1991 IMO Problems/Problem 5
Problem
Let be a triangle and
an interior point of
. Show that at least one of the angles
is less than or equal to
.
Solution
Let ,
, and
be
,
,
, respectively.
Let ,
, and
be
,
,
, respcetively.
Using law of sines on we get:
, therefore,
Using law of sines on we get:
, therefore,
Using law of sines on we get:
, therefore,
Multiply all three equations we get:
Using AM-GM we get:
. [Inequality 1]
Note that for ,
decreases with increasing
and fixed
Therefore, decreases with increasing
and fixed
From trigonometric identity:
,
since , then:
Therefore,
and also,
Adding these two inequalities we get:
.
. [Inequality 2]
Combining [Inequality 1] and [Inequality 2] we see the following:
This implies that for at least one of the values of ,
,or
, the following is true:
or
Which means that for at least one of the values of ,
,or
, the following is true:
Therefore, at least one of the angles is less than or equal to
.
~Tomas Diaz, orders@tomasdiaz.com
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
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