MathDB
Problems
Contests
International Contests
IMO Longlists
1988 IMO Longlists
70
Spanish Sines Inradius Circumradius Inequality
Spanish Sines Inradius Circumradius Inequality
Source: IMO LongList 1988, Spain 1, Problem 70 of ILL
November 9, 2005
trigonometry
geometry
inradius
circumcircle
inequalities
function
geometry unsolved
Problem Statement
A
B
C
ABC
A
BC
is a triangle, with inradius
r
r
r
and circumradius
R
.
R.
R
.
Show that:
sin
(
A
2
)
⋅
sin
(
B
2
)
+
sin
(
B
2
)
⋅
sin
(
C
2
)
+
sin
(
C
2
)
⋅
sin
(
A
2
)
≤
5
8
+
r
4
⋅
R
.
\sin \left( \frac{A}{2} \right) \cdot \sin \left( \frac{B}{2} \right) + \sin \left( \frac{B}{2} \right) \cdot \sin \left( \frac{C}{2} \right) + \sin \left( \frac{C}{2} \right) \cdot \sin \left( \frac{A}{2} \right) \leq \frac{5}{8} + \frac{r}{4 \cdot R}.
sin
(
2
A
)
⋅
sin
(
2
B
)
+
sin
(
2
B
)
⋅
sin
(
2
C
)
+
sin
(
2
C
)
⋅
sin
(
2
A
)
≤
8
5
+
4
⋅
R
r
.
Back to Problems
View on AoPS