# 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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