MathDB
Problems
Contests
International Contests
IMO Longlists
1986 IMO Longlists
5
5
Part of
1986 IMO Longlists
Problems
(1)
There exist P such that X attains a minimum-IMO Long List P5
Source:
8/28/2010
Let
A
B
C
ABC
A
BC
and
D
E
F
DEF
D
EF
be acute-angled triangles. Write
d
=
E
F
,
e
=
F
D
,
f
=
D
E
.
d = EF, e = FD, f = DE.
d
=
EF
,
e
=
F
D
,
f
=
D
E
.
Show that there exists a point
P
P
P
in the interior of
A
B
C
ABC
A
BC
for which the value of the expression
X
=
d
⋅
A
P
+
e
⋅
B
P
+
f
⋅
C
P
X=d \cdot AP +e \cdot BP +f \cdot CP
X
=
d
⋅
A
P
+
e
⋅
BP
+
f
⋅
CP
attains a minimum.
function
geometry proposed
geometry