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France Contests
French Mathematical Olympiad
1992 French Mathematical Olympiad
Problem 1
Problem 1
Part of
1992 French Mathematical Olympiad
Problems
(1)
properties of set of points in plane, vector equality
Source: France 1992 P1
5/13/2021
Let
Δ
\Delta
Δ
be a convex figure in a plane
P
\mathcal P
P
. Given a point
A
∈
P
A\in\mathcal P
A
∈
P
, to each pair
(
M
,
N
)
(M,N)
(
M
,
N
)
of points in
Δ
\Delta
Δ
we associate the point
m
∈
P
m\in\mathcal P
m
∈
P
such that
A
m
→
=
M
N
→
2
\overrightarrow{Am}=\frac{\overrightarrow{MN}}2
A
m
=
2
MN
and denote by
δ
A
(
Δ
)
\delta_A(\Delta)
δ
A
(
Δ
)
the set of all so obtained points
m
m
m
.(a) i. Prove that
δ
A
(
Δ
)
\delta_A(\Delta)
δ
A
(
Δ
)
is centrally symmetric. ii. Under which conditions is
δ
A
(
Δ
)
=
Δ
\delta_A(\Delta)=\Delta
δ
A
(
Δ
)
=
Δ
? iii. Let
B
,
C
B,C
B
,
C
be points in
P
\mathcal P
P
. Find a transformation which sends
δ
B
(
Δ
)
\delta_B(\Delta)
δ
B
(
Δ
)
to
δ
C
(
Δ
)
\delta_C(\Delta)
δ
C
(
Δ
)
. (b) Determine
δ
A
(
Δ
)
\delta_A(\Delta)
δ
A
(
Δ
)
if i.
Δ
\Delta
Δ
is a set in the plane determined by two parallel lines. ii.
Δ
\Delta
Δ
is bounded by a triangle. iii.
Δ
\Delta
Δ
is a semi-disk. (c) Prove that in the cases
b
.
2
b.2
b
.2
and
b
.
3
b.3
b
.3
the lengths of the boundaries of
Δ
\Delta
Δ
and
δ
A
(
Δ
)
\delta_A(\Delta)
δ
A
(
Δ
)
are equal.
vector
geometry
point set