1961 IMO Problems/Problem 2

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Problem

Let $a$, $b$, and $c$ be the lengths of a triangle whose area is S. Prove that

$a^2 + b^2 + c^2 \ge 4S\sqrt{3}$

In what case does equality hold?

Solution

Substitute $S=\sqrt{s(s-a)(s-b)(s-c)}$, where $s=\frac{a+b+c}{2}$

This shows that the inequality is equivalent to $a^2b^2+b^2c^2+c^2a^2\le a^4+b^4+c^4$.

This can be proven because $a^2b^2\le\frac{a^4+b^4}{2}$. The equality holds when $a=b=c$, or when the triangle is equilateral.

 1961 IMO (Problems) • Resources Preceded byProblem 1 1 • 2 • 3 • 4 • 5 • 6 Followed byProblem 3 All IMO Problems and Solutions