Guaranteed Real Roots

Algebra Level 5

Find the largest positive integer nn such that for all real numbers a1a_1, a2a_2, ,an+1\dots , a_{n+1} , the equation an+1x22xa12+a22++an+12+(a1+a2++an)=0 a_{n+1} x^2 - 2x \sqrt{ a_1^2 + a_2^2 + \cdots + a_{n+1}^2} + (a_1 + a_2 + \cdots + a_n) = 0 has real roots.

Details and assumptions

Clarification: a1,a2,,an+1 a_1, a_2, \ldots, a_{n+1} is any set of real numbers. There is no restrictions stated in the problem. They do not need to be integers, nor always positive, nor an arithmetic progression.

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