1997 JBMO Problems/Problem 4
Contents
[hide]Problem
Determine the triangle with sides and circumradius
for which
.
Solutions
Solution 1
Solving for yields
. We can substitute
into the area formula
to get
We also know that
, where
is the angle between sides
and
Substituting this yields
Since
is inside a triangle,
. Substitution yields
Note that
, so multiplying both sides by that value would not change the inequality sign. This means
However, by the AM-GM Inequality,
. Thus, the equality case must hold, so
where
. When plugging
, the inequality holds, so the value
truly satisfies all conditions.
That means so
That means the only truangle that satisfies all the conditions is a 45-45-90 triangle where
is the longest side. In other words,
for all positive
See Also
1997 JBMO (Problems • Resources) | ||
Preceded by Problem 3 |
Followed by Problem 5 | |
1 • 2 • 3 • 4 • 5 | ||
All JBMO Problems and Solutions |