MathDB
Problems
Contests
National and Regional Contests
Turkey Contests
Turkey Team Selection Test
2022 Turkey Team Selection Test
2022 Turkey Team Selection Test
Part of
Turkey Team Selection Test
Subcontests
(7)
9
1
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Tree inequality graph theory
In every acyclic graph with 2022 vertices we can choose
k
k
k
of the vertices such that every chosen vertex has at most 2 edges to chosen vertices. Find the maximum possible value of
k
k
k
.
8
1
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Nice I configuration with orthocenters
A
B
C
ABC
A
BC
triangle with
∣
A
B
∣
<
∣
B
C
∣
<
∣
C
A
∣
|AB|<|BC|<|CA|
∣
A
B
∣
<
∣
BC
∣
<
∣
C
A
∣
has the incenter
I
I
I
. The orthocenters of triangles
I
B
C
,
I
A
C
IBC, IAC
I
BC
,
I
A
C
and
I
A
B
IAB
I
A
B
are
H
A
,
H
A
H_A, H_A
H
A
,
H
A
and
H
A
H_A
H
A
.
H
B
H
C
H_BH_C
H
B
H
C
intersect
B
C
BC
BC
at
K
A
K_A
K
A
and perpendicular line from
I
I
I
to
H
B
H
B
H_BH_B
H
B
H
B
intersect
B
C
BC
BC
at
L
A
L_A
L
A
.
K
B
,
L
B
,
K
C
,
L
C
K_B, L_B, K_C, L_C
K
B
,
L
B
,
K
C
,
L
C
are defined similarly. Prove that
∣
K
A
L
A
∣
=
∣
K
B
L
B
∣
+
∣
K
C
L
C
∣
|K_AL_A|=|K_BL_B|+|K_CL_C|
∣
K
A
L
A
∣
=
∣
K
B
L
B
∣
+
∣
K
C
L
C
∣
7
1
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min of xy+yz+zx+1/x+2/y+5/z
What is the minimum value of the expression
x
y
+
y
z
+
z
x
+
1
x
+
2
y
+
5
z
xy+yz+zx+\frac 1x+\frac 2y+\frac 5z
x
y
+
yz
+
z
x
+
x
1
+
y
2
+
z
5
where
x
,
y
,
z
x, y, z
x
,
y
,
z
are positive real numbers?
6
1
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Interesting integer polynomial construction
For a polynomial
P
(
x
)
P(x)
P
(
x
)
with integer coefficients and a prime
p
p
p
, if there is no
n
∈
Z
n \in \mathbb{Z}
n
∈
Z
such that
p
∣
P
(
n
)
p|P(n)
p
∣
P
(
n
)
, we say that polynomial
P
P
P
excludes
p
p
p
. Is there a polynomial with integer coefficients such that having degree of 5, excluding exactly one prime and not having a rational root?
5
1
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Arcs with equal lengths on a circle, again
On a circle, 2022 points are chosen such that distance between two adjacent points is always the same. There are
k
k
k
arcs, each having endpoints on chosen points, with different lengths. Arcs do not contain each other. What is the maximum possible number of
k
k
k
?
4
1
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Three circles tangent to everything
We have three circles
w
1
w_1
w
1
,
w
2
w_2
w
2
and
Γ
\Gamma
Γ
at the same side of line
l
l
l
such that
w
1
w_1
w
1
and
w
2
w_2
w
2
are tangent to
l
l
l
at
K
K
K
and
L
L
L
and to
Γ
\Gamma
Γ
at
M
M
M
and
N
N
N
, respectively. We know that
w
1
w_1
w
1
and
w
2
w_2
w
2
do not intersect and they are not in the same size. A circle passing through
K
K
K
and
L
L
L
intersect
Γ
\Gamma
Γ
at
A
A
A
and
B
B
B
. Let
R
R
R
and
S
S
S
be the reflections of
M
M
M
and
N
N
N
with respect to
l
l
l
. Prove that
A
,
B
,
R
,
S
A, B, R, S
A
,
B
,
R
,
S
are concyclic.
3
1
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Concurrency wanted, perpendiculars to a line given
In a triangle
A
B
C
ABC
A
BC
, the incircle centered at
I
I
I
is tangent to the sides
B
C
,
A
C
BC, AC
BC
,
A
C
and
A
B
AB
A
B
at
D
,
E
D, E
D
,
E
and
F
F
F
, respectively. Let
X
,
Y
X, Y
X
,
Y
and
Z
Z
Z
be the feet of the perpendiculars drawn from
A
,
B
A, B
A
,
B
and
C
C
C
to a line
ℓ
\ell
ℓ
passing through
I
I
I
. Prove that
D
X
,
E
Y
DX, EY
D
X
,
E
Y
and
F
Z
FZ
FZ
are concurrent.