Two Maximized Constants in an Inequality

Algebra Level 5

Let C1C_1 be the maximum value such that

x2+y2+1C1(x+y)x^2+y^2+1 \geq C_1(x+y)

for all real xx and yy. Similarly, let C2C_2 be the maximum value such that

x2+y2+xy+1C2(x+y)x^2+y^2+xy+1 \geq C_2(x+y)

holds true for all real xx and yy. If C1C2C_1C_2 can be expressed as n\sqrt{n}, find nn.


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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