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Contests
National and Regional Contests
Turkey Contests
Turkey MO (2nd round)
2023 Turkey MO (2nd round)
2023 Turkey MO (2nd round)
Part of
Turkey MO (2nd round)
Subcontests
(6)
6
1
Hide problems
Symmedian involving geometry
On a triangle
A
B
C
ABC
A
BC
, points
D
D
D
,
E
E
E
,
F
F
F
are given on the segments
B
C
BC
BC
,
A
C
AC
A
C
,
A
B
AB
A
B
respectively such that
D
E
∥
A
B
DE \parallel AB
D
E
∥
A
B
,
D
F
∥
A
C
DF \parallel AC
D
F
∥
A
C
and
B
D
D
C
=
A
B
2
A
C
2
\frac{BD}{DC}=\frac{AB^2}{AC^2}
D
C
B
D
=
A
C
2
A
B
2
holds. Let the circumcircle of
A
E
F
AEF
A
EF
meet
A
D
AD
A
D
at
R
R
R
and the line that is tangent to the circumcircle of
A
B
C
ABC
A
BC
at
A
A
A
at
S
S
S
again. Let the line
E
F
EF
EF
intersect
B
C
BC
BC
at
L
L
L
and
S
R
SR
SR
at
T
T
T
. Prove that
S
R
SR
SR
bisects
A
B
AB
A
B
if and only if
B
S
BS
BS
bisects
T
L
TL
T
L
.
5
1
Hide problems
Construction with sets
Is it possible that a set consisting of
23
23
23
real numbers has a property that the number of the nonempty subsets whose product of the elements is rational number is exactly
2422
2422
2422
?
4
1
Hide problems
31 tuplets on a blackboard
Initially given
31
31
31
tuplets
(
1
,
0
,
0
,
…
,
0
)
,
(
0
,
1
,
0
,
…
,
0
)
,
…
,
(
0
,
0
,
0
,
…
,
1
)
(1,0,0,\dots,0),(0,1,0,\dots,0),\dots, (0,0,0,\dots,1)
(
1
,
0
,
0
,
…
,
0
)
,
(
0
,
1
,
0
,
…
,
0
)
,
…
,
(
0
,
0
,
0
,
…
,
1
)
were written on the blackboard. At every move we choose two written
31
31
31
tuplets as
(
a
1
,
a
2
,
a
3
,
…
,
a
31
)
(a_1,a_2,a_3,\dots, a_{31})
(
a
1
,
a
2
,
a
3
,
…
,
a
31
)
and
(
b
1
,
b
2
,
b
3
,
…
,
b
31
)
(b_1,b_2,b_3,\dots,b_{31})
(
b
1
,
b
2
,
b
3
,
…
,
b
31
)
, then write the
31
31
31
tuplet
(
a
1
+
b
1
,
a
2
+
b
2
,
a
3
+
b
3
,
…
,
a
31
+
b
31
)
(a_1+b_1,a_2+b_2,a_3+b_3,\dots, a_{31}+b_{31})
(
a
1
+
b
1
,
a
2
+
b
2
,
a
3
+
b
3
,
…
,
a
31
+
b
31
)
to the blackboard too. Find the least possible value of the moves such that one can write the
31
31
31
tuplets
(
0
,
1
,
1
,
…
,
1
)
,
(
1
,
0
,
1
,
…
,
1
)
,
…
,
(
1
,
1
,
1
,
…
,
0
)
(0,1,1,\dots,1),(1,0,1,\dots,1),\dots, (1,1,1,\dots,0)
(
0
,
1
,
1
,
…
,
1
)
,
(
1
,
0
,
1
,
…
,
1
)
,
…
,
(
1
,
1
,
1
,
…
,
0
)
to the blackboard by using those moves.
3
1
Hide problems
Consecutive numeral block hunting
Let a
9
9
9
-digit number be balanced if it has all numerals
1
1
1
to
9
9
9
. Let
S
S
S
be the sequence of the numerals which is constructed by writing all balanced numbers in increasing order consecutively. Find the least possible value of
k
k
k
such that any two subsequences of
S
S
S
which has consecutive
k
k
k
numerals are different from each other.
2
1
Hide problems
Shouting inversion but has an elegant solution with La Hire
Let
A
B
C
ABC
A
BC
be a triangle and
P
P
P
be an interior point. Let
ω
A
\omega_A
ω
A
be the circle that is tangent to the circumcircle of
B
P
C
BPC
BPC
at
P
P
P
internally and tangent to the circumcircle of
A
B
C
ABC
A
BC
at
A
1
A_1
A
1
internally and let
Γ
A
\Gamma_A
Γ
A
be the circle that is tangent to the circumcircle of
B
P
C
BPC
BPC
at
P
P
P
externally and tangent to the circumcircle of
A
B
C
ABC
A
BC
at
A
2
A_2
A
2
internally. Define
B
1
B_1
B
1
,
B
2
B_2
B
2
,
C
1
C_1
C
1
,
C
2
C_2
C
2
analogously. Let
O
O
O
be the circumcentre of
A
B
C
ABC
A
BC
. Prove that the lines
A
1
A
2
A_1A_2
A
1
A
2
,
B
1
B
2
B_1B_2
B
1
B
2
,
C
1
C
2
C_1C_2
C
1
C
2
and
O
P
OP
OP
are concurrent.
1
1
Hide problems
\frac{n^2+m^2}{m^4+n}=k
Prove that there exist infinitely many positive integers
k
k
k
such that the equation
n
2
+
m
2
m
4
+
n
=
k
\frac{n^2+m^2}{m^4+n}=k
m
4
+
n
n
2
+
m
2
=
k
don't have any positive integer solution.