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National and Regional Contests
Belarus Contests
Belarus Team Selection Test
2024 Belarus Team Selection Test
3.1
3.1
Part of
2024 Belarus Team Selection Test
Problems
(1)
Inequality on radii of incircles
Source: Belarusian TST 2024
10/27/2024
Triangles
A
B
C
ABC
A
BC
and
D
E
F
DEF
D
EF
, having a common incircle of radius
R
R
R
, intersect at points
X
1
,
X
2
,
…
,
X
6
X_1, X_2, \ldots , X_6
X
1
,
X
2
,
…
,
X
6
and form six triangles (see the figure below). Let
r
1
,
r
2
,
…
,
r
6
r_1, r_2,\ldots, r_6
r
1
,
r
2
,
…
,
r
6
be the radii of the inscribed circles of these triangles, and let
R
1
,
R
2
,
…
,
R
6
R_1, R_2, \ldots , R_6
R
1
,
R
2
,
…
,
R
6
be the radii of the inscribed circles of the triangles
A
X
1
F
,
F
X
2
B
,
B
X
3
D
,
D
X
4
C
,
C
X
5
E
AX_1F, FX_2B, BX_3D, DX_4C, CX_5E
A
X
1
F
,
F
X
2
B
,
B
X
3
D
,
D
X
4
C
,
C
X
5
E
and
E
X
6
A
EX_6A
E
X
6
A
respectively. https://i.ibb.co/BspgdHB/Image.jpg Prove that
∑
i
=
1
6
1
r
i
<
6
R
+
∑
i
=
1
6
1
R
i
\sum_{i=1}^{6} \frac{1}{r_i} < \frac{6}{R}+\sum_{i=1}^{6} \frac{1}{R_i}
i
=
1
∑
6
r
i
1
<
R
6
+
i
=
1
∑
6
R
i
1
U. Maksimenkau
inequalities
geometry