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Problems
Contests
National and Regional Contests
Greece Contests
Greece Team Selection Test
2022 Greece Team Selection Test
2022 Greece Team Selection Test
Part of
Greece Team Selection Test
Subcontests
(2)
2
1
Hide problems
circumcenter of EFG lies on (ABC)
Consider triangle
A
B
C
ABC
A
BC
with
A
B
<
A
C
<
B
C
AB<AC<BC
A
B
<
A
C
<
BC
, inscribed in triangle
Γ
1
\Gamma_1
Γ
1
and the circles
Γ
2
(
B
,
A
C
)
\Gamma_2 (B,AC)
Γ
2
(
B
,
A
C
)
and
Γ
2
(
C
,
A
B
)
\Gamma_2 (C,AB)
Γ
2
(
C
,
A
B
)
. A common point of circle
Γ
2
\Gamma_2
Γ
2
and
Γ
3
\Gamma_3
Γ
3
is point
E
E
E
, a common point of circle
Γ
1
\Gamma_1
Γ
1
and
Γ
3
\Gamma_3
Γ
3
is point
F
F
F
and a common point of circle
Γ
1
\Gamma_1
Γ
1
and
Γ
2
\Gamma_2
Γ
2
is point
G
G
G
, where the points
E
,
F
,
G
E,F,G
E
,
F
,
G
lie on the same semiplane defined by line
B
C
BC
BC
, that point
A
A
A
doesn't lie in. Prove that circumcenter of triangle
E
F
G
EFG
EFG
lies on circle
Γ
1
\Gamma_1
Γ
1
.Note: By notation
Γ
(
K
,
R
)
\Gamma (K,R)
Γ
(
K
,
R
)
, we mean random circle
Γ
\Gamma
Γ
has center
K
K
K
and radius
R
R
R
.
3
1
Hide problems
max M, a_{n+1}/a_n >M if a_0+a_1+...+a_{n-1} >= 3 a_n - a_{n+1}
Find largest possible constant
M
M
M
such that, for any sequence
a
n
a_n
a
n
,
n
=
0
,
1
,
2
,
.
.
.
n=0,1,2,...
n
=
0
,
1
,
2
,
...
of real numbers, that satisfies the conditions : i)
a
0
=
1
a_0=1
a
0
=
1
,
a
1
=
3
a_1=3
a
1
=
3
ii)
a
0
+
a
1
+
.
.
.
+
a
n
−
1
≥
3
a
n
−
a
n
+
1
a_0+a_1+...+a_{n-1} \ge 3 a_n - a_{n+1}
a
0
+
a
1
+
...
+
a
n
−
1
≥
3
a
n
−
a
n
+
1
for any integer
n
≥
1
n\ge 1
n
≥
1
to be true that
a
n
+
1
a
n
>
M
\frac{a_{n+1}}{a_n} >M
a
n
a
n
+
1
>
M
for any integer
n
≥
0
n\ge 0
n
≥
0
.