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 |