Consider ``slot" $S_k$, $0\le k\le 1009$, where we say $a\in S_k$ if $v_2(a)=k$ (that is, $2^k\mid\mid a$). There are $1010$ such slots. Now, let the given list be $L=\{a_i:1\le i\le 2021\}$. Since $2^{1010}\nmid a_i$; it follows that $a_i\in S_k$ for a suitable $0\le k\le 1009$. Since $|L|=2021$, and there are $1010$ such slots; pigenhole principle reveals that there exists a $0\le k\le 1009$, and distinct $a,b,c\in L$ such that $a,b,c\in S_k$, hence $v_2(a)=v_2(b)=v_2(c)=k$. Set $a=2^k\cdot m$, $b=2^k\cdot n$, and $c=2^k\cdot t$ with $m,n,t$ odd. Note that
\[
\left|b^2-4ac\right|= 2^{2k}\left|n^2-4mt\right|.
\]Since $n^2-4mt\equiv 3\pmod{8}$, it follows $\left|n^2-4mt\right|\equiv \pm 3\pmod{8}$. Hence, $|b^2-4ac|$ cannot be a perfect square, concluding the proof.