Two Maximized Constants in an Inequality
Algebra
Level
5
Let \(C_1\) be the maximum value such that
\[x^2+y^2+1 \geq C_1(x+y)\]
for all real \(x\) and \(y\). Similarly, let \(C_2\) be the maximum value such that
\[x^2+y^2+xy+1 \geq C_2(x+y)\]
holds true for all real \(x\) and \(y\). If \(C_1C_2\) can be expressed as \(\sqrt{n}\), find \(n\).
This is really a proof problem in disguise. It's a very beautiful problem, and I hope you post a proof rather than just a brute-force solution explaining your answer.
by
Finn Hulse