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.


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