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IMO Longlists
1990 IMO Longlists
55
55
Part of
1990 IMO Longlists
Problems
(1)
Construct triangle ABC - ILL 1990 MON4
Source:
9/18/2010
Given points
A
,
M
,
M
1
A, M, M_1
A
,
M
,
M
1
and rational number
λ
≠
−
1
\lambda \neq -1
λ
=
−
1
. Construct the triangle
A
B
C
ABC
A
BC
, such that
M
M
M
lies on
B
C
BC
BC
and
M
1
M_1
M
1
lies on
B
1
C
1
B_1C_1
B
1
C
1
(
B
1
,
C
1
B_1, C_1
B
1
,
C
1
are the projections of
B
,
C
B, C
B
,
C
on
A
C
,
A
B
AC, AB
A
C
,
A
B
respectively), and
B
M
M
C
=
B
1
M
1
M
1
C
1
=
λ
.
\frac{BM}{MC}=\frac{B_1M_1}{M_1C_1}=\lambda.
MC
BM
=
M
1
C
1
B
1
M
1
=
λ
.
combinatorics proposed
combinatorics