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National and Regional Contests
Moldova Contests
Moldova Team Selection Test
2016 Moldova Team Selection Test
3
3
Part of
2016 Moldova Team Selection Test
Problems
(1)
Find the smallest value of $\frac{AO_1+BO_1}{AB},$
Source: Moldova TST 2016
1/30/2023
Let
A
B
C
ABC
A
BC
be a triangle with
∠
C
=
90
\angle C=90
∠
C
=
90
. The tangent points of the inscribed circle with the sides
B
C
,
C
A
BC, CA
BC
,
C
A
and
A
B
AB
A
B
are
M
,
N
M, N
M
,
N
and
P
.
P.
P
.
Points
M
1
,
N
1
,
P
1
M_1, N_1, P_1
M
1
,
N
1
,
P
1
are symmetric to points
M
,
N
,
P
M, N, P
M
,
N
,
P
with respect to midpoints of sides
B
C
,
C
A
BC, CA
BC
,
C
A
and
A
B
.
AB.
A
B
.
Find the smallest value of
A
O
1
+
B
O
1
A
B
,
\frac{AO_1+BO_1}{AB},
A
B
A
O
1
+
B
O
1
,
where
O
1
O_1
O
1
is the circumcenter of triangle
M
1
N
1
P
1
.
M_1N_1P_1.
M
1
N
1
P
1
.
geometry
circumcircle