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Ecuador Contests
Ecuador Juniors
2018 Ecuador Juniors
3
3
Part of
2018 Ecuador Juniors
Problems
(1)
(PQRS)<= 1/2 (ABCD), quadr. inside a square
Source: 2018 Ecuador Juniors (OMEC) L2 p3
10/24/2022
Let
A
B
C
D
ABCD
A
BC
D
be a square. Point
P
,
Q
,
R
,
S
P, Q, R, S
P
,
Q
,
R
,
S
are chosen on the sides
A
B
AB
A
B
,
B
C
BC
BC
,
C
D
CD
C
D
,
D
A
DA
D
A
, respectively, such that
A
P
+
C
R
≥
A
B
≥
B
Q
+
D
S
AP + CR \ge AB \ge BQ + DS
A
P
+
CR
≥
A
B
≥
BQ
+
D
S
. Prove that
a
r
e
a
(
P
Q
R
S
)
≤
1
2
a
r
e
a
(
A
B
C
D
)
area \,\, (PQRS) \le \frac12 \,\, area \,\, (ABCD)
a
re
a
(
PQRS
)
≤
2
1
a
re
a
(
A
BC
D
)
and determine all cases when equality holds.
geometry
square
geometric inequality
areas