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ITAMO
2022 ITAMO
6
6
Part of
2022 ITAMO
Problems
(1)
Italian Mathematical Olympiad 2022 - Problem 6
Source:
5/7/2022
Let
A
B
C
ABC
A
BC
be a non-equilateral triangle and let
R
R
R
be the radius of its circumcircle. The incircle of
A
B
C
ABC
A
BC
has
I
I
I
as its centre and is tangent to side
C
A
CA
C
A
in point
D
D
D
and to side
C
B
CB
CB
in point
E
E
E
. Let
A
1
A_1
A
1
be the point on line
E
I
EI
E
I
such that
A
1
I
=
R
A_1I=R
A
1
I
=
R
, with
I
I
I
being between
A
1
A_1
A
1
and
E
E
E
. Let
B
1
B_1
B
1
be the point on line
D
I
DI
D
I
such that
B
1
I
=
R
B_1I=R
B
1
I
=
R
, with
I
I
I
being between
B
1
B_1
B
1
and
D
D
D
. Let
P
P
P
be the intersection of lines
A
A
1
AA_1
A
A
1
and
B
B
1
BB_1
B
B
1
. (a) Prove that
P
P
P
belongs to the circumcircle of
A
B
C
ABC
A
BC
. (b) Let us now also suppose that
A
B
=
1
AB=1
A
B
=
1
and
P
P
P
coincides with
C
C
C
. Determine the possible values of the perimeter of
A
B
C
ABC
A
BC
.
geometry