MathDB
Problems
Contests
National and Regional Contests
Turkey Contests
Turkey EGMO TST
2019 Turkey EGMO TST
2019 Turkey EGMO TST
Part of
Turkey EGMO TST
Subcontests
(6)
6
1
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Splitting piles
There are
k
k
k
piles and there are
2019
2019
2019
stones totally. In every move we split a pile into two or remove one pile. Using finite moves we can reach conclusion that there are
k
k
k
piles left and all of them contain different number of stonws. Find the maximum of
k
k
k
.
4
1
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NT about the positive divisors of n and n+1
Let
σ
(
n
)
\sigma (n)
σ
(
n
)
shows the number of positive divisors of
n
n
n
. Let
s
(
n
)
s(n)
s
(
n
)
be the number of positive divisors of
n
+
1
n+1
n
+
1
such that for every divisor
a
a
a
,
a
−
1
a-1
a
−
1
is also a divisor of
n
n
n
. Find the maximum value of
2
s
(
n
)
−
σ
(
n
)
2s(n)- \sigma (n)
2
s
(
n
)
−
σ
(
n
)
.
1
1
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About the subsets of S
A
1
,
A
2
,
.
.
.
,
A
n
A_1, A_2, ..., A_n
A
1
,
A
2
,
...
,
A
n
are the subsets of
∣
S
∣
=
2019
|S|=2019
∣
S
∣
=
2019
such that union of any three of them gives
S
S
S
but if we combine two of subsets it doesn't give us
S
S
S
. Find the maximum value of
n
n
n
.
2
1
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Turkey EGMO TST 2019 P2
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive reals such that
a
b
c
=
1
abc=1
ab
c
=
1
,
a
+
b
+
c
=
5
a+b+c=5
a
+
b
+
c
=
5
and
(
a
b
+
2
a
+
2
b
−
9
)
(
b
c
+
2
b
+
2
c
−
9
)
(
c
a
+
2
c
+
2
a
−
9
)
≥
0
(ab+2a+2b-9)(bc+2b+2c-9)(ca+2c+2a-9)\geq 0
(
ab
+
2
a
+
2
b
−
9
)
(
b
c
+
2
b
+
2
c
−
9
)
(
c
a
+
2
c
+
2
a
−
9
)
≥
0
. Find the minimum value of
1
a
+
1
b
+
1
c
\frac {1}{a}+ \frac {1}{b}+ \frac{1}{c}
a
1
+
b
1
+
c
1
5
1
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Turkey EGMO TST 2019 P5
Let
D
D
D
be the midpoint of
B
C
‾
\overline{BC}
BC
in
Δ
A
B
C
\Delta ABC
Δ
A
BC
. Let
P
P
P
be any point on
A
D
‾
\overline{AD}
A
D
. If the internal angle bisector of
∠
A
B
P
\angle ABP
∠
A
BP
and
∠
A
C
P
\angle ACP
∠
A
CP
intersect at
Q
Q
Q
. Prove that, if
B
Q
⊥
Q
C
BQ \perp QC
BQ
⊥
QC
, then
Q
Q
Q
lies on
A
D
AD
A
D
3
1
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Turkey EGMO TST 2019 P3
Let
ω
\omega
ω
be the circumcircle of
Δ
A
B
C
\Delta ABC
Δ
A
BC
, where
∣
A
B
∣
=
∣
A
C
∣
|AB|=|AC|
∣
A
B
∣
=
∣
A
C
∣
. Let
D
D
D
be any point on the minor arc
A
C
AC
A
C
. Let
E
E
E
be the reflection of point
B
B
B
in line
A
D
AD
A
D
. Let
F
F
F
be the intersection of
ω
\omega
ω
and line
B
E
BE
BE
and Let
K
K
K
be the intersection of line
A
C
AC
A
C
and the tangent at
F
F
F
. If line
A
B
AB
A
B
intersects line
F
D
FD
F
D
at
L
L
L
, Show that
K
,
L
,
E
K,L,E
K
,
L
,
E
are collinear points