= Solution
The <Hasse theorem for elliptic curves> states that
$$
\left|\#E(\mathbb F_p)-(p+1)\right|\leq2\sqrt p.
$$
Let $\pi$ be the <Frobenius isogeny of an elliptic curve> and put $a=p+1-\#E(\mathbb F_p)$. The degree on $\operatorname{End}(E)$ is a positive-definite quadratic form, its associated bilinear form gives $\operatorname{tr}(\pi)=a$, and $\deg\pi=p$. Consequently
$$
\deg([m]+[n]\pi)=m^2+amn+pn^2
$$
for all integers $m,n$. If $a^2-4p>0$, this real quadratic form is indefinite, so by density of rational slopes it is negative at some nonzero integer pair $(m,n)$, contradicting nonnegativity of the degree. Hence $a^2\leq4p$, which is the claimed bound.
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