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Contests
National and Regional Contests
Turkey Contests
Turkey MO (2nd round)
2014 Turkey MO (2nd round)
2014 Turkey MO (2nd round)
Part of
Turkey MO (2nd round)
Subcontests
(6)
4
1
Hide problems
An identity for a circle with tangent lines
Let
P
P
P
and
Q
Q
Q
be the midpoints of non-parallel chords
k
1
k_1
k
1
and
k
2
k_2
k
2
of a circle
ω
\omega
ω
, respectively. Let the tangent lines of
ω
\omega
ω
passing through the endpoints of
k
1
k_1
k
1
intersect at
A
A
A
and the tangent lines passing through the endpoints of
k
2
k_2
k
2
intersect at
B
B
B
. Let the symmetric point of the orthocenter of triangle
A
B
P
ABP
A
BP
with respect to the line
A
B
AB
A
B
be
R
R
R
and let the feet of the perpendiculars from
R
R
R
to the lines
A
P
,
B
P
,
A
Q
,
B
Q
AP, BP, AQ, BQ
A
P
,
BP
,
A
Q
,
BQ
be
R
1
,
R
2
,
R
3
,
R
4
R_1, R_2, R_3, R_4
R
1
,
R
2
,
R
3
,
R
4
, respectively. Prove that
A
R
1
P
R
1
⋅
P
R
2
B
R
2
=
A
R
3
Q
R
3
⋅
Q
R
4
B
R
4
\frac{AR_1}{PR_1} \cdot \frac{PR_2}{BR_2} = \frac{AR_3}{QR_3} \cdot \frac{QR_4}{BR_4}
P
R
1
A
R
1
⋅
B
R
2
P
R
2
=
Q
R
3
A
R
3
⋅
B
R
4
Q
R
4
6
1
Hide problems
A graph with 36 vertices
5
5
5
airway companies operate in a country consisting of
36
36
36
cities. Between any pair of cities exactly one company operates two way flights. If some air company operates between cities
A
,
B
A, B
A
,
B
and
B
,
C
B, C
B
,
C
we say that the triple
A
,
B
,
C
A, B, C
A
,
B
,
C
is properly-connected. Determine the largest possible value of
k
k
k
such that no matter how these flights are arranged there are at least
k
k
k
properly-connected triples.
5
1
Hide problems
Find all n such that two sets are identical
Find all natural numbers
n
n
n
for which there exist non-zero and distinct real numbers
a
1
,
a
2
,
…
,
a
n
a_1, a_2, \ldots, a_n
a
1
,
a
2
,
…
,
a
n
satisfying
{
a
i
+
(
−
1
)
i
a
i
∣
1
≤
i
≤
n
}
=
{
a
i
∣
1
≤
i
≤
n
}
.
\left\{a_i+\dfrac{(-1)^i}{a_i} \, \Big | \, 1 \leq i \leq n\right\} = \{a_i \mid 1 \leq i \leq n\}.
{
a
i
+
a
i
(
−
1
)
i
1
≤
i
≤
n
}
=
{
a
i
∣
1
≤
i
≤
n
}
.
3
1
Hide problems
Prove that the two lines are perpendicular
Let
D
,
E
,
F
D, E, F
D
,
E
,
F
be points on the sides
B
C
,
C
A
,
A
B
BC, CA, AB
BC
,
C
A
,
A
B
of a triangle
A
B
C
ABC
A
BC
, respectively such that the lines
A
D
,
B
E
,
C
F
AD, BE, CF
A
D
,
BE
,
CF
are concurrent at the point
P
P
P
. Let a line
ℓ
\ell
ℓ
through
A
A
A
intersect the rays
[
D
E
[DE
[
D
E
and
[
D
F
[DF
[
D
F
at the points
Q
Q
Q
and
R
R
R
, respectively. Let
M
M
M
and
N
N
N
be points on the rays
[
D
B
[DB
[
D
B
and
[
D
C
[DC
[
D
C
, respectively such that the equation
Q
N
2
D
N
+
R
M
2
D
M
=
(
D
Q
+
D
R
)
2
−
2
⋅
R
Q
2
+
2
⋅
D
M
⋅
D
N
M
N
\frac{QN^2}{DN}+\frac{RM^2}{DM}=\frac{(DQ+DR)^2-2\cdot RQ^2+2\cdot DM\cdot DN}{MN}
D
N
Q
N
2
+
D
M
R
M
2
=
MN
(
D
Q
+
D
R
)
2
−
2
⋅
R
Q
2
+
2
⋅
D
M
⋅
D
N
holds. Show that the lines
A
D
AD
A
D
and
B
C
BC
BC
are perpendicular to each other.
1
1
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An interesting game
In a bag there are
1007
1007
1007
black and
1007
1007
1007
white balls, which are randomly numbered
1
1
1
to
2014
2014
2014
. In every step we draw one ball and put it on the table; also if we want to, we may choose two different colored balls from the table and put them in a different bag. If we do that we earn points equal to the absolute value of their differences. How many points can we guarantee to earn after
2014
2014
2014
steps?
2
1
Hide problems
Find all (x,y,z)
Find all all positive integers x,y,and z satisfying the equation
x
3
=
3
y
7
z
+
8
x^3=3^y7^z+8
x
3
=
3
y
7
z
+
8