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Contests
National and Regional Contests
Turkey Contests
Turkey MO (2nd round)
2015 Turkey MO (2nd round)
2015 Turkey MO (2nd round)
Part of
Turkey MO (2nd round)
Subcontests
(6)
6
1
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2n^2 | a^n-1
Find all positive integers
n
n
n
such that for any positive integer
a
a
a
relatively prime to
n
n
n
,
2
n
2
∣
a
n
−
1
2n^2 \mid a^n - 1
2
n
2
∣
a
n
−
1
.
5
1
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Cyclic quadrilateral
In a cyclic quadrilateral
A
B
C
D
ABCD
A
BC
D
whose largest interior angle is
D
D
D
, lines
B
C
BC
BC
and
A
D
AD
A
D
intersect at point
E
E
E
, while lines
A
B
AB
A
B
and
C
D
CD
C
D
intersect at point
F
F
F
. A point
P
P
P
is taken in the interior of quadrilateral
A
B
C
D
ABCD
A
BC
D
for which
∠
E
P
D
=
∠
F
P
D
=
∠
B
A
D
\angle EPD=\angle FPD=\angle BAD
∠
EP
D
=
∠
FP
D
=
∠
B
A
D
.
O
O
O
is the circumcenter of quadrilateral
A
B
C
D
ABCD
A
BC
D
. Line
F
O
FO
FO
intersects the lines
A
D
AD
A
D
,
E
P
EP
EP
,
B
C
BC
BC
at
X
X
X
,
Q
Q
Q
,
Y
Y
Y
, respectively. If
∠
D
Q
X
=
∠
C
Q
Y
\angle DQX = \angle CQY
∠
D
QX
=
∠
CQ
Y
, show that
∠
A
E
B
=
9
0
∘
\angle AEB=90^\circ
∠
A
EB
=
9
0
∘
.
4
1
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Paintings in an exhibition
In an exhibition where
2015
2015
2015
paintings are shown, every participant picks a pair of paintings and writes it on the board. Then, Fake Artist (F.A.) chooses some of the pairs on the board, and marks one of the paintings in all of these pairs as "better". And then, Artist's Assistant (A.A.) comes and in his every move, he can mark
A
A
A
better then
C
C
C
in the pair
(
A
,
C
)
(A,C)
(
A
,
C
)
on the board if for a painting
B
B
B
,
A
A
A
is marked as better than
B
B
B
and
B
B
B
is marked as better than
C
C
C
on the board. Find the minimum possible value of
k
k
k
such that, for any pairs of paintings on the board, F.A can compare
k
k
k
pairs of paintings making it possible for A.A to compare all of the remaining pairs of paintings.P.S: A.A can decide
A
1
>
A
n
A_1>A_n
A
1
>
A
n
if there is a sequence
A
1
>
A
2
>
A
3
>
⋯
>
A
n
−
1
>
A
n
A_1 > A_2 > A_3 > \dots > A_{n-1} > A_n
A
1
>
A
2
>
A
3
>
⋯
>
A
n
−
1
>
A
n
where
X
>
Y
X>Y
X
>
Y
means painting
X
X
X
is better than painting
Y
Y
Y
.
3
1
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Maximal number of "free" segments
n
n
n
points are given on a plane where
n
≥
4
n\ge4
n
≥
4
. All pairs of points are connected with a segment. Find the maximal number of segments which don't intersect with any other segments in their interior.
2
1
Hide problems
3-variable inequality
x
x
x
,
y
y
y
and
z
z
z
are real numbers where the sum of any two among them is not
1
1
1
. Show that,
(
x
2
+
y
)
(
x
+
y
2
)
(
x
+
y
−
1
)
2
+
(
y
2
+
z
)
(
y
+
z
2
)
(
y
+
z
−
1
)
2
+
(
z
2
+
x
)
(
z
+
x
2
)
(
z
+
x
−
1
)
2
≥
2
(
x
+
y
+
z
)
−
3
4
\dfrac{(x^2+y)(x+y^2)}{(x+y-1)^2}+\dfrac{(y^2+z)(y+z^2)}{(y+z-1)^2} + \dfrac{(z^2+x)(z+x^2)}{(z+x-1)^2} \ge 2(x+y+z) - \dfrac{3}{4}
(
x
+
y
−
1
)
2
(
x
2
+
y
)
(
x
+
y
2
)
+
(
y
+
z
−
1
)
2
(
y
2
+
z
)
(
y
+
z
2
)
+
(
z
+
x
−
1
)
2
(
z
2
+
x
)
(
z
+
x
2
)
≥
2
(
x
+
y
+
z
)
−
4
3
Find all triples
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
of real numbers satisfying the equality case.
1
1
Hide problems
If integer, then square
m
m
m
and
n
n
n
are positive integers. If the number
k
=
(
m
+
n
)
2
4
m
(
m
−
n
)
2
+
4
k=\dfrac{(m+n)^2}{4m(m-n)^2+4}
k
=
4
m
(
m
−
n
)
2
+
4
(
m
+
n
)
2
is an integer, prove that
k
k
k
is a perfect square.