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MAA AMC
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1982 USAMO
3
1982 USAMO #3
1982 USAMO #3
Source:
August 16, 2011
AMC
USA(J)MO
USAMO
geometry
perimeter
function
trigonometry
Problem Statement
If a point
A
1
A_1
A
1
is in the interior of an equilateral triangle
A
B
C
ABC
A
BC
and point
A
2
A_2
A
2
is in the interior of
△
A
1
B
C
\triangle{A_1BC}
△
A
1
BC
, prove that
I.Q.
(
A
1
B
C
)
>
I.Q.
(
A
2
B
C
)
,
\operatorname{I.Q.} (A_1BC) > \operatorname{I.Q.} (A_2BC),
I.Q.
(
A
1
BC
)
>
I.Q.
(
A
2
BC
)
,
where the isoperrimetric quotient of a figure
F
F
F
is defined by
I.Q.
(
F
)
=
Area
(
F
)
[
Perimeter
(
F
)
]
2
.
\operatorname{I.Q.}(F) = \frac{\operatorname{Area}(F)}{[\operatorname{Perimeter}(F)]^2}.
I.Q.
(
F
)
=
[
Perimeter
(
F
)
]
2
Area
(
F
)
.
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