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Rioplatense Mathematical Olympiad, Level 3
2022 Rioplatense Mathematical Olympiad
2022 Rioplatense Mathematical Olympiad
Part of
Rioplatense Mathematical Olympiad, Level 3
Subcontests
(6)
4
3
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Rays, incircle, angles...
Let
A
B
C
ABC
A
BC
be a triangle with incenter
I
I
I
. Let
D
D
D
be the point of intersection between the incircle and the side
B
C
BC
BC
, the points
P
P
P
and
Q
Q
Q
are in the rays
I
B
IB
I
B
and
I
C
IC
I
C
, respectively, such that
∠
I
A
P
=
∠
C
A
D
\angle IAP=\angle CAD
∠
I
A
P
=
∠
C
A
D
and
∠
I
A
Q
=
∠
B
A
D
\angle IAQ=\angle BAD
∠
I
A
Q
=
∠
B
A
D
. Prove that
A
P
=
A
Q
AP=AQ
A
P
=
A
Q
.
Angle = Angle (twice)
Let
A
B
C
D
ABCD
A
BC
D
be a parallelogram and
M
M
M
is the intersection of
A
C
AC
A
C
and
B
D
BD
B
D
. The point
N
N
N
is inside of the
△
A
M
B
\triangle AMB
△
A
MB
such that
∠
A
N
D
=
∠
B
N
C
\angle AND=\angle BNC
∠
A
N
D
=
∠
BNC
. Prove that
∠
M
N
C
=
∠
N
D
A
\angle MNC=\angle NDA
∠
MNC
=
∠
N
D
A
and
∠
M
N
D
=
∠
N
C
B
\angle MND=\angle NCB
∠
MN
D
=
∠
NCB
.
Rioplatense L-1 2022 #4
Let
L
L
L
be the number formed by
2022
2022
2022
digits equal to
1
1
1
, that is,
L
=
1111
…
111
L=1111\dots 111
L
=
1111
…
111
. Compute the sum of the digits of the number
9
L
2
+
2
L
9L^2+2L
9
L
2
+
2
L
.
6
4
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