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International Contests
IMO Longlists
1978 IMO Longlists
5
5
Part of
1978 IMO Longlists
Problems
(1)
Existence of point P, A', B', C' satisfying conditions
Source:
10/14/2010
Prove that for any triangle
A
B
C
ABC
A
BC
there exists a point P in the plane of the triangle and three points
A
′
,
B
′
A' , B'
A
′
,
B
′
, and
C
′
C'
C
′
on the lines
B
C
,
A
C
BC, AC
BC
,
A
C
, and
A
B
AB
A
B
respectively such that
A
B
⋅
P
C
′
=
A
C
⋅
P
B
′
=
B
C
⋅
P
A
′
=
0.3
M
2
,
AB \cdot PC'= AC \cdot PB'= BC \cdot PA'= 0.3M^2,
A
B
⋅
P
C
′
=
A
C
⋅
P
B
′
=
BC
⋅
P
A
′
=
0.3
M
2
,
where
M
=
m
a
x
{
A
B
,
A
C
,
B
C
}
M = max\{AB,AC,BC\}
M
=
ma
x
{
A
B
,
A
C
,
BC
}
.
geometry unsolved
geometry