good if
A
∈
T
A \in T
A
∈
T
,
l
∈
P
l \in P
l
∈
P
and
A
∈
l
A \in l
A
∈
l
. Prove that maximum number of good pairs is no greater than
159
159
159
, and prove that there exits configuration with exactly
159
159
159
good pairs.
2
2
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phi(n,n+1) powers of two- Serbia Mathematical Olympiad 2011
Let
n
n
n
be an odd positive integer such that both
ϕ
(
n
)
\phi(n)
ϕ
(
n
)
and
ϕ
(
n
+
1
)
\phi (n+1)
ϕ
(
n
+
1
)
are powers of two. Prove
n
+
1
n+1
n
+
1
is power of two or
n
=
5
n=5
n
=
5
.
Are there a,b,c? Serbia Math Olympiad 2011
Are there positive integers
a
,
b
,
c
a, b, c
a
,
b
,
c
greater than
2011
2011
2011
such that:
(
a
+
b
)
c
=
.
.
.
2010
,
2011...
(a+ \sqrt{b})^c=...2010,2011...
(
a
+
b
)
c
=
...2010
,
2011...
?
1
2
Hide problems
Easy sequence- Serbia Mathematical Olympiad 2011- Problem 1
Let
n
≥
2
n \ge 2
n
≥
2
be integer. Let
a
0
a_0
a
0
,
a
1
a_1
a
1
, ...
a
n
a_n
a
n
be sequence of positive reals such that:
(
a
k
−
1
+
a
k
)
(
a
k
+
a
k
+
1
)
=
a
k
−
1
−
a
k
+
1
(a_{k-1}+a_k)(a_k+a_{k+1})=a_{k-1}-a_{k+1}
(
a
k
−
1
+
a
k
)
(
a
k
+
a
k
+
1
)
=
a
k
−
1
−
a
k
+
1
, for
k
=
1
,
2
,
.
.
.
,
n
−
1
k=1, 2, ..., n-1
k
=
1
,
2
,
...
,
n
−
1
. Prove
a
n
<
1
n
−
1
a_n< \frac{1}{n-1}
a
n
<
n
−
1
1
.
X, Y, O1, O2 are concyclic
On sides
A
B
,
A
C
,
B
C
AB, AC, BC
A
B
,
A
C
,
BC
are points
M
,
X
,
Y
M, X, Y
M
,
X
,
Y
, respectively, such that
A
X
=
M
X
AX=MX
A
X
=
MX
;
B
Y
=
M
Y
BY=MY
B
Y
=
M
Y
.
K
K
K
,
L
L
L
are midpoints of
A
Y
AY
A
Y
and
B
X
BX
BX
.
O
O
O
is circumcenter of
A
B
C
ABC
A
BC
,
O
1
O_1
O
1
,
O
2
O_2
O
2
are symmetric with
O
O
O
with respect to
K
K
K
and
L
L
L
. Prove that
X
,
Y
,
O
1
,
O
2
X, Y, O_1, O_2
X
,
Y
,
O
1
,
O
2
are concyclic.