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National and Regional Contests
Serbia Contests
Serbia National Math Olympiad
2017 Serbia National Math Olympiad
2017 Serbia National Math Olympiad
Part of
Serbia National Math Olympiad
Subcontests
(3)
3
2
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Combinatorics, Serbia MO 2017
There are
2
n
−
1
2n-1
2
n
−
1
lamps in a row. In the beginning only the middle one is on (the
n
n
n
-th one), and the other ones are off. Allowed move is to take two non-adjacent lamps which are turned off such that all lamps between them are turned on and switch all of their states from on to off and vice versa. What is the maximal number of moves until the process terminates?
Tangents inducing isogonals
Let
k
k
k
be the circumcircle of
△
A
B
C
\triangle ABC
△
A
BC
and let
k
a
k_a
k
a
be A-excircle .Let the two common tangents of
k
,
k
a
k,k_a
k
,
k
a
cut
B
C
BC
BC
in
P
,
Q
P,Q
P
,
Q
.Prove that
∡
P
A
B
=
∡
C
A
Q
\measuredangle PAB=\measuredangle CAQ
∡
P
A
B
=
∡
C
A
Q
.
2
2
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Geometry, Serbia MO 2017
Let
A
B
C
D
ABCD
A
BC
D
be a convex and cyclic quadrilateral. Let
A
D
∩
B
C
=
{
E
}
AD\cap BC=\{E\}
A
D
∩
BC
=
{
E
}
, and let
M
,
N
M,N
M
,
N
be points on
A
D
,
B
C
AD,BC
A
D
,
BC
such that
A
M
:
M
D
=
B
N
:
N
C
AM:MD=BN:NC
A
M
:
M
D
=
BN
:
NC
. Circle around
△
E
M
N
\triangle EMN
△
EMN
intersects circle around
A
B
C
D
ABCD
A
BC
D
at
X
,
Y
X,Y
X
,
Y
prove that
A
B
,
C
D
AB,CD
A
B
,
C
D
and
X
Y
XY
X
Y
are either parallel or concurrent.
Chess combinatorics
Find the maximum number of queens you could put on
2017
×
2017
2017 \times 2017
2017
×
2017
chess table such that each queen attacks at most
1
1
1
other queen.
1
2
Hide problems
Inequality with a+b+c = 1 [Serbia MO 2017, D1, P1]
Prove that for positive real numbers
a
,
b
,
c
a,b,c
a
,
b
,
c
such that
a
+
b
+
c
=
1
a+b+c=1
a
+
b
+
c
=
1
,
a
2
b
+
1
+
b
2
c
+
1
+
c
2
a
+
1
≤
2
−
(
a
2
+
b
2
+
c
2
)
.
a\sqrt{2b+1}+b\sqrt{2c+1}+c\sqrt{2a+1}\le \sqrt{2-(a^2+b^2+c^2)}.
a
2
b
+
1
+
b
2
c
+
1
+
c
2
a
+
1
≤
2
−
(
a
2
+
b
2
+
c
2
)
.
Specific divisors implying perfect square
Let
a
a
a
be a positive integer.Suppose that
∀
n
\forall n
∀
n
,
∃
d
\exists d
∃
d
,
d
≠
1
d\not =1
d
=
1
,
d
≡
1
(
m
o
d
n
)
d\equiv 1\pmod n
d
≡
1
(
mod
n
)
,
d
∣
n
2
a
−
1
d\mid n^2a-1
d
∣
n
2
a
−
1
.Prove that
a
a
a
is a perfect square.