This question comes from a Chinese high school olympiad training program. It seems remarkably more difficult (and indeed, interesting!) than all other problems arising in the same program, especially since an elementary (high-school level) solution is probably available.

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> Show that there exists integers $a,b,c,d,e>2018$ so that the equation $$a^2+b^2+c^2+d^2+e^2+65=abcde$$ is satisfied.
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For what it's worth, here's what I have tried. Rewriting the equation as a quadratic polynomial in $a$,
$$a^2-(bcde)a+b^2+c^2+d^2+e^2+65=0.$$
For there to be integer solutions, the discriminant must be a perfect square. Hence
$$ b^2c^2d^2e^2-4b^2-4c^2-4d^2-4e^2-4\cdot5\cdot13=n^2.$$
I don't however see how I can solve this equation, especially due to the large number of unknowns. Any ideas?

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**Edit:** Ivan Neretin presents an excellent answer by Vieta Jumping, which I'm sure will yield results. However, the training program I mentioned has not discussed such advanced tactics as Vieta Jumping yet, and only covered $\gcd$, $\operatorname{lcm}$, factorisation of polynomials, discriminants, modular arithmetic, divisibility and quadratic residues. Hence despite Ivan's excellent solution, I would still be extremely appreciative of a more elementary solution.
['https://math.stackexchange.com/questions/3015046', 'https://math.stackexchange.com', 'https://math.stackexchange.com/users/496634/']
You are supposed to bruteforce or guess $(a,b,c,d,e)=(1,2,3,4,5)$ or any other small solution, and then go up by [Vieta jumping](https://en.wikipedia.org/wiki/Vieta_jumping). That is, once you have a solution, you rewrite it as a quadratic polynomial in $a$ (just like you did), and since one root is integer, so is the other. Then you do the same to $b,\;c...$ and repeat until the roots are big enough.
*This expands upon @Ivan Neretin 's answer*

This is what is meant by Vieta jumping, when it comes to this problem--we can use Vieta jumping to prove the following claim:

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> Proposition 1: For any $A \in \mathbb{Z}^+$ there is a solution $(a,b,c,d,e)$; $a,b,c,d,e \in \mathbb{Z}^+$ to the equation $a^2+b^2+c^2+d^2+e^2+65 = abcde$ such that $A \leq \min\{a,b,c,d,e\}$.
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*Proof:* We first claim the following:

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> Claim 1: Let us suppose that there is a solution $a^2+b^2+c^2+d^2+e^2+65=120$; $a,b,c,d,e \in \mathbb{Z}^+$, and let us assume WLOG that $a= \min\{a,b,c,d,e\}$. Then
>  there is a solution $(a',b',c',d',e')$; $a',b',c',d',e' \in \mathbb{Z}^+$ s.t. $a'>a$ and $b'=b,c'=c,d'=d,e'=e$.
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[*Proof of Claim 1:* Indeed, iff there exists an integer $a'> a$ is that satisfies the following:

$a'^2 + (b^2+c^2+d^2+e^2)+65 > a'(bcde)$.

Then Claim 1 will follow. However, the above is a quadratic equation in $a'$ of the form $a'^2 - xa' + z$; both $x$ and $z$ positive integers, that has at least one integer solution, namely $a$, and so the other solution is an integer; as $z$ is at least $b^2+c^2+d^2+e^2+65$ $> 4a^2$ and $aa'=z$ with $a$ positive it follows that $a'$ must be strictly greater than $4a$. So indeed Claim 1 does follow. $\surd$ ]

We note that Proposition 1 follows immediately from Claim 1, and the existence of at least one solution $(a\_0,b\_0,c\_0,d\_0,e\_0); a\_0,b\_0,c\_0,d\_0,e\_0 \in \mathbb{Z}^+$ 
such that the equation $a\_0^2+b\_0^2+c\_0^2+d\_0^2+e\_0^2+65 = a\_0b\_0c\_0d\_0e\_0$ holds;
namely $(a\_0,b\_0,c\_0,d\_0,e\_0) =(1,2,3,4,5)$. $\surd$