MathDB
Problems
Contests
National and Regional Contests
India Contests
India IMO Training Camp
2003 India IMO Training Camp
8
8
Part of
2003 India IMO Training Camp
Problems
(1)
a+b+c>r+r_a+r_b+r_c
Source: India tst 2003 p8
3/15/2012
Let
A
B
C
ABC
A
BC
be a triangle, and let
r
,
r
1
,
r
2
,
r
3
r, r_1, r_2, r_3
r
,
r
1
,
r
2
,
r
3
denoted its inradius and the exradii opposite the vertices
A
,
B
,
C
A,B,C
A
,
B
,
C
, respectively. Suppose
a
>
r
1
,
b
>
r
2
,
c
>
r
3
a>r_1, b>r_2, c>r_3
a
>
r
1
,
b
>
r
2
,
c
>
r
3
. Prove that (a) triangle
A
B
C
ABC
A
BC
is acute, (b)
a
+
b
+
c
>
r
+
r
1
+
r
2
+
r
3
a+b+c>r+r_1+r_2+r_3
a
+
b
+
c
>
r
+
r
1
+
r
2
+
r
3
.
geometry
inradius
geometry proposed