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Contests
National and Regional Contests
Greece Contests
Greece JBMO TST
2022 Greece JBMO TST
2022 Greece JBMO TST
Part of
Greece JBMO TST
Subcontests
(2)
2
1
Hide problems
(CEG) tangent to AC
Let
A
B
C
ABC
A
BC
be an acute triangle with
A
B
<
A
C
<
B
C
AB<AC < BC
A
B
<
A
C
<
BC
, inscirbed in circle
Γ
1
\Gamma_1
Γ
1
, with center
O
O
O
. Circle
Γ
2
\Gamma_2
Γ
2
, with center point
A
A
A
and radius
A
C
AC
A
C
intersects
B
C
BC
BC
at point
D
D
D
and the circle
Γ
1
\Gamma_1
Γ
1
at point
E
E
E
. Line
A
D
AD
A
D
intersects circle
Γ
1
\Gamma_1
Γ
1
at point
F
F
F
. The circumscribed circle
Γ
3
\Gamma_3
Γ
3
of triangle
D
E
F
DEF
D
EF
, intersects
B
C
BC
BC
at point
G
G
G
. Prove that: a) Point
B
B
B
is the center of circle
Γ
3
\Gamma_3
Γ
3
b) Circumscribed circle of triangle
C
E
G
CEG
CEG
is tangent to
A
C
AC
A
C
.
3
1
Hide problems
min of \sqrt{2+x}+\sqrt{2+y}+\sqrt{2+z}+\sqrt{x+y}+\sqrt{y+z}+\sqrt{z+x}
The real numbers
x
,
y
,
z
x,y,z
x
,
y
,
z
are such that
x
+
y
+
z
=
4
x+y+z=4
x
+
y
+
z
=
4
and
0
≤
x
,
y
,
z
≤
2
0 \le x,y,z \le 2
0
≤
x
,
y
,
z
≤
2
. Find the minimun value of the expression
A
=
2
+
x
+
2
+
y
+
2
+
z
+
x
+
y
+
y
+
z
+
z
+
x
A=\sqrt{2+x}+\sqrt{2+y}+\sqrt{2+z}+\sqrt{x+y}+\sqrt{y+z}+\sqrt{z+x}
A
=
2
+
x
+
2
+
y
+
2
+
z
+
x
+
y
+
y
+
z
+
z
+
x
.