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Contests
National and Regional Contests
Turkey Contests
Turkey MO (2nd round)
2020 Turkey MO (2nd round)
2020 Turkey MO (2nd round)
Part of
Turkey MO (2nd round)
Subcontests
(6)
6
1
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2021 points on a circle
2021
2021
2021
points are given on a circle. Each point is colored by one of the
1
,
2
,
⋯
,
k
1,2, \cdots ,k
1
,
2
,
⋯
,
k
colors. For all points and colors
1
≤
r
≤
k
1\leq r \leq k
1
≤
r
≤
k
, there exist an arc such that at least half of the points on it are colored with
r
r
r
. Find the maximum possible value of
k
k
k
.
1
1
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Divisibility condition on the set 1, 2, ... , n^2
Let
n
>
1
n > 1
n
>
1
be an integer and
X
=
{
1
,
2
,
⋯
,
n
2
}
X = \{1, 2, \cdots , n^2 \}
X
=
{
1
,
2
,
⋯
,
n
2
}
. If there exist
x
,
y
x, y
x
,
y
such that
x
2
∣
y
x^2\mid y
x
2
∣
y
in all subsets of
X
X
X
with
k
k
k
elements, find the least possible value of
k
k
k
.
2
1
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Four points on AP
Let
P
P
P
be an interior point of acute triangle
Δ
A
B
C
\Delta ABC
Δ
A
BC
, which is different from the orthocenter. Let
D
D
D
and
E
E
E
be the feet of altitudes from
A
A
A
to
B
P
BP
BP
and
C
P
CP
CP
, and let
F
F
F
and
G
G
G
be the feet of the altitudes from
P
P
P
to sides
A
B
AB
A
B
and
A
C
AC
A
C
. Denote by
X
X
X
the midpoint of
[
A
P
]
[AP]
[
A
P
]
, and let the second intersection of the circumcircles of triangles
Δ
D
F
X
\Delta DFX
Δ
D
FX
and
Δ
E
G
X
\Delta EGX
Δ
EGX
lie on
B
C
BC
BC
. Prove that
A
P
AP
A
P
is perpendicular to
B
C
BC
BC
or
∠
P
B
A
=
∠
P
C
A
\angle PBA = \angle PCA
∠
PB
A
=
∠
PC
A
.
5
1
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Polynomial with real coefficients
Find all polynomials with real coefficients such that one can find an integer valued series
a
0
,
a
1
,
…
a_0, a_1, \dots
a
0
,
a
1
,
…
satisfying
⌊
P
(
x
)
⌋
=
a
⌊
x
2
⌋
\lfloor P(x) \rfloor = a_{ \lfloor x^2 \rfloor}
⌊
P
(
x
)⌋
=
a
⌊
x
2
⌋
for all
x
x
x
real numbers.
4
1
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Number of positive divisors of 2p^2+2p+1 where p is prime
Let
p
p
p
be a prime number such that
2
8
p
−
1
2
p
2
+
2
p
+
1
\frac{28^p-1}{2p^2+2p+1}
2
p
2
+
2
p
+
1
2
8
p
−
1
is an integer. Find all possible values of number of divisors of
2
p
2
+
2
p
+
1
2p^2+2p+1
2
p
2
+
2
p
+
1
.
3
1
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Non cyclic inequality
If
x
,
y
,
z
x, y, z
x
,
y
,
z
are positive real numbers find the minimum value of
2
(
x
+
y
+
z
)
(
1
x
+
1
y
+
1
z
)
−
(
1
+
x
y
)
(
1
+
y
z
)
2\sqrt{(x+y+z) \left( \frac{1}{x}+ \frac{1}{y} + \frac{1}{z} \right)} - \sqrt{ \left( 1+ \frac{x}{y} \right) \left( 1+ \frac{y}{z} \right)}
2
(
x
+
y
+
z
)
(
x
1
+
y
1
+
z
1
)
−
(
1
+
y
x
)
(
1
+
z
y
)