Terms overload

Algebra Level 5

Real numbers a1,a2,,a80090 a_1, a_2, \ldots, a_{80090} satisfy

(a1+a2++a80090)2+a12+a22++a80090280090. (a_1 + a_2 + \ldots + a_{80090} )^2 + a_1 ^2 + a_2^2 + \ldots + a_{80090}^2 \leq 80090.

What is the smallest integer value of NN such that for all possible such tuples (a1,...,a80090)(a_1,...,a_{80090}), we have aiN |a_i| \leq N for all ii?

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