Algebra Level 5

Real numbers $a_1, a_2, \ldots, a_{80090}$ satisfy

$(a_1 + a_2 + \ldots + a_{80090} )^2 + a_1 ^2 + a_2^2 + \ldots + a_{80090}^2 \leq 80090.$

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

×