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Problems
Contests
National and Regional Contests
Argentina Contests
Argentina National Olympiad
1994 Argentina National Olympiad
3
3
Part of
1994 Argentina National Olympiad
Problems
(1)
two squares, one given, one wanted
Source: Argentina 1994 OMA L3 p3
5/13/2024
Given in the plane the square
A
B
C
D
ABCD
A
BC
D
, the square
A
1
B
1
C
1
D
1
A_1B_1C_1D_1
A
1
B
1
C
1
D
1
, smaller than the first, and a quadrilateral
P
Q
R
S
PQRS
PQRS
that satisfy the following conditions
∙
\bullet
∙
A
B
C
D
ABCD
A
BC
D
and
A
1
B
1
C
1
D
1
A_1B_1C_1D_1
A
1
B
1
C
1
D
1
have a common center and respectively parallel sides.
∙
\bullet
∙
P
P
P
,
Q
Q
Q
,
R
R
R
,
S
S
S
belong one to each side of the square
A
B
C
D
ABCD
A
BC
D
.
∙
\bullet
∙
A
1
A_1
A
1
,
B
1
B_1
B
1
,
C
1
C_1
C
1
,
D
1
D_1
D
1
belong one to each side of the quadrilateral
P
Q
R
S
PQRS
PQRS
. Prove that
P
Q
R
S
PQRS
PQRS
is a square.
geometry
Squares