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Moldova Contests
Moldova Team Selection Test
2007 Moldova Team Selection Test
2007 Moldova Team Selection Test
Part of
Moldova Team Selection Test
Subcontests
(4)
3
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6 geometry problems out of 8 at our TST so far..
Let
M
,
N
M, N
M
,
N
be points inside the angle
∠
B
A
C
\angle BAC
∠
B
A
C
usch that
∠
M
A
B
≡
∠
N
A
C
\angle MAB\equiv \angle NAC
∠
M
A
B
≡
∠
N
A
C
. If
M
1
,
M
2
M_{1}, M_{2}
M
1
,
M
2
and
N
1
,
N
2
N_{1}, N_{2}
N
1
,
N
2
are the projections of
M
M
M
and
N
N
N
on
A
B
,
A
C
AB, AC
A
B
,
A
C
respectively then prove that
M
,
N
M, N
M
,
N
and
P
P
P
the intersection of
M
1
N
2
M_{1}N_{2}
M
1
N
2
with
N
1
M
2
N_{1}M_{2}
N
1
M
2
are collinear.
an elegant identity in elements of a triangle
Consider a triangle
A
B
C
ABC
A
BC
, with corresponding sides
a
,
b
,
c
a,b,c
a
,
b
,
c
, inradius
r
r
r
and circumradius
R
R
R
. If
r
A
,
r
B
,
r
C
r_{A}, r_{B}, r_{C}
r
A
,
r
B
,
r
C
are the radii of the respective excircles of the triangle, show that
a
2
(
2
r
A
−
r
r
B
r
C
)
+
b
2
(
2
r
B
−
r
r
A
r
C
)
+
c
2
(
2
r
C
−
r
r
A
r
B
)
=
4
(
R
+
3
r
)
a^{2}\left(\frac 2{r_{A}}-\frac{r}{r_{B}r_{C}}\right)+b^{2}\left(\frac 2{r_{B}}-\frac{r}{r_{A}r_{C}}\right)+c^{2}\left(\frac 2{r_{C}}-\frac{r}{r_{A}r_{B}}\right)=4(R+3r)
a
2
(
r
A
2
−
r
B
r
C
r
)
+
b
2
(
r
B
2
−
r
A
r
C
r
)
+
c
2
(
r
C
2
−
r
A
r
B
r
)
=
4
(
R
+
3
r
)
A circle tangent to the circumcircle and two sides
Let
A
B
C
ABC
A
BC
be a triangle. A circle is tangent to sides
A
B
,
A
C
AB, AC
A
B
,
A
C
and to the circumcircle of
A
B
C
ABC
A
BC
(internally) at points
P
,
Q
,
R
P, Q, R
P
,
Q
,
R
respectively. Let
S
S
S
be the point where
A
R
AR
A
R
meets
P
Q
PQ
PQ
. Show that
∠
S
B
A
≡
∠
S
C
A
\angle{SBA}\equiv \angle{SCA}
∠
SB
A
≡
∠
SC
A
2
4
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