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Contests
National and Regional Contests
Turkey Contests
Turkey MO (2nd round)
2022 Turkey MO (2nd round)
2022 Turkey MO (2nd round)
Part of
Turkey MO (2nd round)
Subcontests
(6)
5
1
Hide problems
Circumcenter of $AHO$ and Euler Line
In triangle
A
B
C
ABC
A
BC
,
9
0
o
>
∠
A
>
∠
B
>
∠
C
90^{o}> \angle A> \angle B> \angle C
9
0
o
>
∠
A
>
∠
B
>
∠
C
. Let the circumcenter and orthocenter of the triangle be
O
O
O
and
H
H
H
.
O
H
OH
O
H
intersects
B
C
BC
BC
at
T
T
T
and the circumcenter of
(
A
H
O
)
(AHO)
(
A
H
O
)
is
X
X
X
. Prove that the reflection of
H
H
H
over
X
T
XT
XT
lies on the circumcircle of triangle
A
B
C
ABC
A
BC
.
6
1
Hide problems
Several trips in a school with 2022 students
In a school with
2022
2022
2022
students, either a museum trip or a nature trip is organized every day during a holiday. No student participates in the same type of trip twice, and the number of students attending each trip is different. If there are no two students participating in the same two trips together, find the maximum number of trips held.
2
1
Hide problems
Bounding with factorials and phi function
For positive integers
k
k
k
and
n
n
n
, we know
k
≥
n
!
k \geq n!
k
≥
n
!
. Prove that
ϕ
(
k
)
≥
(
n
−
1
)
!
\phi (k) \geq (n-1)!
ϕ
(
k
)
≥
(
n
−
1
)!
4
1
Hide problems
An easy 3 variable equation
For which real numbers
a
a
a
, there exist pairwise different real numbers
x
,
y
,
z
x, y, z
x
,
y
,
z
satisfying
x
3
+
a
y
+
z
=
y
3
+
a
x
+
z
=
z
3
+
a
x
+
y
=
−
3.
\frac{x^3+a}{y+z}=\frac{y^3+a}{x+z}=\frac{z^3+a}{x+y}= -3.
y
+
z
x
3
+
a
=
x
+
z
y
3
+
a
=
x
+
y
z
3
+
a
=
−
3.
1
1
Hide problems
Cute geometry with bisectors and tangential circumcircles
In triangle
A
B
C
ABC
A
BC
,
M
M
M
is the midpoint of side
B
C
BC
BC
, the bisector of angle
B
A
C
BAC
B
A
C
intersects
B
C
BC
BC
and
(
A
B
C
)
(ABC)
(
A
BC
)
at
K
K
K
and
L
L
L
, respectively. If the circle with diameter
[
B
C
]
[BC]
[
BC
]
is tangent to the external angle bisector of angle
B
A
C
BAC
B
A
C
, prove that this circle is tangent to
(
K
L
M
)
(KLM)
(
K
L
M
)
as well.
3
1
Hide problems
max # of pairs satisying a^2+b>=1/2021
Let
a
1
,
a
2
,
⋯
,
a
2022
a_1, a_2, \cdots, a_{2022}
a
1
,
a
2
,
⋯
,
a
2022
be nonnegative real numbers such that
a
1
+
a
2
+
⋯
+
a
2022
=
1
a_1+a_2+\cdots +a_{2022}=1
a
1
+
a
2
+
⋯
+
a
2022
=
1
. Find the maximum number of ordered pairs
(
i
,
j
)
(i, j)
(
i
,
j
)
,
1
≤
i
,
j
≤
2022
1\leq i,j\leq 2022
1
≤
i
,
j
≤
2022
, satisfying
a
i
2
+
a
j
≥
1
2021
.
a_i^2+a_j\ge \frac 1{2021}.
a
i
2
+
a
j
≥
2021
1
.