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.

Excel in math and science

Master concepts by solving fun, challenging problems.

It's hard to learn from lectures and videos

Learn more effectively through short, interactive explorations.

Used and loved by over 7 million people

Learn from a vibrant community of students and enthusiasts, including olympiad champions, researchers, and professionals.