I found this question:https://mathoverflow.net/questions/420837/chernoff-style-concentration-bound-for-ratio-of-variables.
I want to ask if we get similar thing for the ratio of the sum and the one Gaussian variable.

Given i.i.d. Gaussian random variables $X_1,\dots, X_k$ with $N(0, 1)$. Fix $\epsilon\in (0,1)$, can prove that for any $\delta>0$ there exists $1\le k=k(\epsilon, \delta)$ (some number) so that
$$ 
P\left(\frac{X_1^2+X_2^2+\dots+X_k^2}{X_1^2}>\frac{1}{\epsilon^2}\right)>1- \delta ?
$$

Can we find such $k$?