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Problems
Contests
National and Regional Contests
Turkey Contests
Turkey EGMO TST
2023 Turkey EGMO TST
2023 Turkey EGMO TST
Part of
Turkey EGMO TST
Subcontests
(6)
6
1
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As l varies, circumcircle of XYZ passes through a fixed point
Let
A
B
C
ABC
A
BC
be a scalene triangle and
l
0
l_0
l
0
be a line that is tangent to the circumcircle of
A
B
C
ABC
A
BC
at point
A
A
A
. Let
l
l
l
be a variable line which is parallel to line
l
0
l_0
l
0
. Let
l
l
l
intersect segment
A
B
AB
A
B
and
A
C
AC
A
C
at the point
X
X
X
,
Y
Y
Y
respectively.
B
Y
BY
B
Y
and
C
X
CX
CX
intersects at point
T
T
T
and the line
A
T
AT
A
T
intersects the circumcirle of
A
B
C
ABC
A
BC
at
Z
Z
Z
. Prove that as
l
l
l
varies, circumcircle of
X
Y
Z
XYZ
X
Y
Z
passes through a fixed point.
5
1
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Minimum number of students
In a school there is a person with
l
l
l
friends for all
1
≤
l
≤
99
1 \leq l \leq 99
1
≤
l
≤
99
. If there is no trio of students in this school, all three of whom are friends with each other, what is the minimum number of students in the school?
4
1
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Inequality with polynomial
Let
n
n
n
be a positive integer and
P
,
Q
P,Q
P
,
Q
be polynomials with real coefficients with
P
(
x
)
=
x
n
Q
(
1
x
)
P(x)=x^nQ(\frac{1}{x})
P
(
x
)
=
x
n
Q
(
x
1
)
and
P
(
x
)
≥
Q
(
x
)
P(x) \geq Q(x)
P
(
x
)
≥
Q
(
x
)
for all real numbers
x
x
x
. Prove that
P
(
x
)
=
Q
(
x
)
P(x)=Q(x)
P
(
x
)
=
Q
(
x
)
for all real number
x
x
x
.
3
1
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2xy>1 or yz>1 than find the minimum value
Let
x
,
y
,
z
x,y,z
x
,
y
,
z
be positive real numbers that satisfy at least one of the inequalities,
2
x
y
>
1
2xy>1
2
x
y
>
1
,
y
z
>
1
yz>1
yz
>
1
. Find the least possible value of
x
y
3
z
2
+
4
z
x
−
8
y
z
−
4
y
z
xy^3z^2+\frac{4z}{x}-8yz-\frac{4}{yz}
x
y
3
z
2
+
x
4
z
−
8
yz
−
yz
4
.
2
1
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Equation with primes
Find all pairs of
p
,
q
p,q
p
,
q
prime numbers that satisfy the equation
p
(
p
4
+
p
2
+
10
q
)
=
q
(
q
2
+
3
)
p(p^4+p^2+10q)=q(q^2+3)
p
(
p
4
+
p
2
+
10
q
)
=
q
(
q
2
+
3
)
1
1
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Coincideness of two incenters
Let
O
1
O
2
O
3
O_1O_2O_3
O
1
O
2
O
3
be an acute angled triangle.Let
ω
1
\omega_1
ω
1
,
ω
2
\omega_2
ω
2
,
ω
3
\omega_3
ω
3
be the circles with centres
O
1
O_1
O
1
,
O
2
O_2
O
2
,
O
3
O_3
O
3
respectively such that any of two are tangent to each other. Circumcircle of
O
1
O
2
O
3
O_1O_2O_3
O
1
O
2
O
3
intersects
ω
1
\omega_1
ω
1
at
A
1
A_1
A
1
and
B
1
B_1
B
1
,
ω
2
\omega_2
ω
2
at
A
2
A_2
A
2
and
B
2
B_2
B
2
,
ω
3
\omega_3
ω
3
at
A
3
A_3
A
3
and
B
3
B_3
B
3
respectively. Prove that the incenter of triangle which can be constructed by lines
A
1
B
1
A_1B_1
A
1
B
1
,
A
2
B
2
A_2B_2
A
2
B
2
,
A
3
B
3
A_3B_3
A
3
B
3
and the incenter of
O
1
O
2
O
3
O_1O_2O_3
O
1
O
2
O
3
are coincide.